i

PhD

Achieving Risk Congruence in a Banking Firm

By

Guy Ford

B.Com UNSW, M.Bus (App Fin) UTS, SA Fin, MFP

A dissertation submitted in fulfilment of the requirements for the

degree of Doctor of Philosophy at the University of Western Sydney

December 2005

ii

ABSTRACT One of the reasons for firms decentralising aspects of their operations is to enable

managers to gain specialised knowledge of local conditions. For credit managers in a

banking firm, this may take the form of knowledge of investment opportunities and

the risk profiles of each of these opportunities. In light of principal-agent problems

that arise when information is asymmetrical, the focal point of this dissertation is the

development of incentive-compatible mechanisms that facilitate the free and accurate

disclosure of the private information of managers on the risk profile of investments to

the centre of the bank at the time investment decisions are being implemented. These

mechanisms are required because managers may have strong incentives to

misrepresent their private information when doing so has the potential to favourably

impact on the size of their remuneration. This, in turn, has a direct impact on the

ability of the centre to optimally allocate the capital of the bank and effectively price

risk into bank investments.

The dissertation commences by examining which internal risk measures act to align

the investment decisions of managers in a bank with the risk/return goals of the centre

of the bank. This requires knowledge of the bank risk preference function. It is

initially assumed that managers have developed specialised knowledge of the

opportunity set of available investments, and have no reason to misrepresent this

information to the centre. This assumption is later removed and the implications

assessed. In order to ensure incentive-compatibility between the centre and managers,

a truth-revealing mechanism is employed in the capital allocation process and tied to

the compensation payment function of the bank. This mechanism acts to ensure

managers disclose their private information on the expected risks and returns in the

investments under their control, and facilitates the efficient investment of capital

within the bank.

iii

ACKNOWLEDGEMENTS The completion of this body of research could not have been undertaken without the

input and support of a number of people.

Professor Thomas Valentine supervised the work. He provided constant guidance and

encouragement, and enthusiastically responded to my requests and questions. In

addition to supervising this work, he has become both mentor and colleague.

Katie Pratt provided much encouragement during the early years of this work. When

progress appeared slow, she became a force to fear. She is dearly missed.

Professor Tyrone M. Carlin provided much comical and intellectual stimulus during

the course of this work. During the busy final year of this research, he subtly shielded

me from many administrative burdens, allowing me the momentum to complete this

work. For this I am grateful.

Many academic colleagues offered useful comments when versions of this work were

presented at conferences. While I cannot name all of these individuals, I would like to

single out and thank Dr Neil Esho, who invited me to present versions of the work in

seminars at the Australian Prudential Regulatory Authority. The comments and

suggestions that were received proved highly valuable.

Finally, and most significantly, my immediate family have provided immeasurable

support and sacrificed much while I have been pursuing this work. Agatha Pupaher

encouraged me to follow my instincts and pursue an academic career – I might still be

an unhappy bank clerk if it were not for her. I suspect my children, Rebecca and Luis,

thought for many years that all fathers spent their weekends working on a PhD.

Thank you to both of you for always greeting me at the door with a smile and

bringing me promptly back to earth.

iv

DECLARATION

I hereby certify that this thesis is original, and does not contain without

acknowledgement any material previously submitted for a degree or diploma at any

university, and does not, to the best of my knowledge, contain material previously

published to which due reference has not been made in the text.

……………………………………………….

Guy Ford

December 2005

v

TABLE OF CONTENTS

Item Page Title i Abstract ii Acknowledgements iii Declaration iv Table of Contents v List of Tables xii List of Figures and Illustrations xiv Abbreviations xv Chapter One – Introduction 1 1.1 Introduction 2 1.2 Background 5

1.2.1 Alignment of Economic and Regulatory Capital 5 1.2.2 Capital Allocation 9

1.3 Thesis 11

1.3.1 Incentive-Compatible Risk Measures 12 1.3.2 Agency Problems and Solution 15

1.4 Chapter Overview and Research Questions 16

vi

Chapter Two The Principal’s Dilemma: The Bank Risk Preference Function and Portfolio Selection 20 2.1 Introduction 21 2.2 The Principal’s Dilemma: A Bank Risk Preference Function 26 2.3 Risk-Ranking Criteria 37

2.3.1 First-Order Stochastic Dominance 38 2.3.2 Second-Order Stochastic Dominance 39 2.3.3 Third-Order Stochastic Dominance 40 2.3.4 Non-Expected Utility and Stochastic Dominance 42 2.3.4.1 Prospect Theory 42 2.3.4.2 Prospect Stochastic Dominance 45 2.3.4.3 Markowitz Stochastic Dominance 46 2.3.4.4 Convexity and the Bank Risk Preference Function 49

2.4 Compatibility of Risk Measures with Stochastic Dominance Criteria 54

2.4.1 Criteria for Risk Measures 54 2.4.2 Risk Measures 61 2.4.2.1 Shortfall Probability 61 2.4.2.2 Value-at-Risk 62 2.4.2.3 Expected Shortfall 64 2.4.2.4 First-Order Lower Partial Moment 65 2.4.2.5 Second-Order Lower Partial Moment 67 2.4.3 Summary 68

2.5 Chapter Summary 69

vii

Chapter Three Incentive-Compatible Risk-Adjusted Performance Measurement 71 3.1 Introduction 72 3.2 Determining the Incentive-Compatibility of Risk Measures 75

3.2.1 Portfolio Assumptions 75 3.2.2 Portfolio Risk Profiles 77

3.2.2.1 Portfolio A 78 3.2.2.2 Portfolio B 78 3.2.2.3 Portfolio C 79 3.2.2.4 Portfolio D 80 3.2.2.5 Portfolio E 81 3.2.2.6 Other Considerations 83 3.2.2.7 Summary 84 3.3 Analysis of Risk Measures 85

3.3.1 Analysis of Shortfall Probability 86 3.3.2 Analysis of Value-at-Risk 88 3.3.3 Analysis of Expected Shortfall 93 3.3.4 Analysis of First-Order Lower Partial Moment 97 3.3.5 Analysis of Second-Order Lower Partial Moment 103 3.3.5.1 Spectral Risk Measures 106 3.3.5.2 Distortion Risk Measures 109 3.3.6 Summary 112 3.4 Coherency of Risk Measures 115

3.4.1 Axioms of Coherence 115 3.4.1.1 Axiom 1: Translation Invariance 116 3.4.1.2 Axiom 2: Monotonicity 118 3.4.1.3 Axiom 3: Positive Homogeneity 120 3.4.1.4 Axiom 4: Subadditivity 124 3.4.2 An Alternative Risk Measure: Downside 134 Semi-Deviation 3.5 Internal Risk Measures and Bank Capital 139 3.6 Impact of the Bank Compensation Structure 143

3.6.1 Overview 143 3.6.2 Gains Measured Relative to Portfolio Face Value 146 3.6.3 Gains Measured Relative to Portfolio Expected Value 152 3.6.4 Incorporating Upper Moments 155 3.6.5 Concluding Comments 161 3.7 Chapter Summary 164

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Chapter Four Agency Problems and Risk-Adjusted Performance Measurement 167 4.1 Introduction 168 4.2 Implementation of RORAC Methodologies 171

4.2.1 Sensitivity of Transfer Pricing Assumptions 173 4.2.2 RORAC and Underinvestment 174 4.3 Agency Problems 176

4.3.1 Sources of Agency Problems 177 4.3.1.1 Adverse Selection 178 4.3.1.2 Moral Hazard 179 4.3.1.3 Other Agency Problems 180 4.4 RORAC and Agency Problems 182

4.4.1 Capital Assignment and Information Asymmetries 182 4.4.2 Internal Pricing and Information Asymmetries 190

4.4.2.1 Measurement Horizon 190 4.4.2.2 Funds Transfer Pricing Assumptions 191 4.5 The Internal Hurdle Rate and Bank Risk 193

4.5.1 Overview 193 4.5.2 Does Bank-Specific Risk Matter? 195 4.5.3 Is a Fixed Solvency Standard Consistent with a Fixed Hurdle Rate? 198 4.5.3.1 Merton Model of Default 199 4.5.3.2 Results 200 4.5.4 Compatibility between Hurdle Rate and Bank Risk 203 4.6 Chapter Summary 206

ix

Chapter Five Revealing the Truth: An Internal Capital Market Incorporating an Auction Mechanism 208 5.1 Introduction 209 5.2 Auction Formats 213 5.3 Multi-Unit Auctions 218

5.3.1 Problems with the Uniform Price Format 218 5.3.2 Incentive-Compatible Multi-Unit Auction Format 223 5.4 An Auction Mechanism for Risk Capital 229

5.4.1 Overview 229 5.4.2 Design 230 5.4.3 Example 231 5.5 Qualifications 237 5.6 Chapter Summary 244

x

Chapter Six Whole of Bank Perspective: Dynamics of Target Credit Rating, Hurdle Rates and the Pricing of Bank Assets 246 6.1 Introduction 247 6.2 Loan Pricing Model 250

6.2.1 Overview 250 6.2.2 Model 252 6.3 What Capital Multiplier? 255

6.3.1 Beta Distribution 256 6.4 Dynamics of the Target Credit Rating 259 6.5 Results and Discussion 261

6.5.1 Overview and Assumptions 261 6.5.2 BB-Rated Exposure: Fixed Hurdle Rate 263 6.5.3 Hurdle Rate Revisited 266 6.5.4 BB-Rated Exposure: Leverage-Adjusted Hurdle Rate 267 6.5.5 Varying the Proportion of Retail Funding 269 6.5.6 BBB-Rated Exposure: Leverage-Adjusted Hurdle Rate 272

6.5.7 A-Rated Exposure: Leverage-Adjusted Hurdle Rate 278

6.6 Limitations 281 6.7 Concluding Comments 285 6.8 Chapter Summary 288

xi

Chapter Seven Conclusion 290 7.1 Overview 291

7.2 Key Findings 296 7.3 Areas for Further Research 306 Bibliography 309 Appendices 328

xii

LIST OF TABLES

Table 1.1 Specific Questions for the Study 19 Table 3.1 Probability Distributions: Credit Portfolios A – E 76 Table 3.2 Summary of Risk Profile of Portfolios A – E: Pairwise Rankings 84 Table 3.3 Risk Measures for Portfolios A – E 85 Table 3.4a Compatibility of VaR and Stochastic Dominance: Case 1 91 Table 3.4b Compatibility of VaR and Stochastic Dominance: Case 2 91 Table 3.5a Stochastic Dominance Analysis – Portfolios F and G 102

Table 3.5b First-Order Lower Partial Moment Measures 102 Table 3.6 Second-Order Lower Partial Moment Measures 104 Table 3.7 Summary of Results 112 Table 3.8a Subadditivity for Loans X and Y: Risk Measures 128 Table 3.8b Subadditivity for Loans F and G: Risk Measures 130 Table 3.8c Subadditivity for Loans R and S: Risk Measures 132 Table 3.9 Coherency of Risk Measures 134 Table 3.10 Subadditivity – Downside Semi-Deviation versus LPM2 137 Table 3.11 RAPM (Gains/DSD) – Portfolios A – E 146 Table 3.12 Ranking of Portfolio RAPM under Stochastic Dominance Conditions 150 Table 3.13 Expected Value for RAPM – Risk-Neutral Managers 150

Table 3.14 RAPM (Gain/DSD) – Portfolios A – E 152

xiii

Table 3.15 Ranking of Portfolio RAPM under Stochastic Dominance Conditions 154 Table 3.16 Expected Value for RAPM – Risk-Neutral Managers 154 Table 3.17 Reward to DSD – Portfolios A – E 159 Table 3.18 Portfolios A – E: Distribution of Gains 160 Table 4.1 Constant Probability of Default Scenario 201 Table 4.2 Constant Equity Hurdle Rate Scenario 201 Table 5.1 Second-Price Auction: Payoffs if Bids Above True Value 215 Table 5.2 Second-Price Auction: Payoffs if Bids Below True Value 216 Table 5.3a Uniform Price Auction: No Demand Reduction 220 Table 5.3b Uniform Price Auction: Demand Reduction 222 Table 5.3c Uniform Price Auction: Demand Reduction 222 Table 5.4 Vickrey Multi-Unit Auction 224 Table 5.5 Ascending-Bid Multi-Unit Auction 226 Table 5.6 Ascending Bid Multi-Unit Auction: Demand Reduction 228 Table 5.7 Understate Expected Risk Capital Requirement 234 Table 5.8 Truthful Bid on Expected Risk Capital Requirement 235 Table 5.9 Overstate Expected Risk Capital Requirement 236 Table 6.1 Historical Default Probabilities 1991-2000: All Countries 250 Table 6.2 Capital Multipliers for Expected Losses 257

xiv

Table 6.3 Impact of Increasing Solvency Standard (Fixed Hurdle Rates and BB-Rated Asset) 263 Table 6.4 Bank Credit Spreads: January 2004 265 Table 6.5 Impact of Increasing Solvency Standard (Leverage-Adjusted Hurdle Rates and BB-Rated Asset) 268 Table 6.6 Impact of Increasing Solvency Standard 100% Funded by Rated Debt (Leverage-Adjusted Hurdle Rates and BB-Rated Asset) 270 Table 6.7 Impact of Increasing Solvency Standard (Leverage-Adjusted Hurdle Rates and BBB-Rated Asset) 273 Table 6.8 Impact of Increasing Solvency Standard 100% Funded by Rated Debt (Leverage-Adjusted Hurdle Rates and BBB-Rated Asset) 277 Table 6.9 Impact of Increasing Solvency Standard (Leverage-Adjusted Hurdle Rates and A-Rated Asset) 279

LIST OF FIGURES AND ILLUSTRATIONS

Figure 1.1 Economic Capital 6 Figure 1.2 Portfolio Selection Process 14 Figure 2.1 Prospect Theory Value Function 44 Figure 3.1 Distribution of Portfolio Returns and Investor Classes 108

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ABBREVIATIONS

CAPM Capital Asset Pricing Model

DSD Downside Semi-Deviation

EC Economic Capital

EDF Expected Default Frequency

EL Expected Loss

ES Expected Shortfall

FSD First-Order Stochastic Dominance

IOSCO International Organisation of Securities Commissions

IRB Internal Ratings-Based

LGD Loss Given Default

LPM Lower Partial Moment

MSD Markowitz Stochastic Dominance

PSD Prospect Stochastic Dominance

RAPM Risk-Adjusted Performance Measure

RC Regulatory Capital

ROA Return on Assets

ROE Return on Equity

RAROC Risk-Adjusted Return on Capital

RORAC Return on Risk-Adjusted Capital

SSD Second-Order Stochastic Dominance

TSD Third-Order Stochastic Dominance

UL Unexpected Losses

VaR Value-at-Risk

WT Wang Transform

1

Chapter One

Introduction

“A firm is a command economy in miniature, an island of

authoritarian administration floating in an ocean of market

transactions”

Mark Blaug, 2002

2

1.1 INTRODUCTION

This dissertation examines risk congruence in a large, decentralised banking firm. It

studies the troubling question of how to efficiently align the investment decisions of

managers in the bank with the risk/return goals of the centre of the bank. The central

thesis is that the contemporary approach aimed at achieving such alignment, which

involves the top-down allocation of some proportion of the total bank’s capital against

positions taken by managers, and then remunerating managers based on the return

generated on this capital, serves as a poor mechanism for achieving alignment of

incentives. Indeed, it is argued that this approach leads to outcomes that are against

the best interests of bank stakeholders whom the centre is deemed to represent.

The dissertation sets forth three main propositions:

1. If incentive-compatibility between the actions of managers and the risk/return

preferences of the centre is required, then the risk measure used internally for

assessing the risk-adjusted performance of investments made by managers needs

to diverge from that used for calculating total bank capital, where the latter is

based on achieving a predetermined solvency standard.

2. If managers have private information on expected risks in their investments, and

are expected to act in their own self-interest, then incentive-compatibility

between the centre and managers cannot be achieved without incorporating

some form of a truth-revealing mechanism in the capital allocation and

remuneration processes of the bank.

3. The performance of managers cannot easily be separated from bank-wide

decisions regarding the bank’s target credit rating, funding mix and the hurdle

rate on equity because these factors combine to influence the price of bank

assets, which in turn, impacts on performance metrics upon which managers are

assessed. This needs to be accounted for in the bank’s performance

measurement framework.

3

The basis of the first proposition is that the risk preference function of the centre of

the bank - which embodies the diverse interests of bank owners, depositors, debt

holders and regulators - does not calibrate with the attitude to risk implicit in the

measurement of total bank capital requirements, where capital is linked to a

predetermined solvency standard. The risk preference function of the centre of the

bank is one that is likely to demonstrate non-satiety, risk aversion and a preference for

positive skewness in the distribution of bank returns. This is at odds with the attitude

to risk implicit in a predetermined solvency standard, which is essentially one of risk

neutrality. If banks adopt a policy of spreading their actual capital against risky

positions taken by managers – a full capital allocation policy – then this imposes an

internal risk standard that leads managers to make portfolio decisions that are

suboptimal for the bank. Goal alignment necessitates that the risk measure used for

internal purposes diverge from that used for measuring the total capital requirements

of the bank.

The basis of the second proposition is that managers carry a disincentive to truthfully

reveal their expectations on the distribution of returns in positions when this

information is used by the centre to determine the ex-ante capital that will be allocated

against these positions, which in turn drives the ex-post risk-adjusted performance

measure upon which bonuses to managers are based. If the centre allocates capital

against positions in accordance with historical return volatility, this ignores the

specialised information that managers are likely to possess on the current and

expected volatility in their positions. If the centre allocates too much capital relative

to risk expectations of managers, then managers may be incentivised to take on (and

misrepresent) additional risks in order to meet hurdle rate aspirations. If the centre

allocates too little capital relative to the risk expectations of managers, then managers

are unlikely to reveal this information because a low capital charge will potentially

lead to higher risk-adjusted returns and make hurdle rate aspirations easier to achieve.

In either case, managers acting in their own self interest may lead to the bank being

undercapitalised with respect to the true risk in its books. If banks decentralise their

activities to allow managers to gain specialised knowledge on local risks and

4

opportunities, but managers face incentives to misrepresent this information, then the

performance measurement process must incorporate a truth-revealing mechanism in

order that this specialised knowledge can be appropriately utilised in decisions

regarding the optimal allocation of capital and the measurement and management of

bank-wide risk.

The basis of the third proposition is that changes in the solvency standard (target

credit rating) of a bank have a direct impact on the required return on bank assets,

which has implications for the ability of managers to price assets competitively in

some product markets. If a bank targets a higher credit rating, then the price of bank

assets will rise to the extent that the bank prices its products to achieve a minimum

hurdle rate on equity. This arises because a higher target credit rating requires lower

leverage in the capital structure of the bank, which should lead to lower funding costs

for the bank. The upward pressure on asset prices, however, may not be offset by a

reduction in funding costs, subject to the proportion of retail deposits funding new

business and the sensitivity of retail depositors to changes in the credit rating on bank

debt securities. Further, the philosophy of the centre of the bank with respect to the

relationship between hurdle rates and changes in bank leverage directly influences the

required return on bank assets. These factors interrelate to impact on the risk-adjusted

performance of managers within the bank, yet represent factors that are largely

outside their control.

The remainder of this chapter proceeds as follows. Section 1.2 provides the

background to the study. Section 1.3 describes the central thesis and outlines the

framework for the study. Section 1.4 provides an overview of each chapter and the

key questions addressed therein.

5

1.2 BACKGROUND

1.2.1 Alignment of Economic and Regulatory Capital

Over recent decades, capital adequacy has become the focal point of the prudential

regulation of banking firms. Capital is viewed by bank regulators as a key defence

against financial system instability and a major source of protection for bank

depositors. The requirement that banks hold a minimum level of capital in concert

with the risk in their assets and off-balance sheet activities means that capital has also

served as a regulator of bank asset growth.1

From the perspective of the banking firm, there are two types of capital that must be

measured and managed: ‘economic capital’ and regulatory capital.’ The Basel

Committee of the Bank for International Settlements defines economic capital as the

capital that a bank holds and allocates internally as a result of its own assessment of

risk, while regulatory capital is determined by supervisors on the basis of the Basel

Accord.2 Economic capital is based on the notion that future gains and losses on a

portfolio of credit exposures, over a specified time horizon, can be described by its

probability distribution function. This function forms the basis upon which a bank that

owns the portfolio can assign capital that will reduce the bank’s probability of failure

to a desired confidence level, within a desired time horizon. Economic capital thus

defines risk at a common point (confidence level) in the distribution, where the

confidence level represents the target solvency standard (probability) of the bank.3 In

defining risk in probabilistic terms, economic capital represents a common currency

for risk that allows exposures related to credit risk, market risk and operational risk to

be directly compared across the bank.

1 The Basel Accord of 1988 imposed an 8% capital requirement on assets, adjusted for crude measures of credit risk. In 1995 the Accord was amended to require banks to set aside capital to cover for unexpected losses related to market risk. In the market risk amendment, banks were permitted to use their own models to determine the value-at-risk in their market portfolio, which is the maximum loss that the portfolios are likely to experience with a set probability over a given holding period. 2 Basel Committee on Banking Supervision (2001d), Section 15. 3 For example, a bank that holds sufficient economic capital to protect against losses at the 99.97% confidence level has a 0.03% probability of default, which is about the same solvency standard (default) risk as an AA-rated bond.

6

The solvency standard adopted by a bank forms the link between its internal

assessment of risk and the capital structure of its balance sheet. The economic capital

of the bank is attributed to the difference between the mean of its loss distribution –

expected losses (EL) – and the designated confidence level.4 In this way economic

capital acts to protect the bank against unexpected losses, being downside variations

in the expected loss rate. Figure 1.1 shows a graphical representation of economic

capital within the context of the distribution of portfolio returns for a bank, and

assuming a solvency standard equal to the 99.9% confidence level.

Figure 1.1: Economic Capital

4 Expected losses are typically offset by some combination of margin income and/or provisions. See Bank for International Settlement (2001b), p.40.

Probability

Profits and EL 99.9% Economic capital

Cumulative Distribution Function of Bank Returns

7

In 2004 the Basel Committee of the Bank for International Settlements released a

revised framework for bank capital measurement and standards, which has become

known as Basel II.5 The revised framework was conceived largely as a response to

problems with the original Basel Accord of 1988, and in particular, recognition that

banks had become increasingly able to arbitrage regulatory capital requirements and

exploit divergences between risks measured under the Accord and the true economic

risk in their books.

Under Basel II, banks are permitted a choice between two broad methodologies for

calculating their capital requirements for credit risk. One approach requires banks to

measure credit risk in a standardised manner, supported by external credit

assessments. The alternative approach, which is subject to the explicit approval of the

supervisor of the bank in the country of domicile, allows banks to use their own

internal estimates of various risk components to determine the capital requirement for

a given credit exposure. This approach, known as the ‘Internal Ratings-Based

Approach’ (IRB), is based on measures of unexpected losses and expected losses,

using risk-weight functions to produce capital requirements to cover for unexpected

losses.6 The IRB approach is a point on the continuum between purely regulatory

measures of credit risk and an approach that builds more fully internal credit risk

models developed by banks.7 However, while the revised framework stops short of

allowing the results of such credit risk models to be used for regulatory capital

purposes, the risk weights in the IRB framework are closely calibrated to those used

by ‘sophisticated’ banks in determining their own economic capital requirements. In

this regard, for a given target solvency probability, the risk weights in the IRB

approach are associated with quantifying the volatility of credit losses over a one-year

measurement horizon.8

5 See Bank for International Settlements (2004). An updated version was published in 2005. 6 Bank for International Settlements (2005), p.48. 7 Ibid., p.5 8 Bank for International Settlements (2001b), p.60. The Committee states (p.43) that risk weights are implicitly calibrated so that with a specified minimum probability (the target solvency probability) capital will cover total credit losses.

8

Within the context of the current study, two important observations can be drawn

from the Basel II framework:

1. The IRB approach seeks to make bank regulatory capital requirements for credit

risk approximate economic capital requirements.

2. Regulatory capital requirements have evolved to become directly linked to the

concept of a target solvency probability for a bank.

The second observation follows from the first, given economic capital is measured to

a specified confidence level based on a predetermined solvency standard. This is

reinforced by the Basel Committee, who report that the most important precedent for

indexing capital requirements to measures of risk – and thus to an economic capital

concept – was the Market Risk Amendment to the Accord of 1988, which embodies a

‘Value-at-Risk’ (VaR) methodology to relate capital to a target level of confidence.9

The calibration of risk weights under the IRB approach for credit risk builds upon the

same framework, but with modifications to reflect the characteristics of credit risk.

This means that unexpected losses, and hence the economic capital held by a bank, is

essentially based on a VaR concept of risk.

The implications of these observations for establishing risk congruence within the

banking firm are discussed in the next section.

9 Bank for International Settlements (2001b), p.33.

9

1.2.2 Capital Allocation

Having determined the capital requirements for the bank in the sense of maintaining

capital sufficient to meet a desired solvency standard, the centre of a bank is charged

with the task of apportioning this capital across businesses within the bank in line

with the expected risks in each of their various activities. This process effectively

serves two functions: an ex-ante resource allocation function and an ex-post

performance measurement function. In terms of resource allocation, capital is charged

against disparate activities in order to determine the expected risk-adjusted returns

from these activities, enabling the centre to rank competing uses of capital and direct

the available capital to its most productive uses. In this role, the capital allocation

mechanism also serves as a signalling device to managers, informing them of the risk

implications of each investment decision they are entrusted to make and the impact of

these decisions on the total capital base of the bank.

In terms of the performance measurement function, the risk-adjusted performance of

an investment or activity can be assessed by the centre by comparing the ex-post

profits or gains against the ex-ante capital allocation. This resulting risk-adjusted

performance measure (RAPM) can be compared to a predetermined internal hurdle

rate on economic capital to assess the overall gain to the bank from the activity in

question. Bonuses paid to managers may be linked to the spread between the RAPM

and the hurdle rate, based on the capital invested.10 If profits or gains turn out to be

greater than expected or actual losses lower than expected, then the RAPM should

exceed the hurdle rate, and managers duly compensated in line with the compensation

payment function of the bank.

10 This corresponds to the common concept of ‘Economic Value Added’.

10

The combination of capital allocation and risk-adjusted performance measurement

form a vehicle by which managers are incentivised to make investment decisions that

are congruent with the risk and return objectives of the centre of the bank. The centre,

which acts as an agent for bank stakeholders and a principal to decentralised

managers, can use its position to allocate capital to those activities that are expected to

generate the highest risk-adjusted returns – mindful of bank-wide portfolio

considerations. Managers, in turn, can make pricing decisions that incorporate the

capital being absorbed and the hurdle rate required on capital. Positions carrying

greater risk should receive a higher capital charge, which, in theory, should provide an

efficient pricing signal to managers. For example, if two credit portfolios have the

same face value but one is allocated a higher capital charge than another due to

greater credit risk, then managers will have to set a higher interest rate on the riskier

portfolio in order to achieve the hurdle rate on capital.

This mechanism described above should work well if the risk measure inherent in the

determination of the bank’s economic capital accurately reflects the risk preferences

of bank stakeholders. If it does not, the mechanism may lead to inefficient outcomes.

It is proposed herein that the process of allocating capital and subsequently rewarding

managers based on returns generated on this capital, where the measure of capital is

based on a target solvency standard, does not lead managers to make decisions that

are optimal for bank stakeholders. Indeed, it is argued that internal measures of risk

based on external bank capital requirements have the potential to lead managers to

make decisions that may, perversely, increase the probability of financial distress for

the bank. This arises because the risk attitude implicit in a target solvency standard is

one of risk-neutrality. If bank stakeholders - being creditors, owners and regulators

themselves – have risk preferences that do not conform to a risk-neutral attitude to

losses, then there will be a disjuncture between the risk attitude implicit in the capital

allocation mechanism and the risk preferences of the bank stakeholders. This ‘risk

incongruence’ may lead to inefficient investment decisions within bank firms, in the

sense that managers are guided by capital allocation signals that are not aligned with

the risk preferences of bank stakeholders.

11

1.3 THESIS

The observation that risk attitudes may not be aligned in a banking firm leads to the

central thesis of this dissertation. If incentive-compatibility is to be achieved between

the investment decisions of managers and the risk preferences of the centre of the

bank, then subject to the risk preference function of the centre, risk measures used

within the bank for resource allocation and performance measurement may need to

differ from the measure used to calculate bank capital. This suggests that the

‘assignment’ of risk against positions within a bank may necessarily be unrelated to

the total capital of the bank. This proposition goes against conventional thinking,

which suggests that the total capital held by a bank should be fully allocated across all

businesses and activities (see Merton and Perold (1998), Dowd (1998), Schroeck

(2002) and Perold (2005)), and is based on recognition that measuring risk in terms of

a solvency standard - which is advocated by the Basel Committee and central to the

concept of economic capital - may be considerably misaligned with the actual risk in

bank positions.

It is on this basis that this dissertation examines the question of how to achieve risk

congruence in a decentralised banking firm. We adopt a setting in which the centre of

the bank delegates authority to managers to select and manage credit portfolios. The

centre represents the interests of bank stakeholders. The bank has decentralised

decision-making with respect to portfolio selection in order to allow managers to gain

specialised knowledge on the available set of investment opportunities. This

knowledge includes the expected distribution of returns on the portfolios in the

opportunity set. The centre can only gain access to information on the expected

distribution of returns in each portfolio by investing considerable resources, which is

cost prohibitive. Consequently knowledge of the risks of each portfolio in the

opportunity set is vested in managers. The objective of the centre is to have managers

independently select the portfolios from the opportunity set that the centre itself

would select, if information on the expected distribution of returns on each portfolio

was costlessly available.

12

This setting represents a classical principal-agent relationship, whereby an agent is

engaged by a principal to perform some specific function on behalf of the principal.

This function usually entails the principal delegating some decision-making authority

to the agent. At the heart of agency theory is the notion that agents are rational actors

who seek to maximise their expected utility (Jensen and Meckling, 1976). Agents, by

virtue of their specific knowledge and expertise, gain an advantage over principals

where information is incomplete or costly. This would not be a problem if agents were

dutiful and honest. However if individuals in general are self-seeking and

opportunistic utility maximisers, and their interests differ from those of principals,

then they may misrepresent their information or abilities in order to unduly influence

decisions or outcomes in their favour. The principal’s problem, then, is to devise

appropriate incentive and/or monitoring systems that induce agents to act in

accordance with the desires of the principal. One common mechanism used to align

principal-agent interests is outcome-based compensation, whereby schemes of various

form provide financial rewards based on meeting or exceeding prespecified

objectives. Such incentive schemes will be an attractive option when agents have

significant information advantages and monitoring is costly or difficult.

1.3.1 Incentive-Compatible Risk Measures

We initially assume that managers are inherently trustworthy and have no intention to

misrepresent their private information on portfolio risks. Managers are able to use

their experience and local knowledge to form expectations on the distribution of

returns for each portfolio in the opportunity set, but this information is not available to

the centre. However, once portfolios have been selected by managers, information on

the expected distribution of returns is made freely available to the centre, so that an

appropriate capital charge can be set against each portfolio. This capital charge forms

the basis by which loans in the portfolio can be priced and for measuring the

performance of the portfolios on a risk-adjusted basis.

13

The challenge facing the centre is how to get managers to select the portfolios that are

compatible with the risk/return objectives of the centre. The centre could directly

inform managers of the desired risk/return profile, but this would be cumbersome in a

large decentralised bank, and given the need to describe risk profiles in terms of

probability density functions, somewhat difficult to communicate. The most efficient

solution, it would seem, would be to construct a risk measure that aligns portfolio risk

with the risk preferences of the centre, such that the lowest risk portfolios carry a

lower risk measure. In this setting, this risk measure, which would be used for

determining the capital assignment to each portfolio acts as a signalling mechanism to

managers. Using their information on the expected distribution of returns, managers

can calculate the risk of the each portfolio based on the selected risk measure, which

in turn forms the basis for the denominator of the RAPM. If some proportion of their

remuneration is linked to the ex-post RAPM for the portfolios under their control,

managers should be incentivised to select those portfolios that offer the highest

RAPM. For a given level of expected returns, the portfolios with the highest expected

RAPM would be those carrying the lowest risk.11 This is the basis upon which the risk

measure is deemed to be incentive-compatible.

In order to construct the appropriate risk measure for guiding the portfolio selection

decisions of managers, it is necessary to determine the risk preference function of the

centre of the bank. This is complicated by the fact that the centre is deemed to

represent the interests of a range of bank stakeholders, each of whom can be

considered to carry different (and potentially conflicting) attitudes towards risk. To

the extent that the characteristics of such a function can be described, a mechanism is

needed to evaluate investment alternatives that incorporates the risk preferences of the

centre. Once portfolios can be risk-ordered according to this mechanism, risk measure

candidates can be tested for incentive compatibility. A diagrammatic representation of

this process is presented in Figure 1.2.

11 Expected portfolio returns may be identical, but the distribution of gains may differ subject to the reference point used for measuring gains and losses. This reference point could, for example, be the expected value of the portfolio or the face value of the portfolio. This is later taken into consideration in the incentive-compatible RAPM framework.

14

Stochastic dominance criteria are selected for the risk-ordering mechanism. Stochastic

dominance is an attractive evaluation mechanism because it does not require a full

parametric specification of the preferences of decision-makers – investment

alternatives can be ordered by risk without having to specify the exact form of the

utility function of the investor.12 This suits our needs well given the difficulties in

specifying the exact form of the utility function for the centre of the bank when it

represents a diverse range of stakeholder interests.

Figure 1.2: Portfolio Selection Process

12 Stochastic dominance is consistent with a broad range of economic theories of choice under uncertainty, including expected utility theory and non-expected utility theories, such as prospect theory. This is discussed in chapter two.

Bank Risk Preference

Function

Portfolio Risk-Ordering

Mechanism

Incentive Compatible Risk

Measures

15

1.3.2 Agency Problems and Solution

Next we consider how agency problems impact on the incentive-compatible risk

measurement framework by removing the assumption that managers are trustworthy

and have no intention to misrepresent their private information to the centre. In line

with the predictions of agency theory, it is assumed that managers are self-serving and

opportunistic. Under these conditions, managers are prepared to utilise information

asymmetries to overstate gains and understate risks in their portfolios, in order to

inflate the ex-post RAPM upon which their bonuses are based. The implications are

twofold: an inefficient allocation of resources within the bank, and a high probability

that the bank is undercapitalised with respect to risk. We also question the ability of

internal RAPM to provide a consistent and congruent measure of risk when hurdle

rates used for assessing performance are based on estimates of the cost of equity

capital using the capital asset pricing model (CAPM). This approach assumes that

bank-specific risks do not impose real costs on the bank. It is concluded that an

internal hurdle rate based on the CAPM understates the true cost of economic capital

to the bank.

A solution is devised to remedy these problems. The solution is based on the concept

of an internal market for the allocation of risk capital, and integrates an auction

mechanism with the compensation function to provide a truth-revealing design. In this

design, we draw on the concept of the revelation principle (Myerson, 1979, 1981),

which states that for any mechanism where agents may be induced to be dishonest in

equilibrium, there exists a direct, incentive-compatible mechanism where agents can

be induced to report their information truthfully. The core of the internal market is a

dynamic bidding process under which managers are periodically required to place

bids for the capital needed to support their risky activities. The objective of this

mechanism is to have managers truthfully reveal their private information regarding

current and future volatility in their proposed positions and the expected return on

these positions. In terms of the RAPM framework, payoff structures in the

compensation payment function are designed such that it is in the interests of agents

to truthfully report their risk expectations when bidding for capital. It is proposed that

16

risk congruence between the centre and managers is achieved if managers cannot

personally gain by misrepresenting their information to the centre of the bank. These

properties are demonstrated in the mechanism.

It remains finally to examine the determination of economic capital and its

implications for pricing and performance measurement from a whole-of-bank

perspective. It is proposed that is not possible to separate portfolio pricing decisions at

the level of managers from decisions made by the centre of the bank with respect to

the target credit rating for the bank, the internal hurdle rate and assumptions made by

the centre with respect to diversification across bank assets. This is addressed within

the context of an examination of the conditions under which it is valuable to a bank to

increase its solvency standard. Yet again, the basis for determining the internal hurdle

rate features as a distinguishing factor. The conclusions lead to question the relevance

of remunerating managers based on RAPM, when a significant array of factors

impacting on both the numerator and denominator of the equation are outside of their

control.

1.4 CHAPTER OVERVIEW AND RESEARCH QUESTIONS

The dissertation proceeds as follows. Chapter two provides the theoretical

underpinnings for determining an optimal class of incentive-compatible risk measures

for the banking firm. It addresses the vexing question of what is the relevant risk

preference function for a banking firm by examining the literature on models of the

banking firm and empirical studies on organisational risk-taking. The chapter provides

the framework by which a set of risk measure candidates are assessed for their

compatibility with the risk preference function of the centre.

17

Chapter three provides a practical application of the framework from chapter two by

examining the incentive-compatible properties of the set of risk measure candidates

against five hypothetical credit portfolios. Each portfolio has the same expected value,

but different probability density functions, allowing exclusive focus on risk. The case

for coherency of risk measures, when used as internal allocation and performance

measurement, is examined. The chapter also analyses the impact on incentive

compatibility conditions of the structure of the compensation payment function of the

bank. In particular, consideration is given to portfolio selection when the distribution

of gains is incorporated into the RAPM process. That is, while the expected values of

the portfolios under examination are identical, the distribution of gains is not. An

evaluation tool that incorporates the risk attitude of the centre towards the distribution

of expected gains is provided.

Chapter four examines how agency problems compromise the integrity of the risk

assignment and performance measurement framework in the banking firm. While risk

measures may be congruent with the preferences of the centre, managers may take

advantage of information asymmetries to misrepresent their positions. The chapter

also examines how agency issues impact on the formulation of the internal hurdle rate

in the bank. It is argued that the basis upon which banks typically derive their hurdle

rates underprices the risk in their economic capital.

Chapter five presents a solution to the agency problems identified in chapter four,

based on an internal market in which managers are required to bid for risk capital to

support their activities. The design integrates an auction mechanism into the

compensation payment function. It is demonstrated that this mechanism is incentive-

compatible, subject to a set of general conditions.

18

Chapter six examines how bank-wide decisions regarding hurdle rates, target solvency

standard, loan ratings and funding mix impact on the pricing of bank assets. A number

of scenarios are presented showing the interaction of these factors on the pricing of

bank assets for various bank target credit ratings. It is considered how these factors

impact on the decision-making of managers and the subsequent measurement of their

performance.

Chapter seven provides a conclusion to the dissertation and identifies areas fruitful for

further research.

Table 1.1 provides a summary of the key research questions for this study on a

chapter-by-chapter basis.

19

Table 1.1: Specific Questions for the Study

Questions Chapter

What is the risk preference function of the centre of the bank?

Is there a methodology that can be used to rank portfolios by risk in accordance with the risk preferences of the centre?

Is there a risk measure (or measures) that provide a risk-ordering consistent with the risk preferences of the centre?

Two

Does the internal risk measure need to be coherent in terms of the structural properties identified by Artzner et al (1999)?

Does the structure of the bank compensation payment function impact on incentive-compatibility conditions?

Does the choice of target threshold for gains and losses impact on portfolio selection?

Should the risk attitude of the centre towards the distribution of gains feature in internal risk measures?

Three

How do agency problems impact on the robustness of the risk-adjusted performance measurement framework?

Should internal hurdle rates reflect a total-bank risk perspective or a systematic risk perspective?

Is a fixed hurdle rate consistent with a fixed probability of default (solvency standard)?

Four

Can incentive-compatibility be achieved in the face of information asymmetries between the centre and managers?

Five

How does the target credit rating of the bank influence portfolio selection and pricing?

When is a higher solvency standard beneficial to a banking firm?

Should hurdle rates adjust in line with changes in the target credit rating of a bank?

Six

20

Chapter Two

The Principal’s Dilemma:

The Bank Risk Preference Function and

Portfolio Selection

“A man who seeks advice about his actions will not be grateful

for the suggestion that he maximise expected utility”

A. D. Roy, 1952

21

2.1. INTRODUCTION

This chapter is the first of two that examine the design of incentive-compatible risk-

adjusted performance measures in a banking organisation, where optimality is

achieved through alignment of the goals of the principal and the actions of agents with

respect to investment decisions made by agents. The principal in this setting is

embodied in the notion of the ‘centre’ of the bank (board or asset/liability

management committee), delegated to act in accordance with the risk/return

requirements of the bank’s investors (creditors and owners) and within external

constraints established by bank regulatory authorities (regulators). The agents in this

setting are managers within the bank who are responsible for the selection and

management of bank assets. The focus in these chapters is credit portfolios, given the

considerable proportion of bank assets that comprise credit portfolios. Information on

the expected distribution of returns on these portfolios is asymmetrical and privately

held by managers, who are specialist lending managers. It is cost prohibitive for the

centre to screen the probability distributions of the entire set of portfolios available to

managers for investment – the centre only receives information on the distribution of

returns on the portfolios actually selected by managers. We initially assume that

managers have no incentive to misrepresent this information. Our task is to determine

which risk-adjusted performance measure or measures, when applied, promote an

efficient solution whereby agents select the portfolios that the centre of the bank

would itself select if information on the distribution of portfolio returns was perfectly

available.

Within this setting, there are a number of factors that should be established:

1. The relevant risk preference function for the centre.

2. The risk attitudes implicit in various risk-adjusted performance measures.

3. The question of alignment between the actual economic capital held by the

bank and the risk measure used internally for performance measurement.

4. The influence of targets/aspiration levels on the efficiency of the risk-adjusted

performance measure.

22

These factors are inextricably linked. The risk preference function for the bank

determines the feasible set of portfolios for the centre and establishes the relevant risk

measure for capital allocation, pricing and performance measurement. However,

determining an organisational risk preference function is complicated by the fact that

the centre itself is an agent representing multiple interests – bank owners, creditors,

managers and regulators – each of whom carry potentially conflicting risk attitudes.

As a case in point, in the event that the bank defaults on its debt, some stakeholders

may be less concerned with the magnitude of losses than other stakeholders. We can

consider that the economic impact of default on owners and managers will be largely

invariant to the size of actual losses, with costs to these stakeholders a function of the

event of default itself. Managers face loss of employment regardless of the size of

default, while losses to owners are capped by the institution of limited liability.13 In

contrast, the economic impact of default on regulators and creditors is more directly

related to the size of losses in the event of default. This means risk measures based on

the probability of default are likely to be of more relevance to managers and

stakeholders, while measures linked to losses in the event of default may be more

relevant to regulators and creditors. This has implications for an incentive-compatible

risk-adjusted performance measurement framework in the sense that the centre of the

bank represents stakeholders who may carry different perspectives on risk or tolerance

to unexpected losses. It places focus directly on the question of the appropriate risk

preference function for the centre of the bank.

Directly related to the above is the capital attribution policy of the bank. It is a

premise of this study that the capital attribution policy of the bank impacts on the

efficiency of the internal risk-adjusted performance measure. While all banks must

hold capital equal to the minimum regulatory requirement, actual capital held by

banks is typically linked to a target credit rating, which is in turn determined by the

probability of default. If actual (economic) capital held by the bank is allocated

13 Managers and owners may be concerned with the size of losses if they are to influence the decision to liquidate versus restructure the bank. In the latter case, owners may be able to recover some proportion of their initial investment, and managers retain their employment, depending on the nature of the restructure and the subsequent fortunes of the bank.

23

against the positions/portfolios held by managers, and performance measured against

this capital base, it could be held that bank economic capital is driven more by

external forces such as the views of ratings agencies, or the prerogative of senior

executives in the bank, rather than a disciplined and consistent analysis of risk based

on the full distribution of potential outcomes – both upside and downside.14 Subject

to risk measurement methodologies and the structure of compensation packages used

to remunerate employees within the bank, managers and traders may be incentivised

to take on higher risk portfolios than deemed appropriate by the centre because a

capital charge based on the actual capital held by the bank – in turn based on target

credit rating – is based on default probability and hence invariant to the magnitude of

potential losses. The basis upon which the risk measure is formulated is thus critical to

aligning the interests of principals and agents within the bank. It will be argued that

the goal of incentive-compatibility may not be achieved if a bank mandates that risk-

adjusted performance measures must be linked to actual capital held by the bank. We

argue that to avoid perverse outcomes, it may be entirely appropriate to use a risk

measure for performance evaluation that is different to that used as the basis for

measuring the actual capital held by the bank.15 While this view appears to go against

conventional thinking that the actual capital held by the bank must be allocated across

all businesses and positions, it is based on recognition that the actual capital held by a

bank is largely determined exogenously and may be misaligned with the actual risk in

positions taken by individuals within the bank.

14 This also has implications for the pricing of bank assets, to the extent that the bank prices to earn a minimum hurdle rate on allocated economic capital. 15 The actual capital held by the bank may not match capital allocated to business units due to diversification benefits across business lines, products or portfolios. This will particularly be the case where actual capital matches the regulatory requirement, because regulatory requirements do not capture diversification benefits across businesses. A bank may find that according to its internal models, economic capital may be less than regulatory capital due to diversification across the businesses. In this scenario, some capital may remain unallocated and the bank may be earning less than the hurdle rate while the business units are earning the hurdle rate on allocated capital. In any event, from a performance measurement perspective, managers should not be rewarded for diversification benefits in their businesses (through adjustments to capital charges) unless their actions can be directly attributed to the creation of these benefits. To the contrary, diversification benefits across businesses are more likely due to macroeconomic/global factors or the business mix determined at the centre of the bank, rather than at the level of divisions, business units or individual managers.

24

An additional consideration in the design of an incentive-compatible risk

measurement framework is how the risk attitude of managers may be influenced by

the targets set by the centre, or aspiration levels based on past performance. The

efficiency of the risk-adjusted performance measure may be compromised if the

measure is target dependent, where changing the target may unduly influence

investment decisions, or if it is possible for the risk appetite of managers to change

over the measurement period subject to perceptions of performance relative to

aspiration levels. If targets play a role in guiding the behaviour of employees in

organisations, it is necessary to consider how targets influence risk-taking on the part

of managers in the bank setting described above. A related factor is the structure of

compensation payments made to employees, which typically pay some form of bonus

on the realisation of target. While we may be able to determine a risk measurement

framework that aligns the interests of bank stakeholders, we need to determine if the

manner in which performance is remunerated compromises this framework. If the

payment function upon which managers are remunerated is asymmetric, with the risk

measure in the denominator of the risk-adjusted performance measure based on the

distribution of losses beyond some target threshold while the bonus/reward

component applies only to realisations above target (there is no penalty for

realisations below target), the possibility exists for managers to select portfolios that

are not aligned with the goals of the centre of the bank. In this regard, we consider

how the specific utility functions of managers influence credit portfolio choice.

We separate our examination of these issues into two chapters. The current chapter

provides the theoretical underpinnings for determining the optimal class of risk

measures for the bank, while the following chapter applies this framework using an

example that consists of five credit portfolios. The following chapter also extends the

analysis to consider issues related to the coherency of risk measures and the structure

of the bank compensation payment function.

25

The current chapter has three principal objectives. The first is to determine the

appropriate risk preference function for the centre of the bank in order to determine

the set of feasible portfolios for investment. As discussed, this is complicated by the

fact that the centre must represent the disparate interests and potentially diverging risk

preferences of bank stakeholders. We review literature on models of the banking firm

to gain insight into this question. The second objective of the chapter is to establish a

risk-ordering methodology that allows portfolios to be ranked in a manner that is

consistent with the risk preferences of the centre of the bank. The third objective is to

determine which risk measures, from a given set of candidates, are compatible with

this risk-ordering methodology. To achieve this objective, we seek to categorise risk

measures in terms of their implicit risk attitudes.

The chapter is structured as follows. Section 2.2 considers the relevant risk preference

function for the centre of the bank given the diverse range of stakeholders that the

centre is deemed to represent. We conclude that this function is best characterised by

non-satiety, risk-aversion and positive skewness in the distribution of returns. Section

2.3 discusses the framework by which the risk-ordering of credit portfolios is

established. We determine that portfolios that are dominant by third-order stochastic

dominance principles are compatible with the risk preference function of the centre.

We consider if the bank risk-preference function should include a convex segment,

conversant with the predictions of non-expected utility theory, and draw on recent

empirical studies in this area to gain insight into this question. We also consider the

implications if agents (managers) make decisions in accordance with the predictions

of non-expected utility theory, and draw on recent literature in this area. Section 2.4

provides the framework from which we assess which risk measures are compatible

with the risk preferences of the centre, such that they would lead managers to select

the portfolios that the centre would have them select if it was aware of the full

opportunity set available to managers. We categorise risk measures in terms of their

implicit risk attitudes, and apply the framework to five risk measurement candidates.

Section 2.5 provides a summary of the main findings of the chapter.

26

2.2 THE PRINCIPAL’S DILEMMA: A BANK RISK PREFERENCE

FUNCTION

In order to assess the congruency of a particular risk-adjusted performance measure –

this being the measure which would incentivise agents to select portfolios that are

compatible with the risk/return objectives of the centre – it is necessary to determine a

risk preference function for the centre. A risk preference function is a mathematical

formulation that enables an investor to rank portfolios according to specific

objectives. The most common form of risk preference functions are utility functions,

which can be used to model the subjective risk attitudes of individuals while

satisfying various axioms regarding consistent and rational behaviour on the part of

these individuals. Most of the literature on investment choice under uncertainty

assumes that decision makers are risk-averse. In terms of utility theory, this implies

that decision makers have a utility function that is uniformly concave. Extending this

from the level of individuals to that of an organisation requires researchers to make

the implicit assumption that the risk preferences of individual stakeholders can be

aggregated into a relatively simple and unique organisational utility function. The

question is can a risk preference function be derived for the centre of a banking

organisation given the coalition of potentially conflicting interests – owners, creditors/

depositors, regulators, managers - that the centre is required to take into consideration

when searching for an optimal balance of risk and return in bank assets? The risk

preference function of the bank has to be determined against a backdrop of multi-

dimensional information asymmetries: the private information of managers on the

distribution of portfolio returns, and the potentially disparate risk incentives of

owners, creditors, regulators and managers.

A sizeable literature examines models of the banking firm, motivated largely by

interest in the impact of capital regulation on bank behaviour. In this section we

review this literature in order to gain insight into what may be an appropriate risk

preference function for a bank.

27

Papers that model the banking firm typically start with the assumption that the

banking firm aims to maximise an objective function in terminal wealth, subject to

regulatory constraints that restrict the bank’s opportunity set of assets and liabilities.

From the first derivative of the objective function, more wealth is preferred to less. On

this point, the literature is generally consistent. However, views tend to diverge with

respect to the second derivative of the objective function. Some papers view the

banking firm as an expected value/profit maximiser, consistent with a linear objective

function in terminal wealth, while others view the bank as a risk-averse investor,

consistent with a concave objective function in terminal wealth. This is overlaid with

different views on the relationship between equity investors and bank management.

Some papers view this relationship as unitary, where banks are owned and managed

by the same agent. Risk aversion arises in this context because the owner-manager

cannot completely diversify risk away. Other papers recognise a separation between

owners and managers and conclude that limited liability leads to a risk-seeking

preference on the part of owners, while at the same time managers may be considered

risk-averse to the extent that their wealth is tied to bank-specific human capital. We

evaluate these disparate views below. We also consider the objective function for

bank regulators and bank creditors.

Those papers that employ a linear objective function for the bank do so under the

assumption of frictionless and complete markets, under which investors and

borrowers are able to perfectly diversify their risks and costlessly recapitalise the bank

in the case of insolvency. Bank investors in their model are deemed to be risk-neutral,

seeking to maximise the expected profits of the bank [Hester and Pierce (1975),

Kareken and Wallace (1978) and Crouhy and Galai (1986)] In this context, the

opportunity set of the investor spans that of the bank, and in effect, the bank need not

exist. Any portfolio that the bank selects can be replicated or hedged by the investor.

Bank owners care only about the systematic component of total risk, which is

appropriately priced in their required returns, since they can perfectly diversify their

portfolios to compensate for business risk in the bank.

28

An alternative stream of papers remove the assumption of complete markets and view

banks as risk-averse, expected utility maximisers. In these models, the objective

function for the bank is concave. Papers by Kahane (1977), Koehn and Santomero

(1980), and Kim and Santomero (1988), which are typical of this approach, analyse

risk-taking in banks as a portfolio management problem for a risk-averse owner-

manager whose entire net worth is invested in bank. Risk aversion arises because the

owner-manager cannot completely diversify his risk, and as such, is directly exposed

to the asset portfolio risk and leverage of the bank. These papers find that the

imposition of a fixed capital requirement by regulators forces the bank to reduce its

leverage and reconfigure the composition of its asset portfolio towards riskier assets

as owner-managers aim to compensate for the loss in utility arising from the reduction

in bank leverage.

A bank, however, need not be operated by a single owner-manager for risk-aversion

to be incorporated in the objective function. If there is a separation between bank

owners and management, and the latter is responsible for decision making, the bank

may act in a risk-averse manner to the extent that managers are unable to diversify

their human capital. In an early paper, Shavell (1979) finds that if owners are risk

neutral but managers are risk-averse, under a Pareto optimal incentive contract

managers will not operate to maximise the profits of the firm. If the utility of

managers is directly linked to the returns of the bank, the risk-taking incentives of

managers will decrease and the optimal degree of risk taking is likely to be less than

that desired by bank owners. Later papers by O’Hara (1983) and Benston et al (1986)

show that the costs to managers associated with losing their employment can induce

risk-averse decision-making. Managers seek to reduce the variability of the earnings

stream of the firm to reduce the probability of bankruptcy or if their compensation is

linked to the earnings of the firm (Holmstrom, 1979). Dewatripont and Tirole (1993a)

assert that bank management will act in a risk-averse manner in order to smooth bank

earnings streams because this reduces the probability of interference by external

parties such as creditors, owners and regulators. The basis of their argument is that

managers dislike their projects disrupted or altered because they either enjoy private

benefits whilst their projects are active, or they receive high monetary rewards if the

29

projects they start are pursed. The latter arises because continuation of projects yields

a fatter upper tail for the distribution of profits.

Empirical studies that attempt to measure the risk preference of bank stakeholders are

not large in number. One study, however, does provide evidence that managers in

banks are more risk-averse than owners. Saunders, Strock and Travlos (1990)

examine the relationship between bank ownership structure and risk taking,

hypothesising that managerially-controlled banks take less risk than stockholder-

controlled banks, and that these differences become more pronounced during periods

of financial deregulation. In a similar vein to previously discussed papers, they base

their hypothesis on the proposition that managers will act on a risk-averse rather than

a value-maximising manner to the extent that their wealth is largely in non-

diversifiable human capital that is bank specific. Using capital market measures of

bank risk and the proportion of stock owned by managers (as a proxy for ownership

structure), they find empirical support for the hypothesis that stockholder controlled

banks take more risk than manager-controlled banks. They conclude that regulators

should allocate a greater proportion of their resources toward monitoring stockholder,

rather than managerially-controlled banks – that is, ownership structure should be

used as a criterion for determining examination frequency.

Besanko and Kanatas (1996) present a model of the banking firm where bank

managers own only a fraction of the stock of the bank and take unobservable actions

that maximise their own welfare but which may be against the interests of bank

owners. They analyse the outcomes in a setting of more stringent risk-adjusted

regulatory capital standards, and find that while managers may weight their asset

portfolios towards lower risk-weighted assets (positive asset substitution) under such

standards, they may, at same time, provide less effort in the management of these

portfolios.16 This effort-aversion moral hazard arises in their model because higher

capital standards require the issue of new equity, which in turn dilutes the proportion

of equity held by insiders (managers). They use this as a potential explanation for the

16 In this context, effort is viewed by the Besanko and Kanatas in terms of the commitment of senior management in monitoring and supervising loan officers in their evaluation and screening of loan applicants and the termination of underperforming employees.

30

decline in the stock price of banks when new equity issues are announced. Their

findings support those of Saunders, Strock and Travlos (1990) in so far as banks with

a lower proportion of managerial ownership may be riskier, but the distinguishing

characteristic of their model is that managers provide less effort when their stake in

the bank is diluted, and this in turn increases overall risk for the bank. They conclude

that in certain cases this negative impact on the bank’s solvency arising from less

effort on the part of managers outweighs the asset substitution effect arising from

higher capital standards. The conclusion that managers reduce their effort when their

stake in the bank is reduced, thereby increasing the probability of bank insolvency,

runs counter to the previously discussed view that the costs to managers of losing

their employment can induce risk-averse behaviour in banks.17

Those papers that use a concave bank objective function do so on the basis that

managers who make decisions within the bank are risk-averse. The use of a concave

objective function in models of the banking firm has been criticised by Keeley and

Furlong (1990) and Rochet (1992) on the grounds that the limited liability option of

bank owners should be incorporated into the objective function. Merton (1974) was

the first to recognise that limited liability amounts to an option that allows the owners

of a firm to put the assets of the firm to debtholders when the value of the debt

exceeds that of assets. In addition, Merton (1977) shows that a system of fixed-price

deposit insurance results in a put option subsidy to bank owners, the value of which

increases with bank risk. If risk-insensitive deposit insurance exists or regulators are

perceived to implicitly or explicitly guarantee the value of bank deposits, and limited

liability means bank owners are indifferent to the distribution of losses beyond

insolvency, then shareholder value is maximised by increasing the variance of returns

in bank assets as much as possible.18 In this setting, the payoff to bank owners is a

convex function of the return from investment, implying owners prefer that the bank

acquire higher risk to lower risk assets. Indeed, Rochet (1992) shows that when the

17 If managers reduce their effort when their stake in the bank is low, and increase the risk of bank insolvency in the process, then the current practice of including stock or stock options in the remuneration packages of managers and senior executives in banks may be justified. 18 The incentive for owners to increase the risk of the bank may also be driven by the realisation that debtholders and depositors can only monitor and control owner’s actions imperfectly. Indeed, if depositors believe they are protected by the regulator, they will have little incentive to monitor the actions of managers or bank owners.

31

limited liability of owners is taken into account and bank capital requirements are set

exogenously, the convexity of preferences due to limited liability may dominate risk

aversion.

The ability of bank owners to maximise the value of their limited liability option by

increasing the variance of returns in bank assets depends on the risk preferences of

bank managers and on the constraints imposed by regulators. While the focus of

owners may be volatility risk in the bank, the focus of regulators and bank creditors is

survival risk. If management and ownership are separate, owners must somehow force

management (or the centre of the bank that sets incentive structures for managers) to

operate the bank for their benefit. The question, then, for the centre of the bank is to

what extent should attaining imposed regulatory constraints take precedence over the

preferences of owners?19 This is fundamental to determining the appropriate risk

preference function for the bank.

Fortunately, we may not have to answer this question. If banks possess high franchise

value, the benefits derived by owners and managers may provide sufficient incentive

for bankers to hold capital above the regulatory minimum, and manage and diversify

portfolio risk in order to reduce the probability of insolvency. This suggests that high

franchise value would encourage risk-aversion on the part of owners and managers,

and align their interests with those of regulators. This view is presented by Marcus

(1984) and Keeley (1990), who argue that franchise value restrains moral hazard on

the part of bank owners. Demsetz et al (1996) also observe a positive association

between capitalisation (franchise value) and the propensity of the bank to take risks.

Franchise value is represented by the capitalised stream of above normal profits that

may arise in banks from a number of sources, such as regulatory safety nets, oligopoly

rents, strong customer bases, valuable lending relationships or efficiency gains

harnessed from new technologies. If franchise value is high, banks may have little

need for regulatory requirements to reduce the probability of insolvency. This would

19 In a complete markets setting, it could be argued that the price of bank equity would perfectly incorporate imposed regulatory constraints. It appears that researchers have not attempted to measure the premium, if it exists, that regulatory constraints place on the required return for bank equity.

32

be the case if the value of the franchise to owners exceeds the put option value of

limited liability and deposit insurance.

Milne and Whalley (2001) argue that the basic model of bank moral hazard emerges

when bank franchise value (expected future income) is low. They find that bank

behaviour depends upon the buffer of capital above the regulatory minimum, not the

total level of capital. Banks with low franchise value have low expected earnings or

growth opportunities to protect against a decline in earnings, and as such, have a high

probability of failure. They assert that these banks have little incentive to hold

adequate capitalisation, and are more inclined exploit moral hazard by investing in

riskier bank assets. Conversely, banks with high franchise value have high expected

future earnings and growth opportunities, and have an incentive to maintain

substantial capital buffers to protect the value of the franchise should the bank be hit

by large unexpected losses.

Bigg (2003) presents a contingent claims model of a bank that suggests a U-shaped

relationship between charter (franchise) value and risk. The predictions mirror that of

Milne and Whalley (2001) for low franchise value banks, where potential gains from

exploiting the regulatory safety net outweigh the potential erosion of franchise value

in the event of insolvency. As franchise value rises, banks have a greater incentive to

preserve expected future rents by adopting lower risk strategies. However, Bigg

predicts that highly capitalised banks with low risk of insolvency will not gain from

reducing risk further and will more inclined to increase wealth by engaging in risk-

shifting activities. Bigg tests the predictions of the contingent claims model using data

of ten Australian banks over the period 1992-1997. Using Tobin’s q as a measure of

franchise value and various measures of risk (share price volatility, leverage risk and

portfolio risk), Bigg finds a negative relationship between franchise value and risk for

the lower franchise value banks in the sample, although all banks exhibit strong

positive franchise value over the period of the sample. However, at very high

franchise value banks, which are also more highly capitalised, franchise value is

found to be ineffective in eliminating risk-shifting. These empirical findings support

33

the hypothesis that the relationship between franchise value and bank risk is U-

shaped.

If managers and owners derive equal benefits by preserving or improving the

franchise value of a bank, then there should be incentive compatibility between risk

preferences of owners and the decisions made by managers with respect to the risk in

bank assets. Managers may derive benefits from well-remunerated careers and job

security and as discussed earlier, act in a risk-averse manner. However, if there are

agency conflicts between owners and managers, the latter may still make decisions

that maximise their private benefits (utility) at the expense of bank owners. For

example, managers may lower the price of bank products or services in an attempt to

increase the market share of the bank in order to increase their budgets or perquisites,

while not pricing to adequately cover for risk. Managers may also appropriate part of

the profits of the bank by paying themselves high salaries, recruiting excessive staff,

or by failing to adequately monitor changes in the risk profile of the bank. Credit

officers, for example, may face incentives to refinance delinquent loans or capitalise

unpaid loan balances in order to present a more favourable picture of performance.

Further, managers may become excessively risk-averse when performing above some

predetermined benchmark or target in order to preserve bonuses, while at the same

time becoming risk-seeking when performing below target in order to avoid reporting

losses or missing on bonuses (gambling for resurrection). This changing risk appetite

may be a driven by the structure of compensation contracts presented to managers and

employees. The implications of these agency conflicts for the construction of

incentive-compatible risk measures are examined later in this chapter and in more

detail in the next chapter.

Before concluding this section, it is necessary to consider the objective function for

the bank regulator. Regulators face a trade-off when determining the optimal amount

of capital a bank should hold. Too little capital impairs insolvency, and increases the

value of the implicit call option held by shareholders over the bank’s assets. High

regulatory capital requirements, however, impose costs inefficiencies on banks and

provide incentives for bankers to arbitrage regulations in order to maximise returns on

34

capital. In addition, Dimonson and Marsh (1995) note that high capital requirements

may act as a barrier to entry in banking, restricting competition. A socially optimal

default probability resolves the trade-off between protection against losses and the

preservation of bank efficiency.

Daripa and Varotto (2004) argue that the objective function for the regulator needs to

resolve the trade-off between safety loss and overprotection. There is a ‘loss of safety’

if the actual risk in bank assets exceeds the socially optimal level because the

probability that the bank defaults exceeds the regulatory optimum. Alternatively, there

is an ‘overprotection loss’ if the actual risk falls below the optimal risk. An

overprotection loss principally penalises bank owners if their risk preference function

is convex, while a safety loss penalises depositors and/or regulators, who carry a

concave preference function. The authors propose that a regulatory objective function

allows for the regulator to place different weights on the interests of shareholders and

depositors, although they provide no empirical indication as to the potential size of

these weights. Pointing to the free-rider problem with respect to monitoring banks that

arises when banks have many small and dispersed depositors, Dewatripont and Tirole

(1994) argue that protection against safety loss should be the main goal of bank

regulation. These depositors do not have the information necessary to perform

efficient monitoring. The presence of systemic risk from bank failure also supports

that a greater weight be applied to protection against safety loss.

Jaschke (2002) argues that banking supervision should aim to minimise expected

losses in the event of bankruptcy because depositors, contributors to deposit

insurance, creditors and potentially tax payers are those who must bear the losses that

exceed the capital base of a bank in the event of bankruptcy. In a similar vein,

Guthoff, Pfingsten and Wolf (1998) argue that while it is difficult to derive a formal,

operational objective function for bank regulators, it can be determined that there are

some portfolios that will normatively never be preferred over others by bank

regulators. Using the concept of efficient sets, they assert that a regulatory authority

would always prefer a bank to invest in portfolios that have less weight in the left tail

of the distribution of asset returns because these portfolios are less likely to expose the

35

bank to large losses in the event of default. If the concern of regulators is severity of

potential losses, a regulatory objective function based on a socially optimal

probability of default – in the spirit Daripa and Varotto (2004) – may not be an

appropriate representation of the risk preference function of regulators.

What conclusions can we draw from the literature with respect to the risk preference

function for the centre of the bank?

If there is contention in the literature, it revolves in the main around the risk

preference of bank owners. Those who consider bank shareholders to be risk neutral

(linear objective function) assume that financial markets are frictionless and complete.

While this may be a useful assumption for the purposes of theoretical modelling, it is

not an accurate representation of the markets in which banks operate. When limited

liability and the regulatory safety net are taken into consideration, bank owners may

have a convex risk preference function and prefer higher variance in bank asset

returns. However if the bank possesses franchise value, being the present value of

expected future above normal profits, bank owners may prefer that the bank acts in a

risk-averse manner in order to preserve the associated benefits. In this case the

objective function for the bank would be concave. Much comes down to the extent to

which the value of the bank franchise exceeds the combined put option value of

limited liability, deposit insurance and/or the regulatory safety net. If the value of the

franchise to bank owners exceeds the value of the put option, we can conclude that

owners will be risk averse and a concave preference function applies.20

From the perspective of bank regulators and creditors (depositors and debt-holders),

we conclude that a concave risk preference function also applies. We determine that

regulators and creditors are concerned not with the probability of default when

assessing risk, but rather, expected losses in the event of bank insolvency. This is

20 Longley-Cooke (1998) asserts that for incorporating risk into the measurement of the financial performance of a publicly traded financial institution it is reasonable to use the risk aversion of its shareholders. He shows that analysis of total returns on large company stocks compared to yields on one year Treasury bills, from 1950 to 1995, produces a risk aversion parameter of 5.7 (p.92). Bodie et al (1996) cite that that a broad range of studies place the degree of risk aversion of the representative investor in the range of 2 to 4 (p.187).

36

because the economic impact of default on regulators and creditors is linked to the

size of losses. At the same time the impact of insolvency on bank owners and

managers is less sensitive to the size of losses because the value of equity should

already be minimal upon insolvency and managers are likely to have lost their

employment. Bank owners and managers are thus more likely to be concerned with

unexpected losses up to the predetermined target solvency standard.

37

2.3 RISK RANKING CRITERIA

In situations where there is complete information on preferences, a complete ordering

of alternative investments can be undertaken based on the expected utility function of

the investor. In this setting, those portfolios with the highest expected utility are the

dominant portfolios. The papers reviewed in the previous section that embody

concavity in the objective function of the banking firm typically employ quadratic or

exponential forms of the function. It seems, however, that few banking firms have the

willingness or means by which to parameterise their own utility function - perhaps

reflecting the dominance of regulatory constraints over the risk preferences of owners

and managers. With incomplete information on the exact form of the utility function

for the banking firm, we can only determine a partial ordering of the available

investments.

Stochastic dominance is a generalisation of utility theory that eliminates the problem

of having to explicitly specify the utility function of the investor.21 The central idea of

stochastic dominance is that the decision problem can be simplified by sorting out and

eliminating dominated alternatives. Stochastic dominance converts the probability

distribution of an investment into a cumulative probability curve, which is used to

determine the superiority of one investment over another. Stochastic dominance

criteria provide a set of rules for making choices among risky assets consistent with

the preferences of broad classes of utility functions, obviating the need to know the

precise functional characterisation of the objective function. Different orders of

stochastic dominance correspond to different classes of utility function. We outline

the selection criteria that apply to each order of stochastic dominance below, and

assess the applicability of the assumptions for each order for the risk preferences of

the banking firm established in the previous section.

21 The contemporary notion of stochastic dominance has its roots in papers by Hadar and Russell (1969), Hanoch and Levy (1969) and Rothschild and Stiglitz (1970).

38

2.3.1 First-order Stochastic Dominance

First-order stochastic dominance (FSD) provides a rule for rank-ordering risky

portfolios in a manner consistent with the preferences of investors who prefer more

wealth to less. A portfolio stochastically dominates another portfolio by FSD if

investors receive greater wealth from the portfolio in every ordered state of nature.

This means the only requirement for FSD is that utility functions are increasing: FSD

does not encompass the risk attitude of the investor.

Let F and G represent the cumulative probability distributions of the returns for

portfolios X and Y, and let U(w) refer to the utility of w dollars of wealth. Under the

FSD selection rule, portfolio X will stochastically dominate portfolio Y if

Fx(w) � Gy(w)

for all w with at least one strict inequality.22 Alternatively,

[Gy(w ) - Fx(w)] ✁ 0

for all w with at least one strict inequality. This means the cumulative probability

distribution for portfolio Y always lies to the left of the cumulative distribution for

portfolio X. Further, for investors to prefer more wealth to less, the utility function

must be increasing monotonically. This implies a positive first derivative for the

utility function:

U ✂(w) > 0.

Increasing wealth preference can be considered universal for all utility functions and

representative of the behaviour of the banking firm. Indeed, as mentioned, this

includes investors who are risk-seekers, risk-averters and those who are risk-neutral.

As such, a large proportion of the given set of investment alternatives will be

members of the FSD admissible set, restricting the practical applicability of the FSD

selection rule.

22 For a proof, see Levy (1998) p.48-51 or Martin, et al (1988), p.189-91.

39

2.3.2 Second-order Stochastic Dominance

Second-order stochastic dominance (SSD) assumes that in addition to increasing

wealth preference, investors are risk-averse. Risk aversion can be defined where the

utility function of an investor is increasing and concave, implying a positive first

derivative and a negative second derivative for the utility function:

U �(w) > 0 and U✁(w) < 0

Under the assumption of risk aversion, the expected utility of a risky investment

portfolio is less than the utility of the expected outcome.

Under the SSD selection rule, portfolio X will dominate portfolio Y if

w

[Gy(w ) - Fx(w)] dw ✂ 0

-✄

for all w with at least one inequality.23 This means that in order for portfolio X to

dominate portfolio Y for all risk-averse investors, the accumulated area under the

cumulative probability distribution of Y must be greater than the accumulated area for

X, below any given level of wealth. Unlike FSD, this implies that the cumulative

density functions can cross. Further, a necessary condition for SSD of portfolio X

over Y is that the expected value of portfolio X is greater than or equal to the expected

value of Y.

The assumption that investors are risk averse provides a stronger utility function

constraint than under FSD, and as such, the SSD admissible set is smaller than that

under the FSD criterion.24

23 For a proof, see Levy (1998) p.69-71 or Martin, et al (1988), p.191-2. 24 This has been empirically verified by Levy and Sarnat (1970) and Levy and Hanoch (1970).

40

2.3.3 Third-order Stochastic Dominance

Third-order stochastic dominance (TSD) corresponds to the set of utility functions

where25:

U �(w) > 0, U✁(w) < 0 and U

✁�(w) > 0

The addition of a negative third derivative for the utility function requires the investor

to prefer positive skewness in the distribution of portfolio returns (upside returns will

have a larger magnitude than downside returns, indicating greater probability in the

right tail of the distribution). Using data on the rates of return of mutual funds, Levy

(1998) provides empirical evidence that supports the hypothesis that most investors

prefer positive skewness and dislike negative skewness.26 From the perspective of a

banking firm, a preference for positive skewness can be interpreted as an

unwillingness to accept a small and almost certain gain in exchange for a remote

possibility of the bank defaulting on its debt obligations.

Under the TSD selection rule, portfolio X will dominate portfolio Y if and only if the

following conditions hold:

w t

[Gy(w ) - Fx(w)] dw dt ✂ 0, and

-✄ -✄

EF(x) ✂ EG(x)

for all w with at least one inequality.27 This means a preference for one portfolio over

another by TSD may be due to the preferred investment having a higher mean, a

lower variance or a higher positive skewness.28

25 See Whitmore (1970), Martin et al (1988) and Levy (1998). 26 Levy (1998), p.89-90. 27 For a proof, see Levy (1998) p.92-96. The symbol t arises from the thrice integration of the expression Ew[U(wF)] – Ew[U(wG)]

☎ 0. It indicates that the cumulative of the cumulative of the

cumulative distributon function of F lies above G. See Heyer (2001). 28 Levy (1998), p.97.

41

In addition to positive skewness preference, a rationale for a positive third derivative

for the investor’s utility function is decreasing absolute risk aversion, meaning the

higher the wealth of the investor, the smaller the risk premium that the investor would

be willing to pay to insure a given loss. While this may be the case for bank owners,

this aspect of TSD is less relevant in the current context than the preference for

positive skewness. The unwillingness for the investor to accept a small and almost

certain gain in exchange for a remote possibility of ruin is a property of TSD that

directly conforms to the risk preferences of bank regulators and bank creditors

discussed in section 2.2 of this chapter.

Bawa (1975) shows that for the entire class of distribution functions and for the class

of decreasing absolute risk-averse utility functions, the TSD rule is the optimal

selection rule when distributions have equal means. While in cases where

distributions have unequal means there is no known selection rule that satisfies both

necessary and sufficient conditions for dominance, Bawa shows that the TSD rule

may be used as a reasonable approximation to the optimal selection rule for the entire

class of distribution functions.

TSD represents the most applicable criteria for ranking alternative investment

portfolios in the bank setting given the TSD dominant portfolio embodies risk

aversion and positive skewness preference. If bank owners seek to preserve bank

franchise value, their utility function will display risk aversion (U �(w) < 0). If bank

creditors and regulators are concerned with the size of losses in the event of the bank

becoming insolvent, they will demonstrate a preference for positive skewness in bank

returns (U �✁(w) > 0 ). TSD also applies to the entire shape of the distribution function

of bank returns, and thus allows for non-normality in returns. This is important given

the non-normal distribution of returns that characterise many bank portfolios, and in

particular, loan portfolios. In contrast, the popular mean-variance criterion is only

accurate for ranking portfolios that are normally distributed.29

29 See Elton and Gruber (1995), p.244-5.

42

2.3.4 Non-Expected Utility and Stochastic Dominance

The stochastic dominance criteria considered to this point typify models in economics

and finance that deal with investment decision making under uncertainty in that they

are based on the expected utility paradigm. In short, they assume that the preferences

of investors are characterised by risk aversion across the entire distribution of

outcomes – their utility functions are everywhere concave. However, based on the

observation that individuals exhibit behaviour counter to expected utility theory,

Friedman and Savage (1948) and Markowitz (1952b) theorise that the utility functions

of individuals must include both concave and convex segments.30 In particular,

Markowitz argues that investors make decisions based on perceived changes in their

wealth, and argues that investors are risk-averse for losses and risk-seeking for gains,

except in the case where gains or losses are extreme, where the situation is reversed

and individuals become risk-seeking for losses and risk averse for gains.31 Later

experiments conducted by Kahneman and Tversky (1979) and Tversky and

Kahneman (1992) find that individuals maximise the expected value of a function

with a convex segment for losses and a concave segment for gains, supporting the

earlier theoretical propositions of Friedman and Savage (1948) and Markowitz

(1952b).

2.3.4.1 Prospect Theory

In response to their experimental findings, Kahneman and Tversky (1979) and

Tversky and Kahneman (1992) formulate Prospect Theory and Cumulative Prospect

Theory. The essence of these paradigms is that the preferences of individuals are

defined, not over actual payoffs (as per expected utility theory), but rather over gains

and losses relative to some reference point, so that losses are given a greater utility

weight.

30 Friedman and Savage (1948) base their claim on the observation that individuals simultaneously purchase lottery tickets and insurance policies, implying risk-seeking and risk-averse behaviour. 31 Markowitz reached these conclusions by analysing various hypothetical gambles.

43

The key elements of their paradigms are as follows:

1. Investors base their decisions on change of wealth (x) rather than total wealth,

in contrast to expected utility theory.32

2. Investors employ subjective decision weights rather than objective

probabilities.33

3. Investors maximise the expectation of a value function, V(x), which is S-

shaped. The S-shape of the value function reflects concavity for gains (risk

aversion) and convexity for losses (risk-seeking) on the part of investors:

V�(x) > 0, for all x

✁ 0

V✂(x) ✄ 0, for x < 0 and V✂(x) ☎ 0, for x > 0

The S-shaped value function is shown in Figure 2.1.

4. The value function exhibits loss aversion, which reflects in greater steepness

in the domain of losses than in the domain of gains. In experimental studies,

Tversky and Kahneman (1992) estimate a loss aversion parameter of 2.25,

implying that investors suffer negative utility of around 2.25 times more than

they derive positive utility from gains, where the gains are of equal size to the

losses.

32 This was first postulated by Markowitz (1952b). 33 Tversky and Kahneman (1992) present Cumulative Prospect Theory in response to the drawback that the sum of subjective probabilities under Prospect Theory may total more or less than 1. Cumulative Prospect Theory modifies Prospect Theory by suggesting that individuals conduct a transformation of the cumulative distribution, rather than a transformation of probabilities. See Levy, De Giorgi and Hens (2003), p.5.

44

Figure 2.1: Prospect Theory Value Function

The Prospect Theory Value Function demonstrates convexity for losses and concavity for gains, reflecting risk-seeking behaviour by investors in the domain of losses and risk-averse behaviour by investors in the domain of gains. The greater steepness in the slope in the domain of losses reflects loss aversion on the part of investors, meaning investors are distinctly more sensitive to losses than to gains.

45

2.3.4.2 Prospect Stochastic Dominance

Prospect Stochastic Dominance (PSD), developed by Levy (1998), allows for the

ranking of prospects that correspond to any S-shaped value function that is convex for

returns below the reference point (losses) and concave for returns above the reference

point (gains).

The formal conditions for PSD are as follows. Let F and G be distinct prospects with

cumulative distribution functions F and G. We can say that F dominates G for all S-

shaped utility/value functions [F PSD G] if the following hold34:

0

[G(t) - F(t)] dt � 0 for all y ✁ 0

y

x

[G(t) - F(t)] dt � 0 for all x ✂ 0

0

Prospect stochastic dominance is invariant to the specific details of any value

function, provided the function is S-shaped, with V ✄(x) > 0 for all x ☎ 0, V✆(x) ✂ 0 for

x < 0 and V✆(x) ✁ 0 for x > 0.

Recall that under prospect theory, decisions are based on changes in wealth, while

under expected utility theory decisions are based on total wealth. Levy and Levy

(2002) show that if one prospect dominates another by FSD, SSD and PSD when

outcomes are given in terms of changes in wealth, then the dominance relation holds

in terms of total wealth for any initial level of wealth.35 This means stochastic

dominance principles can be applied in the case of S-shaped value functions.

Appendix 1 provides an example of a PSD-dominating portfolio.

34 A proof of Prospect Stochastic Dominance can be found in Levy (1998). 35 Levy and Levy (2002), p.1338.

46

2.3.4.3 Markowitz Stochastic Dominance

Levy and Levy (2001) criticise the design of the experiments conducted by Kahneman

and Tversky (1979) and Tversky and Kahneman (1992) – which formed the basis of

their prospect theory paradigms - on the grounds that they are biased by the way the

experiments were framed to subjects. Specifically, to test for the shape of the utility

function, subjects were asked questions regarding their choices between alternatives

in which only positive outcomes are possible, and then asked separate questions

regarding their choices between alternatives where only negative outcomes are

possible.36 Levy and Levy (2001, 2002) claim that these experimental questions are

framed unrealistically in the sense that the hypothetical distributions presented to

subjects would virtually never be faced by the subjects. They point out that all

investments in financial markets – stocks, bonds, options, real estate – yield an

uncertain distribution of outcomes covering mixed prospects in both the positive and

negative domain.37 They conclude that the unrealistic framing of alternatives and

biases introduced by a certainty effect38 could largely account for the S-shape value

function found in the experiments by Kahneman and Tversky (1979) and Tversky and

Kahneman (1992).

Levy and Levy (2001, 2002) conduct alternative experiments to test for the shape of

the utility function, correcting for the perceived sources of bias in the experiments of

Kahneman and Tversky (1979) and Tversky and Kahneman (1992). The Levy and

Levy experiments differ in that subjects are requested to choose among investment

portfolios which have both positive and negative outcomes, which are more typical of

investment situations in markets, and all the alternatives presented to subjects have

uncertain outcomes, thus removing biases in responses generated by the certainty

effect. Further, their experiments employ large probabilities (p � 0.25) to remove the

potential for probability distortion to unduly influence responses. These changes allow

Levy and Levy to test whether the risk preferences of investors do conform to an S-

shape value function, as purported by prospect theory.

36 Levy and Levy (2001), p.235. 37 Levy and Levy (2002), p.1337. 38 This refers to gambles where one alternative has a certain outcome.

47

As a result of their experiments, Levy and Levy (2002) reject the S-shaped value

function. Rather, their results support a reverse S-shaped value function with risk

aversion for losses and risk-seeking for gains, as suggested by Markowitz (1952a).

Their experiments suggest risk preferences relative, to a reference point, that are the

exact opposite of the Kahneman and Tversky-type preferences. Further, Levy and

Levy find that subjects make choice according to prospect theory when outcomes are

restricted to either the positive domain or the negative domain, but reject alternatives

that are efficient by PSD when the bets are mixed in terms of both positive and

negative outcomes. This leads Levy and Levy to conclude that support for the S-

shaped value function of Kahneman and Tversky is due more to certainty effects than

investors’ preferences in a realistic setting of mixed investment outcomes.

In response to these findings, and in recognition of the propositions of Markowitz

(1952a), Levy and Levy (2002) develop Markowitz Stochastic Dominance (MSD),

which allows for the ranking of prospects that correspond to any reverse S-shaped

value function. The MSD value function is concave for returns below the reference

point (losses) and convex for returns above the reference point (gains), implying

investors are risk-averse in the domain of losses and risk-seeking in the domain of

gains.

48

The formal conditions for MSD are as follows. Let F and G be distinct prospects with

cumulative distribution functions F and G. We can say that F dominates G for all

reverse S-shaped utility/value functions [F MSD G] if the following hold39:

y

[G(t) - F(t)] dt � 0 for all y ✁ 0

-✂

✂

[G(t) - F(t)] dt � 0 for all x ✄ 0

x

Appendix 1 provides an example of a MSD-dominating portfolio.

It is worth noting that while MSD appears to be symmetrical to PSD, symmetry only

holds under restricted conditions. Specifically, if F and G have the same expected

return, then F dominates G by PSD if and only if G dominates F by MSD.40

Conversely, if F dominates G by PSD and F has a higher expected return than G, then

G cannot dominate F by MSD because having a higher expected return is a necessary

condition for dominance by both rules.

We now consider if the reversal of risk attitude relative to a reference point has

implications for the bank risk preference function.

39 A proof of Markowitz Stochastic Dominance can be found in Levy and Levy (2002), p.1347. 40 Ibid, p.1339.

49

2.3.4.4 Convexity and the Bank Risk Preference Function

In section 2.2 of this chapter it was determined that the bank risk preference function

should be universally concave and characterised by positive skewness. In this section

we assess whether the experimental results that indicate that the utility function for

investors may not be universally concave, with changes in risk attitude occurring

relative to a reference point, have implications for the bank risk preference function.

This assessment is necessary because we use the bank risk preference function to

determine the portfolios that are efficient from the perspective of the centre of the

bank, which in turn forms the basis for assessing the incentive-compatibility of

internal risk-adjusted performance measures. The question to be resolved is should the

bank risk preference function incorporate convex segments relative to a target

reference point? Additionally, if the function is not universally concave, should

convexity be incorporated below the target reference point, as proposed by prospect

theory, or above the target reference point, as proposed by Markowitz?

Since the formulation of prospect theory, a number of researchers have examined

whether risk-taking increases when organisations and managers perform below an

aspiration or reference level. Those studies that examine the firm perspective typically

use empirical data to test for changes in risk, measured as variance in the firm returns,

when the return on equity (ROE) or return on assets (ROA) fall below the figures for

previous periods or fall below industry averages. Those studies that focus on the risk-

attitudes of individuals/managers typically conduct experiments to gauge changes in

risk preferences relative to performance hurdles or internal targets. We briefly review

the results of these studies to assist in evaluating whether or not the bank risk

preference function should contain a convex segment above or below the reference

point that distinguishes gains from losses.

Singh (1986) uses a cross-sectional sample of 64 US and Canadian companies to

investigate the relationship between organisational performance and risk-taking. He

finds that poorly performing organisations engage in more risk taking than

organisations that are performing well.

50

Fiegenbaum and Thomas (1988) find a negative association between risk and return

for firms with ROEs below the industry average level. Conversely, they find a

positive association between risk and return for firms with ROEs above the target

level, both within and across industries. The population in their study covers 47

industries and 2,322 firms. A similar conclusion is reached by Fiegenbaum (1990) in a

study comprising 85 industries and 3,300 firms. He finds a ratio of 3:1 in the variance

of returns for organisations performing below the industry average relative to those

performing above the industry average. Both studies confirm greater risk-taking in the

domain of losses than gains, as predicted by prospect theory. Gooding, Goel and

Wiseman (1996) propose that the reference point for gains and losses should be

elevated above the industry median ROE, on the grounds that firms aspire to perform

above average industry benchmarks. Using data covering a similar period as

Fiegenbaum and Thomas (1988), they find more pronounced differences in the

variance of returns below the higher reference point than above.

In a study of the commercial banking industry, Johnson (1994) examines 142 US

banks over the period 1970-1989 to determine if variability in bank returns is related

to the extent to which banks operate below median levels for ROA and ROE. The

results of her study supports prospect theory among the below-target banks – greater

distances from the target are more often associated with greater variability in rates of

return. Above target, however, the distance from the target is not as strongly

correlated with reduced variability. Thus while above target results may induce risk-

aversion on the part of banks, the reduced variability is not as clearly related to

distance from target.

In terms of stochastic dominance principles, we have established that the centre of the

bank desires that managers select portfolios that are efficient according to third-order

stochastic dominance (TSD) criteria. It has also been established that if agents face

alternative investment prospects that cannot be ranked by TSD, but are efficient by

second-order stochastic dominance (SSD) criteria, then SSD efficient portfolios are

51

preferred.41 Both TSD and SSD assume utility functions that are universally concave,

while TSD adds positive skewness preference.

The studies reviewed above support the notion that organisations become risk-seeking

in the domain of losses and risk-averting in the domain of gains, in accordance with

the behavioural assumptions of prospect theory. However, it is noteworthy that all of

these studies infer the risk attitude of a firm, ex post, based on the historical variance

of returns.42 In the framework of this chapter, our bank centre acts as both principal

for managers within the bank and as agent for external bank stakeholders: regulators,

creditors and owners. In keeping with our view that the preservation of franchise

value is sufficient justification for bank owners to be risk-averse across the full

distribution of returns (both below and above any reference point), we argue that the

ex-ante bank risk preference function should be universally concave. The fact that

bank creditors and regulators face virtually no upside and only potential downside

should the bank become excessively risk-seeking (when performing below

expectation) further supports the notion that the ex-ante bank risk preference function

should be universally concave.43 We conclude that the position of the centre of the

bank must be one that rejects portfolios that dominate according to PSD criteria.

We now consider the implications of reversal of risk attitude about a reference point

from the perspective of the managers within organisations. Recent literature in this

area, which tends to focus on the behaviour of traders and fund managers, also

strongly supports the prospect theory construct of risk-seeking in the domain of losses

and risk-aversion in the domain of gains.

41 Portfolios that are efficient by SSD are automatically efficient by TSD. However TSD efficiency does not guarantee SSD efficiency. Refer Levy (1998). 42 Yee (1997) argues that median industry performance is not as relevant as the reference point as the current performance of a firm relative to its previous performance. He claims that organisations will consider any decrease in ROE as a ‘loss’, even when the ROE of the organization is substantially higher than the industry median. 43 The exception to this could be if the bank is already in severe financial distress, whereby creditors and owners may prefer risk-seeking by the bank in the sense of ‘gambling for resurrection’. We exclude this somewhat extreme scenario from the analysis. Further, the likelihood of regulatory intervention under this scenario is strong, mitigating the potential for excessive risk taking on the part of bank management.

52

Laughhunn, Payne and Crum (1980) conduct experiments to estimate the risk attitude

of 224 managers from the United States, Canada and Europe and find that the

majority of managers are risk-seekers when faced with below-target outcomes. Payne,

Laughhunn and Crum (1980) reach the same conclusion in laboratory experiments

where aspiration levels are changed to reflect both negative and positive monetary

amounts.

Shapira (2001) focuses on the effects of shifts in aspiration level on risk-taking by

managers and bond traders in a large US investment bank and finds that reference

points have a profound effect on risk-taking behaviour. Traders whose cumulated

profits were positive and above target near the end of the measurement period

generally engaged in fewer transactions over the remaining period to not risk their

anticipated bonus. Conversely, traders who were performing below target over the

corresponding period engaged in voluminous activity in an effort to reverse their

position. These findings correspond to risk aversion in the domain of gains and an

increase in risk-taking in the domain of losses. Willman et al (2002) reach similar

conclusions based on interviews with traders and managers in four investment banks

based in London.

Carp (2002) explores the risk attitudes of fund managers relative to benchmark

indexes. Using a large cross-sectional panel of 4,924 equity and 2,682 bond funds, she

finds that managers underperforming relative to their reference points take larger risks

in an attempt to attain targets, while managers meeting or exceeding their targets limit

risk to maintain their superior performance. Kouwenberg and Ziemba (2003)

investigate how the structure of fees paid to hedge fund managers affects their risk-

taking. They find that funds with incentive fees have higher downside risk than funds

without incentive fees, and conclude that performance-related remuneration

encourages excessive risk-taking on the part of fund managers. Locke and Mann

(2000) find that professional futures traders hold losing trades longer than winning

trades and that the average position sizes for losing trades are larger than for winning

trades. This suggests loss aversion on the part of traders when performance is below

expectations, consistent with the predictions of prospect theory.

53

The findings of these studies have significant implications from the perspective of the

centre of the bank, given its preference that managers select portfolios on the basis of

risk aversion across the entire distribution of returns. If we take the case of loan

portfolio managers, aspiration levels may be represented in terms of balances of loans

in arrears, number of loans in arrears, loan closures, targets for loan approvals or

returns above internal hurdle rates. If managers find loans performing below

aspiration levels, risk-seeking behaviour may manifest as failure to take prompt

corrective action (early loan workouts or loan closures), advancing additional funds to

keep underperforming loans liquid (escalation of commitment), misrepresentation of

data on arrears or approvals, underpricing risk in order to maintain approval targets, or

increasing exposure to higher risk loans to achieve profit targets. These actions can be

achieved by managers in the setting described at the beginning of this chapter because

the risk profile of the pool of loans available to the bank is the private information of

managers, and access to this information is cost-prohibitive to the centre of the bank.

If the utility functions of lending managers are S-shaped about an aspiration level, the

implication is they will prefer portfolios that are dominant according to PSD criteria.

This means that incentive-compatible risk measures should act to penalise PSD

portfolios, given the global risk aversion of the centre of the bank, encapsulated in the

concavity of the bank risk preference function. Indeed, the trading maxim ‘cut your

losses and run with gains’,44 which could be applied to the bank risk preference

function, runs counter to the predictions of prospect theory and more closely parallels

the utility function described by Fishburn (1977) that assumes risk neutrality above

the reference point and risk aversion below it. If the utility function of loan portfolio

managers is influenced by the remuneration structure offered to them by the bank or

their business units, this has implications for the determination of incentive-

compatible risk-adjusted performance measures. This is examined in more detail in

chapter three of this thesis.

44 See Willman, et al (2002), p.95 and Locke and Mann (2000), p.3.

54

2.4 COMPATIBILITY OF RISK MEASURES WITH STOCHASTIC

DOMINANCE CRITERIA

2.4.1 Criteria for Risk Measures

In a recent survey of the risk measurement literature, Albrecht (2003) subsumes risk

measures into two broad categories: (1) risk as the magnitude of deviations from a

target, and (2) risk as a measure of the overall seriousness of possible losses. In the

second category, risk is regarded as the capital that must be added to a position to

make it riskless. The two categories are linked in the sense that the first can be used as

a basis for determining the capital requirement in the second.

With respect to the first category, risk measures can be two-sided or one-sided. Since

the work of Markowitz (1952a), variance (standard deviation) has been the traditional

two-sided measure of risk. The theoretical arguments against using the mean-variance

approach for ranking investments centre on the properties of a quadratic utility

function (which exhibits increasing and absolute risk aversion) and a normal

distribution of returns. These two-sided risk measures assign the same weight to both

positive and negative deviations from the expected value, which contradicts the notion

that investors view risk as negative deviations from an expected value. Further,

variance does not capture kurtosis in the underlying distribution of returns, which is

needed if investors wish to incorporate the risk of low probability/high loss events in

their assessment of investment alternatives.

Lower partial moments (LPM) are a general class of risk measures where risk is

measured in terms of negative deviations from a predetermined loss threshold or

target rate of return. For continuous distributions, LPMs are measured as follows:

t

LPM (n, t) = (t – x)n f(x) dx n > 0,

-�

where t is the target rate of return, x are the outcomes of the probability distribution

and f(x) is its density function. The exponential variable n is the degree of the lower

55

partial moment, and represents the weight that an investor places on negative

deviations from the target. The exponential variable thus allows the LPM to describe

below-target risk in terms of the risk tolerance of the investor.

Bawa (1975) provides a proof of the mathematical relationship between lower partial

moments and stochastic dominance for risk tolerance levels of n = 0, 1 and 2, with

higher orders of n corresponding to greater risk aversion on the part of investors.

Fishburn (1977) also provides theoretical support for using lower partial moments to

capture the utility functions of specific investors. Specifically, these authors show that

LPM0 is applicable to all utility functions showing non-satiety (u� > 0), and that this is

analogous to first-order stochastic dominance rules. Further, they show that LPM1 is

consistent with all risk-averse functions (u� > 0, u

✁ < 0), and that this is analogous to

second-order stochastic dominance rules, while LPM2 is consistent for all risk-averse

functions displaying skewness preference (u� > 0, u

✁ < 0, u

✁� > 0), and this

corresponds to third-order stochastic dominance rules.

The findings of Bawa (1975) and Fishburn (1977) are significant in our search for risk

measures that are compatible with the risk preference function of the centre of the

bank because they indicate consistency between specific risk measures and nth order

stochastic dominance criteria. The strong relationship between risk measures based on

lower partial moments and stochastic dominance concepts is significant given that

stochastic dominance criteria apply to a general class of utility functions and make no

assumptions regarding the distribution of portfolio returns. Our earlier conclusion that

third-order stochastic dominance (TSD) represents the most applicable criteria for

ranking alternative investment portfolios in the bank setting suggests that lower partial

moments of order n > 1 may be a more relevant family of risk measures for the centre

of the bank. The key research question is would risk measures based on higher order

lower partial moments lead managers/agents in banks to select the portfolios that the

centre would have them select in the presence of perfect information? In order to

address this question we will shortly examine a range of risk measures within the

general class of lower partial moments.

56

Drawing on the findings of Bawa (1975) and Fishburn (1977), we begin with the

following definition:

A risk measure �(X) is consistent with nth order stochastic dominance, and

consequently expected utility maximisation, when portfolios can be ranked by nth

order stochastic dominance. Specifically, for portfolios X1 and X2:

X1 SD(n) X2 ✁ �(X1) ✂ �(X2)

This means that if one portfolio X1 stochastically dominates another portfolio X2, the

risk measure for X1 will be lower than for X2 and the expected utility deriving from

portfolio X1 will exceed that from X2.45 Further, a risk measure that is consistent with

(n+1)th order stochastic dominance is also consistent with nth order stochastic

dominance. This means a risk measure consistent with higher-order stochastic

dominance will be more applicable than a risk measure consistent with lower-order

stochastic dominance.

Next, we establish criteria under which a risk measure �(X) incorporates losses in the

tail of a distribution beyond a specified threshold. Using the definition of the Bank for

International Settlements (2000), ‘tail risk’ arises when the probability of returns for

one investment has a greater risk of larger losses than another, where both

distributions have equal means.46 Tail risk is applicable whenever bank stakeholders

are concerned with the size of losses in the event that losses exceeded a predetermined

level. We established earlier that bank creditors, depositors and regulators are more

likely to be concerned with the size of losses in the event of default (tail risk), whereas

managers and owners are likely to be more concerned with the probability of losses

given their personal losses are linked to default itself, and are invariant to the size of

losses.

45 See Levy (1998), theorem 3.2, for a proof. 46 Bank for International Settlements (2000), p.8.

57

With respect to a risk measure �(X) and the degree of tail risk in the distribution of a

portfolio, we put forward the following proposition:

If beyond a specified loss threshold, a portfolio X1 has lower expected losses than

another portfolio X2, then the risk measure �(X1) should be lower than the risk

measure �(X2).

This requires that our risk measure(s) should differentiate among portfolios in terms

of their tail risk. It remains to establish how a risk measure can be categorised in terms

of its ability to capture specific degrees of tail risk in a loss distribution.

First, let us consider the most restrictive case. If an investor is concerned only that the

probability of loss for one portfolio X1 is less than the probability of loss for another

portfolio X2, then the cumulative distribution function for portfolio X1 will always lie

to the right of the cumulative distribution function for portfolio X2. In other words, the

cumulative distribution functions for the portfolios under consideration should not

cross. As discussed earlier, this implies a positive first derivative for the utility

function of the investor.

Under these conditions we can say for a risk measure �(X), at a given loss threshold

(t), that �(X1) < �(X2) if:

F1(x) ✁ F2(x), for all x, x ✁ t

where F1(x) and F2(x) are the cumulative distribution functions for X1 and X2. Notably,

this is consistent with portfolio X1 dominating portfolio X2 in terms of first-order

stochastic dominance principles (FSD).47 Risk measures that fit into this category we

denote ‘Type 1’ risk measures.

Next, consider the case of an investor who is concerned not only with the probability

of losses, but also the size of losses should a certain loss threshold (t) be exceeded. If

an investor is concerned that the average losses for portfolio X1 be less than the

47 Refer section 3.1 in this chapter.

58

average losses for portfolio X2, for a given loss threshold (t), then the accumulated

area under the cumulative distribution function for portfolio X1 must be less than the

accumulated area under the cumulative distribution function for portfolio X2. Unlike

the first case above, this means that the cumulative density functions can cross, and

this in turn is consistent with portfolio X1 dominating portfolio X2 according to second

order stochastic dominance principles (SSD).48 Further, in addition to increasing

wealth preference, this implies the investor is risk-averse (negative second derivative

for the utility function).49

Under these conditions we can say for a risk measure �(X), at a given loss threshold

(t), that �(X1) < �(X2) if:

t t

(t – x) f1(x) dx ✁ (t – x) f2(x) dx for all x, x ✁ t

-✂ -✂

where f1(x) and f2(x) are the density functions of X1 and X2. Risk measures that fit into

this category we denote ‘Type 2’ risk measures.

Finally, consider an investor who perceives a low probability of a large loss to be

riskier than a high probability of a small loss, even when expected losses are the same.

In addition to increasing wealth preference and risk aversion on the part of the

investor, this condition also encapsulates a preference for positive skewness in the

portfolio distribution, and is consistent with a negative third derivative for the utility

function of the investor.

Drawing on this, let two portfolios, X1 and X2, have equal means and equal average

losses beyond a given loss threshold (t), but portfolio X2 has a small probability of

large losses beyond the threshold while portfolio X1 has a larger probability of small

losses beyond the threshold. If an investor displays a preference for positive skewness

48 Refer section 3.2 in this chapter. 49 Ibid.

59

in the distribution of portfolio returns, we can say for a risk measure �(X), that �(X1) < �(X2) if the following holds:

t t

(t – x)(n-1) f1(x) dx ✁ (t – x)(n-1) f2(x) dx for all x, x ✁ t

-✂ -✂

where f1(x) and f2(x) are the density functions of X1 and X2 and n > 2.

This condition employs the lower partial moment of degree (n-1) to penalise large

deviations from the loss threshold more than smaller deviations from the loss

threshold. This differs from the previous condition, which implicitly assumed that

investors have a linear response to losses beyond the threshold. Where n = 2, this

condition is equal to the previous condition. When n =3, this condition places a

quadratic penalty on deviations below the loss threshold50, and is consistent with

portfolio X1 dominating portfolio X2 according to third order stochastic dominance

principles (TSD).51 Risk measures that fit into this category, where n = 3, we denote

‘Type 3’ risk measures.

50 Higher order powers (n > 3) place even larger penalties on wider deviations from the loss threshold. For example, n = 4 places a cubic penalty on deviations below the loss threshold. 51 Refer section 3.3 in this chapter.

60

We summarise each risk measure category as follows:

Risk Measure Category

Characteristics

Stochastic Dominance

Compatibility

Implicit Risk Attitude

Type 1

The investor is concerned with probability of loss beyond a given threshold

FSD

Non-satiety

Type 2

Investor is concerned with probability of losses and the average size of losses beyond a given threshold

SSD

Non-satiety and risk aversion

Type 3

Investor is concerned with probability of losses, the average size of losses and larger deviations more than smaller deviations from a given threshold

TSD

Non-satiety, risk aversion and positive skewness

61

2.4.2 Risk Measures

In this section we examine a number of downside risk measures to assess their

compatibility with the principles established in the preceding section. In the next

chapter we extend the analysis to examine the degree of dependency of the risk

measure on the choice of the target level. We also examine how the compensation

scheme paid to agents in a bank may compromise the efficacy of the risk-adjusted

performance measure.

The downside risk measures under examination are based on statistics of the loss

distribution of a portfolio over some predetermined time horizon. The selected

measures, which are somewhat contemporary measures of downside risk, are the

shortfall probability, value at risk (VaR), expected shortfall, first-order lower partial

moment (LPM1) and the second-order lower partial moment (LPM2)52. In the next

chapter we consider measures related to LPM2: spectral risk measures and Wang

Transformation distortion risk measure.

2.4.2.1 Shortfall probability

Shortfall probability is the probability that the return on a portfolio will fall below the

prespecified target level. Shortfall probability is measured by the zero-order lower

partial moment (LPM0), which is the integral of the unweighted return distribution. It

is defined as follows:

t t

LPM (0, t) = (t – x)0 f(x) dx = f(x) dx = F(t)

-� -�

where F(t) is the cumulative density function for expected returns below the target (t).

Shortfall probability is the relevant risk measure for an investor who is only interested

in the probability of falling short of the prespecified target return, ignoring the extent

or severity of the shortfall should it eventuate. Fishburn (1977) demonstrates that this 52 The second-order lower partial moment is also known as the shortfall variance.

62

measure is consistent with a risk-seeking utility function; specifically the order n < 1

characterises an investor who is willing to gamble at fair odds in an attempt to

minimise the extent to which returns fall short of the target.53

2.4.2.2 Value-at-Risk

Value-at-risk (VaR) is defined as the loss that will not be exceeded over a certain time

period with a specified confidence level (�). Put differently, the VaR of an investment

is the loss that will be exceeded only with a given probability (1 - �) over a specified

measurement period. VaR is closely related to shortfall probability through the

cumulative distribution function. If the VaR is designated as the benchmark for

measuring shortfall, the probability that losses exceed the VaR level corresponds to

the shortfall risk measure.54 In terms of our earlier defined criteria, VaR and shortfall

probability correspond to Type 1 risk measures.

If V is the value of an investment at the end of the designated measurement period, we

define Vp such that

Vp

Prob (V ✁ Vp) = dF(V) = 1 - �

-✂

where F(V) is the cumulative distribution function of V. Thus the value of the

investment will be below Vp with a probability of (1 - �). In defining the VaR, it

remains to determine the benchmark such that outcomes below this benchmark are

regarded as losses. This benchmark, for example, may be the initial value of the

investment or the expected value of the investment. If the expected value of the

investment is determined to be the benchmark, the VaR is defined as follows:

VaR ✄ = E(V) - Vp

53 This is demonstrated later in the chapter. 54 See Schroder (1997) for a mathematical derivation.

63

In the bank loan setting, the expected value of the loan can be considered the

appropriate benchmark because this determines the cut-off point for expected losses.

For example, if the maximum value of a loan is determined to be $100 and the

expected value of the loan is $99, expected losses are equal to $1. The expected value

of $99 then forms the benchmark for determining unexpected losses. To further the

example, if the distribution of loan values indicates there is a 1% probability that the

value of the loan will be lower than $60 (ie Vp = $60) over the designated time

horizon, then the unexpected loss is equal to $39. We can conclude that the VaR 99% is

$39, and there is a corresponding 1% probability that losses will exceed the VaR of

$39 over the measurement period. If the bank wanted to hold economic capital such as

to achieve a target credit rating on its senior-rated debt equal to a 1% probability of

default, it would need to set aside $39 in economic capital to support the loan

(excluding any diversification contribution of the loan on the existing portfolio).

It is worth noting that VaR based on an expected value benchmark is unaffected by a

constant shift in the entire distribution of returns. In the above example, an economic

recession may cause expected losses to rise, but not the VaR. In this sense, the

efficacy of the VaR measure is reduced because it is less sensitive to weak economic

conditions which would induce a decline in returns under all states of nature – there is

no change in the magnitude of the risk measure despite a larger absolute loss at the

given confidence interval.

The preceding example shows how the determination of economic capital in a bank is

consistent with the VaR concept of estimating the distance between expected and

unexpected outcome. The VaR confidence level is scaled to the critical threshold level

for determining the amount of economic capital deemed necessary to protect the bank

against adverse events. In the desire that banks monitor and manage the size of lower-

tail outcomes so that the probability of financial distress is low, regulators have

adopted VaR as the standard for measuring risk for determining bank capital

adequacy. Szegö (2002) notes that the second consultative paper of the new Basel

Accord assumes the VaR concept as the risk measure for deriving minimum capital

standards, and requires in its solution that the risk of each loan must be portfolio

64

invariant.55 Further, the Accord requires that regulatory capital for each loan must be

correlated to its marginal contribution to the VaR.56

2.4.2.3 Expected Shortfall

Expected shortfall is defined as the conditional expectation of loss given that the loss

is beyond the VaR level.57 Thus, by definition, expected shortfall measures losses

beyond the VaR level. In terms of our earlier criteria, expected shortfall corresponds

to a Type 2 risk measure.

If x is the profit/loss of a portfolio X, with positive values of x representing profits and

negative values (–x) representing losses, and VaR�(X) is the VaR at the (✁) percent

confidence level, expected shortfall at the (✁) percent confidence level (ES�) is

defined as follows:

ES�(X) = VaR�(X) + (1 - ✁)-1 E {max [–x – VaR�(X), 0]},

where E( ) is the expected value operator.

This definition shows that ES�(X) is more sensitive to the severity of losses exceeded

VaR�(X) given expected shortfall is calculated by taking the expected value of all

losses which are greater than or equal to VaR�(X). If investors (and regulators) are

concerned not with the potential loss that would occur at a specified confidence level,

but rather, the severity of losses beyond the VaR level, the expected shortfall may be

considered to be a more suitable measure of risk than the VaR.

An example showing the calculation of VaR and expected shortfall for alternative

portfolio distributions is provided in Appendix 2.

55 Szegö (2002), p.1258. 56 Ibid, p.1259. 57 Other names for expected shortfall in the literature include tail conditional expectation, tail VaR, conditional VaR and conditional loss.

65

2.4.2.4 First-Order Lower Partial Moment

As defined earlier, lower partial moments measure risk in terms of deviations below a

loss threshold or target rate of return. In section 2.4.1 the general class of lower partial

moments, for a continuous distribution, was represented as follows:

t

LPM (n, t) = (t – x)n f(x) dx n > 0,

-�

The first-order lower partial moment (LPM1) has a power of n = 1, and thus measures

the weighted average deviation from the target level. LPM1 is related to the expected

shortfall risk measure in that it provides the expected loss relative to the loss threshold

or benchmark return (t). Appendix 2 provides an example to show that LPM1 and

expected shortfall provide an identical risk measure ✁(X) when the losses are

measured in terms of the same loss (✂) quantile. Like expected shortfall, LPM1

corresponds to a Type 2 risk measure in terms of our earlier defined criteria.

While LPM1 and expected shortfall produce the same measure of risk for losses

beyond the loss threshold, the LPM1 risk measure has a significant advantage over

expected shortfall in that while expected shortfall (like VaR) is usually measured in

terms of a specific loss quantile, the LPM1 can be calculated on the basis of deviations

from zero (t = 0). This enables the full distribution of losses to be taken into

consideration in the risk measure, rather than expected losses beyond the loss

threshold. If investors are concerned with all losses (or below target returns), rather

than those that are greater than the loss threshold, then the lower partial moment class

of risk measures, with t = 0, is a more complete measure of risk. The implications of

this in the search for optimal risk measures in the bank principal/agent setting are

identified in the next chapter of this thesis.

While the focus of this chapter is incentive-compatible risk measures from the

perspective of investment decisions within a bank, it is worth highlighting the

implications of the above from the perspective of external regulatory risk measures. If

66

regulators are concerned only with protection against bankruptcy, then a VaR measure

may be appropriate. If regulators, however, are concerned that a bank be sufficiently

capitalised to cover the size of losses in the event of bankruptcy (losses beyond the

predetermined loss threshold), then expected shortfall is a more relevant risk measure

for regulatory purposes. But what of smaller losses below the loss threshold, being

losses that occur with a less than (�) confidence interval? A regulatory risk measure

based on VaR or expected shortfall implicitly assumes that losses that are less than the

threshold are self-insured by the bank or that the bank can efficiently recapitalise in

the event that it needs to raise equity to cover these unexpected losses. If, however, a

bank frequently suffers losses less than the threshold, it may find insurance or

recapitalisation costly, particularly in times of economic slowdown where loan losses

are likely to be larger and equity recapitalisation more expensive.

The lower partial moment class of downside risk measures incorporate all losses into

the measure, both above and below the predetermined confidence interval, when the

target is set at (t = 0). Risk measures that focus either on the probability of losses

(VaR) or extreme losses (expected shortfall) fail to capture the likely systemic impact

of banks incurring frequent losses below the predetermined loss threshold. If

regulators determine that bank capital requirements should be based on either of the

above measures of unexpected losses, then they are ignoring the potential for loss in

confidence in the banking system if banks do incur frequent unexpected losses below

the threshold and subsequently find it difficult to recapitalise. In these circumstances,

LPM1 (or a higher order LPM risk measure) is likely to be a more appropriate than

VaR or expected shortfall for determining bank capital requirements.58

58 Later in the chapter we will see that higher-order lower partial moments have other properties that make them attractive as a regulatory risk measure for capital adequacy purposes.

67

2.4.2.5 Second-Order Lower Partial Moment

With respect to the general class of lower partial moment risk measures, as the power

function n increases, larger deviations from the threshold are penalised more than

smaller deviations. The second-order lower partial moment (LPM2) places a quadratic

penalty on deviations below the threshold (n = 2). Formally, LPM2 represents the

semi-variance or lower partial variance (Markowitz, 1959), and the square root of the

lower partial variance represents the downside standard deviation. By placing a larger

penalty on larger losses, LPM2 corresponds to a Type 3 risk measure in terms of our

earlier defined criteria.

Like LPM1, when the target level from which deviations are measured is set to cover

all losses (t = 0), LPM2 captures in the risk measure smaller unexpected losses that are

not included in measures based on loss thresholds linked to predetermined confidence

intervals (such as VaR and expected shortfall). This means that unlike VaR and

expected shortfall, the LPM2 risk measure does not create an incentive for portfolio

managers to take actions that increase the cumulative distribution function for losses

that are smaller than the loss threshold. Such actions would increase the risk of the

portfolio but not be captured in risk measures that base losses relative to a loss

threshold. Managers may be motivated to take such actions to increase the risk-

adjusted return on their portfolios and increase their remuneration where bonuses are

linked to such measures. Risk measures that are based on the full distribution of losses

do not entice such gaming on the part of managers. This will be demonstrated in the

next chapter.

The use of the semi-variance as a basis for portfolio optimisation (asset allocation)

decisions has been the subject of ongoing research. Markowitz (1959) agues that the

semi-variance as a risk measure tends to produce more efficient portfolios than

portfolios based on variance as the risk measure. Further, semi-variance can be

calculated relative to an investor-specific benchmark. For a summary of empirical

research on the use of semi-variance for asset allocation, see Narwocki (1992).

68

2.4.3 Summary

Any risk measure that is based on the general class of lower partial moments is a

suitable candidate for an incentive-compatible risk measure because of the

mathematical relationship between lower partial moments of order n and stochastic

dominance criteria. The five risk measures reviewed above have been selected

because they are derivations of the general class of lower partial moments. By

analysing the risk measures in terms of lower partial moments we can determine the

risk attitude implicit in each measure.

Shortfall probability is measured by the zero order lower partial moment (LPM0) and

is consistent with a risk-seeking utility function. VaR can be interpreted as a special

case of shortfall probability – fixing the probability of the LPM0 gives the

corresponding VaR measure. VaR is a suitable risk measure for an investor having a

positive first derivative (non-satiety), but risk-aversion on the part of the investor is

not a necessary condition. Expected shortfall and LPM1 produce the same measure of

risk for losses when the target threshold is identical. They only differ in that expected

shortfall typically corresponds to a � percent confidence level while LPM1 can be

based on negative deviations from any target level. The power function of one

indicates that these measures are consistent with a risk-neutral utility function. The

quadratic power function of LPM2 means large deviations from the threshold are

penalised more than smaller deviations in the risk measure, consistent with a risk-

averse attitude to losses.

In the next chapter we apply these risk measures to five credit portfolios to assess

their incentive-compatibility properties with respect to the risk preference function of

the centre. We also examine the coherence of the risk measures in terms of Artzner et

al (1999), and assess the relevance of the coherency axioms for internal risk measures

where the aim of the measures is to promote portfolio selection on the part of

managers that is consistent with the objectives of the centre.

69

2.5 CHAPTER SUMMARY

The main findings of this chapter are summarised as follows:

1. The risk preference function for a bank is likely to be concave. Bank creditors

and regulators are concerned with not only with the probability of the bank

defaulting, but also the size of loses in the event of default. This makes creditors

and regulators risk-averse. When limited liability and the regulatory safety net

are taken into consideration, bank owners may have a convex risk preference

function, but if bank possesses franchise value, owners may prefer that the bank

acts in a risk-averse manner in order to preserve the associated benefits. In this

case the objective function for the bank would be concave. If the value of the

franchise to bank owners exceeds the value of the put option associated with

limited liability, we can conclude that owners will also be risk-averse.

2. Stochastic dominance is a generalisation of utility theory that allows portfolios

to be ranked without having to explicitly specify the utility function of the

investor. Different orders of stochastic dominance correspond to different

classes of utility function. Third-order stochastic dominance (TSD) embodies

non-satiety, risk aversion and a preference for positive skewness in the

distribution of returns. This makes it the most applicable criteria for ranking

credit portfolios in the bank setting because it best characterises the risk

preference function of the centre.

3. The risk preference function of the centre of the bank should not incorporate

convex segments around a reference point, despite the results of empirical

studies that suggest organisations become risk-seeking in the domain of losses

and risk-averse in the domain of gains. This is typically an ex-post response to

performance below some target level. The preservation of franchise value is

sufficient justification, ex-ante, to assume the bank risk preference is universally

concave across the full distribution of expected returns.

70

4. Empirical studies suggest that individuals exhibit a reversal of risk attitude about

an aspiration level that reflects a reference point. This means managers may

prefer portfolios that are dominant by prospect stochastic dominance (PSD)

criteria. Consequently for risk measures to be incentive-compatible, they should

act to penalise portfolios that dominate by PSD criteria.

5. A strong mathematical relationship exists between risk measures based on lower

partial moments and stochastic dominance criteria, and for this reason, risk

measures based on the general class of lower partial moments are suitable

candidates for incentive-compatible risk measures. By analysing the risk

measure in terms of lower partial moments, the implied risk attitude in the

measure can be inferred.

71

Chapter Three

Incentive-Compatible Risk-Adjusted

Performance Measurement

“Orthodox economic theory has little to offer in terms of

understanding how nonmarket organisations, like firms, form

and function. This is so because traditional theory pays little or

no attention to the role of information, which evidently lies at

the heart of organisations. The recent development of

information economics, which explicitly recognises that agents

have different information, is a welcome invention.”

Bengt Holmstrom, 1982

72

3.1 INTRODUCTION

In this chapter we apply the framework developed in chapter two and analyse the

incentive-compatible properties of the risk measures that conform to the general class

of lower partial moments. These measures were identified in section 2.4.2. To give

structure to the analysis, we use an example based on the probability distributions of

five credit portfolios which represent a set of alternatives available for investment.

Managers, who must select portfolios from the opportunity set, have specialised

knowledge that enables them to form expectations on the distribution of returns for

each portfolio in the set. We assume that this information is not available to the

centre, which only receives information on the expected distribution of returns for

portfolios actually selected by managers. This information must be provided to the

centre in order that a risk measures can be assigned to each portfolio, which is later

used to measure the risk-adjusted performance of the portfolio. We assume in this

chapter that managers have no incentive to misrepresent this information to the

centre.59

The portfolios that form the investment opportunity set have the same expected value,

initially allowing us to examine the risk of the portfolios independently from expected

return. The portfolios increase in risk in accordance with stochastic dominance

principles. If managers are remunerated on the basis of the risk-adjusted performance

measure (RAPM) for the portfolios under their control, they should be incentivised to

select the portfolios that have the lowest risk measure in the denominator. In the first

instance, risk measures that fulfil this requirement will be those that increase in size

for portfolios that are dominated in accordance with third-order stochastic dominance

(TSD) conditions. Our aim is to test for the risk measures that promote congruency

between the goals of the centre of the bank and the decisions of managers with respect

to the selection of credit portfolios.

59 We consider the impact of relaxing this assumption in later chapters.

73

Artzner et al (1999) present and justify a set of four desirable structural properties for

measures of risk, which they argue should hold for any risk measure which is to be

used to effectively regulate or manage risks. They call measures that satisfy these

properties ‘coherent’. The four axioms that characterise coherent risk measures are

translation invariance, monotonicity, positive homogeneity and subadditivity. We find

that risk measures that are incentive-compatible in terms of stochastic dominance

principles are not necessarily coherent. This leads us to question if risk measures need

to be coherent, in keeping with Artzner et al (1999), in order to be incentive-

compatible. We examine this question and determine that risk measures that fail to

meet at least two coherency axioms – positive homogeneity and subadditivity – may

lead managers to make suboptimal investment decisions. This is despite the fact that

these measures conform to risk-ordering according to stochastic dominance

principles. This leads us to extend our search to risk measures that are both coherent

and compatible with stochastic dominance.

Finally, in the search for internal risk measures that align the interests of the centre

and managers, we investigate how the structural form of a bank’s compensation

payment function impacts on the portfolio selection decisions of managers. If banks

focus solely on the risk measure that forms the denominator of the RAPM equation,

they may miss the fact that the distribution of gains in the numerator of the equation is

also likely to influence portfolio selection by managers. If the payment function is

asymmetrical, with bonuses paid only upon the realisation of gains, then it is

reasonable to assume that the ranking of portfolios will be influenced by both the

distribution of gains in the numerator and the distribution of losses in the denominator

of the RAPM. We find that if the distribution of gains is not equal across the portfolio

opportunity set, then in some circumstances it is not be possible to determine which

portfolios managers will select without specific knowledge of their utility functions.

This adds a further complication to the design of an incentive-compatible risk

measurement framework. If the centre of the bank is charged with managing both

downside risk and upside potential, and the goal is to have managers select the

portfolios that the centre itself would select, then the risk-attitude of the centre with

74

respect to the distribution of returns above the target threshold must be incorporated

into the RAPM for managers.

These issues are the focus of this chapter, which as has three principal objectives.

First, we need to determine if there is a set of risk measures that conform to the risk-

ordering of portfolios according to stochastic dominance principles. These measures

should be consistent with TSD given the risk preference function of the centre

discussed in the previous chapter. Second, it is necessary to ascertain if the risk

measures need to be coherent in order for them to be incentive-compatible. Third, we

investigate how the structure of the compensation payment function of the bank

impacts on the incentive-compatible characteristics of the RAPM.

The rest of the chapter is structured as follows. Section 3.2 introduces the five credit

portfolios that form the basis of the study in this chapter and identifies their risk-

ordering according to stochastic dominance criteria. This captures ordering by FSD,

SSD, TSD, PSD and MSD. Section 3.3 presents calculations and discussion by each

risk measure for the five portfolios. Section 3.4 examines the axioms the coherency of

each of the risk measure candidates, and assesses the relevance of coherency within

the specific principal-agent setting of this chapter. Section 3.5 considers whether the

risk measure used for internal purposes should be aligned with the basis for measuring

total bank capital. Section 3.6 investigates how the structure of the bank’s

compensation function impacts on incentive-compatibility conditions, and

recommends adjustments to the RAPM to reflect the attitude of the centre to upside

volatility. Section 3.7 presents a summary of the main findings of the chapter.

75

3.2 DETERMINING THE INCENTIVE-COMPATIBLITY OF RISK

MEASURES

In this section we examine the probability distributions of the five sample credit

portfolios that form the basis of the study to determine their congruence with

stochastic dominance principles.

3.2.1 Portfolio Assumptions

The probability distributions for the five sample credit portfolios under examination

are presented in Table 3.1. The portfolios are designated A – E.

The probability distributions are based on the expected market value of the portfolio

at the end of one year60. The face value of each portfolio is $100 and the credit

portfolios are assumed to each have the same duration. The expected value of each

portfolio is equal at $98.99, and losses are measured as downside deviations from this

expected value.61 For the purposes of this analysis, the expected value of each

portfolio is the same in order to isolate differences in the risk of the portfolios.

60 The one year period is in keeping with the measurement horizon over which a bank’s solvency standard is determined for the purposes of calculating economic capital under the Bank for International Settlement’s (Basel II) revised framework for the international convergence of capital measurement and capital standards. 61 Expected losses are thus equal to $100 - $98.99 = $1.01. For accounting purposes, this is typically treated as a provision in the profit and loss statement of the bank.

76

Table 3.1

Probability Distributions: C redit Portfolios A - E

Portfolio A B C D E

Market Value

$0

30 0.4% 0.5%

60 1% 1% 1%

90 2.5% 0.6% 0.5%

97 5% 7.5% 5% 5% 10%

98 15% 10% 10% 10% 6%

99 30% 32.5% 30% 40% 25%

100 40% 40% 40% 41% 53%

101 5% 5% 5% 2% 4%

102 3% 3% 3% 1%

103 1% 1% 1%

104

105

106 2.5% 1%

Expected value $98.99 $98.99 $98.99 $98.99 $98.99

Probability distributions for the credit portfolios correspond to the expected market

value of the portfolio at the end of one year. Market values have been used as the

measurement basis, as opposed to accounting profit or loss, for a number of reasons.

If we take the view that bank stakeholders are interested primarily in the real

economic impact of decisions made within the bank, then performance should be

measured in terms of the market values of bank assets and equity. In this regard,

accounting measures of profit may be less meaningful to the extent that they may be

distorted by transfer pricing assumptions and arbitrary internal cost allocations.

Further, a market value basis for measurement allows the performance of credit

portfolio managers to be judged on both an upside and downside basis, creating a

more efficient alignment of incentives between principal and agent. We expand on

this point below.

77

Subsequent to origination, the market value of a credit portfolio will change as

external market valuations of benchmark instruments change and the credit quality of

borrowers change - borrowers may migrate to lower internal credit ratings

(corresponding to a decline in the market value of the portfolio) or higher internal

credit ratings (corresponding to an increase in the market value of the portfolio).62 If

the performance of credit portfolios is measured on the basis of accounting income,

such portfolios will have virtually no upside, particularly if they are priced to earn a

minimum hurdle rate on allocated capital. Here accounting income on the portfolio

can only turn out to be more than expected if actual loan losses ex-post turn out to be

less than provisioned. If managers are remunerated on this basis, a perverse incentive

may be created whereby managers deliberately overstate expected losses in order to

realise bonuses in the expectation that actual write-offs turn out to be less than

provisioned. If expected losses are overstated, this may result in inflated loan prices

and render the bank uncompetitive in some markets.63 If a mark-to-market basis is

used for measuring gains and losses on credit portfolios, greater scope exists to

reward managers for positive credit migrations in their portfolios, creating stronger

incentives to actively manage portfolios for the upside. It is for these reasons that we

model the risk-adjusted performance of the portfolios on a market value basis.

3.2.2 Risk Profiles of Portfolios

Portfolios A to E are increasing in risk according to specific stochastic dominance

criteria outlined below. As discussed above, the portfolios have been constructed to

ensure they have equal expected values, allowing analysis to be undertaken purely on

their differentiation by risk.

62 Estimating credit quality migrations is typically based on ratings transition matrices, which show the probability of a given credit-quality borrower moving from one ratings band to another within a one-year period. These matrices are published by ratings agencies and are derived from historical patterns, to which smoothing techniques are applied. See Day (2003) chapter 15 and Bessis (2002) chapter 49 for more detailed descriptions and examples. 63 For further discussion related to marking-to-market credit portfolios, see Kealhofer (2002).

78

3.2.2.1 Portfolio A

Portfolio A represents the base portfolio. As will be shown below, portfolio A is the

most efficient portfolio on a risk-return basis because it has the same expected value

as the other portfolios but is the dominant portfolio for a risk-averse investor in terms

of both second-order stochastic dominance (SSD) and third-order stochastic

dominance (TSD) criteria.

3.2.2.2 Portfolio B

Portfolio B is created from portfolio A through a mean-preserving spread, a concept

developed by Rothschild and Stiglitz (1970). These authors show that if two

portfolios X and Y have probability density functions f and g, and g is obtained from f

by taking some of the probability weight from the centre of f and adding to each tail of

f in such a way as to leave the mean unchanged, then Y is a riskier prospect than X.64

Further, Rothschild and Stiglitz (1970) prove that the following statements are

equivalent for X and Y:65

Y is a mean-preserving spread of X

X dominates Y by second order stochastic dominance [X SSD Y]

Table 1 in Appendix 3 shows how portfolio B is created from portfolio A through a

mean preserving spread in the style of Rothschild and Stiglitz (1970).

Table 2 in Appendix 3 shows that portfolio A cannot be ranked against portfolio B by

first-order stochastic dominance (FSD) because the cumulative probability

distribution functions of the portfolios cross. However, the table shows portfolio A

dominates B by SSD because the sum of the cumulative probability distribution

functions of the portfolios do not cross. Table 4 in Appendix 4 shows that portfolio A

also dominates portfolio B by TSD.

64 Rothschild and Stiglitz (1970), p.226. 65 Ibid, p.237.

79

These findings indicate that portfolio A would be preferred over portfolio B by any

non-satiated, risk-averse investor who prefers positive skewness in the distribution of

returns. This fits the risk preference function of the centre of the bank as determined

in the previous chapter. Given that the portfolios have the same expected value, and

portfolio A is less risky than B by SSD and TSD, it follows that any risk measure that

is compatible with the preference function of the centre will show a lower value for

portfolio A than B:

3.2.2.3 Portfolio C

Portfolio C is created as a mean-preserving spread of portfolio B, as shown in Table 1

of Appendix 4. This means portfolio B dominates C by SSD. Further, portfolio A

must also dominate portfolio C because portfolio B is created as a mean-preserving

spread of A, as discussed in 5.2.2. We can conclude that portfolio C is riskier than

both portfolios A and B, with A dominating both portfolios.

Table 2 of Appendix 4 shows that portfolios A, B and C cannot be ranked by FSD

because the cumulative probability distributions of the portfolios cross. Table 3 of

Appendix 4 confirms that portfolios A and B dominate C by SSD because the sum of

the cumulative distribution functions of portfolios A and B do not cross C. Table 4 of

Appendix 4 shows that portfolio A also dominates portfolios B and C by TSD, and

confirms portfolio B dominates C by TSD. 66 Dominance by TSD criteria occurs

because the sum of the cumulative probabilities of each of the distribution functions

do not cross.

66 Recall (footnote 28) that portfolios that are efficient by TSD are automatically efficient by SSD. However SSD efficiency does not guarantee TSD efficiency. Refer Levy (1998)

A SSD B

A TSD B

� (A) < � (B)

80

These findings indicate that portfolio A would be preferred over portfolios B and C,

and portfolio B would be preferred over portfolio C, by any non-satiated, risk-averse

investor who prefers positive skewness in the distribution of returns. Given the risk

preference function of the centre of the bank, and the fact that the three portfolios

have the same expected value, we can determine that portfolio C is riskier than B, and

B is riskier than A. It follows that a risk measure that is compatible with the

preference function of the centre will show a lower value for portfolio A than B, and a

lower value for portfolio B than C:

3.2.2.4 Portfolio D

Table 1 of Appendix 5 shows that portfolio D cannot be ranked against A, B, or C by

FSD because the cumulative probability distributions of the portfolios cross.

Portfolio D has a extreme loss in the lower tail of its distribution and is not a mean

preserving spread of portfolios A, B or C. This means it cannot be ranked against

these portfolios by SSD criteria. This is confirmed in Table 2 of Appendix 5, which

shows that the sum of each of the cumulative distribution functions of portfolios A, B,

C and D cross. Consequently we cannot determine if portfolio D is riskier than A, B

or C by SSD.

We can, however, rank portfolio D against portfolios A, B and C by TSD criteria.

Portfolio D is dominated by A, B and C. This is confirmed in Table 3 of Appendix 5,

which shows that the sum of the cumulative probabilities of each of the distribution

functions do not cross. We can infer that portfolio D would not be preferred over

portfolios A, B and C by any non-satiated, risk-averse investor who prefers positive

skewness in the distribution of returns. It follows that a risk measure that is

A SSD B � A SSD C � B SSD C

A TSD B � A TSD C � B TSD C

✁ (A) < ✁ (B) < ✁ (C)

81

compatible with the preference function of the centre of the bank must show a higher

value for portfolio D than A, B and C. Further, the risk measure must be lower for

portfolio A than B, B than C and C than D:

3.2.2.5 Portfolio E

Portfolio E has an extreme value in the lower tail of its distribution, and has a higher

probability of a larger loss than portfolio D. Table 1 of Appendix 6 shows that

portfolio E cannot be ranked against A, B, C or D by FSD because the cumulative

probability distributions of the portfolios cross. Table 2 of Appendix 6 also shows that

portfolio E cannot be ranked against A, B, C or D by SSD because the sums of the

cumulative probability distribution functions cross for these portfolios. Consequently

we cannot determine if portfolio D is riskier than A, B or C by SSD.

We can, however, rank portfolio E against the other portfolios by TSD. This is shown

in Table 3 of Appendix 6, where the sum of the cumulative probabilities of the

portfolios do not cross. The positive difference between the sum of the cumulative

probabilities for portfolio E and portfolios A, B, C and D indicates that portfolio E is

dominated by A, B, C and D by TSD, meaning E is riskier than the other portfolios

for investors who are non-satiated, risk-averse and prefer positively skewed

investment portfolios. Again, it follows that a risk measure that is compatible with the

preference function of the centre of the bank must show a larger value for portfolio E

than the other portfolios, with the smallest value for the risk measure being for A, and

ascending through each of portfolios B to E:

A TSD B � A TSD C � B TSD C

A TSD D � B TSD D � C TSD D

✁ (A) < ✁ (B) < ✁ (C) < ✁ (D)

82

Table 2 of Appendix 6 also shows that portfolio E dominates portfolios A, B and D by

prospect stochastic dominance (PSD) criteria. This arises because the differences in

the sum of the cumulative probability between portfolio E and portfolios A, B and D

are positive in the domain of losses (the market value of the portfolio falls below the

expected value) and negative in the domain of gains (the market value falls above the

expected value).67 The positive difference reflects greater cumulative weight in the

left side of the tail, and thus indicates greater risk in the domain of losses, while the

negative difference reflects greater cumulative weight in the right side of the tail and

thus indicates less risk in the domain of gains. This is conversant with the precepts of

prospect theory.

The observation that portfolio E is dominant by PSD is significant. If managers within

the bank have a utility function that is convex in losses and concave in gains, they will

have a preference for PSD efficient portfolios.68 This reflects loss aversion on the part

of managers, and indicates that should returns fall below the loss threshold, managers

will be inclined to take greater risks in an attempt to recover their positions. This

could be to reduce the likelihood of penalties for performing below expectations, or to

increase the likelihood of receiving bonuses or other forms of compensation for

performing above the reference point.

Given our bank risk preference function, the centre will not want managers to select

portfolios that dominate by PSD. Indeed, if there were portfolios that could not be

ranked by SSD or TSD criteria, but which could be ranked by PSD and MSD criteria,

we determine that the centre should reject the PSD-efficient portfolios given their

67 Recall the expected value of each portfolio is the same at $98.99. 68 Refer to section 3.4 of this chapter.

A TSD E � B TSD E � C TSD E � D TSD E

✁ (A) < ✁ (B) < ✁ (C) < ✁ (D) < ✁ (E)

83

characterisation of risk-taking in the domain if losses. In this scenario, the PSD

efficient portfolio should have a larger value for any risk measure:

It is difficult to accept that the centre of a bank would have a risk-based preference for

MSD-efficient portfolios, given such portfolios carry a convex segment in the domain

of gains. However, it seems probable that the centre would be more willing to accept

risk-seeking in the domain of gains (than the domain of losses) provided the exposure

is not so significant as to increase the potential for actual values to drop below the

target reference point should outcomes turn out to be less than favourable.

3.2.2.6 Other considerations

A final consideration arises with respect to determining the ranking of portfolio

distributions that are not stochastically dominant under any of the definitions provided

in this chapter. The benefit of the stochastic dominance approach is that portfolios can

be ranked according to risk attitudes of investors without having to formulate specific

utility functions. If portfolios cannot be ranked according to stochastic dominance

criteria, dominant portfolios can only be determined by estimating the respective

expected utilities of the available portfolios. This would require a specific formulation

of the utility function for the centre of the bank, or in turn, for each stakeholder

interest that the centre is deemed to represent. A specific formulation of the utility

function of a banking firm is yet to appear in the literature.69

69 Guthoff, Pfingsten and Wolf (1998) and Yamai and Yoshiba (2002) provide examples showing expected utility calculations for a bank based on alternative investment portfolios, but do not provide a theoretical basis or justification for these utility functions as they apply to a banking firm. Their examples serve to demonstrate how the ranking of investment portfolios may be preference-dependent.

A TSD E � B TSD E � C TSD E � D TSD E

E PSD A � E PSD B � E PSD D

✁ (A) < ✁ (B) < ✁ (C) < ✁ (D) < ✁ (E)

84

3.2.2.7 Summary

Table 3.2 provides a summary of the results of this section. It shows the dominant

portfolios according to the various stochastic dominance criteria reviewed earlier in

the chapter, where a dominant portfolio is a lower risk portfolio in terms of the

specifications of the order/category of stochastic dominance under examination. For

example the table shows that portfolio A dominates B and C by SSD, and B, C, D and

E by TSD. Portfolio A also dominates portfolio E by MSD.

Table 3.2

Summary of Risk Profile of Portfolios A - E: Pairwise Rankings

Ranking Criteria FSD SSD TSD PSD MSD

Dominating Portfolio

Portfolio A B, C B, C, D, E E

Portfolio B C C, D, E E

Portfolio C D, E

Portfolio D E E

Portfolio E A, B, D

In summary, from the perspective of the centre of the bank, the lowest risk portfolio is

portfolio A, followed by B, C, D and E. This order of ranking also calibrates with

portfolio efficiency/dominance considerations given each of the five portfolios has the

same expected value. The congruence of these results with stochastic dominance

principles (via mean-preserving spreads and extreme tail losses) provides evidence

that the conclusions are not dependent on the sample distributions selected. Incentive-

compatibility between the centre and managers in the bank, in terms of portfolio

selection, will in the first instance be determined in line with these considerations. In

the second instance, we will examine the subset of incentive-compatible risk measures

within the framework of coherency, in keeping with Artzner et al (1997) and Artzner

et al (1999).

85

3.3 ANALYSIS OF RISK MEASURES

In this section we present calculations of the risk for each portfolio using the

categories of risk measures defined earlier in the previous chapter,70 and examine the

congruence of these risk measure with the risk of the portfolio as determined under

stochastic dominance principles. The calculations are presented in Table 3.3.

Table 3.3: Risk Measures for Portfolios A - E

Probability Distributions

Portfolio A B C D E

Market Value

0

30 0.4% 0.5%

60 1% 1% 1%

90 2.5% 0.6% 0.5%

97 5% 7.5% 5% 5% 10%

98 15% 10% 10% 10% 6%

99 30% 32.5% 30% 40% 25%

100 40% 40% 40% 41% 53%

101 5% 5% 5% 2% 4%

102 3% 3% 3% 1%

103 1% 1% 1%

104

105

106 2.5% 1%

Expected value 98.99 98.99 98.99 98.99 98.99

Risk measures

1. LPM0 (98.99) 0.210 0.185 0.185 0.160 0.170

2. VaR 99% 38.99 38.99 38.99 8.99 8.99

3. ES 99% 38.99 38.99 38.99 32.99 38.99

4. LPM1 (98.99) 0.6379 0.6382 0.8132 0.5284 0.6483

5. LPM1 (99.01) 0.6451 0.6451 0.8199 0.5356 0.6542

6. LPM2 (98.99) 15.547 15.597 17.519 19.819 24.657

7. Spectral (98.99) 15.465 15.473 16.089 158.509 198.090

8. WT 99% 20.206 20.244 21.619 27.557 29.347

9. WT 99.97% 33.974 33.983 34.555 54.986 56.251

70 We include additional measures, to be discussed shortly.

86

The expected value of each of portfolios A to E is $98.99. In the bank setting in this

chapter, expected losses are measured as the difference between the $100 face value

of each portfolio and the expected value of the portfolio. Losses beyond this threshold

represent unexpected losses, and form the focus for the measurement of downside

risk.71

3.3.1 Analysis of Shortfall Probability (LPM0)

Shortfall probability was earlier defined as the probability that the return on a

portfolio will fall below a prespecified target level. Shortfall probability is measured

by the zero-order lower partial moment (LPM0).

Table 3.3 shows the shortfall probability for each portfolio (an extract is provided

below) where the loss threshold is the expected value of the portfolio of $98.99

[LPM0 (98.99)]. The shortfall probability for portfolio A is 0.21 (there is a 21%

probability that the market value of the portfolio will be lower than the target level,

which corresponds to the expected value of the portfolio), 0.185 for portfolios B and

C, 0.16 for portfolio D and 0.17 for portfolio E.

Portfolio A B C D E

LPM0 (98.99) 0.210 0.185 0.185 0.160 0.170

First, it is apparent that the probability of shortfall risk measure is not congruent with

SSD or TSD because the lowest risk portfolio according to stochastic dominance

principles, portfolio A, has the largest risk measure in terms of shortfall probability.

Indeed, a manager who wishes to maximise the risk-adjusted return on a credit

portfolio would find portfolio A the least attractive investment if shortfall probability

is used as an internal risk measure. In fact a manager who is remunerated against a

shortfall probability risk measure is more likely to select portfolio D or E because

they have the lowest values for the risk measure, yet these are the highest risk

portfolios according to stochastic dominance criteria. For a given level of return,

71 Recall that banks are required to provision for expected losses in their financial statements.

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portfolio A will produce the lowest risk-adjusted performance metric, while portfolio

D (and then E) will generate the highest risk-adjusted performance metrics. Further,

shortfall probability fails to discriminate between the riskiness of portfolios B and C,

yet we have determined that C is riskier than B according to SSD and TSD criteria.

A second observation is that the evaluation of portfolio risk according to shortfall

probability tends to induce behaviour on the part of managers that is consistent with a

risk-seeking utility function. Perversely, managers will have a risk-based preference

to select the least attractive portfolios from the perspective of the centre of the bank,

given these portfolios generate the highest expected risk-adjusted returns. This arises

because the lower partial moment of order n < 1 gives less weight to larger deviations

from the reference point.72

We can conclude that if shortfall probability is used as an internal risk measure within

a bank, and managers are remunerated on the basis of the risk-adjusted return of the

portfolios under their responsibility, then managers are more likely to select portfolios

that are not congruent with the risk preference function of the centre. Shortfall

probability identifies only the probability that losses will exceed a certain less

threshold, and not the size of losses. It also places less weight on larger deviations

from the loss threshold, thereby encouraging risk-seeking behaviour on the part of

managers. Shortfall probability is not an incentive-compatible risk measure for bank

stakeholders who are risk-averse and who prefer positive skewness in the distribution

of returns.

72 A lower partial moment of order n = 1 gives equal weight to larger deviations below the reference point, and thus does not differentiate among portfolios with larger expected losses. This is a weakness of the expected shortfall measure, which will be examined later in this section.

88

3.3.2 Analysis of Value-at-Risk

Value-at-risk (VaR) was earlier defined as the loss that will not be exceeded over a

certain time period with a specified confidence level (�), or the loss that will be

exceeded only with a given probability (1 - �), over a specified measurement period.

VaR is closely related to shortfall probability through the cumulative distribution

function. For example, if we consider portfolio A, the 1% shortfall probability

corresponds to a portfolio loan market value of $60. At the portfolio expected value of

$98.99, the VaR at a 99% confidence level for portfolio A is $98.99 - $60 = $38.99.

This demonstrates the relationship between the VaR and shortfall probability risk

measures.73

Table 3.3 shows VaR calculations for a 99% confidence interval for each portfolio (an

extract is provided below) where the basis for measuring losses is the expected value

of the portfolio of $98.99.74 The VaR (99%) for portfolio A is $38.99, as demonstrated

above. The VaR (99%) for portfolios B and C is also $38.99, while the VaR (99%) for

portfolios D and E is $8.99.

Portfolio A B C D E

VaR 99% 38.99 38.99 38.99 8.99 8.99

The results indicate that like shortfall probability, VaR is not incentive-compatible

with the risk preference function of the centre of the bank. The figures suggest that

VaR is not congruent with SSD or TSD criteria because portfolios A, B and C carry

the same VaR metric at the 99% confidence level, indicating the portfolios carry the

same risk, while we have previously shown that portfolios A to C are increasing in

risk in terms of SSD and TSD. Of potentially greater concern is that the VaR for

portfolios D and E at the 99% confidence level is actually lower than for portfolios A,

B and C, yet portfolios D and E are the riskiest portfolios in terms of SSD and TSD

criteria. This arises because VaR (99%) fails to capture the greater tail risk in portfolios

73 The VaR (99%) of $38.99 indicates there is 99% probability that losses on the portfolio will be less than $38.99 over the specified time horizon, and a 1% probability that losses will be greater than or equal to $38.99 over the specified time horizon. 74 Refer to Appendix 7 for calculations.

89

D and E – both portfolios have a small probability of larger losses than portfolios A, B

and C. Further, the VaR (99%) fails to identify the greater risk in portfolio E relative to

D. These results arise because VaR fails to incorporate losses beyond the

predetermined loss threshold (the selected confidence interval).

The theoretical basis for these results lies in the conditions under which the VaR

measure will provide a ranking of portfolios that will not be consistent with stochastic

dominance principles. Tables 3.4a and 3.4b provide calculations to support the

discussion that follows.

Consider first the compatibility of VaR and FSD. If one investment F shows FSD

over another investment G, this indicates that F will be preferred to G for an investor

who seeks more over less – the utility function for the investor is everywhere

increasing in portfolio F compared to G.75 This requirement holds regardless of the

risk attitude of the investor.

Table 3.4a compares the loss payoffs on two portfolios, X and Y, for which X FSD Y.

A non-satiated investor would prefer portfolio X over Y according to FSD principles.

The table shows that for X FSD Y, the VaR for portfolio X is lower than the VaR for

portfolio Y except for those confidence levels where the cumulative density functions

coincide. At the confidence levels of 99% and 95%, the cumulative density functions

do not touch and the VaR of portfolio X is lower than the VaR of portfolio Y. At the

99% level, the VaR of X is $5 and the VaR of Y is $6. At the 95% level, the VaR of X

is $2 and the VaR of Y is $3. At these points the VaR measure is consistent with the

ranking of the portfolios according to FSD. At the 97% confidence level, however, the

cumulative density functions coincide (X1 = X2 = 3%) and the VaR of each portfolio

is equal at $4.76 At this point, the VaR measure fails to differentiate between the risk

profile of the portfolios, and for this reason, we can state that the VaR risk assessment

may not be compatible with FSD principles. The example also highlights that the

75 Refer section 3.1 for a more formal representation. 76 Incidentally, the same results hold for the probability of shortfall measure (LPM0). At the loss threshold of $4, LPM0 (X) = LPM0 (Y) = 0.03.

90

ranking of alternative investment opportunities based on the VaR measure is not

independent of the selected confidence interval.

Now consider the compatibility of risk assessment according to VaR and SSD. Both

Tables 3.4a and 3.4b indicate that the ranking of alternative investment opportunities

according to the VaR measure may contradict the risk ranking given by SSD criteria.

Table 3.4a shows that X SSD Y, indicating that X will be preferred to Y by an

investor who prefers more over less and who is risk-averse in utility. Yet we have

seen above that VaR provides an identical value at the confidence level where the two

cumulative density functions coincide, and thus fails to identify the greater risk in

portfolio Y. In Table 3.4b, two portfolios, S and T, are compared for FSD and SSD.

The table shows that the portfolios cannot be ranked by FSD because the cumulative

density functions cross. The portfolios can be ranked, however, by SSD with portfolio

S dominating portfolio T.

Despite the fact that S SSD T, Table 3.4b shows that the cumulative density functions

for the portfolios coincide at a confidence level of 97%, with the result that the 97%

VaR of the portfolios is identical at $4. Again, this shows that at certain confidence

levels the VaR risk assessment fails to discriminate among the risk of the portfolios,

despite the fact that portfolio T is riskier than S by SSD. If VaR is used within a bank

as the basis for measuring risk among alternative investments, a manager may be

misled into selecting the higher risk portfolio in the belief that the risks of the

portfolios are identical.

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Table 3.4a

Compatibility of VaR and Stochastic Dominance: Case 1

Probability

Cumulative Probability

Difference

Sum of Cumulative Probability

Difference

Portfolio X Y X1 Y1 (Y1 – X1) X2 Y2 (Y2 – X2)

Payoffs

-7 1% 1% 1% 1% 1%

-6 1% 1% 1% 2% 1% 1% 3% 2%

-5 2% 1% 3% 3% 0% 4% 6% 2%

-4 1% 2% 4% 5% 1% 8% 11% 3%

-3 1% 1% 5% 6% 1% 13% 17% 4%

-2 4% 5% 9% 11% 2% 22% 28% 6%

VaR 99% 5 6 X FSD Y X SSD Y

VaR 97% 4 4

VaR 95% 2 3

Table 3.4b

Compatibility of VaR and Stochastic Dominance: Case 2

Probability

Cumulative Probability

Difference

Sum of Cumulative Probability

Difference

Portfolio S T S1 T1 (T1 – S1) S2 T2 (T2 – S2)

Payoffs

-7 1% 1% 1% 1% 1%

-6 1% 1% 1% 2% 1% 1% 3% 2%

-5 2% 1% 3% 3% 0% 4% 6% 2%

-4 2% 1% 5% 4% -1% 9% 10% 1%

-3 1% 1% 6% 5% -1% 15% 15% 0%

-2 4% 5% 10% 10% 0% 25% 25% 0%

VaR 99% 5 6 No FSD S SSD T

VaR 97% 4 4

VaR 95% 3 2

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Table 3.4b also shows the VaR of the portfolios at the 95% confidence level is $3 for

portfolio S and $2 for portfolio T, signalling that portfolio T has less risk than S. The

lower VaR measure for portfolio T arises at the 95% confidence level because at this

point the cumulative density function for portfolio S exceeds that of T. The ranking

of the portfolios according to VaR completely contradicts the risk assessment of the

portfolios according to SSD. If managers in the bank are remunerated in terms of the

risk-adjusted return on their portfolios, and VaR is used as the risk basis for the

measure, then at the 95% confidence level they may be more inclined to select the

lowest reported VaR portfolio, particularly if the expected returns on the portfolios are

the same. In the current case, this means managers may select portfolio T over S given

its lower VaR, yet this portfolio is dominated by S according to SSD. At the 99%

confidence interval, the VaR of S exceeds the VaR of T. This is in keeping with the

result in Table 3.4a, and arises because the cumulative density function for portfolio T

exceeds S at this point.

The results in Table 3.4b confirm that the ranking of portfolios according to the VaR

measure is not congruent with the ranking of portfolios under stochastic dominance

principles, as embodied in the risk preference function of the centre of the bank. With

respect to portfolios S and T, we have three different VaR signals within three

different confidence levels. At the 99% confidence level, portfolio T is highest risk

portfolio, while at the 95% level, portfolio S is the highest risk portfolio. At the 95%

confidence level the portfolios have the same VaR and this signals equal risk. Thus,

depending on the desired confidence level, managers may make investment decisions

that are not aligned with the interests of the centre of the bank. We can conclude that

VaR is not an incentive-compatible risk measure.

There are additional considerations arising from these observations. If VaR is used

within the bank as the basis for measuring and allocating capital against credit

portfolios, and pricing is based on achieving a target return on economic capital, then

the failure of VaR to discriminate against portfolio risk may lead to an incorrect

pricing of risk. This could result in the bank being undercapitalised relative to the true

risk in its book. Further, managers may be enticed to select portfolios where risk is

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undercapitalised because this may enable them to report larger risk-adjusted returns

on their portfolios, subject to the pricing of such portfolios, and thus increase the

probability of receiving bonuses or other rewards. In other circumstances, managers

may be able to deliberately manipulate their reported VaR to misrepresent reported

risk and achieve high risk-adjusted returns on their investments. These issues are

addressed in more detail in the chapter that follows.

3.3.3 Analysis of Expected Shortfall

Expected shortfall was earlier defined as the conditional expectation of loss given that

the loss is beyond the VaR level. This means that expected shortfall measures losses

beyond the VaR level, and in particular, extreme losses that could prove catastrophic

for the bank. The ability of expected shortfall to capture tail risk, as well as recognise

portfolio diversification (subadditivity), has led to a number of authors to recommend

expected shortfall as a better alternative than VaR for both regulatory and internal risk

management purposes. These authors include Artzner, et al (1999), Rockafellar and

Uryasev (2000), Acerbi et al (2001), and Acerbi and Tasche (2002). Despite these

recommendations, our analysis reveals that expected shortfall is not an incentive-

compatible risk measure for internal risk management purposes.

Table 3.3 shows expected shortfall calculations for a 99% confidence interval ES (99%)

for each portfolio (an extract is provided below) where the basis for measuring losses

is the expected value of the portfolio of $98.99.77 The extract also includes VaR

measures for comparison. The ES (99%) of portfolios A, B, C and E is equal at $38.99,

while the ES (99%) for portfolio D is lower at $32.99:

Portfolio A B C D E

VaR 99% 38.99 38.99 38.99 8.99 8.99

ES 99% 38.99 38.99 38.99 32.99 38.99

77 Refer to Appendix 7 for calculations.

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The results immediately suggest that the ES risk measure is not congruent with the

risk ranking of the portfolios according to stochastic dominance principles, which

have risk increasing in terms of TSD from portfolio A to E.

Consider first the compatibility of ES with SSD. A number of authors claim that the

ranking of investments according to ES is consistent with the ranking of investments

according SSD [see Barbosa and Ferreira (2004), Yamai and Yoshiba (2002)]. Our

study, however, demonstrates that there are circumstances where ES will not provide

a ranking of investments that is consistent with SSD. Specifically, portfolios A, B and

C all carry the same ES (99%) of $38.99, despite the fact that we have shown A SSD B,

A SSD B and B SSD C. This implies that a manager may inadvertently select a

dominated portfolio that does not maximise the utility of the bank, as expressed by the

risk preference function of the centre.

Next consider the compatibility of ES with TSD. We know that for portfolios A, B

and C, ES (99%) is not consistent with TSD because the portfolios carry the same value

for the risk measure, yet we have shown that A TSD B, A TSD C and B TSD C. This

also corresponds to the SSD case above. Now consider portfolio D, which cannot be

ranked by SSD, but which is dominated by portfolios A, B and C by TSD. We have

ascertained that a portfolio that is dominated by TSD should have a larger risk

measure:

We find, however, that the ES (99%) for portfolio D is actually lower at $32.99 than for

the preceding portfolios, implying that portfolio D is less risky than portfolios A, B

and C. This implies that a rational manager seeking to select either the minimum risk

portfolio or the portfolio with the highest risk-adjusted return is more likely to select

the portfolio that is TSD dominated. Further, if the centre of the bank (or regulator)

placed an internal cap on the expected shortfall for a particular position, then at the

99% confidence level, the perverse situation could arise whereby managers within the

A TSD B � A TSD C � B TSD C

A TSD D � B TSD D � C TSD D

✁ (A) < ✁ (B) < ✁ (C) < ✁ (D)

95

bank are forced to invest in the TSD-dominated portfolio.78 Finally, our calculations

show that portfolio E has the same ES (99%) as portfolios A, B and C, despite portfolio

E having the largest tail risk of all the portfolios under consideration: recall A TSD E,

B TSD E, and C TSD E. Again, in terms of TSD criteria, ES (99%) fails to discriminate

among the higher risk in the portfolios, with the exception of portfolios D and E,

where D TSD E and the ES (99%) for D is lower than the ES (99%) for E.

There are two main factors that explain these results.

The failure of ES (99%) to correctly rank portfolios A, B and C by SSD arises because

the mean preserving spread that created portfolios B and C (and which guarantees

they will be dominated by portfolio A by SSD) occurs within the range of the

portfolio distribution that is to the right of the loss threshold, and thus only impacts on

portfolios values that are above the �-quantile of the distribution.79 This is shown in

Appendix 3, Table 1 for portfolio B, and Appendix 4, Table 1 for portfolio C. As was

found to be the case with the VaR measure of risk, this result confirms that the risk

ranking of investment opportunities according to ES is not independent of the selected

confidence interval. This also implies that ES does not differentiate among portfolios

that may have greater losses below the �-quantile of the distribution. We will see

shortly that a risk measure that has a close resemblance to ES, the first-order lower

partial moment (LPM1), does take into account the full range of losses when the target

threshold is set to cover the full distribution of portfolios losses.

The second factor explaining the inconsistency of the risk rankings of ES with SSD

and TSD is that the risk attitude implicit in ES is not risk-aversion, but rather, risk

neutrality. This arises because the ES risk measure gives equal weights to loss

quantities below the �-quantile. This means that it makes no difference as to the

dispersion of losses in the left tail of the distribution - if the average of the losses are

equal, the ES of the portfolios will also be equal. This clearly runs counter to the

78 In the current context, this would occur if a cap of $33 was placed on the expected shortfall. This would exclude portfolios A, B and C, despite these being TSD-dominating portfolios. 79 Indeed it is for this reason that expected shortfall is also referred to as ‘conditional VaR’ or ‘tail conditional expectation’ in the risk management literature.

96

characteristics of risk measures that are compatible with TSD, as discussed in section

2.3 of the previous chapter, being that they should discriminate among investments

where there are larger deviations from the loss threshold then smaller deviations, even

though the average losses beyond the threshold may be equal.

The implicit risk-neutral attitude in the ES measure is apparent in the loss

distributions of portfolios C and E. It has already been established that C dominates E

by TSD, yet the ES (99%) for portfolios D and E are equal at 38.99. However,

inspection of the loss distributions at the 99% confidence level reveals that portfolio E

has a smaller probability of larger deviations from the loss threshold than portfolio D.

More specifically, the mean deviations for the portfolios are as follows:

Portfolio D: [(98.99 – 60) x 1%] = 0.3899

Portfolio E: [(98.99 – 90) x 0.5%] + [(98.99 – 30) x 0.5%] = 0.3899

Portfolio D carries a 1% probability of a deviation of $38.99, while Portfolio E carries

a 0.5% probability of a deviation of $8.99 and a 0.5% probability of a deviation of

$68.99. The potential for the larger deviation from the loss threshold in portfolio E is

not captured in the ES risk measure.

The risk-neutral attitude implicit in the ES measure also indicates that like VaR, ES

may lead managers to engage in manipulations in the lower tail of the distribution,

such that a portfolio with greater tail risk may be misrepresented as one with equal or

lower risk. This action may lead to an overstatement of the risk-adjusted return on the

portfolio, with the intention to increase the probability or size of bonuses paid to

managers.

We can conclude that if ES is used as an internal risk measure within a bank, and

managers are remunerated on the basis of the risk-adjusted return of the portfolios

under their responsibility, then managers may be induced to select portfolios that are

not congruent with the risk preference function of the centre because ES fails to

adequately differentiate risks in portfolios. Like VaR, we have shown that ES may

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actually assign a lower value to portfolios that are dominated by TSD. These

problems arise first because the ES measure places an equal weight on losses that

exceed the threshold, regardless of the size of the deviation from the threshold, and

second because the measure fails to recognise losses that are less than the threshold

but which nonetheless are increasing in risk in terms of stochastic dominance

principles.80 ES is not an incentive-compatible risk measure for bank stakeholders

who are risk-averse and who prefer positive skewness in the distribution of returns,

and thus should be rejected from the set of acceptable internal risk measures.

3.3.4 Analysis of the First-Order Lower Partial Moment

The first-order lower partial moment (LPM1) measures the weighted average

deviation from the target level. Earlier we showed that LPM1 is closely related ES

because it captures the weighted average of losses that occur beyond the prespecified

loss threshold – indeed, if the threshold for ES and LPM1 is the �-quantile, then these

measures assign the same value to the risk measure. The distinguishing characteristic

of LPM1 relative to ES is LPM1 is not restricted to a particular quantile and thus

allows for a target that covers all or part of the distribution of losses. This, in part,

makes the measure superior to VaR and ES with respect to congruence with the risk

preference function of the centre of the bank. Like VaR and ES, however, LPM1

displays characteristics – risk neutrality in losses and target dependence - that cause it

to fail as an incentive-compatible risk measure. We examine these below.

Table 3.3 shows LPM1 values for the portfolios (an extract is provided below) where

the basis for measuring losses is the expected value of the portfolio of $98.99. The

LPM1 measures for portfolios A to E, respectively, are 0.6379, 0.6382, 0.8132, 0.5284

and 0.6483:

80 Earlier we identified that small but regular unexpected losses that are less than the ✁-quantile may cause problems for the bank if it forced to frequently recapitalise to make up for these losses. These source of problems are related to transaction costs and negative market signals. The latter may make it difficult for the bank to recapitalise on favourable terms.

98

Portfolio A B C D E

LPM1 (98.99) 0.6379 0.6382 0.8132 0.5284 0.6483

The first observation is the ranking of portfolios A, B and C for LPM1 with the

ranking of these portfolios according to SSD and TSD is aligned. This is in contrast to

the VaR (99%) and ES (99%), which assigned equal values for portfolios A, B and C and

thus failed to identify the dominance of portfolio A over B and C, and portfolio B

over C. Our second observation, however, is less encouraging – portfolios D and E

show lower values for LPM1 than portfolio C, suggesting that these portfolios are less

risky than portfolio C, but this is not congruent with the ranking of the portfolios

according to TSD.81 Further, portfolio D has the lowest LPM1 value of the five

portfolios, incorrectly signalling that it is the lowest risk portfolio. This could entice

managers within the bank to select portfolio D on the basis that it is the lowest risk

portfolio, with potentially the highest risk-adjusted return. This decision would not be

aligned with the best interests of the centre of the bank and the stakeholders it is

deemed to represent.

Why does the LPM1 ranking of portfolios concur with SSD and TSD for portfolios A,

B and C? Consistency in the risk-ordering occurs because LPM1 set at the expected

market value of the portfolios measures losses as deviations below $98.99, and this

captures all losses below the expected value, rather than those that fall below a

predetermined �-quantile. This means that intermediate losses, being those losses that

fall between $98.99 and the �-quantile value, are also captured in the risk measure.

For portfolios B and C, this means the area of the distribution impacted by the mean-

preserving spread calculations is recognised in the risk measure. Thus, for LPM1, the

following holds:

81 The risk ordering of portfolios D and E, however, matches the ranking according to TSD.

99

Why is LPM1 inefficient with respect to the risk ranking of portfolio D relative

portfolios A, B and C? In this case, problems arise because LPM1 fails to recognise

the greater tail risk in portfolio D relative to A, B and C - like expected shortfall,

LPM1 embodies a risk-neutral stance with respect to the deviation of losses beyond

the threshold. The order of one for the power function means LPM1 also only captures

average losses beyond the threshold, and thus does not distinguish between large

losses with a small probability of occurrence and smaller losses carrying a large

probability of occurrence. This contradicts the assumption of risk aversion and

positive skewness preference inherent in risk-ordering by TSD.

Portfolios D and E carry the largest tail risk of the portfolios – despite having the

lowest probability of losses of the five portfolios,82 both portfolios D and E have

larger extreme losses in their return distributions. Specifically, the probability that the

market value of the portfolio will drop to $60 is 0.40% for portfolio D and 0.50% for

portfolio E – corresponding to a large below target deviation of $68.99 for both

portfolios. Yet LPM1, by failing to place greater weight on larger deviations from the

threshold, does not materially differentiate among the portfolios carrying large tail

risk. A good example of this arises in a comparison of portfolio A, the lowest risk

portfolio by TSD, with portfolio E, the highest risk portfolio by TSD. Portfolio A has

a 1% probability of losses of $38.99, while portfolio E has a 0.5% probability of

losses of $8.99 and a 0.5% probability of losses of $68.99. At the 1% confidence

level, LPM1, like ES, treats these loss positions as essentially equal, despite the larger

extreme loss in portfolio E:

82 Shortfall probability for portfolio D is 16% and for E is 17% - refer shortfall probability measures in Table 3.

A SSD B � A SSD C � B SSD C

A TSD B � A TSD C � B TSD C

LPM1(A) < LPM1(B) < LPM1(C)

100

Portfolio A: [(38.99 x 1%)] = 0.3899

Portfolio E: [(68.99 x 0.5%) + (8.99 x 0.5%)] = 0.3900

This again reflects the risk-neutral risk attitude implicit in LPM1, and explains its lack

of congruence with risk-ordering by TSD principles.

We must also reject the efficiency of LPM1 on the basis that it displays target

dependence. Just as we identified different risk values for VaR and ES based on the

selected confidence level, we find that LPM1 can also fail to correctly identify riskier

portfolios depending on the target set for defining losses. Consider the impact when

the target is set at $99.01, which is only slightly higher than the expected value of the

portfolio at $98.99. Calculations for LPM1 under this scenario are shown in Table 3.3,

and an extract of the relevant figures is presented below:

Portfolio A B C D E

LPM1 (98.99) 0.6379 0.6382 0.8132 0.5284 0.6483

LPM1 (99.01) 0.6451 0.6451 0.8199 0.5356 0.6542

The figures show that LPM1 gives the same value for portfolios A and B, despite the

fact that A dominates B by SSD. The result is that a small change in the target level

produces risk measures that incorrectly deem the portfolios to be of equal risk. This

result occurs because the alternative target of $99.01 happens to lies outside that area

of the return distribution impacted by the mean-preserving spread that relates portfolio

B to portfolio A.83 This also represents a point at which the cumulative probability

density functions of the portfolios are equal, as shown for these portfolios in Table 2

of Appendix 4.84 This is of significance because when the cumulative distribution

functions cross, the LPM1 measure may give a portfolio risk-ordering that is

83 Refer Appendix 3, Table 1. 84 At the market value of $99, the cumulative density function for portfolios A and B (columns A1 and B1) is equal at 51%. However, note that while for portfolios B and C (columns B1 and C1) the cumulative density functions at a market value of $97 are equal at 8.5%, the LPM1 for the portfolios is not equal at this threshold. This occurs because the threshold of $97 lies within the area of the distribution impacted by the mean-preserving spread that relates portfolio C to B. We can say that if the cumulative distribution functions are equal at some target or threshold level, risk-assessment compatibility between LPM1 and SSD may not hold for some target threshold levels.

101

inconsistent with SSD, depending of the choice of target level from which losses are

measured. This is demonstrated in the following example.

Table 3.5a presents loss distribution data for two portfolios, F and G. The expected

losses for the portfolios are equal. The table shows that the portfolios cannot be

ranked by FSD, but portfolio F dominates G by SSD. Table 3.5b presents LPM1

calculations for portfolios F and G at various target loss thresholds. For example, at

the loss threshold target of $4, the LPM1 for portfolio F is 0.240 and the LPM1 for

portfolio G is 0.250, indicating G is riskier than F. Referring to Table 3.5a, there are

four points at which the cumulative density functions for the portfolios are equal

[F(w) = G(w)]: $9, $6, $2 and $1. The corresponding LPM1 for the portfolios at these

target points, as shown in Table 3.5b, indicate that G is riskier than F at the target

points of $9 and $6, but G and F have equal risk at the target points of $2 and lower.

This again demonstrates the target dependence of the LPM1 risk measure. It shows

that for some target loss thresholds, LPM1(F) < LPM1(G), and the LPM1 risk measure

is compatible with the ranking of the portfolios by SSD. However, at other target loss

thresholds, LPM1(F) = LPM1(G), and consistency between SSD and the LPM1 risk

measure does not hold.

We can conclude that LPM1 fails as an incentive-compatible risk measure because

under certain conditions it fails to provide a risk-ranking of portfolios that is

compatible with SSD and TSD principles. If LPM1is used as an internal risk measure

within the bank, managers may be induced to select portfolios that are not congruent

with the risk preferences of the centre of the bank. We have found that these

conditions relate to the target dependence and implicit risk-neutral risk attitude of the

LPM1 risk measure. The problem of target dependence can be overcome if the target

threshold is set to cover all portfolio losses, rather than those occurring beyond some

target level or �-quantile. This, however, may not be practical in the bank setting,

particularly if the risk measure is used as the basis for internally allocating capital

against risky positions taken by managers or business units because internal capital

allocation is typically linked to a desired solvency standard for the bank. The

implications of the implicit risk-neutral attitude of the LPM1 measure are that

102

managers may inadvertently select portfolios that have large tail risk, thus increasing

the risk exposure of the bank without a compensating increase in capital allocation or

pricing.

Table 3.5a: Stochastic Dominance Analysis - Portfolios F and G

Probability Distribution

Cumulative Probability

Difference

Sum of Cumulative Probability

Difference

Portfolio F G F1 G1 (G1 – F1) F2 G2 (G2 – F2)

Loss

10 1% 1.5% 1% 1.5% 0.5% 1% 1.5% 0.5%

9 1% 0.5% 2% 2% 0% 3% 3.5% 0.5%

8 1% 1.5% 3% 3.5% 0.5% 6% 7% 1%

7 1% 0% 4% 3.5% -0.5% 10% 10.5% 0.5%

6 2% 2.5% 6% 6% 0% 16% 16.5% 0.5%

5 2% 2.5% 8% 8.5% 0.5% 24% 25% 1%

4 1% 0% 9% 8.5% -0.5% 33% 33.5% 0.5%

3 1% 1% 10% 9.5% -0.5% 43% 43% 0%

2 1% 1.5% 11% 11% 0% 54% 54% 0%

1 1% 1% 12% 12% 0% 66% 66% 0%

Expected loss 0.66 0.66 No FSD F SSD G

Table 3.5b: First-Order Lower Partial Moment Measures

Target LPM1(F) LPM1(G) Target LPM1(F) LPM1(G)

9 0.010 0.015 4 0.240 0.250

8 0.030 0.035 3 0.330 0.335

7 0.060 0.070 2 0.430 0.430

6 0.100 0.105 1 0.540 0.540

5 0.165 0.165 0 0.660 0.660

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3.3.5 Analysis of the Second-Order Lower Partial Moment

We earlier defined the second-order lower partial moment (LPM2) as a risk measure

that places a quadratic penalty on deviations below the threshold. The quadratic power

function acts to place a greater penalty, in terms of the value of the risk measure, in

larger deviations from the threshold than smaller deviations. By placing a larger

penalty on larger losses, LPM2 overcomes the problems associated with the implicit

risk-neutral attitude of ES and LPM1 because the order n > 1 implies a risk-averse

attitude to losses.85

Table 3.3 shows LPM2 values for the portfolios A to E (an extract is provided below)

where the expected value of the portfolio of $98.99 is set as the threshold upon which

losses are measured. The LPM2 measures for portfolios A to E, respectively, are as

follows:

Portfolio A B C D E

LPM2 (98.99) 15.547 15.597 17.519 19.819 24.657

These results indicate that LPM2 provides a portfolio risk-ordering that corresponds to

the risk-ordering according to TSD principles. The LPM2 values for the portfolios

increase in ascending order from A to E:

85 Appendix 8 provides an example to demonstrate how the order (n) of the LPM measure corresponds to a specific risk attitude on the part of investors.

A TSD E � B TSD E � C TSD E � D TSD E

LPM2(A) < LPM2(B) < LPM2(C) < LPM2(D) < LPM2(E)

104

Given our earlier finding that TSD represents the most applicable criteria for ranking

investment portfolios for an investor who is risk-averse and who has a preference for

positive skewness in the distribution of returns, we can conclude that LPM2 is

congruent with the risk preference function of the centre of the bank. Congruence

arises because the LPM2 risk measure includes all losses that occur beyond the target

level (which includes the entire distribution of losses if the target threshold is set at

zero) and it places greater weight on larger deviations from the target threshold, even

though average losses may be the same.

An assessment of the incentive-compatibility of LPM2 rests, then, with consideration

of whether or not the measure is target dependent. Target dependence in the LPM1

measure arises because, as discussed, losses are treated equally in the calculation of

the LPM1 regardless of their size. However, if a portfolio F dominates a portfolio G

by SSD, and LPM1(F) = LPM1(G), then it must always hold that LPM2(F) < LPM2(G)

because the quadratic power function in LPM2 places a greater weight on larger

deviations from the loss threshold. This is where compatibility between SSD and

LPM2 arises: a risk-averse investor will place greater emphasis on larger tail losses

than smaller tail losses. We conclude that LPM2 is not target dependent.

Table 3.6 shows LPM2 calculations for portfolios F and G. The figures show that

unlike the case for LPM1, LPM2(F) < LPM2(G) independent of the target level from

which loss deviations are measured.

Table 3.6: Second-Order Lower Partial Moment Measures

Target LPM2(F) LPM2(G) Target LPM2(F) LPM2(G)

9 0.010 0.015 4 0.960 1.030

8 0.050 0.065 3 1.530 1.615

7 0.140 0.170 2 2.290 2.380

6 0.300 0.345 1 3.260 3.350

5 0.560 0.615 0 4.460 4.550

105

We can conclude that that LPM2 is an incentive-compatible risk measure because it

provides a portfolio risk ranking that will always be equal to that which accords to

both SSD and TSD principles. The measure provides a consistent risk-ranking

because it captures all losses that are beyond the threshold (does not rely on a specific �-quantile), places a greater weight on larger losses, and is not target dependent.

The LPM2 measure is thus increasing in risk, and if used as the basis for measuring

risk in bank credit portfolios, should entice managers to reject portfolios that are

dominated by TSD and invest in portfolios that the centre of the bank would have

them select if information on the expected distribution of portfolios was perfectly

available.

At this point one might question the justification for using a quadratic power function

when any LPM of order n > 1 will place a greater weight on larger deviations from

the loss threshold.86 The order of the LPM reflects the risk aversion on the part of the

investor, with higher orders reflecting larger aversion to tail losses. Possibly the only

significant aspect of the quadratic power function is that LPM2 represents the measure

of downside variance which was first espoused by Markowitz. This leads us to

conclude that LPM2 should be recognised as one of a general class of risk measures

that place larger weights on larger losses. In this regard, we now examine two recently

developed classes of risk measure which have arisen largely out of the insurance

literature, and assess their incentive-compatibility for the bank case in this chapter.

These are ‘spectral risk measures’ and ‘distortion risk measures’.

86 Refer Appendix 8.

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3.3.5.1 Spectral Risk Measures

The concept of a ‘spectral risk measure’, developed by Acerbi (2002), is closely

related to LPMn where n > 1. Spectral risk measures allow for different weights

)( p�

to be given to different p-confidence level ‘slices’ of the left tail of the

distribution of returns. Similarities arise with LPMn because the weight function is

increasing in risk, indicating that larger losses have a larger risk-aversion weight.

The formal expression for the class of risk measures✁M that are weighted averages of

the quantiles of the distribution of returns on a portfolio is as follows: ✁M =

1

0

)( dpqp p

✂

where )( p�

is a weighting function, p is probability and qp is the p-quantile of the

distribution. The risk aversion function�

is a weighting function that reflects an

investor’s subjective risk aversion, and is said to be an admissible risk spectrum if it is

positive, decreasing and |✄| = 1.87 In the case of VaR, which does not take into

account tail losses, the function )( p�

is zero everywhere but for p = ☎, where it

becomes infinite. In the case of expected shortfall, the function )( p�

is uniform in

losses beyond the threshold and zero for other losses.

There are at least two ways in which the concept of spectral risk can be applied to the

bank setting that has been the focus of this chapter. In the previous chapter we

examined the difficulties in determining the risk preference function for the centre of

a bank when the stakeholders that it is deemed to represent have conflicting risk

attitudes. It was determined that depositors and senior debtholders are likely to be

highly risk-averse and concerned with the size of losses in the event of default, while

diversified bank owners and managers are likely to be less risk-averse and possibly

87 Acerbi (2002), p.1510. The fact that the risk-aversion function is decreasing reflects the assignment of larger weights to bigger losses. Acerbi does not specify a form for the risk-aversion function.

107

risk-seeking, subject to the franchise value of the bank.88 Further, we considered the

impact on the risk preference function if bank stakeholders faced an S-shaped utility

function that reflects risk-aversion in gains but a risk-seeking attitude in the domain of

losses. The spectral risk approach provides a framework by which some of these

difficulties can be addressed. Specifically, by allowing for different degrees of risk

aversion (or risk-neutral or risk-seeking attitudes) among different classes of

stakeholders, tailored risk measures can be derived and used as the basis for

determining the capital that should be internally allocated against portfolios and other

risky positions.

In this regard, Figure 3.1 shows the distribution of returns for a hypothetical bank

credit portfolio and the ranking of different investor classes with respect to absorbing

the impact of losses on the portfolio. Any decline in the market value of the portfolio

is initially absorbed by profits (expected returns) and provisions for expected losses

(EL). The target solvency standard for the bank has been set at confidence level of

99.9%, meaning that there is a 0.1% probability that the bank will default on Class A

investors over some specified time horizon. In a bank setting, Class A investors are

likely to be depositors and senior debt holders, with depositors ranking ahead of

senior debt holders. Class A investors carry the largest aversion to risk in the bank.

We could consider Class B investors to be junior debt holders, Class C represented by

subordinated debt or preferred stock, and Class D investors are ordinary shareholders

in the bank. It is apparent from the diagram that Class D investors incur the greatest

risk of losses in capital on their initial investment, while Class A investors carry the

greatest protection against losses in capital. To apply the concept of spectral measures

of risk to this example, different weights )( p�

would be assigned to each of the p-

confidence levels as they apply to the each class of investors. For example, Class A

investors might be assigned a zero weighting reflecting their concern only with

probability of losses, rather than the size, while the weighting function for Class B

investors might be slightly greater than one, reflecting the lower risk aversion of this

investor class.

88 It was determined that if the franchise value of the bank is high, owners and managers are likely to be risk-averse.

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Figure 3.1: Distribution of Portfolio Returns and Investor Classes

To provide a concrete example, we consider how LPMs can be modified to reflect the

aversion of different classes of bank investors to tail risk. Specifically, the modified

risk measure would involve changing the order of the LPM in concert with the

expected size of loss deviations from the target threshold, with a larger order

(weighting) applying to larger losses. Table 3.3 shows such a spectral risk measure for

each of the portfolios A to E, based on LPM calculations where the order of the power

function is increased as the size of losses increases. An extract is provided below:

Portfolio A B C D E

Spectral (98.99) 15.465 15.473 16.089 158.509 198.090

Probability

Profits and EL 99.9% Capital allocation

Class A Class B Class C Class D

109

These calculations are based on increasing the order of the LPM from 1 for small

losses above the expected loss threshold of 98.99 through to 2.5 for the most extreme

losses relative to the loss threshold. Calculations are provided in Appendix 9.

The large penalties implicit in the spectral risk measure for the portfolios carrying

large tail risks - portfolios D and E - are apparent. If we compare the spectral risk

measure with the figures for LPM2, it is notable that the spectral risk measure is lower

for the portfolios A, B and C given the lower weighting the spectral measure applies

to intermediate losses in the distribution (a constant quadratic power function applies

across the full range of losses in the LPM2 measure), but significantly larger for the

portfolios carrying a low probability of extreme losses.

Portfolio A B C D E

Spectral (98.99) 15.465 15.473 16.089 158.509 198.090

LPM2 (98.99) 15.547 15.597 17.519 19.819 24.657

The large spectral risk measure value for the high tail risk portfolios would act to

discourage investment in these portfolios on the part of managers, in line with the risk

preferences of the centre. This risk measure is congruent with the risk preference

function of the centre, and the main appeal of spectral risk measures is their capacity

to incorporate different degrees of aversion to risk subject to the size of losses beyond

the threshold and/or different risk classes of bank investors.

3.3.5.2 Distortion Risk Measures

Distortion risk measures are derived from distortion functions which transform the

cumulative distribution function for a risky investment in such as way that higher

weight is given to larger losses. By distorting probabilities to reflect higher risk

aversion to large losses, distortion risk measures are similar in concept to spectral risk

measures discussed above. Risk measures based on distorted probabilities are worthy

of attention in light of the large volume of experimental studies, discussed earlier in

this chapter, that indicate individuals consistently violate expected utility by

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subjectively distorting probabilities when making risky decisions.89 Distortion risk

measures were developed by Wang (1996) and Wang (2000) to address pricing

problems in insurance. In this section we consider the application of a recently

developed distortion risk measure – the ‘Wang Transform (WT)’ – to the

measurement of credit portfolio risk and assess the incentive-compatibility of the

measure within the bank setting of this chapter.

For a loss variable X with discrete distribution F, the WT measure [Wang (2002)]

calculates the expected values for a risky investment using risk-adjusted probabilities

generated by the transformation function F* (x):

F* (x) = �

[�-1(F(x)) –

✁]

where �

(.) denotes the cumulative normal distribution, F(x) denotes the cumulative

distribution function for the objective probability measure, and ✁ represents a

parameter for risk-aversion.90 Wang shows the risk-aversion parameter can be set

such that ✁ =

�-1(✂), where ✂ is a predetermined security or confidence level. This is

useful in the current context because it allows the measure to be aligned with the

target solvency standard of the bank. The risk measure ✄(X) is calculated as the

expected value under F* :

✄(X) = WT☎ = E*[X]

Table 3.3 shows WT risk measures for portfolios A -E at confidence levels of 99%

and 99.97% (an extract is provided below) with the latter selected to reflect the target

solvency standard that typically applies to AA-rated banks.91

89 Yaari (1987) proposes that every decision is based on a probability distribution that is adjusted according to the risk-aversion of the individual making the decision. Tversky and Kahneman (1992) incorporate distortion of the cumulative distribution function in their Cumulative Prospect Theory paradigm. 90 The transformation function takes the original cumulative probability p and transforms this through a standard normal inverse transformation to obtain

✆-1(p). The risk-aversion parameter ✝ is then

subtracted and the resulting expression again transformed through a standard normal transformation to achieve the distorted cumulative probability. 91 Calculations for each portfolio for the 99% confidence level are shown in Appendix 10.

111

Portfolio A B C D E

WT 99% 20.206 20.244 21.619 27.557 29.347

WT 99.97% 33.974 33.983 34.555 54.986 56.251

These calculations indicate that the WT risk measure is congruent with the risk

preferences of the centre of the bank. The risk measures for the portfolios rise in

accordance with the risk ranking determined under SSD and TSD principles. The

calculations show that the risk measures also rise as the bank becomes less tolerant to

losses, as determined under the higher 99.97% confidence level. Further, the risk

measures for the portfolios with the largest tail risks, D and E, increase more than

proportionately under the higher confidence level. We can conclude that the WT risk

measure is an incentive-compatible risk measure within the framework established in

this chapter.

The incentive-compatibility of the WT risk measure arises for two reasons. First, the

measure captures all information regarding the loss distribution for risky investment

portfolios, and thus includes intermediate losses that are below the �-quantile VaR

and ES. Second, the measure places a larger penalty on larger deviations from the

target loss threshold, thus adequately accounting for low-probability high-severity tail

losses.92

92 We drew the same conclusion for the LPM risk measure where n > 1 and the target threshold covered all losses in the distribution.

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3.3.6 Summary

In this section we analysed various risk measures to determine their congruence with

the risk preferences of the centre of a bank that exhibits non-satiety, risk aversion and

positive skewness in the distribution of returns. The discussion incorporated five

credit portfolios which were increasing in risk in terms of TSD, which correlates with

the risk preferences of the centre. The five main risk measures analysed were the

probability of shortfall (LPM0), VaR, ES, LPM1 and LPM2. We also examined two

recently-developed risk measures that are closely aligned to LPM2 – spectral risk

measures and distortion risk measures.

The results are presented in Table 3.7.

Table 3.7: Summary of Results

Risk measures

Portfolios

Congruence

A B C D E

1. LPM0 (98.99) 0.210 0.185 0.185 0.160 0.170 Not congruent

2. VaR 99% 38.99 38.99 38.99 8.99 8.99 Not congruent Not independent of the confidence level

3. ES 99% 38.99 38.99 38.99 32.99 38.99 Not congruent Not independent of the confidence level

4. LPM1 (98.99) 0.6379 0.6382 0.8132 0.5284 0.6483

5. LPM1 (99.01) 0.6451 0.6451 0.8199 0.5356 0.6542

Not congruent and dependence on target threshold

6. LPM2 (98.99) 15.547 15.597 17.519 19.819 24.657 Congruent

7. Spectral (98.99) 15.465 15.473 16.089 158.509 198.090 Congruent

8. WT 99% 20.206 20.244 21.619 27.557 29.347 Congruent

9. WT 99.97% 33.974 33.983 34.555 54.986 56.251 Congruent

113

The results indicate that of the five main risk measures analysed only LPM2 is

congruent with the risk preferences of the centre. Congruence was also found for

spectral risk and WT risk measures, which are similar in concept to LPM2.

Our analysis reveals at least four reasons why a measure may fail to align with a risk

ranking according to SSD and TSD principles. First, a measure may be dependent on

the target or loss threshold for measuring losses, meaning it gives a different risk-

ordering depending on the selected target or confidence level. This is evident in VaR,

ES and LPM1. Second, a measure may not capture losses beyond the threshold. This is

evident in shortfall probability, which reveals only the probability of losses, and VaR,

which reveals losses only up to a predetermined confidence level. Third, a measure

may not distinguish between high severity and low severity losses beyond the

threshold, meaning the measure may be risk-neutral in large losses. VaR ignores all

losses beyond the threshold, and ES and LPM1 weight large losses the same as small

losses and thus expose the investor to tail risk. Shortfall probability assumes a risk-

seeking attitude to losses. Finally, a risk measure may fail to capture the full range of

losses in the distribution, and thus ignore losses below the �-quantile or target level.93

This is evident in quantile-VaR and ES, but not LPM1 when the target for measuring

losses is set at zero. LPM2, however, qualifies on all four points provided the target

from which losses are measured is set at zero.

We can also classify the five risk measures in terms of their compatibility with

stochastic dominance criteria of specific order, as defined in the previous chapter.

Shortfall probability (LPM0) is classified as a type 1 risk measure, in terms of our

definition, given its exclusive focus on the probability of losses, and is this compatible

with FSD only. VaR can also be classified as a type 1 risk measure because it fails to

consider losses beyond the threshold. VaR is generally compatible with FSD, but we

have shown cases where it will not correspond with portfolios ranked by FSD subject

to the selected confidence level. Consequently shortfall probability and VaR are

suitable risk measures for investors whose utility functions are characterised by a

preference only for more over less (non-satiety). Expected shortfall and LPM1 are 93 We explained in 4.2.4 how intermediate losses below the ✁-quantile can cause problems for banks when recapitalisation following losses is costly.

114

classified as type 2 risk measures because they incorporate expected losses beyond the

measurement threshold. This means these measures are suitable for investors whose

utility functions are characterised by the non-satiety and risk-aversion. While type 2

risk measures are compatible with SSD, we have also shown cases where both ES and

LPM1 fail to order investment in line with SSD criteria. This arises because of the

target dependence of ES and LPM1 and the failure of these measures to capture losses

that are below the loss threshold. Finally, LPM2 can be classified as a type 3 risk

measure because it puts a higher penalty on larger deviations from the threshold than

smaller deviations. This is compatible with investors who, in addition to non-satiety

and risk aversion, prefer positive skewness in the distribution of returns. The utility

functions of such investors exhibit decreasing absolute risk aversion. We showed that

LPM2 does not suffer from target dependence and captures intermediate losses when

the target threshold is set at zero.

This leads to support for the use of LPM2 as the basis for measuring the risk in

positions within a bank. Of the five risk measures analysed, only LPM2 is incentive

compatible in the sense that managers can be expected to select portfolios that are

aligned with the risk preferences of the centre of the bank. We also could consider the

class of risk measures closely related to LPM2 - spectral risk measures and distortion

risk measures - for this purpose as both are incentive-compatible with the risk

preferences of the centre. Spectral risk and distortion risk can be classified as type 3

measures given they allow for larger weight to be given to low-frequency high-

severity losses, and thus apply to investors who prefer positive skewness in the

distribution of returns.

Our analysis, however, is only partially complete. In the next section we examine the

coherence of the risk measures, in terms of Artzner (1997). Specifically we consider

the extent to which the axioms of coherency are desirable when the objective is to

design risk-adjusted performance measures that promote goal congruence between

principal and agent in a bank.

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3.4 COHERENCY OF RISK MEASURES

3.4.1 Axioms of coherence

Artzner et al (1999) present and justify a set of four desirable structural properties for

measures of risk, which they argue should hold for any risk measure which is to be

used to effectively regulate or manage risks. They call measures that satisfy these

properties ‘coherent’. In this section we examine the properties of coherent risk

measures, and determine if coherency is a desirable characteristic for risk measures

used within the bank where the aim is to create incentive compatibility between the

risk preferences of the centre of the bank and the investment decisions of credit

portfolio managers.

Let X and Y represent the random outcomes of two risky assets and �(X) and �(Y)

represent risk measures for these investments. Further, r f represents the risk-free rate

of return, and ✁ and ✂ are positive numbers. The four axioms that characterise

coherent risk measures, translation invariance, monotonicity, positive homogeneity

and subadditivity, are represented as follows:

Axiom 1 Translation Invariance �(X + ✁ rf ) = �(X) – ✁

Axiom 2 Monotonicity �(X) ✄ �(Y) if X ☎ Y

Axiom 3 Positive Homogeneity �(✂X) =

✂ �(X)

Axiom 4 Subadditivity �(X + Y) ☎ �(X) + �(Y)

We discuss each in turn, and their implications for the opportunity set of incentive-

compatible risk measures.

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3.4.1.1 Axiom 1: Translation Invariance

Artzner et al (1999) define risk in terms of the variability of the future value of a

position due to market changes or uncertain events. In determining the desirability of

a risk measure, they focus on the random variables on the set of states of nature at a

future date, interpreted as possible future values of positions or portfolios currently

held.94

Consider an investor who desires fixed future wealth of Z. In order to achieve this

objective, the investor can invest in either a risky asset X, or a combination of the

risky asset and an amount � in a non-risky investment that guarantees a certain

outcome equal to (� rf ). The translation invariance axiom implies that investment in

the risk-free asset today reduces the amount that needs to be invested in the risky asset

where the objective is to achieve a certain future portfolio value. As the risk-free

investment provides a guaranteed profit, it reduces the potential losses arising from

investment in the risky asset by exactly the amount invested in the risk-free asset:

✁(X + � rf ) = ✁(X) – �

In terms of capital requirements, the axiom indicates that if a risk-free investment of �

is added to a risky portfolio, then the capital requirement should decrease by the

amount �. In the extreme case where the amount invested in the riskless asset is set

such that � = ✁(X), then ✁(X + ✁(X) r f ) = 0. This justifies a capital requirement equal to ✁(X) to cover the risk of loss, rendering the position in asset X acceptable without

further capital injection.

The risk measures examined in this study that characterise risk as the overall

seriousness of potential losses (VaR and ES) are translation invariant. If a risk-free

asset is added to a risky portfolio and the profit and loss distribution reconstructed to

reflect the addition, each profit and loss value will be reduced by the amount added.

Consequently both the VaR and the ES for the portfolio, at the designated loss

94 Artzner et al (1998), p.206.

117

threshold, will be lower in accordance with the translation invariance axiom.95

However, for those risk measures examined in this study that represent risk as the

magnitude of deviations from a prespecified target (shortfall probability, LPM1 and

LPM2), translation invariance in terms of the risk-free condition of Artzner et al

(1999) is not achieved.96 For the class of LPMn risk measures, the requirement that the

risk measure value decrease by the amount of the investment in the risk-free asset will

hold only if the cash flow from the risk-free asset is matched by the value of the worst

loss of the risky asset.

The failure of our recommended incentive compatible risk measure, LPM2, to

conform to the translation invariance axiom is not of consequence. Barbosa and

Ferreira (2004) claim that the translation invariance axiom is too restrictive in the

sense that it is not necessary that the risk measure value decrease by the exact amount

of the investment in risk-free asset.97 If a risk measure is decreased (increased) when a

position in a risk-free asset is added (withdrawn) from a risky portfolio, then the risk

measure motivates investment in risk-free assets, regardless of whether the measure

declines by the exact amount of the risk-free asset.

It is also worth noting that a number of other authors place less restrictive conditions

on the property of translation invariance. In a similar vein to Artzner et al (1999),

Pedersen and Satchell (1998) specify axioms that represent desirable properties of a

financial risk measure. Their axioms cover non-negativity, homogeneity, subadditivity

and shift-invariance. The shift-invariance axiom of Pedersen and Satchell (1998)

differs from translation invariance axiom of Artzner et al (1999) in that the former

makes the risk measure invariant to the addition of a constant to the random variable:

�(X + ✁) ✂ �(X) for all real ✁

95 See Tasche (2002) for proof that VaR is translation invariant and Acerbi and Tasche (2002a, 2002b) for proof that ES is coherent. 96 See Barbosa and Ferreira (2004) p.25 for a proof that LPMn does not satisfy the translation invariance axiom. 97 Barbosa and Ferreira (2004), p.6.

118

This indicates that the risk measure may be either unchanged or decreased by the

addition of a constant, and is thus less restrictive than the translation invariance of

Artzner et al (1999). Gaivoronski and Pflug (2001) specify the translation invariance

condition for a risk measure as follows:

�(X + t) = �(X) for all real t

In this interpretation of the translation invariance condition, the risk of a portfolio

cannot be changed by adding to the portfolio a fixed sum of riskless money. This is in

keeping with the general results for the LPM categories of measures, where adding a

constant to a random variable results in a new random variable with the same

deviation around the mean.

3.4.1.2 Axiom 2: Monotonicity

The monotonicity axiom says that if a portfolio X is always worth less than a portfolio

Y in terms of all possible outcomes, then the risk measure of X should be greater than

the risk measure of Y. From an economic perspective, the axiom implies that

portfolios embodying higher potential losses should report a larger risk measure and

require more risk capital.

The shortfall probability measure does not conform to the monotonicity axiom

because it assigns a larger risk measure to the least risky portfolio, reflecting the

convex attitude to risk implicit in the measure. VaR, ES and LPM1 do conform to the

monotonicity axiom, as proven by various authors.98 Szego (2002) asserts that

monotonicity rules out any semi-variance type of risk measure, where we note that

LPM2 represents the semi-variance when losses are measured as deviations below the

expected value of the portfolio.99 In contrast, Barbosa and Ferreira (2004) claim the

measure LPMn,t satisfies monotonicity when n > 0. This clearly captures LPM2. We

examine this conflict below.

98 See Tasche (2002), Acerbi and Tasche (2002a, 2002b) and Supremo (2001). 99 Szego (2001), p.1260.

119

The monotonicity axiom implies that returns can be used to determine the risk ranking

of instruments or portfolios. Indeed, monotonicity suggests that if one portfolio Y has

greater losses than another portfolio X, such that the probability of observing an

outcome below any threshold is lower for portfolio X, then portfolio X should

stochastically dominate portfolio Y.

The problem that arises is risk measures that are coherent in terms of Artzner et al

(1999) are generally not consistent with first or second-order stochastic dominance

principles, and are never likely to be consistent with third-order stochastic dominance

principles. This has been shown to be the case in section 3.3 of this chapter, where

only the LPM2 risk measure provided a risk-ranking of portfolios that was consistent

with the ranking of the portfolios by both SSD and TSD. In particular, we showed that

the VaR, ES and LPM1 measures were not increasing with the risks of the portfolios,

and indeed, in some cases were lower as the risk of the portfolios increased.100 Yet

these measures are monotonic increasing in terms of Artzner et al (1999), and ES and

LPM1 are deemed coherent risk measures.101

The inconsistency of coherent risk measures with TSD arises because TSD dominance

of one risky portfolio over another implies dominance in the third-distribution

function, which as shown in 4.1, implies the expectation of squared profits and losses

at each point in the distribution. More specifically, we showed in 4.1 that if an

investor exhibits non-satiety, risk aversion and a preference for positive skewness in

the distribution of returns, then �(X1) < �(X2) if the following holds:

t t

(t – x)(n-1) f1(x) dx ✁ (t – x)(n-1) f2(x) dx for all x, x ✁ t,

-✂ -✂

where f1(x) and f2(x) are the density functions of X1 and X2 and n > 2. Recall that this

condition employs the lower partial moment of degree (n-1) to penalise large

100 We ignore shortfall probability from this discussion given the underlying risk attitude in this measure is one of risk-seeking. 101 See Tasche (2002), Acerbi and Tasche (2002a, 2002b) and Supremo (2001).

120

deviations from the loss threshold more than smaller deviations from the loss

threshold. At n = 3 there is a quadratic penalty on deviations below the loss threshold,

which was shown to be consistent with portfolio X1 dominating portfolio X2 according

to TSD. This is inconsistent with coherent risk measures, which do not place a larger

penalty on larger deviations from the target threshold.102

The lack of consistency between coherent risk measures and stochastic dominance

principles implies that if we restricted our internal risk measures to only those

measures that are coherent, then LPM2 would be omitted from our list of candidates

and we would have no risk measure that is concurrently coherent and incentive-

compatible with the risk preference function of the centre. If we are restricted to the

condition that risk measures are coherent in terms of Artzner et al (1999), then the set

of acceptable risk measures would allow managers to select portfolios that are

dominated by TSD. This contradicts the risk-preference function of the centre. For

this reason, we suggest that the monotonicity axiom be replaced by the stronger

condition that the risk measure provides a risk-ranking that is consistent with TSD

principles, where the axioms are to be applied to risk measures for use within the

banking firm.103

3.4.1.3 Axiom 3: Positive Homogeneity

The positive homogeneity axiom indicates that if an investor purchases the same risk

twice (identical portfolios), then the risk should be doubled. This suggests the risk

measure should not be influenced by the size of the position, and for all � ✁ 0, the risk

is scalar multiplicative.104 For positive homogeneity to hold there should be no

diversification effect across portfolios with identical payoff distributions.

102 As pointed out by Barbosa and Ferreira (2004) p.22, coherent risk measures involve the expectation of non-squared profit and losses. 103 This is in keeping with De Giorgi (2005), who includes SSD as a property that reward and risk measures should satisfy for portfolio selection. 104 It holds that the liquidity of the position should not be influenced by the size of the position. If this is the case, positive homogeneity will not hold.

121

The shortfall probability measure is not positive homogenous. The probability of loss

of two combined portfolios that are identical is exactly the same as the probability of

loss of each of the individual portfolios. Shortfall probability thus does not exhibit

scalar multiplicativity. Supremo (2001) shows that VaR and ES are positive

homogenous – the VaR and ES for identical portfolios is doubled when the portfolios

are summed. Similarly, the LPM1 for identical portfolios that are combined into one

portfolio is equal to the sum of the LPM1 for the individual portfolios. LPM2,

however, is not positive homogenous owing to the quadratic power function – the

LPM2 measure for two identical combined portfolios will always exceed the sum of

the LPM2 measures for each of the individual portfolios. This will hold for any LPMn

measure where losses are measured as deviations from the expected value and for

which n > 1. Further, the LPMn measure for two or more combined identical

portfolios will be lower than the sum of the LPMn measures for each of the individual

portfolios when 0 < n < 1.

Appendix 11 shows the failure of shortfall probability and LPM (where 0 < n < 1 and

n > 1) to meet the positive homogeneity axiom. The appendix also shows that LPM1 is

positive homogenous.

The significance of these results is that the incentive-compatible risk measure LPM2

fails to meet the positive homogeneity axiom and is thus not a coherent risk measure

in terms of Artzner et al (1999). While we have found that the failure of LPM2 to meet

the translation invariance axiom is not of major consequence and that the

monotonicity axiom should be replaced with the stronger condition of congruence

with TSD, we must assess if the failure of LPM2 to meet the positive homogeneity

axiom is to its detriment for use as an incentive-compatible risk measure within the

bank.

Whether or not the failure of LPM2 to meet the positive homogeneity axiom is of

consequence depends to some extent on how the performance of credit portfolios is

measured within the bank. If performance is measured on the basis of individual

loans, then the fact that LPM2 is not positive homogenous should be of little concern

122

because a credit portfolio manager will be judged on the basis of the sum of the

individual loans that make up the portfolio under the control of the manager. This

would not seem an unrealistic assumption subject to the extent to which loans are

priced and managed on an individual basis. If, however, the performance of a credit

portfolio manager is based on the aggregated portfolio of loans under the control of

the manager, meaning risk measures are based on the portfolio rather than the sum of

the individual loans in the portfolio, then the use of LPM2 would overstate the risk of

the portfolio when the loans carry no diversification benefits.105 In this case, the

failure of LPM2 to meet the positive homogeneity axiom is of significance. If the risk

measure overstates the risk of the portfolio when loans of identical risk are added to

the portfolio, the credit portfolio manager will have a greater propensity to reject

loans where the risk of the portfolio will be significantly (and incorrectly) overstated.

This means loans that are valuable to the bank may be rejected.

Some may take the view that it is appropriate for a credit portfolio manager to resist

adding loans to a portfolio where the risks are positively correlated, and in this regard,

the penalty placed on the portfolio when using LPM2 serves to make this risk measure

attractive. We argue that the opposite is the case. Many credit portfolio managers in

large banks will specialise in a certain loan type, region or industry. Specialisation

such as this offers information advantages and other cost economies, and is desirable.

Under these conditions, the credit manager will face very few opportunities where

loans can be written that provide significant diversification benefits. Indeed, it is

likely that most loans will be positive correlated in terms of the distribution of returns.

Diversification across loan portfolios is more likely to be a higher-level function

within the bank because it is at the head office or business unit level that

diversification strategies are determined and diversification opportunities more easily

identifiable. Thus accountability for diversification across loan portfolios or regions

rests at a higher level within the bank than that of the line manager – unless there are

105 The same result holds for any LPM measure where n > 1.

123

opportunities to add loans with uncorrelated loss distributions to the portfolio.106 In

the case where loans of identical loss distributions are added to a portfolio, and where

the performance of managers is measured on a portfolio basis, we argue that the risk

of the portfolio should be scalar multiplicative and positive homogeneity must hold. If

a manager purchases the same risk twice, the risk of the portfolio should be doubled.

This will not be the case if LPM2 is used as the basis for measuring risk within the

bank.107 If, however, the performance of managers is measured on an individual loan

basis, then the failure of LPM2 to meet positive homogeneity is of little consequence.

Prior to concluding this section, it is worth noting that some authors believe that the

positive homogeneity assumption that risk increases proportionally to the initial

wealth placed on a position does not reflect investors’ perceptions of risk. De Giorgi

(2005) quotes cases of laboratory experiments that suggest decision-makers become

more risk averse with a larger net payoff (positive and negative).108 In this case it

would not be appropriate to impose that a risk measure satisfies the property of

positive homogeneity.

The decision to measure the performance of managers on a portfolio or individual

loan basis depends, to a large extent, on the degree to which the manager faces

opportunities to reduce the risk of the portfolios under their responsibility through

diversification. It is with this in mind that we assess the final axiom for a coherent risk

measure: subadditivity.

106 This suggests that skilled managers should be better able to identify potential diversification benefits among the opportunity set of loans available to them. Such managers should be rewarded for these skills through the risk measure. 107 Note that a risk-seeking manager would have a preference to use a LPM risk measure of order 0

� n

< 1 because the risk of the portfolio is understated when loans of identical risk are added to a portfolio. As shown in Appendix 11, the risk of the portfolio in these cases is less than the sum of the individual loans. Given there are no diversification benefits when loans of identical risk are combined in a portfolio, the LPM measure with 0

� n < 1 sends an incorrect signal regarding the true underlying risk

of the portfolio. 108 De Giorgi (2005), p.905.

124

3.4.1.4 Axiom 4: Subadditivity

A risk measure � is said to be subadditive when the risk of the combined position of

two investments, X and Y, is less or equal to the sum of the risk of the individual

portfolios: �(X + Y) ✁ �(X) + �(Y)

Subadditivity embodies the notion that portfolio diversification results in a reduction

in risk when there is less than perfect positive correlation in the returns in the

individual investments that comprise the portfolio. The subadditivity axiom implies

that the act of combining uncorrelated risks in a portfolio should never increase the

risk measure or the capital requirement.

Artzner et al (1999) contend that subadditivity is a natural requirement for a risk

measure for a number of reasons:

1. If a risk measure fails to incorporate diversification benefits then an individual

will have an incentive to establish two separate trading accounts, one for each

risk, in order to lower the overall margin requirement. In this vein, a credit

portfolio manager within a bank could act to lower the apparent credit risk of a

portfolio (and any subsequent capital requirements) by artificially splitting the

portfolio into smaller holdings or individual credits.

2. A banking institution may have an incentive to break up into various

subsidiaries in order to reduce the overall regulatory capital requirement if the

risk measure used for determining minimum capital requirements does not

reward diversification benefits. In this case the non-subadditive risk measure

encourages regulatory capital arbitrage.

3. At the business unit level, a bank can allocate capital among managers or trading

desks in the knowledge that the global risk for the unit is less than the sum of

125

local risks at the line level. Subadditivity of the selected risk measure ensures

risk management can be decentralised in this way.

Some authors argue that these arguments do not hold for certain types of risk.109 For

example, consider the case of two catastrophe bonds for which the risks are

independent. One bond is linked to earthquake in City A and the other linked to

earthquake in City B, and both cities are in different geographic regions. If an

earthquake occurs, the payment of interest and principal to the holder of the bond is

reduced or eliminated. Should an investor place their available investment funds into

one bond, or diversify and put equal amounts into two bonds? The subadditivity

property would suggest that investment in two bonds is more appropriate because the

likelihood of earthquakes occurring in both cities at the same time is highly remote.

However, if the investor is concerned with the probability of default, then a portfolio

comprising investment in both bonds will have a higher default probability than an

investment in a single bond. This arises because the probability that an earthquake

occurs in at least one city is larger than the probability that earthquake occurs in either

city.110 Such an example suggests that the relevance of coherency axioms depends on

which characteristics of risk are relevant to those responsible for managing risk.

While the probability of default may be the element of risk of most interest for some

investors, we have established that the aspect of risk most relevant to bank

stakeholders is the size of loss in the event of default.

In the case of the banking firm, subadditivity considerations are relevant for the

determining the relevant internal measure of risk. If a credit portfolio manager

identifies an opportunity to add a loan to the portfolio that has diversification benefits

in losses, then it is in the interests of the centre of the bank that such loans are

obtained. If, however, the risk measure used to assess the performance of the portfolio

manager is not subadditive, then there will be cases where it is against the interests of

the credit manager to add the loan to the portfolio, despite the real underlying benefits

109 See Rootzen and Kluppelberg (1999) and Yamai and Yoshiba (2002b). 110 If the probability of earthquake in City A is 1% and the probability of earthquake in City B is 1%, then an investment in one bond will have a 1% probability of default, while alternatively an investment in two bonds will have a default probability of approximately 2%.

126

to bank stakeholders. In terms of the subadditivity axiom, this will be the case where

the risk measure for the diversified portfolio is larger than the sum of the individual

credits that make up the portfolio. If credit portfolio managers are to be encouraged to

use or develop their skills to identify loans that provide diversification benefits to the

bank, then for incentive-compatibility, it is a requirement that the risk measure is

subadditive.

If the performance of credit portfolio managers is measured on the basis of individual

loans that make up their portfolios, then there will be little incentive for managers to

identify loans that provide diversification benefits for the bank. This is because the

risk measure used for performance measurement will provide no recognition or

reward for identifying individual credits that provide such benefits. It is for this reason

that we argue that the performance of credit portfolio managers should be assessed on

a portfolio basis. Incentive-compatibility considerations require that managers add

loans to their portfolios that are optimal from the perspective of the centre of the bank

and the stakeholders that it represents.

Which risk measures from the set of candidates are subadditive?

In order to address this question, we examine three hypothetical loan portfolio

distributions that combine two individual loans possessing diversification benefits in

the domain of losses. These portfolios are presented in Tables 3.8a, 3.8b and 3.8c.

Each table shows the value of the individual loans under various states of nature and

the probability distribution for each loan. The tables also show the risk measures for

the individual loans, the sum of the risk measures for the individual loans, and the risk

measures for the portfolio that combines the individual loans. In terms of Artzner et al

(1999), subadditivity holds if the risk measure for the portfolio is less than or equal to

the sum of the risk measures for the individual loans.

Table 3.8a considers two loans, X and Y. The loans have a face value of $100, an

expected value of $98 and are diversified in the domain of losses. There a five states

of nature. Under the first state of nature, loan X has a value of $50 and loan Y has a

127

value of $90, both with equal probability of occurrence. Under the second state of

nature, X has a value of $90 and Y a value of $100, while X has a value of $90 and Y

a value of $50 under the third state. Under the fourth state, X has a value of $100 and

Y a value $90. Under the fifth state both loans have a value of $100. Given these

values under each state of nature, it is clear that the loans are diversified in the domain

of losses (where losses are measured as shortfalls below the expected value). Risk

measure calculations for the individual loans and the portfolio comprising the loans

are presented at the bottom of the table. The risk measures are VaR and ES at the 95%

confidence level, LPM of orders n = 0, 0.5, 1 and 2,111 the downside semi-deviation

(DSD)112 and the Wang Transform at the 95% confidence level.

The results in Table 3.8a show that both VaR and LPM2 fail subadditivity because for

these measures, the value of the risk measure for the portfolio exceeds the sum of the

risk measures for the individual loans. If the performance of credit managers within

the bank was to be assessed on a portfolio basis, managers would in this case not add

loan Y to a portfolio with a payoff replicating loan X because the risk measure would

for the portfolio overstates the true risk. These measures place a penalty on

diversification, which is clearly not optimal from the perspective of the centre of the

bank. Of particular concern is the finding that the only measure that we have found to

be incentive-compatible with the centre of the bank in terms of expected utility,

LPM2, is not subadditive. If the use of LPM2 acts to discourage managers from

seeking and adding loans into their portfolios that provide risk-reducing benefits, then

the risk measure must fail our overall test of incentive-compatibility between

managers and the centre of the bank. We return to this issue, and explore a viable

alternative risk measure at the end of this section.

111 For LPM calculations, losses are measured as deviations below the expected loan value. This is consistent with earlier calculations in the chapter. 112 The downside semi-deviation (DSD) is the square root of the semi-variance (LPM2).

128

Table 3.8a

Subadditivity for Loans X and Y: Risk Measures Loan X

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 50 3% 98.0 48.0 30.0 0.208 1.440 69.120

2 90 2% 98.0 8.0 36.0 0.057 0.160 1.280

3 90 3% 98.0 8.0 0.085 0.240 1.920

4 100 2% 98.0

5 100 90% 98.0

Sum 100% 64.0 66.0 0.349 1.840 72.320

Loan Y

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

3 50 3% 98.0 48.0 30.0 0.208 1.440 69.120

4 90 2% 98.0 8.0 36.0 0.057 0.160 1.280

1 90 3% 98.0 8.0 0.085 0.240 1.920

2 100 2% 98.0

5 100 90% 98.0

Sum 100% 64.0 66.0 0.349 1.840 72.320

Portfolio (X+Y)

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 140 3% 196.0 56.0 70.0 0.224 1.680 94.080

3 140 3% 196.0 56.0 70.0 0.224 1.680 94.080

2 190 2% 196.0 6.0 0.049 0.120 0.720

4 190 2% 196.0 6.0 0.049 0.120 0.720

5 200 90% 196.0

Sum 100% 124.0 140.0 0.547 3.600 189.600

Risk Measures

Risk measure Investment X Investment Y Sum X, Y Portfolio Comments

VaR 95% 8.000 8.000 16.000 56.000 Fails subadditivity

ES 95% 32.000 32.000 64.000 56.000 Subadditive

LPM0 0.080 0.080 0.160 0.100 Subadditive

LPM0.5 0.349 0.349 0.699 0.547 Subadditive

LPM1 1.840 1.840 3.680 3.600 Subadditive

LPM2 72.320 72.320 144.640 189.600 Fails subadditivity

DSD 8.504 8.504 17.008 13.770 Subadditive

WT 95% 21.027 21.027 42.054 30.644 Subadditive

129

Now consider Table 3.8b, which shows risk measures for two individual loans, F and

G, and the portfolio that combines these loans. These loans also have a face value of

$100, but their expected value of $98.3 is higher than the previous case. These loans

also have lower volatility than for the previous case, but also offer similar

diversification benefits in the domain of losses. In the first state of nature, loan F has a

value of $50 while loan G has a value of $100. In the second state, loan F has a value

of $90 while loan G has a value $100, and in the third state loan F has a value of $100

while loan G drops to $50. In the fourth state, F has a value of $100 and G has a value

of $90, and both loans have a value of $100 in the fifth state.

In the case of loans F and G, only the VaR risk measure fails subadditivity, with the

VaR of the portfolio exceeding the sum of the VaR of the individual loans. The

shortfall probability measure (LPM0) is weakly subadditive, in the sense that the risk

measure for the portfolio matches the sum of the risk measures for the individual

loans. While this risk measure does not penalise portfolio diversification, it is notable

that the measure fails to reward diversification. If a risk measure does not reward

diversification, this may act as a disincentive to credit portfolio managers to actively

seek loans with diversification benefits, and at the very least, make them indifferent

about adding such loans to a portfolio. We assert that such indifference is not

congruent with the risk objectives of the centre of the bank.

It is worth noting that the LPM2 risk measure does reward diversification in the case

of loans F and G, with the value of the risk measure for the portfolio being less than

the sum of the value of the risk measures for the individual loans. This indicates that

for certain portfolio distributions, the measure may encourage diversification. Its

failure, however, to consistently recognise diversification benefits rules it out as an

incentive-compatible risk measure in our bank setting.

130

Table 3.8b

Subadditivity for Loans F and G: Risk Measures Loan F

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 50 3% 98.3 48.3 30.0 0.208 1.449 69.987

2 90 2% 98.3 8.3 36.0 0.058 0.166 1.378

3 100 3% 98.3

4 100 2% 98.3

5 100 90% 98.3

Sum 100% 56.6 66.0 0.266 1.615 71.364

Loan G

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

3 50 3% 98.3 48.3 30.0 0.208 1.449 69.987

4 90 2% 98.3 8.3 36.0 0.058 0.166 1.378

1 100 3% 98.3

2 100 2% 98.3

5 100 90% 98.3

Sum 100% 56.6 66.0 0.266 1.615 71.364

Portfolio (F+G)

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 150 3% 196.6 46.6 75.0 0.205 1.398 65.147

3 150 3% 196.6 46.6 75.0 0.205 1.398 65.147

2 190 2% 196.6 6.6 0.051 0.132 0.871

4 190 2% 196.6 6.6 0.051 0.132 0.871

5 200 90% 196.6

Sum 100% 106.4 150.0 0.512 3.060 132.036

Risk Measures

Risk measure Investment F Investment G Sum F, G Portfolio Comments

VaR 95% 8.300 8.300 16.600 46.600 Fails subadditivity

ES 95% 32.300 32.300 64.600 46.600 Subadditive

LPM0 0.050 0.050 0.100 0.010 Weakly subadditive

LPM0.5 0.266 0.266 0.532 0.512 Subadditive

LPM1 1.615 1.615 3.230 3.060 Subadditive

LPM2 71.364 71.364 142.728 132.026 Subadditive

DSD 8.448 8.448 16.895 11.491 Subadditive

WT 95% 20.418 20.418 40.836 25.670 Subadditive

131

Finally, consider the case of Table 3.8c, which shows risk measures for two individual

loans, R and S, and the portfolio that combines these loans. These loans have a face

value of $100, but different expected values, and offer diversification benefits in the

domain of losses. In the first state of nature, loan R has a value of $50 while loan S

has a value of $90. In the second state, loan R has a value of $90 while loan S has a

value $50. In the third, fourth and fifth states of nature, both loans have the identical

values of $96, $98 and $100 respectively.

In the case of loans R and S, the only measure that fails subadditivity is LPM2.

However, it is worth noting that VaR, ES and LPM1 are all weakly subadditive and

hence fail to reward the diversified portfolio through the risk measure. In this

particular scenario, it is observed that the VaR measure is subadditive because

diversification across the individual loans occurs outside the 95% confidence level

upon which the VaR is based. That is, diversification occurs only in the first and

second states of nature, which cover the 4% of the cumulative losses of the loans,

starting from the largest loss. In a similar vein, ES fails to reward portfolio

diversification (by assigning a lower risk value to the diversified portfolio than for the

sum of the individual risk values) because diversification occurs outside the 95%

confidence threshold. This serves as a reminder of the target dependence of risk

measures based on a predetermined confidence level.

132

Table 3.8c

Subadditivity for Loans R and S: Risk Measures Loan R

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 50 3% 98.24 48.24 25.0 0.208 1.447 69.813

2 90 1% 98.24 8.24 15.3 0.029 0.082 0.679

3 96 2% 98.24 2.24 31.7 0.030 0.045 0.100

4 98 4% 98.24 0.24 0.020 0.010 0.002

5 100 90% 98.24

Sum 100% 72.0 0.287 1.584 70.595

Loan S

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 90 3% 99.04 9.04 45.0 0.090 0.271 2.452

2 50 1% 99.04 49.04 8.3 0.070 0.490 24.049

3 96 2% 99.04 3.04 32.0 0.035 0.061 0.185

4 98 4% 99.04 1.04 0.041 0.042 0.043

5 100 90% 99.04

Sum 100% 85.3 0.236 0.864 26.729

Portfolio (R+S)

State

X

P(X)

E(X)

Losses Weighted

Tail

[E(X)-X] 0.5 P(X)

[E(X)-X] P(X)

[E(X)-X] 2 P(X)

1 140 3% 197.28 57.28 70.0 0.227 1.718 98.430

2 140 1% 197.28 57.28 23.3 0.076 0.573 32.810

3 192 2% 197.28 5.28 64.0 0.046 0.106 0.558

4 196 4% 197.28 1.28 0.045 0.051 0.066

5 200 90% 197.28

Sum 100% 157.3 0.394 2.448 131.863

Risk Measures

Risk measure Investment R Investment S Sum R, S Portfolio Comments

VaR 95% 2.240 3.040 5.280 5.280 Weakly subadditive

ES 95% 26.240 13.740 39.980 39.980 Weakly subadditive

LPM0 0.100 0.100 0.200 0.100 Subadditive

LPM0.5 0.287 0.236 0.523 0.394 Subadditive

LPM1 1.584 0.864 2.448 2.448 Weakly subadditive

LPM2 70.595 26.729 97.324 131.863 Fails subadditivity

DSD 8.402 5.170 13.572 11.483 Subadditive

WT 95% 20.066 20.517 40.583 31.065 Subadditive

133

Let us consider the general findings regarding the subadditivity of the risk measures

studied in this chapter.

To prove that a risk measure is not subadditive we need only provide a counter

example for each risk measure where the subadditivity condition fails. The preceding

examples show that VaR and LPM2 fail the subadditivity condition. For both these

measures, we have shown that portfolio diversification may increase the VaR for the

portfolio, when the true position is one of lower risk for the portfolio.113

The risk measures that conform to the subadditivity condition in our examples are ES,

LPM (order 0 � n � 1), DSD and the Wang Transform (WT). Artzner, et al (1997,

1999) provide proofs that ES is subadditive, and Acerbi, Nordio and Sirtori (2001)

extend the work of Artzner et al (1997, 1999) to show that ES is subadditive in cases

where the underlying profit/loss distributions are discontinuous. We have identified

cases where ES is weakly subadditive in the sense that it does not penalise

diversification, but at the same time, the risk measure does not reward diversification

by presenting a lower risk value for the diversified portfolio. This occurs when

diversification in the distribution of returns occurs outside of the confidence level for

the measurement of the shortfall. As discussed above, if the true underlying risk of the

portfolio is less than the sum of the risks of the individual loans that make up the

portfolio, then a risk measure should reflect this. Incentive-compatibility requires that

managers are incentivised to seek out loans that are risk-reducing when combined in a

portfolio, and to add them to the portfolio. If the portfolio risk measure does not

reflect the lower risk, then the incentive to diversify is reduced.

Our results confirm the findings Barbosa and Ferreira (2004) that the LPMn categories

of risk measures are subadditive when the order is 0 ✁ n ✁ 1. The result for LPM1 is

not surprising given this measure closely resembles ES, the only difference being ES

is based on a ✂ confidence level while LPM1 is typically based on deviations below

the expected value for the portfolio. Although ES and LPM1 are coherent risk

113 The only time that VaR is consistently subadditive is for elliptical distributions. See Yamai and Yoshiba (2002a), p.108 and Embrechts, McNeil and Straumann (1999), p.12-13.

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measures, we have previously discarded them from the acceptable list of incentive-

compatible risk measures because they embody risk-neutrality in losses and thus fail

to rank portfolios consistently in terms of stochastic dominance principles, and in

particular, TSD. Similarly, we have shown that LPMn of order 0 � n < 1 embodies a

risk-seeking attitude on the part of investors, which again does not conform to the

risk-preference function of the centre of the bank. Thus the subadditivity of the LPM

risk measure within these specifications is not of consequence.

Table 3.9 summarises our results on the coherence of the five risk measure candidates

selected for this study.

Table 3.9: Coherence of Risk Measures

Translation Invariance

Monotonicity

Positive Homogeneity

Subadditivity

1. Shortfall probability No No No Yes

2. VaR Yes Yes Yes No

3. Expected Shortfall Yes Yes Yes Yes

4. LPM1 No Yes Yes Yes

5. LPM2 No Partial No No

3.4.2 An Alternative Risk Measure: Downside Semi-Deviation

We have determined that while LPM2 is an incentive-compatible risk measure in

terms of stochastic dominance principles, it is not a coherent risk measure. In terms of

coherence, the main areas of concern are its failure to meet the properties of positive

homogeneity and subadditivity. We have shown that if LPM2 is used for measuring

risk within the bank, under particular conditions, its failure to meet these properties

implies that credit portfolio managers have no incentive to add loans to portfolios that

carry risk-reducing benefits.

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Despite the failure of LPM2 to encourage managers to construct diversified portfolios,

all is not lost. Our examples in Tables 3.8a-3.8c indicate that the downside semi-

deviation (DSD), which is the square root of LPM2, is subadditive. Barbosa and

Ferreira (2004) and Fisher (2002) find that DSD meets the broader requirements for a

coherent risk measure.114 If DSD provides a risk ranking of portfolios that matches the

risk-ordering of portfolios in terms of TSD, then this, combined with coherency,

indicates that the DSD risk measure fulfils our requirements for an incentive-

compatible risk measure. We examine this proposition in the remainder of this

section.

First, let us consider the subadditivity of DSD in more detail.

To examine the subadditivity properties of DSD, we take the distributions for loans X

and Y in Table 3.8a, and vary the payoff under states of nature 3 and 1 for loan X and

Y respectively, such that the degree of diversification across these loans changes.

Specifically, we vary the payoff under these states from $110 to $0, which has the

effect of reducing the degree of diversification from payoff $110 to $50, and then

increasing the degree of diversification from payoff $50 to $0. We then measure

subadditivity in the DSD risk measure by comparing the value of the DSD for the

portfolio against the sum of the DSD of the individual loans that comprise the

portfolio, for each payoff level from $110 to $0. We also include measures for LPM2

under the same conditions. The results are presented in Table 3.10.

Column 1 in Table 3.10 shows the variable payoff for the loans under the states of

nature outlined above. The specific case in Table 3.8a is represented by the data at the

$90 payoff. Column 2 shows the DSD for the portfolio comprising loans X and Y,

and column 3 shows the sum of the DSD for each individual loan. If the risk measure

is subadditive, the value at each payoff in column 2 should be lower than the same for

114 Barbosa and Ferreira (2004) refer to DSD as the root lower partial moment. Like LPM2, DSD is not translation invariant, but for the similar reasons presented for LPM2, this is not considered to impact significantly on the incentive-compatibility of the DSD risk measure. Like LPM2, the DSD risk measure decreases when a risk-free asset is added to the portfolio, although not necessarily by the exact amount of the investment in the risk-free asset. This is considered not to reduce the incentive for managers to invest in risk-free assets, meaning the failure of the measure in terms of strict observance of the translation invariance axiom is not of major consequence.

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column 3. Column 4 represents the difference between columns 2 and 3, with a

negative value indicating the risk measure is subadditive. Further, larger values in

column 4 should correspond to greater diversification benefits across the portfolio.

Columns 5 to 7 include the corresponding data for LPM2.

Table 3.10 shows that DSD carries highly desirable characteristics for an incentive-

compatible risk measure. First, column 4 shows that DSD always rewards

diversification: the DSD for the diversified portfolio is always less than the sum of the

DSD for the individual loans. Second, the reward becomes lower as the portfolio

becomes less diversified, and vice-versa. This is evidenced by the narrowing of the

gap between the figures in columns 3 and 4 as the payoff variable moves from $100 to

$50, and then a widening of the gap as the payoff variable moves from $50 to $0. The

payoff of $50 represents the point where the diversification benefits across the loans

are at their lowest.115 At this point, the DSD for the portfolio almost matches the sum

of the DSD for the individual loans.116 Third, the DSD is increasing as the downside

risk of the portfolio increases – as the payoff variable under the designated state of

nature gets smaller and the downside risk increases, the DSD gets larger. At the same

time, diversification benefits across the loans are still captured because the DSD for

the portfolio is lower than the sum of the DSD of the individual loans, encouraging

credit portfolio managers to add diversified loans to their portfolios wherever

possible.

115 At the $50 payoff, the lowest value for the portfolio under states 1 and 3 is $100. This matches the lowest value for a non-diversified portfolio comprising double the investment in Loan X or in Loan Y. From a downside risk perspective, this indicates that at a payoff variable of $50, diversification benefits across the loans are at their minimum. As the payoff variable moves lower than $50, diversification benefits again begin to increase as the minimum value for the portfolio exceeds the minimum value for the sum of the individual loans that make up the portfolio. The same holds for values greater than $50 under the designated states of nature. 116 The DSD measures do not match at this point because a slight diversification benefit remains under state of nature 2 – here the payoff for the portfolio is $190, compared to $180 for a non-diversified portfolio comprising double the investment in Loan X or Y.

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Table 3.10: Subadditivity – Downside Semi-Deviation (DSD) versus LPM2

1 2 3 4 5 6 7

Portfolio X Payoff

(State 3)117

DSD Portfolio (X +Y)

DSD (X) +

DSD (Y)

Subadditivity (2 – 3)

LPM2

Portfolio (X +Y)

LPM2 (X) +

LPM2 (Y)

Subadditivity (5 – 6)

110 9.2 17.0 -7.8 85 145 -60

100 11.5 16.9 -5.4 132 143 -11

90 13.8 17.0 -3.2 190 145 45

80 16.1 17.8 -1.7 258 158 100

70 18.3 19.1 -0.7 337 182 155

60 20.6 20.9 -0.2 426 218 208

50 22.9 23.0 -0.1 526 265 261

40 25.2 25.4 -0.2 637 323 314

30 27.5 28.0 -0.5 758 393 366

20 29.8 30.8 -0.9 890 473 417

10 32.1 33.6 -1.5 1033 566 467

0 34.4 36.6 -2.1 1186 669 517

Note from columns 5 to 7 that LPM2 fails to reward diversification except for payoff

values where the diversification impact is very large. This corresponds to payoffs of

$110 and $100. At all other values, LPM2 penalises diversified portfolios by assigning

a larger risk measure to the diversified portfolio than for the sum of the DSD of the

individual loans. This result arises because the impact of the quadratic power function

in LPM2 swamps any diversification benefits that may be captured by the risk

measure, except for payoffs where the diversification benefits are substantial. It could

be asserted that these are payoffs that are perhaps the most unlikely to occur, at least

consistently, in practice.

117 This also corresponds to the payoff under state 1 for portfolio Y.

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Having established that DSD is a desirable risk measure from the perspective of

subadditivity118, we now consider whether the measure is congruent in terms of third-

order stochastic dominance (TSD) criteria. We have shown previously that the LPM2

risk measure provides a risk-ranking of portfolios that is consistent with TSD. More

generally, it was shown in the previous chapter that the measure LPMn is consistent

with stochastic dominance of order (n +1). Given the DSD is calculated as the square

root of the semi-variance (LPM2), we must conclude that the DSD is consistent with

TSD because DSD is a positive monotone transformation of LPM2. This means that

whenever portfolios can be ranked by TSD, the DSD is consistent with expected

utility maximisation for an investor who is non-satiated, risk-averse and who prefers

positive skewness in the distribution of returns.

If we return to the five portfolios examined earlier in the chapter (portfolios A – E),

and calculate the DSD for the portfolios (where losses are based on negative

deviations from the expected value), we obtain the following:

Portfolio A B C D E

DSD (98.99) 3.943 3.949 4.186 4.452 4.966

This example confirms that the DSD risk measure increases as the risk of the

portfolios increase in terms of TSD.

We conclude that DSD meets all our requirements for an incentive-compatible risk

measure for use within a banking firm. The DSD meets our requirements for

coherency, and it ranks portfolios in accordance with the risk preference function of

the centre of the bank.119 We have shown that if risk-adjusted performance measures

within the bank use DSD as the basis for measuring risk, then credit portfolio

managers will have strong incentives to select loans and construct portfolios that

match those that the centre would have them select.

118 Appendix 12 shows that DSD also meets desirable property of positive homogeneity. 119 Note that other measures that examined in 6.5 that were found to be incentive-compatible in terms of TSD are also coherent. Wirch and Hardy (2001), Wang (2002) and Balbas, Garrido and Mayoral (2002) show that distortion risk measures (including the Wang Transform) are coherent. Acerbi (2002) and Cheng, Liu and Wang (2004) show that spectral risk measures are coherent.

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3.5 INTERNAL RISK MEASUR ES AND BANK CAPITAL

The total capital held by a bank, over and above the minimum regulatory requirement,

is largely driven by the target credit rating of the bank.120 The target credit rating is

linked to the probability of the bank defaulting on its senior debt. This implies that for

a given level of risk, a bank that desires a higher credit rating on its senior debt will

increase its economic capital. The question we examine in this section is to what

extent should the calculation of economic capital of the bank be based on the internal

measure of risk, which we have determined to be DSD? Further, if economic capital is

calculated using DSD, how can the resulting measure be interpreted with respect to

the probability of default for the bank?

In the previous chapter we reviewed the literature on the objective function of a bank

and established that the determination of bank capital requirements in terms of a

solvency standard linked to the probability of bank default is not an appropriate

representation of the risk preference function of bank stakeholders. Nonetheless, we

cannot ignore that the VaR measure has become the regulatory standard for

determining minimum capital requirements.121 This is most likely due to the fact that

VaR encapsulates the key characteristics for insurance against default – it measures

the size of losses at a given confidence level – allowing the bank to determine the

capital needed to keep the probability of default below the desired confidence

threshold. However, from the internal perspective of the bank, we have demonstrated

that using VaR as the basis for risk-adjusted performance measurement will not

guarantee that managers make investment decisions that are aligned with the interests

of bank stakeholders. Put simply, when used as an internal risk measure, VaR is

dangerous.

We have shown that the DSD risk measure is coherent in terms of Artzner et al (1997)

and compatible with the expected utility of investors who are risk-averse and desire

120 The relationship between bank capital, target credit rating and the pricing of credit facilities is examined in chapter six. 121 See Yamai and Yoshiba (2002b), p.61.

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positive skewness in returns. DSD is not concerned with the probability of default, but

rather, the size of expected losses and their probability of occurrence - DSD places a

quadratic penalty on downside deviations from expected values, consistent with a

desire for positive skewness. If regulators or ratings agencies deem that bank capital

should be measured in terms of the default probabilities, then we firmly argue that the

internal risk measure must diverge from the external measure of risk, where the

objective is to achieve a disciplined and consistent analysis of risk based on the entire

distribution of potential outcomes. VaR may be desirable for determining economic

capital based on default probabilities, but it is not desirable as an internal risk

measure. Goal congruency between principal and agents within the bank demands the

use of incentive-compatible risk measures. If performance measurement is based on a

VaR assessment of risk (or other measures that have been shown to fail our

requirements), we have shown that investment decisions within the bank will be

inefficient. Indeed, we have presented a strong case to suggest that a bank using such

measures will be undercapitalised with respect to risk.122

If VaR is inappropriate for measuring risk, but DSD is appropriate, can a bank base its

capital requirements on DSD? The Bank for International Settlement’s revised

guidelines for bank capital requirements, commonly referred to a Basel II, allow for

banks to use their own internal models for determining capital requirements.123 While

it is beyond the scope of this thesis to develop a capital allocation methodology based

on alternative risk measures, it would not seem unreasonable to assume that DSD

could form a bottom-up basis for determining the aggregate capital needs of a

particular bank. However, it is important to recognise that such a basis for capital

measurement would be likely to result in a more conservative estimate of capital,

given its focus on the entire distribution of losses.124 Further, it must be acknowledged

that the resulting capital measure would be unlikely to bear any relationship to the

122 This issue is examined in further detail in chapter four of this thesis. 123 This is captured in Pillar One of the Bank for International Settlements revised capital requirements under the ‘internal ratings approach’ (foundation and advanced versions). Refer to Bank for International Settlements, (2004). 124 Given the coherence of DSD, the measure may result in a more accurate representation of diversification across credit portfolios and other bank businesses. This may act to partially offset the higher capital requirement likely to arise if DSD is used as the basis for determining external capital requirements.

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capital required to keep the probability of the bank defaulting at some prespecified

level.

Is this a bad thing? Should we be concerned if the basis for measuring risk within the

bank differs from the risk basis for determining external capital? While alignment is

desirable, we argue that it is acceptable to use different measures where the objectives

differ. It may not be desirable, or possible, for a risk measure to satisfy competing

objectives – different measures may be needed to meet the specific requirements

under consideration. We have shown that the objectives of insuring against default

and aligning incentives do not allow for a common risk measure.

With respect to the internal measurement of risk, we assert that the overriding

objective is to use risk measures that align the interests of the diverse group of bank

stakeholders (creditors, depositors, owners, regulators) - represented by the centre of

the bank - and credit portfolio managers. It does not necessarily follow that the actual

allocation of capital across portfolios should be based on these risk measures. Our

argument is that a risk-adjusted performance measure that forms the basis for

determining bonuses and other forms of compensation to managers must, in the

presence of information asymmetries, be incentive-compatible with the objective

function of the bank. This risk measure need not reflect the actual capital held or

allocated against the portfolio, particularly when external capital requirements are

based on other objectives, such as achieving a desired external credit rating. The bank

can apportion its actual capital against various portfolios in order to insure the

portfolios against default, within the predetermined confidence level. However, the

apportionment of actual capital held by the bank, when based on risk measures that

are not coherent or compatible with TSD principles, may result in investment

decisions that are inefficient with respect to the risk-preference function of the centre

of the bank. It is for this reason that we argue that the risk-basis for allocating actual

capital can and must differ from the risk-basis for measuring the performance of

managers within the bank.

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In chapter four we examine in more detail how measuring risk-adjusted performance

on the basis of actual capital allocated to a portfolio or position may induce managers

to act against the best interests of the bank. In chapter five, we devise a solution to this

problem based on an internal market for risk capital.

We now turn to an investigation of how the structure of the compensation payment

function of the bank impacts on portfolio selection by managers, and assess the

implications for incentive-compatibility between the centre of bank and credit

portfolio managers.

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3.6 IMPACT OF THE BANK COMPENSATION STRUCTURE

3.6.1 Overview

Our focus to this point has been on the appropriate measurement of risk within the

banking firm, where the objective has been to align the interests of the centre of the

bank with credit portfolio managers. The implicit assumption has been that managers

will select the portfolio with the lowest value for the risk measure when deciding

among competing portfolios carrying the same expected value, on the basis that this

will provide them with highest risk-adjusted performance measure. If management

compensation is linked to the risk-adjusted performance measure, then managers

should be incentivised to select the portfolios that carry the lowest risk for a given

expected return. We now consider how the structural form of the management

compensation function in the bank may impact on our incentive-compatibility

conditions.

The optimal compensation function requires a balance between risk sharing (between

managers and owners) and incentives. Risk-averse managers prefer a larger

component of their compensation to be fixed, such that losses realised by the firm that

are associated with random events are borne by owners. Conversely, owners can be

considered to be less risk-averse than managers to the extent that a smaller fraction of

their wealth is tied to the performance of the firm than for managers who are

employed by the firm.125 Consequently optimal risk sharing between managers and

owners suggests that the compensation structure should incorporate a significant fixed

salary component. Offset against this, however, is the problem that fixed salaries may

not provide strong incentives for managers to increase their effort to achieve greater

output for owners, particularly where greater effort incurs personal costs for managers

that result in reduced utility. In order to encourage greater effort on the part of

managers, and compensate them for potentially reduced personal utility, the

compensation function should include a pay-for-performance component. The optimal 125 The implicit assumption is that owners are able to diversify their risks more effectively than managers, who tend to derive a large proportion of their income from single firms.

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compensation function should thus trade incentive compensation (to increase effort)

against a fixed salary (to promote efficient risk sharing).

Holmstrom and Milgrom (1987) show that the optimal compensation function

consists of a variable component based on performance and a fixed component that is

independent of performance. Their optimal contract is linear and of the form

Compensation = � z + ✁

where ✂ is the ratio of pay-to-performance, z is the basis upon which the bonus is

determined and ✄ is the fixed component or base salary. While there are various

formulations of the optimal pay-to-performance ratio in the principal-agent

literature,126 the common elements are the responsiveness of output to increased

effort by the agent, the degree of risk-aversion of the agent, the level of risk that is

beyond the control of the agent (noise in the performance measure) and the aversion

of the agent to effort. These feature in the following representation of the optimal pay-

to performance ratio127:

rvc

m☎✆1

✝

where m is the marginal contribution of agent effort to output, r is the risk-aversion of

the agent, v is noise in the performance measure and c is the effort-aversion of the

agent. The expression shows that the greater the responsiveness of firm output to

effort, the greater the power of incentives in the compensation contract. Conversely,

the greater the aversion of the agent to risk and effort, the lower the power of

incentives in the compensation contract. Finally the more noise there is in the

performance measure, the poorer to job it does in tracking the outcome of interest, and

the lower the power of incentives in the compensation contract.

126 See Campbell (1995), Varian (1992), Holmstrom and Milgrom (1987), Milgrom and Roberts (1992), Besanko, Dranove and Stanley (1996) and Salanie (1997). 127 See Besanko, et al. (1996), p.657.

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In the bank setting, we can express the linear compensation function as follows:

Compensation = � RAPM + ✁

.

In this formulation we use � to designate the ratio of pay-to-performance128, RAPM is

the risk-adjusted performance measure upon which any bonus is determined, and ✁

remains the fixed component of salary. In the current context, this compensation

function indicates that credit portfolio managers receive a base salary plus bonus

linked to some proportion � of the RAPM of the portfolios under their management.129

Let us return to the example of five portfolios (A - E) examined earlier in the chapter,

and using DSD as the basis for measuring portfolio risk, assess the likely impact of

the above formulation of the compensation structure on the investment decisions of

credit portfolio managers. In keeping with the basis for measuring losses on the

portfolios, this being the expected market value of the portfolio at the end of the

measurement period, we use the market value of the portfolio as the basis for

measuring gains.130 The RAPM for the bank is thus determined as follows:

DSD

valuemarketinGainRAPM ✂

If the portfolio makes losses over the period, the numerator of this equation has a zero

value and no bonus is paid. This represents the typical asymmetrical compensation

function.

128 This is to avoid confusion with the use of the symbol ✄ earlier in the chapter within the context of risk tolerance levels. 129 This may be based on directly on the RAPM, or incorporate adjustments to the RAPM such as the excess return above the predetermined hurdle rate or the risk-free rate. For the purposes of this chapter we assume that performance is assessed on RAPM relative to a minimum hurdle rate. Issues related to the selection of the relevant hurdle rate are examined in chapter four. 130 See discussion in section 3.2.1 justifying the measurement of portfolio performance on the basis of changes in market value rather than accounting profit or returns. Each portfolio has a face value of $100, and gains are initially measured relative to this figure. An alternative would be to measure gains relative to the $98.99 expected value of each portfolio. The difference between the face value of $100 and the expected value of $98.99 represents expected losses, and downside risk measures have been assessed relative to the expected value of each portfolio because expected losses are provisioned for in the accounting statements of the bank. It should be noted that an upgrade in credit rating should also be accompanied by a reduction in the expected losses on the portfolio, moving the expected value of the portfolio closer to the face value. The relationship between credit rating, expected losses and unexpected losses is examined in detail in chapter six.

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3.6.2 Gains Measured Relative to Portfolio Face Value

Table 3.11 presents the RAPM for each of the five portfolios A to E, and the

probability distribution that applies at each level of gains in market value. Gains are

measured in terms of increases in the market value of the portfolio relative to the $100

face value of the portfolio. In the case of portfolio A, for example, a $1 gain in market

value (relative to the portfolio face value) measured against the DSD risk measure of

$3.943 gives a RAPM of 25.4%. This RAPM has a 5% probability of occurrence,

based on the distributions provided in Table 3.1 in this chapter.

Table 3.11: RAPM (Gain/DSD) – Portfolios A to E (Gains measured relative to face value of portfolio)

Portfolio A Portfolio B Portfolio C Portfolio D Portfolio E

Gain RAPM Prob RAPM Prob RAPM Prob RAPM Prob RAPM Prob

0 91.0% 91.0% 88.5% 97.0% 95.0%

1 25.4% 5.0% 25.3% 5.0% 23.9% 5.0% 22.5% 2.0% 20.1% 4.0%

2 50.7% 3.0% 50.6% 3.0% 47.8% 3.0% 40.3% 1.0%

3 76.1% 1.0% 76.0% 1.0% 71.7% 1.0%

4

5

6 143.6% 2.5% 134.8% 1.0%

Observe first from Table 3.11 that the probability of gain in market value for each

portfolio appears relatively small. This reflects the fact that loan portfolios tend to be

characterised by a high probability of small losses – expected losses that are

provisioned for in the profit and loss statement of the bank. If we measured gains

relative to the expected value of the loans, rather than face value, then the

probabilities of gains are significantly larger (refer Table 3.1).131 Further, any increase

in the market value of a loan will be associated with an upgrade in the internal credit

131 For example, in the case of portfolio A, the cumulative probability of gains in excess of the expected value of $98.99 is 79%. Comparable figures for portfolios B to E, respectively, are 81.5%, 81.5%, 84% and 83%.

147

rating assigned to the loan. An upgrade would occur if credit managers, upon annual

review, assess that the probability of default on the loan is lower – factors guiding this

could be improved financial performance of the borrower, lower firm leverage,

change of management, improved economic conditions, etc.

It is appropriate at this point to note that if gains in loan market values are linked to

the re-rating of loans by credit managers within the bank, there may be scope for

credit managers to engineer an increase in their annual bonus by upgrading loans

under their control, even where such upgrades may not be fully justified. If

information on the performance or prospects of individual loans is largely the private

domain of credit managers, the incentive to overstate positive prospects, in order to

upgrade the internal loan credit rating, may be significant. Credit managers, for

example, may be inclined to be selective in the use of data on the performance of the

borrower, accentuating positive information while ignoring or understating negative

information. For this reason, the data and models used by credit managers to assess

default probability and subsequently rate (and re-rate) loans under their control

require close scrutiny within the bank, and quality judgements on the part of credit

managers should require evaluation by parties who do not have a pecuniary interest in

the performance of loans.

Let us return to the figures in Table 3.11. Given the RAPM for the portfolios, we ask

how a credit manager would rank the five portfolios? Will incentive-compatibility

conditions be preserved when we move from ranking portfolios solely on risk to

ranking portfolios on the basis of both risk and potential upside? Driving these

questions is the observation that while the risk measure that forms the denominator of

the RAPM is based on downside deviations from the target value, only gains in

portfolio value feature in the numerator of RAPM. Earlier in this chapter we were able

to rank portfolios solely on the basis of stochastic dominance principles because the

expected value of each portfolio was identical. However, when we introduce an

asymmetrical compensation function that pays bonuses only on the realisation of

gains, then the distribution of gains for each portfolio will feature in the investment

decisions of managers. We need to assess the implications of this from the perspective

148

of the centre of the bank, and determine if the asymmetric compensation function will

lead managers to make investment decisions that are consistent with the desired

risk/return profile of the bank.

We consider how a credit manager will rank the portfolios by their RAPM, where risk

in the denominator is measured by the DSD and gains are measured relative to the

face value of each portfolio. An examination of Table 3.11 indicates, with the

exception of portfolios A and B, that it is not immediately certain how a credit

manager will rank the portfolios without making assumptions regarding the attitude of

the manager towards risk. In the case of portfolios A and B, observe that the portfolios

have identical distributions for gains relative to portfolio face value, with an 9%

probability that market value gains on the portfolio will be greater than zero. For each

point on the distributions in gains, portfolio A generates a higher RAPM than

portfolio B, and we can conclude that portfolio A will be preferred ahead of B. We

cannot, however, draw similar conclusions regarding portfolios C, D and E because

the distributions in gains are not identical – portfolio C, for example, has lower

RAPM but a higher probability of gains than portfolios A and B, and while D has a

lower probability of gains, it has, like C, a small probability of a large RAPM. These

uneven distributions make it not possible to rank the portfolios without incorporating

assumptions regarding the risk attitudes of managers within the bank. For this

purpose, we will initially use stochastic dominance principles. While this will allow

us to incorporate different risk attitudes of managers without having to specify the

precise form of their utility functions, we face the potential problem that stochastic

dominance allows only for a partial ranking of risky prospects, and as such, may leave

some portfolios unranked. Within the context of RAPM, we later recommend a

solution that allows for comparison against all risky prospects.

149

We first assume that the manager prefers more over less,132 but make no other

assumptions regarding the risk attitude of the manager. This allows us to use FSD

principles to rank the portfolios, as FSD is consistent with investors concerned only

that the probability of loss of one portfolio is less than another.133 The relevant

calculations for portfolio RAPMs under FSD are provided in Appendix 13, and the

results are summarised in Table 3.12 below. We find that the portfolio combinations

that can be ranked by FSD are A/B, A/E, B/E, C/D and C/E.134 However there are

portfolio combinations that cannot be ranked by FSD because the cumulative

distribution functions of the portfolios cross. These portfolios are A/C, A/D, B/C, B/D

and D/E. As mooted, we cannot achieve a discrete ranking of each of the portfolios

under FSD. This means we cannot determine how managers in the bank will rank

each of the portfolios without making assumptions regarding their risk attitudes, and

even this provides no guarantee that no portfolios remain unranked.

We now examine the outcomes for the ranking of portfolios when we incorporate the

risk preferences of managers. We first consider the case where the manager is risk-

neutral, and then the case where the manager is risk-averse. The risk-neutral manager

will rank the portfolios based on the expected value for the RAPM. The results are

presented in Table 3.13. We observe that portfolio C has the highest expected RAPM

at 6.93%, followed by portfolio A (3.55%), B (3.54%), D (1.80%) and E (1.21%).

Except for portfolio C, the other portfolios are ranked in the same order as

corresponding to the DSD risk measure. The dominance of portfolio C arises because

it carries the potential for an extreme gain in market value of $6, corresponding to a

large RAPM. For the risk-neutral manager, the potential for a large gain outweighs the

higher risk of portfolio C relative to A and B. While portfolio D also carries the

potential for an extreme gain, this is offset by the fact that the probability of gains for

the portfolio is the lowest of the five portfolios.

132 For all strictly increasing utility functions, the manager will prefer a portfolio X over Y where the expected utility of X is greater than the expected utility of Y. 133 Refer section 2.4.1 of chapter two. 134 The expression A/D refers to the ranking of portfolio A against D, and so on. The first portfolio (here A) will be the dominant portfolio if stochastic dominance applies.

150

Table 3.12: Ranking of Portfolio RAPM under Stochastic Dominance Conditions

(Gains measured relative to face value of portfolio)

First-Order Stochastic Dominance (FSD): Portfolios by RAPM

Portfolio combinations which cannot be ranked by FSD

A/C A/D B/C B/D D/E

Portfolio combinations which can be ranked by FSD

A dominates B, E B dominates E C dominates D, E

Second-Order Stochastic Dominance (SSD): Portfolios by RAPM

Portfolio combinations which cannot be ranked by SSD

D/E

Portfolio combinations which can be ranked by SSD

A dominates B, D, E B dominates D, E C dominates A, B, D, E

(Refer Appendix 13 for supporting calculations)

Table 3.13

Expected value for RAPM – Risk-neutral managers (Gains measured relative to face value of portfolio)

Portfolio A B C D E

Expected RAPM 3.55% 3.54% 6.93% 1.80% 1.21%

Ranking 2 3 1 4 5

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Now consider the case where managers are risk-averse. We use SSD principles to

determine how risk-averse managers may rank the portfolios, given SSD is consistent

with a negative second derivative for the utility function of the investor.135 The

relevant calculations for portfolio RAPMs under SSD for are also provided in

Appendix 13, and the results are summarised in Table 3.12. We find that only one

portfolio combination cannot be ranked by SSD – this being D/E.136 Significantly,

portfolio C stands out as the dominant portfolio for a risk-averse manager, dominating

each of the other four portfolios by SSD. The dominance of portfolio C is again

attributed to the potential for a very high RAPM given the probability of a large gain

in the value of the portfolio. This reinforces the risk/reward trade-off implicit in

RAPM, where portfolio gains are measured against portfolio risks.

In addition to the dominance of portfolio C, we expect the risk-averse manager to

select portfolio A ahead of B, D and E, and portfolio B ahead of D and E given the

dominance of these portfolios by SSD. We can thus determine that the risk-averse

manager will rank portfolio C first, followed by portfolio A and then portfolio B. As

indicated, we cannot determine the ranking of portfolios D and E by RAPM. In the

case of portfolios D and E, this implies the investment decision on the part of

managers will be driven by their specific utility functions. These cannot be

determined without some form of consultation with managers.

While the results under SSD are better than for those under FSD in the sense that we

can rank more portfolios under the former, we again cannot determine a discrete

ranking of the five portfolios by RAPM, thwarted by the limitation that stochastic

dominance allows only for a pairwise ranking of risky prospects. As previously

indicated, we develop a potential solution to this problem shortly.

135 Refer section 4.1 of this chapter. 136 This arises because the sum of cumulative probability of the distribution functions for portfolios D and E cross.

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3.6.3 Gains Measured Relative to Portfolio Expected Value

We now assess the ranking of portfolios on the part of managers when gains are

measured relative to the expected value of the portfolios ($98.99) rather than the face

value of the portfolios ($100). We undertake this in recognition that some banks may

consider expected value as the appropriate base to measure and remunerate gains,

reflecting that gains in market value have a positive impact on reducing expected

losses that will have been provisioned for in the profit and loss statement.

Table 3.14 presents the RAPM for each of the five portfolios A to E, and the

probability distribution that applies at each level of gains in market value relative to

the expected value of the portfolio. In the case of portfolio A for example, there is a

40% probability that the market value of the portfolio will be $100.137 Given the

expected value of $98.99, this corresponds to a gain in market value of $1.01. The

RAPM at a gain of $1.01 against the DSD risk measure for the portfolio of $3.943 is

25.6%.

Table 3.14: RAPM (Gain/DSD) – Portfolios A to E (Gains measured relative to expected value of portfolio)

Portfolio A Portfolio B Portfolio C Portfolio D Portfolio E

Gain RAPM Prob RAPM Prob RAPM Prob RAPM Prob RAPM Prob

0 21.0% 18.5% 18.5% 16.0% 17.0%

0.01 0.2% 30.0% 0.2% 32.5% 0.2% 30.0% 0.2% 40.0% 0.2% 25.0%

1.01 25.6% 40.0% 25.6% 40.0% 24.1% 40.0% 22.7% 41.0% 20.3% 53.0%

2.01 51.0% 5.0% 50.9% 5.0% 48.0% 5.0% 45.2% 2.0% 40.5% 4.0%

3.01 76.3% 3.0% 76.2% 3.0% 71.9% 3.0% 60.6% 1.0%

4.01 101.7% 1.0% 101.5% 1.0% 95.8% 1.0%

5.01

6.01

7.01 167.5% 2.5% 157.5% 1.0%

137 Refer Table 3.1 in this chapter.

153

The use of expected value as the basis for measuring gains provides a much larger

distribution of gains. We observe from the table that the probability of loss is now

measured as negative deviations from the expected value. The change in the

distribution of gains will have an impact on how managers rank the portfolios. For

example, while an unbiased ranking between portfolios A and B was possible when

gains were measured relative to the face value of portfolios, observe from the table

that we cannot determining the risk ordering of these portfolios without making

assumptions regarding the risk preferences of managers. This arises because while

portfolio A has higher expected RAPM than portfolio B at each level of gains, relative

to B, it also has a lower overall probability of gains. We cannot make a judgement

over which portfolio will be selected without incorporating the risk attitude of

managers.

Let us consider the ranking of the portfolios by managers when risk attitudes vary. To

do this we will consider the ranking under conditions of FSD, SSD and risk-neutrality

on the part of managers. The relevant calculations for portfolio RAPMs under FSD

and SSD are provided in Appendix 14 and the results are summarised in Table 3.15

below. First, none of the five portfolios dominates under conditions of FSD when we

measure gains relative to expected portfolio value. This arises because the cumulative

probability distributions of each pairwise combination of portfolios cross. This

reinforces that we cannot draw any firm conclusions of how managers may act

independent of their attitude to risk.

The results for the risk-neutral manager are presented in Table 3.16. Again, we

observe that portfolio C has the highest expected RAPM at 19.43%, followed by

portfolio A (16.18%), B (16.16%), E (13.06%) and D (11.87%). In this case, the

ranking of portfolios D and E is reversed, reflecting that D has a higher overall

probability of gains when gains are measured relative to the expected value of the

portfolio. Portfolio C again dominates when managers are risk-neutral, partially

reflecting one large outlier observation for the RAPM (refer Table 3.14).

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Table 3.15: Ranking of Portfolio RAPM under Stochastic Dominance Conditions

(Gains measured relative to expected value of portfolio)

First-Order Stochastic Dominance (FSD): Portfolios by RAPM

Portfolio combinations which cannot be ranked by FSD

A/B A/C A/D A/E B/C B/D B/E D/C C/E D/E

Portfolio combinations which can be ranked by FSD

No portfolio dominates by FSD

Second-Order Stochastic Dominance (SSD): Portfolios by RAPM

Portfolio combinations which cannot be ranked by SSD

A/B A/D A/E B/D B/E C/D C/E D/E

Portfolio combinations which can be ranked by SSD

C dominates A, B

(Refer Appendix 14 for supporting calculations)

Table 3.16

Expected value for RAPM – Risk-neutral managers (Gains measured relative to expected value of portfolio)

Portfolio A B C D E

Expected RAPM 16.18% 16.16% 19.43% 11.87% 13.06%

Ranking 2 3 1 5 4

155

Finally, if we assume managers are risk-averse, we find that most portfolio

combinations cannot be ranked by SSD when we use the expected value of the

portfolio as the basis for measuring gains. The only conclusion we can draw is that

risk-averse managers will have a preference for portfolio C over portfolios A and B

because C dominates A and B by SSD. Again we find that we cannot obtain a discrete

ranking among the five portfolios on the part of managers without knowing their

specific utility functions.

3.6.4 Incorporating Upper Moments in Investment Decisions

We have found that DSD is an incentive-compatible risk measure when the risk

preference function of the centre of the bank embodies non-satiety, risk aversion and a

preference for positive skewness in the distribution of returns. We have also found

that if managers are compensated on the basis of an asymmetric payment function that

pays bonuses linked to positive RAPM outcomes but no bonus when hurdles are not

achieved, then it is generally not possible to predict how managers will choose among

alternative investment prospects without knowing the exact specification of their

utility functions. While, in the current case, we have five portfolios that carry the

same expected value, the distribution of gains for each portfolio differs considerably.

Should potential upside be a concern for the centre of the bank if the prime objective

is to protect bank stakeholders against adverse outcomes? This is a critical question,

and one on which the literature appears largely silent. If the internal risk measure

implemented by the bank achieves alignment between the risk preferences of the

centre and the portfolio decisions of managers, where risk is defined in terms of the

downside outcomes, can we ensure incentive-compatibility between the centre and

managers when the expected upside distribution of returns is incorporated into the

investment decisions of managers?

156

To answer this question requires assessment of the risk attitude of the centre with

respect to potential portfolio gains. One view is that if credit portfolios are priced to

earn the required return on economic capital, then bank stakeholders should be

satisfied that they are earning a return that adequately compensates them for downside

risk.138 In this context, subsequent increases in the market value of credit portfolios

represent a direct gain to stakeholders, and as such, should be a little concern to the

centre of the bank. The basis of this view is that the centre is charged with managing

downside risk. An alternative view is that the centre of the bank, acting as an agent for

bank stakeholders, is charged with managing both risk and return. Here the centre

should govern the investment decisions of managers to ensure that returns to

stakeholders are maximised, while at the same time ensuing risk is appropriately

assessed and incorporated into pricing and capital requirements. If we are to adopt the

first view, then we can be satisfied that DSD is the appropriate risk basis for

determining RAPM within the bank, and our observation that an asymmetric

compensation function will lead managers to select portfolios in accordance with their

specific utility functions is of little consequence. If we are to adopt the second view,

then it is necessary to incorporate into our framework a portfolio risk-ranking

mechanism that allows for the preferences of the centre with respect to the right tail of

the distribution of portfolio returns.

Luce and Weber (1986) present a conjoint expected risk model which allows for the

separation of the distribution of returns in terms of upside and downside probabilities.

Their model quantifies the perceived risk of a random variable as a linear combination

of the probability of positive and negative outcomes, and the probability of a zero

outcome. Additionally, they incorporate the possibility that upside and downside

variability in returns may have a different effect on perceived riskiness by allowing

for the conditional expectation of positive and negative outcomes to each be raised to

some power function. Sortino, van der Meer and Plantinga (1999) present an ‘upside-

potential ratio’ which measures the upside potential for a random variable against the

downside variance.

138 The question of the appropriate basis for determining the required rate of return in the bank setting is examined in the next chapter.

157

These models provide intuition as to how the risk attitude of the centre of the bank

with respect to gains can be incorporated into a framework for the ranking of credit

portfolios. Drawing on the concept of the upside-potential ratio of Sortino el al

(1999), portfolios can be ranked in terms of the ratio of upside gains to downside

losses, where the benchmark for gains and losses is set at some predetermined loss

threshold. In the analysis that follows, and in keeping with the earlier sections of this

chapter, we will set the expected portfolio value as the benchmark. The resulting ratio,

which can be interpreted as the shadow price per unit of risk for gains in portfolio

market value, will permit a discrete ranking of portfolios because a single measure

will apply to each portfolio. There is, however, one major limitation to the upside-

potential ratio of Sortino et al (1999) – returns above the benchmark are weighted

linearly. This is a significant limitation because the measure does not allow for

varying risk attitudes to outcomes that are above the target threshold. In the bank case,

the risk attitude of the centre may be influenced by the magnitude of positive

deviations from the target threshold. With this in mind, we refine the model of Sortino

el al (1999) by incorporating a power function for upside outcomes, in keeping with

Luce and Weber (1986) and Farinelli and Tibiletti (2003).

For a benchmark/target level t and a power function of order n, let below-target

deviations in the left tail of the distribution be defined as follows:

�n-,t(X) = E 1/n[{( X – t)-} n]

Note that with n = 2 and t = expected value of the portfolio, this expression represents

the downside semi-deviation (DSD). Conversely, for a target level t and a power

function of order n, let above-target deviations in the right tail of the distribution be

defined as follows:

�n+,t(X) = E 1/n[{( X – t)+} n]

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These measures represent normalised lower and upper partial moments.139 Raising

partial moments of order n to the power of 1/n ensures �n-,t and �n

+,t are coherent risk

measures, as discussed in section 3.4 of this chapter.140

For a random portfolio X, the ratio of �n+,t(X) to �n

-,t(X) represents the ratio of reward

to downside risk, where the order n for each of the upper and lower partial moments

incorporates the differential risk attitude of the centre to the magnitude of positive and

negative deviations from the target threshold. Let us apply this measure to each of the

five portfolios that have been examined in this chapter. For �n-,t(X), we apply a power

function of n = 2 in order to use DSD as the measure of downside risk. In the case of �n+,t(X), we allow the order n to vary in order to assess the impact of different attitudes

on the part of the centre to positive deviations on the ranking of portfolios. The results

are presented in Table 3.17.

First, let us consider the case where the centre of the bank has a preference for small

rather than large positive deviations above target. This would be the case if the centre

was adverse to upside volatility in returns, preferring moderate but more consistent

portfolio gains, as opposed to large but less frequent gains.141 In order to place greater

emphasis on smaller positive deviations from the target, the order n for �n+,t(X) should

be low and less than unity. In Table 3.14, we use n = 0.1 to reflect a strong preference

for small portfolio gains, and n = 0.5 to assess the impact of a more moderate

preference for smaller portfolio gains.

In the case of n = 0.1, we find that portfolio E is the highest ranked portfolio, followed

by C, B, A and D. It turns out that if the centre has an aversion to upside volatility

from the target threshold, the portfolio with the highest downside risk by DSD has the

largest ratio of upside per unit of risk. This is not a general rule, but rather, reflects the

specific distribution of gains for E relative to the other portfolios. Table 3.18 shows

the cumulative distribution of gains for each of the five portfolios. It shows that

139 For further discussion, see Farinelli and Tibiletti (2003). 140 See Fischer, T. (2002). This was discussed in 3.4.2 within the specific context of LPM2 and DSD risk measures– while LPM2 is not coherent, its transformation, DSD, is coherent. 141 This could be the preferred position for the centre of the bank if large but infrequent gains impacted unrealistically on the expectations of bank shareholders.

159

portfolio E has the largest cumulative probability of gains: it has an 83% probability

of gains, and all of these gains are $3.01 or less. Thus while portfolio E has the largest

downside risk, this is more than offset by its high probability of smaller gains. The

ranking of upside to risk changes markedly when we reduce the intolerance to larger

gains and apply an order of n = 0.5 to positive deviations from the target. The highest

ranked portfolio is now portfolio C, and portfolio E drops to four. The key factor

behind C achieving the highest ranking is that it has a cluster of small gains and one

extreme gain of $7.01, occurring with a 2.5% probability. The lower intolerance to

large positive deviations, associated with an increase in the order to 0.5, plus the

cluster of smaller gains, are the factors contributing to the high ranking for portfolio

C.

Table 3.17: Reward to DSD – Portfolios A to E (Upside and downside deviations measured relative to expected portfolio value)

Order n Portfolio A Portfolio B Portfolio C Portfolio D Portfolio E

Upside 0.1 0.0239 0.0299 0.0367 0.0268 0.0508

Downside 2.0 3.9430 3.9493 4.1855 4.4519 4.9656

Ratio 0.0061 0.0076 0.0088 0.0060 0.0102

Rank 4 3 2 5 1

Upside 0.5 0.3306 0.3335 0.4111 0.2569 0.3990

Downside 2.0 3.9430 3.9493 4.1855 4.4519 4.9656

Ratio 0.0838 0.0844 0.0982 0.0577 0.0804

Rank 3 2 1 5 4

Upside 1.0 0.6379 0.6382 0.8132 0.5284 0.6483

Downside 2.0 3.9430 3.9493 4.1855 4.4519 4.9656

Ratio 0.1618 0.1616 0.1943 0.1187 0.1306

Rank 2 3 1 5 4

Upside 2.0 1.0211 1.0211 1.5070 0.9952 0.8904

Downside 2.0 3.9430 3.9493 4.1855 4.4519 4.9656

Ratio 0.2590 0.2586 0.3601 0.2236 0.1793

Rank 2 3 1 4 5

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Table 3.18: Portfolios A to E: Distribution of Gains (Gains measured relative to expected value of portfolio)

Portfolio A Portfolio B Portfolio C Portfolio D Portfolio E

Gain Prob

Cum Prob

Prob

Cum Prob

Prob

Cum Prob

Prob

Cum Prob

Prob

Cum Prob

0.01 30.0% 30.0% 32.5% 32.5% 30.0% 30.0% 40.0% 40.0% 25.0% 25.0%

1.01 40.0% 70.0% 40.0% 72.5% 40.0% 70.0% 41.0% 81.0% 53.0% 78.0%

2.01 5.0% 75.0% 5.0% 77.5% 5.0% 75.0% 2.0% 83.0% 4.0% 82.0%

3.01 3.0% 78.0% 3.0% 80.5% 3.0% 78.0% 1.0% 83.0%

4.01 1.0% 79.0% 1.0% 81.5% 1.0% 79.0%

5.01

6.01

7.01 2.5% 81.5% 1.0% 84.0%

Next consider the case where the centre of the bank is indifferent to the magnitude of

positive deviations to the target threshold. To embody a risk-neutral attitude to the

size of gains, we set the order n = 1.142 Table 3.14 shows that Portfolio C retains the

highest ranking based on return to DSD, followed by portfolios A, B, E and D. Again

the dominant factor in the ranking of portfolio C is the existence of an extreme gain of

$7.01 with 2.5% probability.

Finally, consider the case where the centre of the bank has a strong preference for

large positive deviations from the target threshold. In this case, we set the order n = 2,

noting that an even greater preference for positive outcomes can be achieved by

setting the order n > 2. Table 3.14 shows that the ranking of the five portfolios

remains largely unchanged, with portfolio C dominating, followed by A, B, D and E.

Interestingly this ranking closely resembles that of ranking of the portfolios on a

downside risk basis (that is, by TSD), with the only difference being that portfolio C

ranks ahead of portfolio A. The factor that distinguishes portfolio C, when upside

potential is incorporated into the ranking mechanism, is the moderate probability of a

142 This corresponds to the upside-potential ratio of Sortino et al (1999).

161

very large gain. While portfolio D also has the potential to realise an extreme gain of

the same magnitude, although with lower probability, it achieves a lower overall

ranking because it also carries an exposure to an extreme loss.143 Portfolio C does not

carry the potential for extreme losses.

3.6.5 Concluding Comments

We have demonstrated how the ratio of reward to DSD allows for a discrete ranking

of credit portfolios. The ratio signals the shadow price for risk by measuring the trade-

off of upside potential against downside risk, and in our formulation of the ratio,

allows for investor risk attitudes to the size of gains to be incorporated into the

evaluation. In the bank setting, if the risk attitude of the centre with respect to the

magnitude of gains is of relevance, then the key question is can the ratio be

incorporated into the performance compensation system such as to increase the

likelihood that managers will select the portfolios that the centre would have them

select?

The difficulty in answering this question lies in the in the observation that the ratio of

reward to DSD requires ex-ante knowledge of the expected distribution of returns,

both upside and downside. In the principal-agent setting of this chapter, we have

assumed that managers have private knowledge on the distribution of returns which is

not freely available to the centre of the bank. While the centre may be able to express

its risk preferences with respect to upside gains, there is no guarantee that managers

will select portfolios that conform to these preferences, and in this regard, we have

shown that in terms of RAPM, the selection of credit portfolios by managers will be

largely driven by their specific utility functions. The problem lies somewhat in the

nature of RAPM themselves, which match ex-ante data on risk against ex-post data on

gains or profits. If managers are remunerated on the basis of RAPM, then it is difficult

to form a nexus between the ratio of reward to DSD, which uses ex-ante data on

143 Refer Table 3.

162

portfolio distributions to rank portfolios, and RAPM, which use ex-post data on

performance to remunerate managers.

This leads to a broader question, and one that forms the focus of the next chapter of

this study. We have seen that DSD is a congruent risk measure to the extent that it

assigns a value for risk to a portfolio that matches the risk preferences of the centre -

portfolios that the centre would deem riskier are assigned a higher risk value when

DSD is used to measure risk. If the centre is largely indifferent to the magnitude of

gains on portfolios, and DSD is used as the basis for measuring risk within the bank,

we can expect that incentive-compatibility between the centre and managers will be

upheld because managers should select portfolios based on the highest expected

RAPM. Portfolios with lower risk, as measured by DSD, will have potentially larger

RAPM, increasing the likelihood that bonuses will be paid to managers.144

However, if information on the expected distribution of returns is the private domain

of managers, then an accurate estimate of the DSD can only be assigned to portfolios,

ex-ante, if managers are prepared to disclose their private information on the

distribution of returns. But managers may have little incentive to reveal this

information if the compensation structure of the bank is structured such that the data

they provide may later work against them. While management expectations may be

genuinely based on the best information they have at the time regarding portfolio risk,

if it turns out their information is not accurate, or factors beyond their control impact

on the performance of the portfolios under their responsibility, then the act of

disclosing an ex-ante estimate of DSD may not be desirable if this risk value is

assigned to the portfolio over the measurement period used for determining RAPM.

Indeed, managers may be incentivised to misrepresent their private information to the

centre in order to be assigned lower ex-ante risk measures for their portfolios.

144 Although we have seen in this section that the distribution of gains may mean that the lowest risk portfolio by DSD is not the preferred choice on the part of managers or the centre of the bank.

163

If the centre cannot rely on managers to accurately disclose their private information

on the distribution of portfolio gains and losses, then an alternative is to ignore the

private information of managers and the DSD risk measure for RAPM on the basis of

historical portfolio distributions. The problem with this approach is the historical

distribution of returns may have little bearing on the expected returns on prospective

portfolios. If the centre assigns a risk measure (and subsequently allocates capital) to

credit portfolios on the basis of historical return distributions, this may create perverse

incentives on the part of managers depending on whether or not the historical

distribution matches the ex-ante distribution expected by managers. An assignment of

risk – in terms of a DSD risk measure - that exceeds management expectations could

encourage managers to reveal their expectations, but there is no way for the centre to

verify the validity of the data provided by management. Further, if the assignment of

risk is too high relative to the expectations of managers, managers may respond by

either rejecting investment in the portfolio or by engaging in higher risk-taking in

order to improve the expected RAPM and consequent bonus. Conversely, a lower

assignment of risk relative to manager expectations may leave the bank

undercapitalised relative to the true risk in the portfolios. Managers, however, should

not be concerned because a lower risk assignment will increase the probability of a

higher RAPM, and consequent bonus.

Our main proposition is that measuring and rewarding the performance of managers

on the basis of the RAPM for their credit portfolios will be of little use in a bank if the

RAPM is based on an ex-ante risk measure and managers have weak incentives to

accurately disclose their private information on portfolio risk. In the principal-agent

setting, managers have information on expected risk that is not available to the centre.

The fact that management bonuses are typically linked to ex-post RAPM means

managers may have an incentive to not disclose accurate information on expected risk

when engaging in portfolio selection, in order to favourably influence their potential

performance bonus. The central question, then, is can we get managers to reveal their

private information regarding the expected distribution of gains and losses on their

portfolios? We address this question in the next chapter.

164

3.7 CHAPTER SUMMARY

The main findings of this chapter are summarised as follows:

1. A mark-to-market market basis is used for measuring gains and losses on credit

portfolios for the purposes of determining the numerator of the RAPM because

it provides greater scope to reward managers for positive credit migrations in

their portfolios, creating stronger incentives to actively manage portfolios for the

upside. Accounting measures do not capture upside for credit portfolios and may

create incentives for managers to misrepresent expected losses.

2. Shortfall probability, VaR, expected shortfall and LPM1 are not incentive-

compatible risk measures given the risk preferences of the centre of the banking

firm. The use of these measures in the dominator of a RAPM may induce

managers to select portfolios that are dominated in terms of TSD criteria. VaR,

expected shortfall and LPM1 also display target dependence, meaning changes

in the target loss threshold for these measures can impact on the risk-ordering of

portfolios.

3. The LPM2 risk measure is compatible with TSD and provides a risk-ordering of

portfolios that is incentive-compatible. The measure also provides a consistent

risk-ordering independent of the target threshold. Two recently developed risk

measures - spectral risk measures and the Wang Transform - are also incentive-

compatible with the risk preference function of the centre. These measures bear

close resemblance to LPM2 in the sense of placing larger weights on deviations

from the loss threshold.

4. Although LPM2 is consistent with stochastic dominance principles, it is not a

coherent risk measure. LPM2 fails each of the four axioms of a coherent risk

measure.

165

5. The failure of LPM2 to meet the axioms of translation invariance and

monotonicity is not of consequence in terms of goal congruence between the

centre and managers. The failure of LPM2 to meet the axiom of positive

homogeneity is not of significance if performance is measured and remunerated

on an individual loan basis, but is relevant if managers are measured and

remunerated on a portfolio basis.145 This is because the LPM2 for two identical

loans that are combined in a portfolio will always exceed the sum of the LPM2

of the individual loans. This may lead managers to reject loans that are valuable

to the bank. The failure of LPM2 to the meet the axiom of subadditivity is

relevant because the measure does not reward portfolio diversification and thus

does not encourage managers to seek-out and add loans to their portfolios that

provide risk-reducing benefits. This is also against the best interests of the centre

of the bank.

6. Despite the problems associated with the lack of coherence of LPM2, the

downside semi-deviation (DSD), which is the square-root of the LPM2, is an

incentive-compatible risk measure because it is coherent and conforms to the

risk-ordering of portfolios in terms of TSD. For this reason DSD is

recommended as the internal measure of risk in the denominator of RAPM

equation.

7. If regulators or ratings agencies deem that total bank capital should be measured

in terms of a bank solvency standard, which in turn is based on the probability of

the bank defaulting on its debt, then the internal risk measure must diverge from

the external measure of risk, where the objective of the internal measure is to

achieve a disciplined and consistent analysis of risk based on the entire

distribution of potential outcomes. It may not be desirable or possible for a

single risk measure to meet competing objectives.

145 This is where loans with different distribution of returns are combined in a portfolio. In our setting, for example, this would be the case if loan portfolios A and B (or any other combination) were combined to create a third portfolio.

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8. The structural form of a bank’s compensation payment function may impact on

incentive-compatibility conditions. If this payment function is asymmetrical,

with bonuses paid only upon the realisation of gains, then the ranking of

portfolios will be influenced by both the distribution of gains in the numerator

and the distribution of losses in the denominator of the RAPM. If the

distribution of gains is uneven, then it may not be possible to determine which

portfolios managers will select without specific knowledge of their utility

functions.

9. The way gains are measured in the compensation payment function of the bank

may also impact on the ranking of portfolios by managers. Gains in market

value can be measured relative to the face value of loans, or the expected value

of loans. The choice influences the ranking of loan portfolios in the study. The

expected value may be the most appropriate given the gains in market value will

have a positive impact on reducing expected losses that will have been

provisioned for ex-ante.

10. If the centre is charged with managing both risk and return, as opposed to only

managing downside risk, then the RAPM upon which managers are remunerated

should incorporate the preferences of the centre with respect to right tail of the

distribution of returns. A reward to risk ratio, where the numerator measures

upper partial moments in the distribution of returns,146 allows portfolios to be

ranked in accordance with the attitude of the centre towards variability in upside

returns. Such a ratio, which signals the shadow price for risk, relies on ex-ante

knowledge of the distribution of returns, which may not be freely disclosed by

managers.

146 The numerator measures downside risk using the DSD.

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Chapter Four

Agency Problems and Risk-Adjusted

Performance Measurement

“All human endeavours are constrained by our limited and

uncertain knowledge – about external events, present, and

future; about the laws of nature, God and man; about our own

productive and exchange opportunities; about how other people

and even we ourselves are likely to behave. Economists have of

course always recognised the all-pervasive influence of

inadequate information, and its correlate risk, on human

affairs”

Jack Hirshleifer, 1992

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4.1 INTRODUCTION

This chapter identifies agency problems that are likely to arise when risk-adjusted

performance measures (RAPM) are used for resource allocation decisions and employee

remuneration in banking organisations. In this chapter we use the term ‘return on risk-

adjusted capital’ (RORAC) in place of the general term ‘RAPM’ in order to highlight the

specific methodology applied in banks whereby the profits on a position are measured

against a capital charge that incorporates the expected risk in the position.147 In the

previous chapter it was concluded that the most appropriate basis for measuring risk

within the banking firm is the downside semi-deviation of the position or portfolio. In

this chapter the focus is not on risk measurement, but rather, the application of the

RAPM framework in banks and the agency problems that manifest. In the RAPM

framework for this chapter, capital is allocated by the centre of the bank against positions

taken by managers and business units on the basis of a statistical measure of unexpected

losses. This capital allocation could represent an actual attribution of capital held by the

bank, or as suggested in the previous chapter, some notional measure of capital based on

the risk measurement methodology employed by the bank.148 At the end of the

measurement period, the profits or gains in the position are measured against the

allocated capital to determine the RORAC for the position. If the RORAC measure

exceeds some predetermined hurdle rate,149 managers may be entitled to a bonus. The

basis upon which risk in positions is measured may be based on historical distributions

of gains and losses or the private information of managers with respect to expected

distributions of gains and losses.

147 RORAC measures profit in the numerator against an internal measure of the capital at risk in the denominator. This contrasts with RAROC, which adjusts for risk in the numerator (rather than the denominator) against regulatory capital in the denominator. See Matten (2000), p.147. 148 It was discussed in the previous chapter that capital held by a bank is typically based on a target credit rating, which in turn is related to the probability that the bank defaults on its senior debt. It was argued in the chapter that this concept of risk differs from the concept embodied in the risk preference function of the centre of the bank. As such, incentive compatibility between the centre of the bank and managers requires that the measure of risk used internally may not align with the actual capital held by the bank. For this reason, the capital assigned against a position for the purposes of risk-adjusted performance measurement may necessarily be notional, rather than an assignment of actual capital held by the bank. 149 The determination of the hurdle rate also has implications for incentive-compatibility between principal and agent in the bank. This is examined in section 3.5 of this chapter.

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In the preceding chapter we assumed that managers had no reason to misrepresent to the

centre of the bank their private information regarding the expected distribution of gains

and losses in positions available to them. In the current chapter we must relax this

assumption. When managers have private information available to them, which is

difficult for the centre to screen, and which forms the basis upon which bonuses are later

assessed, they may have an incentive to misrepresent this information in order to

increase the probability of receiving a bonus, or to influence the size of bonus. The result

is the bank may be undercapitalised with respect to risk, given the incentive for managers

to understate potential losses in order to achieve a lower risk assignment, thereby

increasing the RORAC measure for their positions and any related bonus.

A potential safeguard against misrepresentation by managers as described above is for

the centre to compare the ex-post actual distribution of returns for a particular position

against the expected distribution of returns advised by managers ex-ante. However, if the

actual distribution of returns on a position turns out differ from the expected distribution

it could be difficult for the centre to assess if this is due to a poor risk assessment by

managers, misrepresentation by managers, or factors beyond the control of managers.

Further, consider a scenario under which a position incurs no variability in returns ex-

post – does this imply that the actual capital utilised over the measurement period is zero

and the resulting RORAC infinite? It is our view that regardless of the actual ex-post

variability in returns, the act of originating a risky position results in the absorption of

capital to protect the bank against under-provisioned losses - capital that might otherwise

have been deployed elsewhere in the bank. In this regard, the capital assigned to a

position represents the capital actually utilised by the position, regardless of the ex-post

distribution of returns.

In light of information asymmetries and associated screening difficulties, the simple

solution, it seems, is for the centre to ignore the private information of managers and

assign risk capital against positions based on risk measures drawn from historical return

distributions. In this regard, it is somewhat unfortunate that the benefits of decentralising

knowledge into the hands of specialist managers are lost if private information cannot be

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incorporated into the assessment of risk and the calculation of risk capital (real or

notional) to be assigned against positions. The seed of the problem lies in the notion that

protecting against large unexpected losses in a bank requires an ex-ante assessment of

risk which is difficult to relate to ex-post outcomes. Exacerbating the problem is the

possibility that managers change their risk attitudes and actions subsequent to the risk

capital assignment, depending on the perceived performance of the position relative to

target or aspiration levels.

The alternative to using the private information of managers – the assignment of risk

capital against positions based on the historical distribution of returns - presents a

different set of problems for the centre of the bank. As foreshadowed, if the risk measure

based on the historical distribution of returns turns out to exceed the risk measure based

on the expectations of managers, managers may be incentivised to reject the investment

opportunity at hand or alter their behaviour by taking greater risks in order to increase

the probability of achieving a high RORAC and subsequent bonus. We cite evidence of

this in this chapter. Conversely, if the historical distribution of returns provides a risk

assignment that is lower than management expectations on risk, then managers face little

incentive to reveal this information because the lower risk value implies a greater

probability of a larger RORAC and subsequent bonus. In this case, the position is

undercapitalised with respect to risk.

We examine these problems, and related issues, in this chapter. The rest of the chapter

proceeds as follows. Section 4.2 examines the application of RORAC methodologies in

banks generally. Section 4.3 considers agency problems in the banking firm where the

principal is the centre of the bank and the agents are employees/managers within the

bank who are expected to act in accordance with the desires of the centre. Section 4.4

examines specific agency problems related to the use of RORAC methodologies for

remunerating managers within the bank. Section 4.5 examines the relevant basis for

determining the hurdle rate for performance measurement against RORAC in a banking

firm. We challenge the standard assumption that the hurdle rate should reflect the market

cost of equity capital to the bank, and argue that this is inconsistent with the risk concept

implicit in the measurement of economic capital for the bank. The implications for

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pricing and performance measurement are considered. Section 4.6 provides a summary

of the main conclusions of the chapter.

4.2 IMPLEMENTATION OF RORAC METHODOLOGIES

RORAC models can be viewed as an extension of funds transfer pricing systems, which

are used by banks to measure the profits of their business units or specific positions.

Funds transfer pricing systems unbundle interest rate risk from credit and funding risks

in positions by requiring all transactions booked in business units to be offset with a

central risk management unit at terms that match the duration of the underlying

transactions.150 The isolation and transfer of interest rate risk in positions to a risk

management unit provides for a more efficient measure of business unit performance

because business unit profits will be more reflective of factors that are within the control

of the units. A lending unit, for example, has little control over funding or interest rate

risk, but significant control over credit risk. Consequently only net income corresponding

to credit spreads should be used to assessing the performance of the unit.

RORAC models enhance the performance measurement process by comparing the net

income of business units or positions against the risk capital allocated assigned to the

unit or position to cover for unexpected losses. RORAC models measure the risk

inherent in the activities of business units (these include fee-based activities, trading

activities and traditional lending, funds gathering and balance sheet gapping functions),

and charge the units for the capital deemed necessary to support these activities. This

process aims to make business line managers more accountable for the amount of

investor capital they are essentially putting at risk, and provides a basis for comparing

disparate transactions and business across the bank.

150 For more details on the operation of funds transfer pricing systems, refer Crandon and McCarthy (1991).

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By permitting banks to measure the risk/return impact of incremental transactions,

RORAC models can also be used to guide resource allocation decisions. Specifically,

when capital or funding is scarce, those opportunities providing the highest expected

RORAC may win the right to be allocated capital and other resources. To the extent that

the RORAC measure for a business or position is higher than a predetermined hurdle

rate, then the business can be judged to be creating value for bank stakeholders. At the

same time, if the net income of a business line is disproportionate to the potential risk,

management can use this information to adjust pricing or to devote lower resources to

the business. In this way RORAC models can be used in banks to determine factors such

as product pricing and entering or exiting business lines. Risk capital charges can also be

worked into underwriting, reporting and management compensation schemes.

Before considering agency problems in general terms, let us consider two issues that

need to be taken into consideration when RORAC models are used for performance

measurement and resource allocation. The first is the sensitivity of the net income

measure to transfer pricing assumptions. The second relates to the potential for

underinvestment when RORAC forms the basis for remunerating managers.

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4.2.1. Sensitivity of Funds Transfer Pricing Assumptions

The numerator of the RORAC equation is the profit margin of the business unit or

transaction, which, as discussed, is derived using a funds transfer pricing methodology

based on a matched-duration marginal cost of wholesale funds for the bank.151 While

much research has tended to focus on the measurement of economic capital which

comprises the denominator of the RORAC equation, there are a number of factors that

render the RORAC measure highly sensitive to the measurement of profit that comprises

the numerator of the equation. First, banks tend to be more highly leveraged than other

types of businesses, and as such, the amount of capital allocated against a position tends

to be small in relation to revenues and costs used to measure profit. Second, to the extent

that product volumes may be large but margins small, the greater the sensitivity of the

RORAC measure to the profit term in the numerator. Given some transactions may be

based on only a few basis points, errors or assumptions used to measure profit may lead

to major swings in the measured RORAC.

The above indicates that RORAC may be highly sensitive to assumptions regarding the

determination of funds transfer prices. In this regard, a number of theoretical difficulties

arise in setting funds transfer prices in financial institutions. One relates to the notion that

funds transfer prices should be based on the marginal cost of wholesale funds. For a

lending unit, a transfer price based on funding costs implies that profit for the unit will be

measured as the difference between the yield on loans and the transfer price of funds of

matched duration, less operating costs for the unit. Similarly, for a funding unit, profit

will be based on the difference between the transfer price of funds and the actual funding

rate of matched duration, less operating costs for the unit. For the treasury unit, profits

will be based on the spread between the transfer price charged to the lending unit and the

transfer price paid to funding unit.152 The use of the marginal cost of wholesale funds for

setting the transfer price implies that the marginal opportunity (opportunity cost) for the

bank rests on the funding side – in essence, once a loan request is approved the bank

151 Interest rate swap rates available to the bank, at its credit rating, would be a benchmark for the wholesale cost of funds for the bank at various terms. 152 Given the marginal cost of wholesale funds is used to determine these transfer prices, a positive margin in the treasury unit provides an indication that it is taking a position on interest rates in the banking book

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enters the market to raise the required funds, provided equity capital is sufficient to

maintain the desired leverage of the bank. However, for a financial intermediary,

marginal opportunities can swing between assets and liabilities, depending on whether

the bank at a particular point in time has excess loan demand or excess funding. This, in

turn, is driven by broader economic and competitive factors. In this context, there may be

some periods where transfer prices should be based on marginal wholesale funding costs,

and others where transfer prices should be based on benchmarks for asset yields. While

determining the wholesale cost of funds for the bank may be relatively easy given the

liquidity of interest rate swap markets, determining an opportunity cost benchmark with

respect to bank investment opportunities is likely to be a complicated exercise given the

range of options potentially available to the bank in terms of borrower credit ratings and

loan markets.

An additional problem relates to deriving funds transfer prices for products with

embedded options, such as non-maturity demand deposits and fixed-rate loan portfolios

with high prepayment risk. Measuring the duration for products with embedded options,

(which are likely to characterise a large percentage of bank products) for the purposes of

determining appropriate funds transfer prices is difficult given there may be no clear

indication of the relevant duration of the products at their origination. The result is the

transfer price that will be used for pricing and performance measurement is likely to be

heavily dependent on assumptions regarding future customer behaviour.

4.2.2 RORAC and Underinvestment

One problem with measuring performance using a rate-of-return measure such as

RORAC is that it may encourage business unit or portfolio managers to reject

investments that are value-increasing for the organisation but which reduce average

returns for the unit or portfolio. It is a general proposition that a firm’s value will be

maximised if managers within the firm acquire assets that generate returns in excess of

some hurdle rate that is typically linked to the cost of capital for the firm. However, if the

remuneration of managers is linked to the returns generated by business units or

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portfolios under their control, without taking into consideration the size of the portfolios

or portfolios, then sub-optimal behaviour on the part of managers may eventuate.

For example, suppose a hurdle rate of 12% is established and a portfolio is generating an

average return on investment of 15%. If the portfolio manager receives a bonus based on

portfolio return, subject to return being in excess of the hurdle rate, then the manager

may reject new investment opportunities that generate expected returns that are less than

15% but greater than 12% because such investments would lower the overall return on

the portfolio. Similarly the manager may act to divest assets in the portfolio that are

earning returns less than 15% but greater than 12%, in order to increase the overall return

on the portfolio. Such actions, while increasing the return on the portfolio and the

potential bonus to the portfolio manager, will reduce overall value for the entity as a

whole given the notion that any return in excess of the hurdle rate is value-increasing for

the entity. A bank that remunerates portfolio or business unit managers on the basis of

the size of the RORAC may experience such sub-optimal behaviour on the part of these

managers given RORAC is a return on investment measure.153

An additional measurement consideration is that a hurdle rate approach tends to view

transactions independently of customer relationships, diversification and other

potential synergies across transactions and business units. While an individual

transaction may not meet a firm-wide hurdle rate, it may generate overall value to the

firm as a result if it has risk-reducing benefits or it leads to revenue in other business

units. To ensure managers are encouraged to invest in assets that are value-enhancing

for the bank, a RORAC measure would need to capture such interactions across

products and business units. This, however, may be a complicated task and brings into

question the basis upon which hurdle rates are determined. It may be the case that a

hurdle rate that is based on the cost of capital for the overall bank may not be

incentive-compatible from the perspective of specific bank stakeholders. This is

examined in the section 4.5 of this chapter.

153 To overcome this problem, banks should subtract a capital charge from risk-adjusted income, where the capital charge is based on the capital allocated to the portfolio and the minimum hurdle rate for the bank. This produces a residual profit figure that represents the value added to the overall entity from the portfolio or business unit.

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4.3 AGENCY PROBLEMS

Several authors advocate the use of RORAC for performance measurement in banks on

the basis that it promotes goal congruence between the centre of the bank and managers

within the bank by aligning realised returns with the risks taken to achieve these returns.

Mussman (1996) argues that the methodology fosters consistency between the objectives

of business units and that of central management by ‘applying a consistent risk/return

criterion to business units and individual transactions as it does to the organisation as a

whole’.154 Punjabi and Dunsche (1998) state that RORAC measures provide scope not

only to capture portfolio level effects of transactions, but they also correctly motivate

lending actions and relationship strategy.155

We argue that RORAC models may not achieve these outcomes due to the existence

of agency problems in banks. If the basis upon which risk is estimated for the

denominator of RORAC is historical data on volatility, and managers have a good

understanding of where historical estimates of volatility understate current or

expected volatility, then it is possible for managers to exploit this information

asymmetry and select investments where the risk estimate used for RORAC is less

than the true estimate in the eyes of the manager. This will allow managers to evade

risk limits and take on greater risk than permitted or desired by the centre, and

increase the potential to achieve a high RORAC and associated bonus. Managers may

also desire to evade risk limits to exploit convexities in the compensation payment

function or because they are gambling to resurrect a position that is incurring losses.

154 Mussman (1996), p.7. 155 Punjabi and Dunsche (1998), p.21.

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A possible solution to this problem would be for the centre to base risk measures for

RORAC on estimates of volatility provided by managers, rather than using data based

on historical volatility. However managers may have little incentive to reveal their

private information, particularly when this information is used to derive a measure of

risk that forms the basis upon which performance will later be judged. Further, and

worse, managers may be incentivised to misrepresent this information in order to

achieve a more favourable risk capital allocation and increase the size of any potential

bonus. Such misrepresentation of information may lead to the bank being

undercapitalised with respect to risk.

In the next section we consider agency problems in general in banking. This is

followed by specific consideration of how agency problems can erode the usefulness

and reliability of risk-adjusted performance measures and lead to adverse

consequences for the overall banking entity.

4.3.1 Sources of Agency Problems

The two main forms of agency problems within organisations are adverse selection and

moral hazard. We briefly review each within the general banking context.

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4.3.1.1. Adverse Selection

Adverse selection is a problem caused by pre-contractual information asymmetries,

and arises when the incentives of agents are not compatible with the objectives of the

principal. Adverse selection refers to the tendency for agents with private information

to take actions that are detrimental to the principal.

The various agents within an organisation will typically have divergent interests

related to factors such as market share, monetary compensation, perquisites, status,

and the quality of work. In addition to the pursuit of these objectives, these players

may also have different attitudes towards risk. The private information possessed by

these players gives them some measure of power to pursue their own goals, which

may be in conflict with corporate goals. In the bank context, if managers have private

information about their costs, revenue capacity, risks and/or investment opportunities

that cannot be directly observed by the centre of the bank, they may have an incentive

to misrepresent this information in order to influence the transfer prices that they

receive or are charged, risk capital that is assigned to their positions, or to influence

their bargaining position within the bank.

The process of integrating the strategic plan of a bank with an operational plan for the

balance sheet presents a typical example of the problem of goal incongruence between

the centre of the bank and its business units. Strategy objectives are set at the top of

the organisation and it is the role of the centre to implement plans for achieving these

objectives. Information is communicated to business units, which in turn are required

to provide the centre unit with forecast data on volumes and margins for the various

businesses under their control. It is then the task of the bank treasury to produce a

balance sheet plan that encompasses the strategic objectives of the centre and the

forecasts of the business units within constraints linked to capital targets, regulatory

requirements, funding needs and interest rate risk exposures. To achieve these overall

objectives, it may be necessary for the treasury unit to place constraints on business

unit products or volumes, which divisional management may regard as preventing

them from carrying out their functions effectively. Conflict arises to the extent that the

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divisional goals may work against the broader treasury goals. For example, a business

unit may be devoting considerable resources to a product that is profitable at the

business unit level, but which increases the overall risk profile of the banking entity.

This could arise if the product being expanded at the business unit level reduced

diversification benefits at the entity level.

4.3.1.2 Moral Hazard

Moral hazard is a problem caused by post-contractual information asymmetries, and

arises when agents have incentives to deviate from contracts with the principal and

take self-interested actions because the principal has insufficient information to assess

whether the contract was honoured.

There are two aspects of the moral hazard problem that are applicable to financial

institutions such as banks. The first is that the information that flows from managers

to the centre will usually be conveyed after investments or transactions have been

implemented. In business units requiring fast decisions, such as in trading operations,

decentralisation of decision-making may be essential to take advantage of short-term

market opportunities. Delays in information to the centre may mean positions are

undercapitalised over significant periods of time if risk limits are breached.

Information asymmetries related to intra-day transactions may invite opportunistic

behaviour on the part of dealers or their managers who are looking to increase their

exposures relative to capital assigned to trading desks in order to increase their trading

profits and reported RORAC.

The second issue has been previously raised. The compensation payment function for

managers in banking firms tends to be non-linear, with employees participating in

gains but not losses. Consequently negative outcomes in high-risk activities may not

be borne by dealers or managers, but by the bank as a whole. This may encourage

dealers or managers to breach limits and engage in high risk transactions, given the

potential for higher short-term rewards and subsequent bonuses. While the possibility

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of an extreme loss may be remote, the potential for such outcomes will be correlated

with the increased appetite for risk on the part of managers.

4.3.1.3 Other Agency Problems

‘Influence activities’ are a form of moral hazard. They arise in organisations where there

is a high degree of central authority, particularly in respect to resource allocation

decisions. In pursuit of their own interests, managers or business units may devote a

considerable amount of resources in an attempt to influence decisions of the centre to

their benefit. For example, a business unit may consume a significant amount of time,

effort and physical resources in putting a case to the centre that it has better investment

opportunities than other business units in order to receive a larger allocation of funds and

other resources. Indeed, influence activities may accumulate over many periods in an

attempt to ensure favourable outcomes. In reality the investment opportunities in the unit

may be no better than in other units. Influence activities incur costs to the extent that

resources expended to influence decisions of the centre represent a cost that brings no

real offsetting gain to the organisation. From the whole of bank perspective, these

resources may have been deployed more profitability in other activities. Further, in

addition to resources wastefully consumed in influence activities, funds may be diverted

from other units and activities that had the potential to earn higher returns for the bank.

Thus there are further costs to the organisation in terms of an inefficient resource

allocation.

Agency costs may also manifest in terms of an increased cost of co-ordination within the

organisation. In general, coordination costs are linked to the degree of specialisation of

the work performed by managers or business units within an organisation. They arise

when each manager or unit usually has partial or incomplete information about the rest

of the organisation. The value of the organisation may be influenced by the degree to

which information can be transferred across business units in order to utilise synergies

within the organisation that result in lower costs and/or increased revenues. Agency

problems, however, may restrict the free flow of such information. For example, suppose

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a corporate bond desk obtains information that could be valuable to a corporate finance

unit, and benefit the overall organisation. If competition for internal resources is strong,

the manager of the bond desk may choose not to pass this information to the corporate

finance unit if there is a perception that the finance unit will gain at the expense of the

trading desk. This would be the case if the information led to improved performance in

the finance unit, and as a consequence, the bonus pool available to the bond desk is

reduced. Alternatively, there may be a perception on the part of the bond desk that

stronger current performance in the finance unit will lead to future resources being

diverted to the finance unit at the expense of the trading desk.

Further, traders may hold private knowledge regarding the risk profile of securities in

their portfolios, and their cooperation with the risk management unit is essential to

develop an effective risk management strategy for the bank. However risk managers may

be seen by dealers as a threat to achieving larger trading profits, rather than as partners

who seek to reduce the costs and exposures of the trading operation. Traders may have a

strong incentive to misrepresent information to risk managers in order to achieve a lower

capital assignment for the trading unit, allowing the true risk to be larger than the

expected risk.

Agency problems may be exacerbated in banking organisations due to the potentially

long time frame between executing a poor decision and the realisation of negative

outcomes. In the context of lending activities, poor decisions associated with loan

approval and origination may take many years to surface as losses or for loans to be

characterised as under-performing. If the compensation formula for the bank

motivates managers to achieve sales targets or short-term profits, then less emphasis

may be placed on monitoring the performance of existing loans. The lack of interest in

monitoring and corrective action on the part of a loan portfolio manager may be

intensified if it is the same manager who originated the loans in the portfolio, given

the reporting of under-performing loans may be perceived to reflect negatively on the

manager who originated the assets. The manager may fear penalty from the centre if

the highlighting of under-performing loans is attributed to poor judgement or weak

credit analysis on the part of the originating manager. Under these circumstances, the

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manager may choose to take no corrective action which could limit eventual losses to

the bank arising from default, or somewhat worse, provide further funding in order to

present a favourable cash position on the part of the distressed borrower. This

represents a gambling for resurrection strategy. From the perspective of bank

stakeholders, the bank may be carrying a high proportion of underperforming assets

that may threaten its capital base.

4.4 RORAC AND AGENCY PROBLEMS

In this section we focus specifically on how agency problems in banks can work to

corrupt RORAC performance measures and potentially lead to situations where banks

are undercapitalised with respect to risk.

4.4.1 Capital Assignment and Information Asymmetries

In the previous chapter it was concluded that the downside semi-deviation (DSD) is

the most appropriate internal measure of risk for the denominator of RAPM when the

objective is to achieve goal congruency between the centre of the bank and managers

within the bank. Central to the choice of DSD is the assumption that the risk

preference function of the centre of the bank embodies non-satiety, risk aversion and a

preference for positive skewness in the distribution of expected returns. This also

corresponded to a risk-ranking of investment alternatives in alignment with third-

order stochastic (TSD) dominance principles. Based on our findings in that chapter,

we assert that an internal RORAC measure will be compromised if a risk measure is

used in the denominator that does not conform to TSD principles and the coherency

axioms as established in that chapter.

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The value-at-risk (VaR) methodology has become the industry and regulatory

standard for assessing risk in financial institutions. This can be traced to the

endorsement of VaR for determining regulatory capital requirements for market risks

in banks by the Basel Committee of the Bank for International Settlements in 1996 in

the publication of the Amendment of the Capital Accord (Bank for International

Settlements, 1996). This amendment gave banks the option to determine regulatory

capital requirements for market risks using VaR estimates derived from their own

internal risk measurement models, and in doing so, established VaR as the preferred

regulatory measure of market risk. The commitment to the VaR methodology was

confirmed in 1999 when the Basel Committee and the International Organization of

Securities Commissions (IOSCO) Technical Committee jointly issued a report calling

for banks to publicly disclose summary VaR information in their annual reports (Bank

for International Settlements, 1999).

While regulatory endorsement for VaR has centred on the assessment of market risks

in banks, broader market endorsement has evolved on the part of ratings agencies,

where the basis for determining the solvency standard of a bank is the probability

associated with the bank defaulting on its senior debt securities over a specified time

horizon.156 If a bank is targeting a specific credit rating, then it is expected to hold

economic capital such sufficient to cover unexpected losses up to the (1 – �) percent

solvency threshold. This corresponds to a VaR risk assessment, given VaR models

measure the loss that will be exceeded with a specified probability over a specified

time horizon. This suggests a propensity for banks to use VaR methodologies in their

internal models as the basis for determining economic capital requirements. If this is

the case, then there is a high probability that RORAC measures in use within banks

are based on a VaR-type risk measure in their denominator.

This use of VaR for internal risk measurement has potentially serious consequences.

We demonstrated in the previous chapter that when VaR is used as the basis for

measuring risk, credit portfolio managers can be expected to select portfolios that do

not align with the risk preferences of the centre. Basak and Shapiro (2001) show, in a

156 This time horizon is typically one year.

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partial equilibrium framework, that an agent with his VaR capped optimally chooses

to insure against intermediate loss states (those that occur with (1- �) percent

probability) while incurring losses that occur in the worst states. This suggests agents

ignore the (�) percent loss states that are not included in the computation of VaR, and

supports our findings in the previous chapter. The intuition is that when agents are

remunerated on the basis of a RORAC measure that uses VaR as the base measure for

risk, it is optimal to incur losses in those states against which it is the most expensive

to insure. Basak and Shapiro demonstrate that while the probability of a loss is fixed,

the optimal behaviour on the part of agents lead to a larger loss (when a large loss

occurs) than would have resulted if the bank had not engaged in risk management

based on VaR measures.

If the remuneration of a trader or asset manager is linked to a RORAC based on VaR

in the denominator, it is possible for the individual to engage in positions that

manipulate the VaR measure without a compensating reduction in underlying risk.

This could be achieved by entering into positions that produce small gains with a

reasonably high probability and a large loss with lower probability. This position will

have a low VaR if the probability of a large loss is sufficiently low, increasing the

RORAC on the position and the size of the bonus to the trader.157 At the same time,

the bank is exposed to a large loss that is not captured in the VaR measure, and is thus

undercapitalised with respect to risk.

157 Danielsson (2001) shows how such a position can be established by a combination short call options and long put options, with the call options written at an exercise price corresponding to the VaR at the beginning of the period and the exercise price on the put options corresponding to the expected VaR at the end of the period.

185

Next consider gaming of the risk measure used in the RORAC by managers using

their private information. This information may relate to knowledge of estimation

errors in the risk measure when historical data is used to estimate future return

distributions. While banks may use a variety of models to derive and estimate future

distributions, all models tend to suffer from the small size of the data set used for the

estimation. This arises because the extreme events that cause very large losses are by

definition rare and will tend not to be included in the data. The result is that the risk

estimate will largely reflect outcomes under normal market conditions, and not

capture the potential for correlations across markets to increase significantly during

extreme conditions. If a limited number of observations are made in the lower tail of

the distribution, estimation errors may be large. If managers have a good

understanding of the estimation errors in the risk estimate they can exploit any bias by

selecting positions where the risk estimate is lower than the actual risk being taken.

Given the expected return on the position should be determined by the true risk being

taken, the manager will on average expect to generate higher profits for a lower risk

measure, providing compensation for the true risk being taken. The resulting higher

RORAC measures will increase the size of the potential bonus to be received by the

manager.

A manager may also have private information on the expected distribution of returns,

based on a detailed knowledge of local market conditions or experience gained in

managing specific bank asset portfolios. The potential for adverse outcomes depends

to some extent on the process by which the centre of the bank assigns capital to

positions. In the scenarios discussed above, managers selected positions where they

knew the capital assigned to a position would be less than reflected in the true risk of

the position. It would generally be the case that capital is assigned prior to a position

being originated so, for instance, pricing can be set to earn the required hurdle rate on

the assigned capital. In this case managers were assumed free to select the positions

where risk true risk matched or was higher than that based on historical distributions.

It may be the case, however, that a manager is more restricted in the selection of

portfolios given a lower number of alternatives in a particular period or pressures to

achieve targets with respect to business volumes, etc. Under these circumstances

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managers are less free to choose among alternative investments, and indeed, may be

forced to accept positions where they expect risk to be lower than that implied by

historical distribution of returns.158 This means the expected RORAC will be lower

than the ‘true’ RORAC and the position may fail to earn the bonus that the manager

believes should accrue to the position.

Let us consider how the manager may react, and the implications for the bank, under

circumstances where expected risk is lower than that implied by historical data, and

consequently the capital assignment is considered by the manager to be excessive

relative to the underlying risk. We present four options:

1. Reject the position

2. Price the position to incorporate the higher capital charge

3. Increase risk in the position or substitute with a riskier position (asset

substitution moral hazard)

4. Reveal private information on the expected distribution of returns to the centre

of the bank.

The first option may be limited by the number of investment alternatives available in

the current period and/or pressures to meet sales targets. If the position is rejected on

the basis of the expected low RORAC, then the bank will be under-investing to the

extent that a position expected to add value to the banking entity will not be pursued

by managers.

The second option will arise if the bank prices its business to earn a minimum hurdle

rate on assigned capital. The higher capital charge will increase the profit required to

achieve the minimum hurdle rate, and thus increase the price of the business to the

customer of the bank. This option may be adopted by the manager, but the higher

price may force the business to be lost to a competitor. If the manager absorbs the

price into a lower margin, the RORAC and potential bonus will be lower.

158 It is assumed, however, that all positions are priced such as to at least meet the minimum hurdle rate. This will limit the size of the bonus paid to managers where compensation is linked to performance outcomes in excess of the hurdle rate.

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The basis of the third option is that managers will believe that their ability to meet

aspiration levels for RORAC will be restricted if the capital assignment is too large

relative to the expected risk in the position. In order to make up for the over-allocation

of risk, and increase the expected RORAC in the position, the manager may be less

conservative in the management of risk in the position or substitute it for a position

that carries greater risk. The third option thus presents the most potential damage to

the bank, subject to the degree of risk taken by the manager relative to the perceived

excess capital assignment. If any change in risk is not adequately compensated for in

the capital assignment, the bank may be undercapitalised with respect to risk.

If we examine the results of empirical studies on the behaviour of individuals when

they perceive themselves to be operating below aspiration levels, it is not unrealistic

to assume that individuals will change their risk attitude in order to achieve their

objectives. In chapter two of this thesis we reported on the results of a number of

empirical studies that conclude that individuals reverse their risk attitude from one of

risk-aversion to one of risk-seeking when confronted with the likelihood of

performance below aspiration. This is in keeping with basic premise of prospect

theory, being that individuals are risk-averse in the domain of gains and risk-seeking

in the domain of losses (Kahneman, and Tversky, 1979). If managers face restricted

choice on the range of investment alternatives and are assigned capital against

positions that they believe is high relative to their expectations on risk, they may

display behaviour consistent with an S-shaped utility function, as described in chapter

two of this thesis. Despite the cushion associated with the over-allocation of capital

relative to risk expectations, this is an undesirable outcome in the sense that a

gambling strategy on the part of managers may expose the bank to extreme losses that

are not covered by its capital base.

Theoretical support for this proposition is provided by Berkelaar and Kouwenberg

(2002). In a general equilibrium setting, they find that while in most cases VaR-based

risk management tends to reduce stock return volatility, with the stock return

distribution displaying a thin left tail and positive skewness, in very bad states

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managers switch to a gambling strategy that adversely amplifies default risk through a

heavier left tail of the return distribution.159 In keeping with the basic propositions of

prospect theory, Berkelaar and Kouwenberg conclude that an agent working under a

VaR-based capital constraint tries to limit losses most of the time, but starts taking

risky bets once wealth drops below the reference point upon which gains and losses

are measured. This closely resembles the optimal strategy of loss-averse agents with

the utility function described by prospect theory.

We assert that the tendency for managers to select or manage positions such that the

risk estimate in the capital assignment is less than the ‘true’ risk will exist whenever

the capital assignment is based on historical return distributions, independent of the

risk measure being used to determine the capital assignment. The motivation for this

behaviour is the exploitation of convexities in the compensation payment function, in

order to maximise the RORAC measure upon which bonuses (or resource allocation

decisions) are based. If the range of investment opportunities in the current period is

large, managers can use their private information regarding errors in the risk estimate

to choose positions in which risk is understated relative to current volatility and

expectations of future volatility. If the range of investment opportunities in the current

period is small, there may be circumstances where managers take positions where the

expected risk, according to their private information, is less than that embodied in the

capital assignment. Regardless of the risk measure used for the capital assignment,

managers in these positions may be encouraged to adopt a gambling strategy in order

to increase the probability that aspirations levels for RORAC are achieved.

In circumstances where historical distributions are such that the capital assignment

reflects a lower assessment of risk than that expected by managers, we can expect that

managers will not signal this information to the centre. If the true risk is higher than

the capital assignment, the potential for achieving a high RORAC and consequent

bonus is enhanced.

159 Berkelaar and Kouwenberg (2002), p.141.

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Stoughton and Zechner (1998) demonstrate that whenever there is asymmetric

information about a business unit’s investment opportunity set the optimal capital

allocation will embody a risk premium. They show that the existence of asymmetric

information increases the price of risk to the business unit, making capital allocations

more sensitive to risk-taking. They conclude that in the presence of asymmetric

information, the centre will assign more capital for a given position than in the presence

of perfect information on the return distribution of the position. This, in turn, may

encourage managers to use their private information to evade risk limits and increase

underlying risk of the position, in order to achieve aspiration levels with respect to

RORAC.160

This leads us to the fourth option, which is for managers to reveal to the centre their

private information with respect to expected risk in order to achieve a capital

assignment that matches the true risk being taken. This could limit the adverse

consequences of using historical data to assign capital to positions, and increase the

returns to decentralisation by incorporating the specialised information of managers

with respect to expected risk. The key question, however, is how can managers

credibly convey this information to the centre, particularly when information

asymmetries may be significant and information verification difficult? The centre

may expect managers to misrepresent their information on the expected distribution

returns in order to achieve a ‘favourable’ assignment of capital and increase the size

of the RORAC upon which performance will be measured. In the next chapter in this

thesis we develop an incentive-compatible mechanism by which it is in the best

interests of managers to truthfully reveal their private information with respect to

expected risk.

160 Hermalin (1993) shows that a manager may be incentivised to invest in highly volatile assets if the personal abilities of the manager are under scrutiny. The basis of this proposition is that highly volatile assets create more noise in the performance assessment and thus deflect focus away from the manager in periods where investments may be underperforming relative to market expectations.

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4.4.2 Internal Pricing and Information Asymmetries

In this section we briefly consider how information asymmetries may impact on the

measurement of RORAC, and how managers can take advantage of this to present an

inflated measure of performance. Our main focus remains on circumstances where the

overall banking firm may be undercapitalised with respect to risk.

4.4.2.1. Measurement Horizon

Long duration credit portfolios tend to carry greater credit risk than shorter portfolios at

origination given the longer time frame over which financial conditions may deteriorate.

This phenomenon is recognised in capital markets as a term default premium, where for

an identical borrower, credit risk spreads on long-term debt exceed those on short-term

debt. From a risk capital perspective, this suggests long-duration portfolios should carry a

larger capital assignment than short-duration portfolios to reflect the larger credit risk.

However, target solvency standards, which in part determine the economic capital

requirements for a bank, are typically measured in terms of one-year default rates – the

assumption being markets are sufficiently liquid for a bank to recapitalise in the event of

default on its senior debt.161 This may encourage managers to invest in long-duration

loans at the expense of short-duration loans, if term default premiums are built into the

pricing of a credit portfolio but are not captured in the capital assignment for the

portfolio. The result should be higher RORAC results for longer-duration credit

portfolios. Whether or not this leads to the bank being undercapitalised relative to risk

depends on the speed and ability of the bank to recapitalise in the event of default, and the

profit ‘buffer’ provided by the term default premium. As discussed in chapter two of this

thesis, a bank that defaults on its senior debt may find insurance or recapitalisation costly,

particularly in times of economic slowdown where losses may be larger and

recapitalisation expensive. It is not unreasonable to assume that default will also reduce

the confidence of retail depositors and precipitate a liquidity crisis for the bank.

161 This is examined in more detail in chapter five.

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Alternatively, if the capital assignment does incorporate a larger capital charge for

longer-duration credit portfolios to reflect the term default premium, there is a case for

the capital assignment to decrease over time given to the extent that default potential

declines with residual life. Interest rates on credit facilities, however, may not incorporate

the declining term default premium as the residual life of the facility declines. If the

capital assignment methodology of the bank recognises the lower capital requirement

over time but at the same time there is no compensating change in the rate on credit

facilities, the RORAC on the facilities will tend to be increasing over time. This will

signal improved performance on the part of credit portfolio managers over time, when

this may not be an accurate representation of performance. However while the RORAC

measure may be inflated, in this case there is no threat of the bank being undercapitalised

with respect to risk – it is more the case that the bank is overcompensating for risk as the

residual life of the loan decreases.

4.4.2.2 Funds Transfer Pricing Assumptions

Under an upward-sloping yield curve environment, retail funding units can be expected to

receive a high transfer price on their call deposits in recognition that the effective

duration of these funds tends to significantly exceed their contractual maturity. The

relative insensitivity of retail call deposits to changes in interest rates has led to claims

that their effective duration may be as long as ten years.162 If this is incorporated into the

funds transfer pricing system, and the yield curve is upward-sloping, the funding unit of

the bank will earn a considerably higher internal profit margin than had the deposits been

priced assuming their contractual maturity. This may create an incentive on the part of

funding units to overstate the effective duration of their retail deposit book in order to

achieve a higher RORAC result.

In a similar vein, business units within the bank can be expected to have better

information than the centre regarding the expected duration and liquidity of their

162 Rose, S. (1993), p.6.

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positions. For example, lending units should have better knowledge than the centre

regarding prepayment patterns for various facilities, and hence their effective duration.

Similarly, funding units should have better knowledge than the centre on the effective

duration of their retail call deposits.163 If the yield curve is upward sloping, lending units

may have an incentive to overstate expected prepayments in order to reduce the duration-

matched transfer price charged by the centre for the use of funds. A funding unit may

have an incentive to overstate the duration of call deposits in order to receive a higher

transfer price for their funding to the centre. In the former case the bank may be

underpricing its fixed-rate loans with respect to risk, while in the latter the bank may be

failing to adequately capture its funding exposure on its deposit book.

163 Rose (1993), ibid.

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4.5 THE INTERNAL HURDLE RATE AND BANK RISK

4.5.1 Overview

When banks internally apportion their total capital against investments and other

activities they are essentially rationing the risk that can be sustained from the given pool

of capital. In order to assess the risk-adjusted contribution made by business units or

specific investments to the overall value of the bank, the typical approach is to compare

the single period performance measure (here RORAC) against an internal hurdle rate that

embodies the compensation required by investors for the systematic risk of the bank.164 If

the bank decrees that the capital of the bank must be apportioned against all risky banking

activities, and that the sum apportioned be used for the calculation of the RORAC for

these activities, then the resulting measure incorporates the expected utilisation of capital

by the risky activity over the planning period.165 In this case, banks are essentially

requiring that the internal RORAC measure adjusts the risk of an activity to the total

capital base of the bank. This raises the question of what is the appropriate basis for

determining the hurdle rate that is used to measure the contribution of a risky activity to

the overall performance of the bank.

A centrepiece of risk-adjusted performance measurement in banking is that a uniform

hurdle rate be used to assess the contribution of risky activities (Zaik, et al, 1996),

justified on the grounds that the total capital held by the bank adjusts for risk by changes

in leverage. In turn, the level of capital issued by a bank is determined on the basis of a

solvency standard that equates the probability of default with the target credit rating of

the bank on its senior debt, the intuition being that the financial distress costs associated

with high leverage can be mitigated by controlling the probability of default. If bank

owners are the contributors of this capital, and these owners are assumed to hold well-

diversified portfolios within which the unsystematic risk of individual assets is

164 For example, see Zaik et al (1996). 165 In chapter two of this thesis we presented a case for using a risk measure for performance assessment that does not correspond to the actual capital held by the bank. This was established on the grounds that incentive-compatibility been achieved between the risk preferences of the centre and investment decisions of managers.

194

unimportant, then a case can be established for using a hurdle rate that corresponds to the

compensation required by bank owners for the covariance of returns of the bank to a

broader market portfolio. This is consistent with a hurdle rate linked to the return required

by bank owners under the Capital Asset Pricing Model (CAPM) derived by Sharpe

(1964).

Based on the above, we can summarise the key underpinnings of RORAC performance

methodologies as follows:

1. Total bank capital is consistent with a fixed (target) probability of default.

2. Capital is apportioned across the risky activities, typically using a VaR-based

methodology for measuring risk.

3. A uniform hurdle rate is used to measure performance contribution of risky

activities, and this is contingent upon the bank maintaining a fixed probability of

default.

4. The hurdle rate is consistent with a CAPM-approach.

In this section we question the premise that a predetermined solvency standard for

determining bank capital is consistent with a fixed hurdle rate that reflects the bank’s cost

of equity capital. Using an option-pricing framework based on the Merton model of

default, we show that this will not always be the case. We then assess the circumstances

under which it is legitimate to compare the RORAC for an activity with a market-driven

hurdle rate based on the CAPM. If RORAC is a single factor model that captures

concerns with total risk in the denominator, but then compares this to a hurdle rate that

reflects only systematic risk factors, then we must question the ability of the RORAC

methodology to provide a consistent and congruent basis measuring and pricing risk

within the bank. We also consider how hurdle rates might be constructed to reflect the

‘true’ cost of economic capital to the bank.

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4.5.2 Does Bank-Specific Risk Matter?

If the internal hurdle rate is a CAPM-based return required by bank owners, then the

implicit assumption is that investors are well-diversified and the unsystematic risk that

is specific to the bank is unimportant. This implies that the total risk of an individual

bank is of no consequence to its owners, because asset specific risk can be diversified

away. Yet the capital they contribute is based on bank-specific risk, given the notion

that the capital requirements of the bank are set to cover for a predetermined

probability of default. Is there a misalignment of incentives if the hurdle rate is based

on systematic risk, but the capital requirement is based on bank-specific risk? We

begin by examining whether risk management in a bank, and the pricing of risk,

should be based on systematic risk or bank-specific risk.

While its is generally accepted that that banks perform information transformation and

portfolio management functions, in principle, these functions fall under the

Modigliani-Miller theorem on the irrelevance of pure financing decisions. In a world

of perfect capital markets – one without taxes, financial distress costs or agency

conflicts between stakeholders – the pricing of specific risks in a bank portfolio is

irrelevant and the management of specific risks does not add value to shareholders.

The neoclassical world predicts that banks and other financial intermediaries would

not exist, because all market agents would contract directly with each other in

complete, costless, capital markets.

However, in a world where capital markets are not frictionless, total risk matters and

has to be taken into consideration when pricing risk and valuing investment decisions.

Shapiro and Titman (1986) show that financial distress costs and the existence of non-

linear tax schedules result in an inverse relationship between the total risk of a firm

and its expected cash flow. This makes reductions in total risk valuable even for

perfectly diversified shareholders. Froot and Stein (1998) show that the more costly it

is for a bank to raise external funds, the more risk-averse the bank will be with respect

to fluctuations in its internal wealth. Specifically, the bank will act to manage its

specific risks to ensure that sufficient internal funds are available for reinvestment in

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profitable projects. Central to their model is the assumption that the transaction costs

associated with raising new external funds are convex, reflecting the greater opacity

of bank assets which makes them difficult for outside investors to screen and value.

They conclude that the contribution of a business unit or a single transaction to the

overall volatility of the cash flows of the bank needs to be considered when valuing

investment decisions – in short, the pricing of specific risks matters when allocating

funds to competing sources within the bank.

A further body of literature indicates that information asymmetries within firms lead

to agency problems that provide incentives to manage firm specific risk (Jensen and

Meckling (1976) and Tufano (1998)). If financial distress costs are not trivial and

firms are nearing bankruptcy, managers, acting in the interests of shareholders, may

have a propensity to overinvest in risky projects in order to restore value to equity.

While holders of equity have the potential to recover value, debtholders are not

compensated for the extra risk that is taken and are subsequently transferring value to

equity. Conversely, managers may underinvest in profitable but low value projects

when it appears that a more than proportionate amount of value will accrue to

debtholders. Finally, management of specific risk may be favourable if high risk

pressures management towards short-sighted decisions, resulting in indirect financial

distress costs when trading partners or prospective employees, fearing future default,

are less likely to deal with the firm on favourable terms (Narayanan (1985)).

These studies indicate that banks should price total risk into their investment

decisions, and that the management of total risk is relevant. This means the economic

capital of a bank is a total risk measure that should reflect both systematic and bank-

specific risks. Indeed, the agency costs of high firm-specific risk may be of

heightened importance in the context of banking given the relative ease with which

changes in the riskiness of bank assets and funding books can be implemented, the

lack of transparency that remains in some bank asset portfolios (despite increasing

regulatory oversight), and the fact that there may be a considerable time lag between

the origination of a credit position and the time at which losses on the position

materialise.

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RORAC models that compare risk-adjusted returns based on total risk (systematic

plus firm-specific risk) to a hurdle rate that requires compensation only for systematic

risk implicitly assume that debtholder risks and shareholder risks are proportional. If

we consider the RORAC equation, the denominator, economic capital, represents the

amount of capital the bank deems necessary to achieve a sufficient level of protection

against adverse circumstances. This represents a debtholder perspective on risk.

Debtholders should care about the total risk of the bank because all risks impact on

the probability of default and the size of losses will be linked to the left tail of the

distribution of bank returns. Given their exposure to the total risk of the bank’s assets,

we argue that debtholders should value risk diversification within the bank portfolio at

every level, provided the costs of diversification strategies do not exceed the risk

reduction benefits.

In contrast, the neoclassical view implicit in the CAPM states that shareholders are

concerned only with systematic risk because they can diversify away firm-specific

risk at much lower cost than the bank. Thus while the focus of bank debtholders is the

probability of loss and the tail of the distribution of losses, the neoclassical view has

bank owners more concerned with the correlation of returns on the existing bank

portfolio with the broader ‘market’ portfolio. If the cost of economic capital is to be a

hurdle rate based on systematic risk, and bank debtholder and shareholder risks to be

compatible, then systematic risk and total risk in a bank must be proportional. This

means that as bank-specific risk changes, the leverage of the bank must adjust if the

bank is to maintain a fixed solvency standard and uniform internal hurdle rate. We

examine this proposition using a model of default based on Merton (1974).

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4.5.3 Is a Fixed Solvency Standard Consistent with a Fixed Hurdle Rate?

The RORAC methodology asserts that uniform hurdle rate can be used bank wide to

value prospective positions because the amount of economic capital required adjusts

for risk by changing the leverage of the positions. We have seen that the industry

standard for determining economic capital is VaR, which summarises the worst loss

over a target horizon with a given level confidence – the quantile of the projected

distribution of gains and losses over the target horizon. The intuition underlying the

VaR-approach for determining economic capital is that financial distress costs are

associated with high leverage and can be mitigated by controlling the probability of

default.

In this section we consider the ability of a bank to maintain a target probability of

default and a uniform equity hurdle rate when risk is measured in terms of the

expected volatility in the value of the bank’s assets. We draw on Crouhy et al (1999)

and use a Merton model of default to determine the capital structure for the bank that

is consistent with a predetermined expected default probability. We vary the volatility

of the bank’s asset value and, using an options framework, estimate the optimal

capital structure that is in keeping with the predetermined probability of default. As

the expected return on the bank’s assets changes in proportion to changes in volatility

assumptions, we calculate the required return on equity (hurdle rate) to ascertain the

degree to which it is invariant to changes in the risk of the bank’s assets. We then

conduct a similar analysis, but keep the required return on equity constant, in

accordance with the RORAC performance methodology assumption of a uniform

hurdle rate, and examine the extent to which the probability of default is invariant to

changes in the risk of bank assets as the capital structure of the bank is altered to

maintain the constant return on equity.

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4.5.3.1 Merton Model of Default

Merton (1974) demonstrates how the value of a firm’s equity can be considered

equivalent to the value of a European (Black-Scholes) call option written on the firm’s

underlying asset values. The premise of this approach is shareholders own the right to

default on the firm’s debt and will exercise this right when the value of the firm’s

assets falls below that of equity. Debtholders charge for the put option they effectively

provide on the firm’s assets by demanding a spread over the risk-free rate on the funds

they lend to the firm. The probability of default is a function of the firm’s net asset

value distribution and its current net asset value. The Merton model assumes the risk-

free term structure is flat, the firm issues only pure discount debt, and asset values

follow a geometric Brownian motion. The exercise price on the option is the par value

of the firm’s debt. If the distribution of returns for bank assets is known, the model

can be used to determine the economic capital required to maintain a constant

probability of default. We apply this framework to determine the likelihood that the

hurdle rate on capital remains constant as capital adjusts to maintain a fixed

probability of default.

We employ the example of a bank that carries assets with a market value of $1 billion.

The required return on these assets (ra) is determined by:

ra = r f +[[(�✁A)/✁m] (rm – r f)]

where � is the correlation coefficient between the rate of return on the bank’s assets

and the rate of return on the market portfolio, ✁A and ✁m are the standard deviations of

the returns on the bank’s assets and the market portfolio (respectively), rm is the return

on the market portfolio and r f is the risk-free rate of return.166 We assume that the

solvency standard for the bank is based on a one-year default probability, and thus

employ one-year time horizon (T) for the determination of economic capital. We

166 This is in keeping with Crouhy, et al (1999).

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follow the approach of Merton (1974) and assume that there is only one class of

equity and one class of debt, and employ risk-neutral default probabilities.167

Ong (1999) shows that the risk-neutral expected default probability (the probability

that the value of assets at time T will fall below the default point) can be determined

as follows:168

prob (default) = N(-d2)

where N(.) is the cumulative standard normal distribution and

d2 = (ln[A0/DT] + [r f – 0.5�2)]T) / �✁T

where A0 is the market value of the bank’s assets, DT is the face value of debt at time

T, r f is the risk-free rate, � is the constant volatility of the bank’s assets and T is the

time to maturity.

4.5.3.2 Results

Our results are shown in Tables 4.1 and 4.2.

Table 4.1 shows the required return on equity when the probability of default is kept

constant at 1%. It is clear that as the volatility of the bank’s assets changes, the

required return on equity changes. Thus keeping the probability of default constant

appears inconsistent with a constant expected rate of return on equity as bank asset

values become more or less volatile, challenging the premise that RORAC can be

directly compared with a hurdle rate based on the bank’s cost of equity capital.

167 Under the risk neutral probability measure the expected return on all securities is the default free rate of interest. Crouhy et al (2000) shows that the risk neutral probability of default, after adjusting for the price of risk, is higher than the actual probability of default where the latter is based on probabilities observed in the market place from historical data. 168 Ong (1999), p.86.

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Table 4.1

Constant Probability of Default Scenario

Case A B C D

Default probability 1.0% 1.0% 1.0% 1.0%

Asset volatility (�A) 10.0% 20.0% 30.0% 40.0%

Optimal bank debt/equity ratio169 372.0% 160.0% 90.0% 57.0%

Required return on assets 7.1% 9.2% 11.3% 13.4%

Cost of debt 5.0% 5.0% 5.0% 5.0%

Equity hurdle rate 14.9% 15.9% 17.0% 18.2%

Assumptions

Risk-free rate of return is 5%, expected market risk premium is 7% and time horizon is one year

Standard deviation of market portfolio (�m) is 10%

Correlation coefficient (✁) between return on bank asset and return on the market portfolio is 30%

Asset beta is (✁✂A))/ ✂

m

Table 4.2

Constant Equity Hurdle Rate Scenario

Case A B C D

Equity hurdle rate 19.0% 19.0% 19.0% 19.0%

Asset volatility (�A) 10.0% 20.0% 30.0% 40.0%

Optimal bank debt/equity ratio170 567.0% 233.0% 122.0% 67.0%

Required return on assets 7.1% 9.2% 11.3% 13.4%

Cost of debt 5.0% 5.0% 5.0% 5.0%

Default probability 5.9% 4.8% 3.3% 1.9%

Assumptions Risk-free rate of return is 5%, expected market risk premium is 7% and time horizon is one year

Standard deviation of market portfolio (�m) is 10%

Correlation coefficient (✁) between return on bank asset and return on the market portfolio is 30%

Asset beta is (✁✂A))/ ✂

m

169 To preserve a 1% probability of default on debt. 170 To preserve the required return on equity at the uniform hurdle rate of 19%.

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This result reflects that fact that when the return on the bank’s risk assets is less than

perfectly correlated with the return on the market portfolio, an increase in the variance

of the rate of return on the bank’s risky asset will increase both systematic risk and

bank-specific risk. Table 4.1 depicts how the capital structure of the bank changes as

asset volatility changes, with higher volatility increasing the value of the call option

on the bank’s assets and resulting in a larger economic capital requirement (lower

leverage) for the fixed 1% probability of default. The figures show that the

relationship between greater risk, lower leverage and a fixed required return on equity

is not proportional.

Table 4.2 shows the impact on the bank’s probability of default if the required return

on equity is held constant and we vary the volatility of the rate of return on the bank’s

assets as above. Here we assume the bank aims to achieve a degree of leverage such

that the expected rate of return on the equity, the hurdle rate, is constant at 19%.

Using the Merton model, we compute the probability of default for changes in the

volatility of assets while altering the capital structure of the bank to keep the required

equity return constant. The table shows that the probability of default does not remain

constant with changes in the risk of the bank’s assets when the expected rate of return

on equity is kept constant. Thus, despite the fact that leverage adjusts with changes in

bank-specific risk, a uniform hurdle rate is not necessarily consistent with a constant

probability of default. We conclude that RORAC cannot be required to compensate

for risk as the market does, and as such, the cost of economic capital is not a hurdle

rate based on systematic risk. We discuss how consistency between economic capital

and the hurdle rate on capital can be achieved in section 4.5.4.

An additional complication with the RORAC approach is that it implicitly requires

that economic capital is synonymous with actual capital invested by bank owners.

This is not necessarily the case. We have defined economic capital in terms of its

industry standard – that being the level of capital necessary to guarantee the solvency

of the bank at a predetermined confidence level that is consistent with the target credit

rating of its senior debt. This means both shareholders and junior debt holders can

provide economic capital. Thus only capital invested by bank owners can be required

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to earn a CAPM-based hurdle rate. Setting economic capital equal to cash capital

invested by shareholders ignores the default risk of other tranches.

Are there valid conditions under which the RORAC can be compared to a CAPM-

based hurdle rate? Wilson (1992) provides insight into this question. He shows that

the risk-adjusted performance measure will only lead to unbiased results if no real

equity cash investment is necessary in a transaction or position, as would apply in the

case of self-financing portfolios or in bank activities that require no capital outlay,

such as swap transactions. This situation arises because the investment decision rule

considers only the risk of a position, and not the capital actually invested in the

position. Wilson considers the example of a trader who has an asset with a positive

mark-to-market value and a risk-free investment opportunity. The trader can achieve

an infinite risk-adjusted return measure by liquidating the portfolio and investing the

proceeds in the risk-free asset. This strategy will generate positive realised returns

with no capital at risk, implying an infinite risk-adjusted return. The bias arises

because the capital invested in the position is not incorporated into the performance

measure.

4.5.4 Compatibility Between the Hurdle Rate and Bank Risk

We concluded in the previous section that the cost of economic capital is not a

CAPM-based hurdle rate. When the DSD is used to measure bank risk, economic

capital takes into account a bank’s concern with total risk that makes a bank behave as

if it were risk-averse, consistent with the risk preference function of the centre of the

bank derived in chapter two of this thesis. The hurdle rate used for benchmarking

performance within the banking firm should capture any additional costs associated

with bank-specific risk. With the exception identified in the previous section, a

CAPM-derived hurdle rate will understate the true cost of economic capital. If firm-

specific risks impose real costs on a bank, a hurdle rate based on CAPM may lead

banks to underprice risk or direct funds to investments that generate lower returns

than expected. If the true cost of economic capital for a specific bank investment is

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linked to both a market beta (systematic risk) and an internal beta (contribution of the

investment to bank asset volatility), then this should be incorporated into the hurdle

rate to achieve consistency between risk and the pricing of risk.

Froot and Stein (1998) present a two-factor model which indicates that the hurdle rate

for a transaction should reflect the price of market risk and the contribution of the

transaction to the overall volatility of the cash flows that cannot be hedged in an

external market. This latter is measured as

[(rp – r f)/�p2] �i,p

where rp is the return on the existing portfolio of bank assets, rf is the risk-free

benchmark return, �p2 is the variance of returns on the existing portfolio and �i,p

is the covariance of the individual transaction (i) with the bank portfolio (p). The

expression in brackets represents the unit cost for volatility of the bank’s portfolio of

non-hedgeable cash flows.

Merton and Perold (1993) argue that risk capital is the smallest amount that can be

invested to insure the value of the firm’s net assets against a loss in value relative to

the risk-free investment of those net assets. This embodies a notion that economic

capital is kept in a separate ‘pool’ and invested in risk-free assets so that it is available

for unexpected losses up to a prespecified confidence level. In this framework, (1) the

actual cash capital invested by bank owners is required to earn a CAPM-determined

required return, and (2) the economic cost of risk capital to the bank is the cost of

asset insurance related to information asymmetries and agency costs. In other words,

information and agency costs (bank-specific risk) require a bank to pay a price for

insuring its net assets that is above an actuarial fair market value. This represents the

cost of economic capital that is over and above the CAPM-based hurdle rate that

reflects only systematic risks.

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Schroeck (2002) combines elements of the models of both Froot and Stein (1998) and

Merton and Period (1993) to develop a two-factor model that defines the hurdle rate

(in terms of required profit) for new transactions in a bank as follows:

Hurdle rate = (re x IC) + (� EC)

where re is the CAPM-determined required rate of return, IC is the actual invested

shareholder capital, � is a parameter representing the financial distress costs of the

bank and EC is economic capital required to support the transaction. The second

expression, (� EC), represents the cost of total risk to the bank arising from an

incremental transaction. Schroeck does not specify how � is determined. An

appropriate starting point might be the risk premiums implicit in agency credit ratings.

As the probability of default increases for a bank, its credit rating will be downgraded

and the market yield on its traded debt will increase. Changes in market yields may

give some indication as to a market assessment of financial distress costs. If the costs

of distress decline as bank credit ratings increase, historical ratings migration data

may prove useful for assessing financial distress costs that differentiate bank-specific

risk from systematic risk.

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4.6 CHAPTER SUMMARY

The main findings of this chapter are summarised as follows.

1. If managers (agents) in a banking firm have a good understanding of where

historical estimates of volatility understate current or future volatility, and this

private information is difficult for the centre of the bank (principal) to screen,

then it is possible for managers to exploit this information asymmetry in order to

achieve favourable outcomes with respect to resource allocation decisions and

bonus payments.

2. If the centre assigns risk capital to positions on the basis of historical volatility

and this exceeds managers’ estimates of current or future volatility, managers

may reject profitable investment opportunities or increase risk-taking in order to

achieve aspiration levels for the RORAC. If managers believe the capital

assignment does not reflect current or future volatility, they are unlikely to

reveal this information because a higher capital charge will reduce the potential

RORAC. In either case, managers’ desires to achieve high bonuses may lead the

bank to be undercapitalised with respect to risk.

3. If risk capital is assigned to positions on the basis of achieving a target solvency

standard for the bank, which in turn is based on a predetermined probability of

default, the optimal behaviour of managers is to incur losses in those states

against which it is the most expensive for them to insure. Further, managers may

change their attitude to risk subject to the performance of a position relative to a

reference point upon which gains and losses are measured.

4. The use of a CAPM-based methodology to determine the internal hurdle rate for

measuring performance understates the true cost of economic capital to the

bank. Further, a uniform hurdle rate for pricing business is not consistent with a

constant probability of default when bank returns are less than perfectly

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correlated with the return on the market portfolio. The internal hurdle rate

should capture the additional costs to investors associated with bank-specific

risks.

A major premise of this chapter has been that agency problems may create an

environment where managers have little incentive to reveal their private information

on the expected distribution of returns when this forms the basis upon which

performance may later be judged. In the next chapter we develop a framework for an

internal capital market in a banking firm where it is optimal for managers to reveal the

truth regarding their expectations on risk.

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Chapter Five

Revealing the Truth:

An Internal Market Mechanism for Allocating

Risk Capital

“It should not be surprising that game theory has found ready

application in economics. The dismal science is supposedly

about the allocation of scarce resources. If resources are

scarce, it is because more people want them than can have them.

Such a scenario creates all the necessary ingredients for a

game.”

Ken Binmore, 1992

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5.1 INTRODUCTION

It is somewhat ironic that organisations decentralise to allow managers to develop

superior information about their local conditions, and then need to develop mechanisms

to encourage managers to truthfully reveal this information. This situation arises

because managers may have an incentive to misrepresent their private information

when used within the organisation for compensation, planning and control purposes.

For example, by overstating the expected profits or understating the expected risks in

their activities managers may be able to achieve a more favourable allocation of

resources from central planners. Similarly by understating actual risks absorbed or

failing to bring to account under-performing assets, managers may be able to unduly

influence measures upon which their performance is assessed.

In the presence of information asymmetries the centre cannot assume that the managers

will carry out instructions or reveal all information; and is consequently compelled to

design incentive-compatible mechanisms or contingent contracts to induce managers,

acting in their own self-interest, to reveal all relevant private information. In

implementing these mechanisms, the organisation cannot expect to achieve first-best

profit maximisation (that is, profits it would realise if there did not exist any

information asymmetries) - the organisation can only expect second-best profit

maximisation because a part of the first-best profit is absorbed in incentives created to

achieve goal congruence between managers and the centre. This implies that a part of

first-best profits are diverted into the creation and implementation of the incentive

structure (see Amershi and Cheng, 1990).

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In this chapter we assume that the centre of the bank determines the total capital

requirements of the bank using the downside semi-deviation risk (DSD) measure,

which was shown in chapter three to be coherent and incentive-compatible given the

risk preference function of the centre.171 It is the responsibility of the centre to allocate

this capital to those activities or business units that are expected to generate the highest

returns per unit of risk.

In the setting of chapters two and three the centre did not have information on the

expected distribution of returns of the full set of investments available to managers, but

this information was available to managers based on their specialist knowledge of local

investment opportunities and their inherent risk profiles. The centre only received

information on the return distribution of investments actually selected by managers,

which managers freely disclosed to the centre in order that a risk measure could be

assigned to the portfolio for the purposes of measuring its subsequent performance.

Incentive-compatible conditions were achieved by establishing a risk-adjusted

performance measurement framework in which it became in the best interests of

managers to select the portfolios that the centre itself would select, given its risk

preference function, if information on the full set of investment opportunities was freely

available. The risk measure – DSD - served as the vehicle by which managers could be

guided to select the preferred portfolios. Managers had no incentive to misrepresent to

the centre their private information on the expected distribution of returns on the

invested portfolios.

In chapter four we identified agency problems inherent in this risk-adjusted

performance measurement framework and concluded that there were a number of

conditions under which managers could be expected to use their private information on

current and expected volatility to engineer higher RORAC performance measures than

would apply if the true expected distribution of returns were incorporated in the

denominator of the RORAC equation. In the current setting we assume that the

171 This implies that the total capital of the bank will not necessarily match that based on a target probability of default. It was shown in chapter two that such a measure, if used internally, will lead managers to make investment decisions in some circumstances that are inefficient in the sense of not being aligned with the risk preferences of the centre.

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conditions identified in chapter four hold, and the centre cannot be assured that

managers will truthfully reveal their private information on the expected distribution of

portfolio returns. Thus while the DSD risk measure can be considered incentive-

compatible in the sense that managers can be expected to select portfolios that that are

aligned with the risk preference function of the centre, we now incorporate the

likelihood that managers have a propensity to misrepresent their private information in

order to achieve more favourable personal outcomes. When there are competing

demands for capital within the bank, managers have an incentive to misrepresent

expected risks in order to favourably influence the capital allocation decisions of the

centre – for a given position or activity managers may understate risks or overstate

profits in order to inflate the expected RORAC and increase the size of any bonus that

may be linked to this measure. This rent-seeking behaviour will also arise when

managers derive private benefits from the size of their businesses or growth in the size

of their business units. In a competitive internal environment, higher expected returns

will increase the probability that the unit will be awarded the level of capital required to

support its planned activities.

The temptation for managers to misrepresent their private information arises because

risk capital is used for both planning and control within a bank – it serves as an ex-ante

measure of value and an ex-post measure of performance. The amount of capital

allocated to a business unit not only affects the amount of risk that can be taken but also

directly influences its potential for growth and competitive position. If information

asymmetries are large, it is possible that managers will allow their vested interest in

performance measures influence their reporting of risk. The scope for misrepresentation

is particularly high in banking book activities, where the fundamental opacity and

illiquidity of assets means information asymmetries may be significant. The same holds

for trading activities, where the centre will typically rely on information on market

opportunities and expected volatility when determining the size of trading activities and

exposure limits.

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It has been identified that if managers cannot be expected to reveal their private

information on expected risks in positions, then the centre may have little option other

than to use data based on the historical distribution of returns as the basis for

determining the expected risk in positions and for allocating capital against the

positions. However, as discussed, if historical volatility proves not to be an accurate

representation of current or future volatility, managers can exploit this information to

evade risk limits and select portfolios for which the true risk is greater than the

measured risk. If pricing capacity is aligned with the true risk in portfolios, these

investments should produce inflated risk-adjusted returns. In these circumstances the

concern for the centre is that the bank will be undercapitalised with respect to risk. If

current or expected volatility is lower than historical data, but capital is allocated on

the basis if historical data, then managers may be compelled to take additional

(hidden) risks in order to increase the probability of achieving the aspiration level for

the RORAC. Again, the bank may be undercapitalised with respect to risk.

In this chapter we develop a potential solution to these problems. The centrepiece of

this solution is an internal market for the allocation of risk capital, based on an auction

mechanism. Auction mechanisms have developed for the distribution of single or

multiple items when sellers have uncertain information regarding their value. The

core of the internal market examined in this chapter is a dynamic bidding process

under which portfolio or business unit managers in the bank are required to

periodically place bids for the capital needed to support their risky activities over a

specified planning period in the bank. Our goal is to have managers truthfully reveal

their private information regarding current and future volatility in their proposed

positions and the expected return on these positions. We draw on the concept of the

revelation principle (Myerson, 1979, 1981), which states that for any mechanism

where agents may be induced to be dishonest in equilibrium, there exists a direct,

incentive-compatible mechanism where agents can be induced to report their

information truthfully. The key to the revelation principle is the design of payoff

structures for agents such that it is in their interests to report truthfully – they cannot

gain by misrepresenting their information to the principal. The remainder of the

chapter proceeds as follows. In section 5.2 we examine which auction formats induce

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bidders to truthfully reveal their preferences. In section 5.3 we consider the

application of these formats to the distribution of multiple units of a divisible

commodity, as would apply to bank capital. In section 5.4 we examine the specific

design of the auction mechanism for allocating capital in the banking firm, and

demonstrate the truth-revealing properties of the mechanism. Section 5.5 examines a

number of structural issues related to the auction mechanism in the bank setting and

presents qualifications. Section 5.6 provides a summary of the main findings in the

chapter.

5.2 AUCTION FORMATS

An auction is defined as a market institution with an explicit set of rules determining

resource allocation and prices on the basis of bids from market participants.172 An

auction can generally be classified in terms of one of four basic structures: ascending-

price auction, descending-price auction, first-price sealed-bid auction and second-price

sealed-bid auction.

The ascending-price auction is a sequential bidding structure under which the standing

bid wins the item unless another higher bid is submitted. Bidders can submit more than

one bid, and they observe all previous bids. The descending-price auction is a

sequential bidding structure under which a standing price is gradually lowered,

typically by means of an exogenous counting device, until stopped by a bidder. The

first-price sealed-bid auction is a simultaneous bidding structure under which the

bidder who submits the highest bid is awarded the object at their bid price. The main

difference between the ascending-price auction and the first-price sealed-bid auction is

that bidders in the former can choose to revise their bids in light of information gained

by observing rival bids. Typically under the first-price sealed-bid auction each

participant can only submit one bid. The second-price sealed-bid auction is a

simultaneous bidding structure under which the bidder who submits the highest bid is

172 For recent surveys of auction theory, see McAfee and McMillan (1987), Milgrom (1989) and Janssen (2004).

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awarded the object, but pays the highest competing bid (the second highest bid or the

highest losing bid).173

A fundamental prescription for effective auction design, identified by Vickrey (1961) is

that an auction should be structured so that the price paid for the object by a winning

participant is as independent as possible of that player’s own bids. This is because there

will be less incentive for participants to shade their bids if the price that a winning

participant pays is determined by competitors’ bids alone. The incentive to shade bids

will arise when a participant fears that by winning the auction they will pay a price that

is above that of the market consensus for the object. The second-price sealed-bid

auction format meets this prescription – in the auction of a single good, participants

submit bids in written form, the highest bid wins, and the winner pays the second-

highest amount bid. It can be shown that the second-price sealed-bid auction structure

induces “truth-telling” for bidders with independent private values, that is, it is a

dominant strategy for each bidder to reveal the maximum price they are willing to pay

for the good.174. The key to this result is that the choice of bid determines only whether

or not a participant wins; the amount paid by the winner is beyond that participant’s

control.

A second prescription for effective auction design, identified by Milgrom and Weber

(1982) is that an auction should be structured in an open-fashion so that information is

revealed and can be maximised by participants at the time bids are placed. The

ascending-price auction format meets this prescription. Indeed, the ascending-price

auction format and the second price-sealed bid auction format are equivalent in a

number of respects. With independent private values, the dominant strategy in the

ascending auction is for a bidder to remain bidding until the price reaches the private

value placed on the item – the price at which the bidder is indifferent between

winning and not winning the auction. Given the second-highest bidder will withdraw

173 The second-price sealed-bid auction is also known as the ‘Vickrey auction’ after the economist who proposed it. See Vickrey (1961). 174 In the case of independent private values, each bidder has independent information that permits him to know with certainty the value he places on the object being sold. This value is not affected by information regarding the values placed on the object by other bidders. This is in contrast to the common-value case, where the object is typically purchased for resale. Here no individual bidder is certain of the value of the object, possessing only an estimate of its value.

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from the auction when her value is reached, the bidder with the highest value will win

the auction at the price equal to the value of the second-highest bidder. This is

equivalent to the design of the second-price sealed-bid auction, where the winning bid

pays the price of the second-highest (losing) bid.

The attraction of the second-price sealed-bid auction in the bank setting is that the

dominant strategy is for a participant to bid her true value, independent of the actions

of other participants in the auction. From the perspective of the centre of the bank, this

design allows for the private information of managers with respect to expected risk-

adjusted returns to be revealed to the centre, allowing for an optimal allocation of

capital based on the ‘true’ expected distribution of portfolio returns.

First, let us consider the truth-revealing properties of the second-price sealed-bid

auction. To illustrate that the dominant strategy in a second-price sealed-bid auction is

for a participant to bid true valuation, consider the case where a participant places a

bid that does not equal true valuation. Let vi be the true value of the item to a

participant i, let bi denote the bid that participant i considers making, and let hj be the

highest of all bids received other than that of participant i. Table 5.1 examines the

payoffs to participant i should they bid above the true value, vi.

Table 5.1

Second-Price Auction: Payoffs if Bid Above True Value

Scenario Payoff to participant i

bi > vi > hj Gain (vi - hj) regardless of bidding either bi or vI

bi > hj > vi Lose (hj – vi)

hj > bi > vi Zero (hj wins auction)

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If the participant bids truthfully and wins the auction, the payoff is the difference

between the participant’s value and the second-highest losing bid (ie (vi - hj)). If we

compare this to the payoffs should the participant bid above true value in Table 5.1, it is

clear that bidding above true value is not a dominant strategy. The participant cannot

improve upon bidding truthfully (maximum payoff remains at (vi - hj)), and at the same

time is exposed to a possible negative payoff should the second highest bid exceed the

participant’s true value. In this case raising the bid above value causes the participant to

win the auction, but at a price that exceeds the value of the object. The loss to the

participant is (hj - vi).

Now consider the case in Table 5.2 where the participant bids below true value vi. It is

again clear that this bidding strategy cannot improve upon bidding truthfully: the

maximum payoff of (vi - hi) matches that of bidding truthfully. In general no participant

has an incentive to bid less than true value because to do so would reduce the

probability of winning the auction without reducing the price that would be paid should

the participant place the highest bid.

Table 5.2:

Second-Price Auction: Payoffs if Bid Below True Value

Scenario Payoff to participant i

vi > bi > hj Gain (vi - hj) regardless of bidding either bi or vi

vi > hj > bi Zero (hj wins auction)

hj > vi > bi Zero (hj wins auction)

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The preceding analysis shows that a bid equal to a participant’s true value is a dominant

strategy for the second-price sealed-bid auction.175 The Nash equilibrium that arises

from all bidders playing their dominant strategy of truth-telling in the second-price

auction design represents a Pareto-efficient allocation because the winning (highest) bid

comes from the participant who values the item most highly. Klemperer (2004) likens

this auction format to that where an auctioneer maximises social surplus and sets prices

so that each participant’s net profits equal her contribution to social surplus – the

participant pays a price for those items she wins equal to her declared value for those

items less the total social surplus achieved by the allocation plus the social surplus that

the auctioneer could have achieved if that participant had not been present.176

In addition to truth-revealing properties, the second-price sealed-bid auction has two

other advantages over other auction formats. First, it has been shown that the second-

price sealed-bid auction replicates the outcome of the ascending bid auction with

small increments without the need to bring participants together.177 Second, the

second-price sealed-bid auction presents a simple bidding strategy to the participant,

which is rationally to bid the true value placed on the item, without the need to take

into consideration the number of participants in the auction or their valuations placed

on the item.178

175 Vickrey (1961) also showed that in the case of independent private values, the expected revenue from each type of the four auction formats (for a single good) is the same. This is known as the revenue-equivalence theorem. 176 See Klemperer (2004), p.63. 177 While this may be achieved with an online auction format, participants are still required to be present at the time the auction is being conducted. This is not the case with the second-price sealed-bid auction. 178 A key element of the second-price sealed-bid auction is the setting of a reserve price by the seller. Consider the case of the New Zealand government’s auction of electromagnetic spectrum bandwidth to companies in 1990 using the second-price sealed-bid design. McMillan (1994) describes that one company that bid NZ$100,000 for the spectrum ended up paying the second-highest bid of NZ$6. In another case a company that bid NZ$7 million paid the second-highest bid of NZ$5,000. These situations caused significant opportunity losses for the New Zealand government which received revenue from the auction that was NZ$214 million below forecast.

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5.3 MULTI-UNIT AUCTIONS

The discussion thus far has been largely restricted to auctions that involve a single,

indivisible item being sold at a one-time event. However, in the bank setting that is

the focus of this chapter it is necessary to consider a multi-unit auction framework, as

risk capital in a banking firm is divisible and distributed to business units on a

repeated, periodic basis. In this sense an auction mechanism for the allocation of risk

capital in a banking firm has many of the attributes of auctions for Treasury securities,

where discrete multiple-units are distributed at periodic tender. Recent interest in

multi-unit auctions can also be linked to the Federal Communications Commission

auctions for spectrum rights in the United States.179

The typical format for a multi-unit auction is bidders may submit more than one bid

for the divisible commodity. Participants submit a demand curve for the item, with

bids consisting of both prices and the desired number of units of the commodity at

each price level. In the ascending-price auction for multiple-units, the quantity bid for

at each price is compared to the amount offered. If the total value of bids submitted at

a given price is greater than the amount offered, the seller raises the price until the

amount bid for by the remaining participants no longer exceeds the amount offered.

Participants who remain until the last round receive the full award.

5.3.1 Problems in the Uniform Price Auction Format

Milgrom (2004) argues that the second-price sealed-bid auction can be readily

extended to sales of multiple identical items. However, to duplicate the result that

participants bid the highest prices they are willing to pay, Milgrom (2004) asserts that

the auction rule must award items at a uniform price equal to the highest rejected

bid.180 The basis of this argument is that in an ascending-bid auction where seven

identical items are offered for sale, the items will be awarded to the seven bidders

179 See Cramton (1998) and McMillan (1994). 180 A similar uniform pricing rule has been applied to the sale of Treasury securities in the United States. See Chari and Weber (1992) and Ausubel and Cramton (1998a).

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with the highest values for prices approximately equal to eighth highest value. For this

outcome to be replicated with the second-price sealed-bid auction, the seven winning

bids must pay the eighth bid.181 It must be noted, however, that the uniform price

auction, as described, does not share the desirable properties of the second-price

sealed-bid auction in situations where participants are interested in more than one unit

of the divisible item. Vickrey (1961) demonstrates that truthful demand revelation

will occur in a second-price sealed-bid auction for multiple items only when each

participant can submit one bid. Vickrey writes:

“…only in this way is it possible to ensure that each bidder will be

motivated to put in a bid at the full value of the article to himself, thus

assuring an optimal allocation of resources.”182

Allocative inefficiencies arise in uniform-price auctions for multiple units of a

divisible commodity because participants may carry strong incentives to reduce their

demand – participants may be able to increase their profits by either reducing the

price bid for each unit after the first (shading bids) or reducing the total number of

units demanded at or above any price. This arises because there is a positive

probability that bids on second or later units determine the price paid on other units

that are won – the consequences of a later bid influencing the price paid for an earlier

bid become greater the more units a participant wins. Given continuing to bid for two

or more units raises the price paid for the first unit, there is an incentive for the

participant to bid less than true value on later units in order to reduce the price paid on

earlier units. 183 In making the decision to restrict the number of units that are bid, a

participant must anticipate where prices are likely to go as a function of demand.

Allocative inefficiency results because high-value bidders have greater incentive to

shade their bids more than low-value bidders, leading to the possibility that low-value

bidders will inefficiently win units that have greater value placed on them by high-

value bidders. 181 Recall the outcome equivalence of the ascending-bid auction and the second-price sealed-bid auction discussed in section 5.2. 182 Vickrey (1961), p.26. 183 Ausubel and Cramton (1998b) present a general theory of demand reduction in multiple unit auctions. They show that if at least one bidder has downward-sloping demand, any Nash equilibrium is guaranteed to have bid reduction.

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An illustrative example showing the incentives for demand reduction in a uniform-

price auction follows. Consider a situation where six units of a divisible commodity

are to be auctioned using a sealed-bid uniform-price auction. There are three

participants – Bidders A, B and C – each of who are interested in four units of the

commodity. Table 5.3a shows the marginal valuations placed by each bidder – Bidder

A, for example, expresses a value (bid) of $15 for the first unit she acquires, $14 for

the second unit, $10 for the third unit, etc.

Table 5.3a

Uniform Price Auction: No Demand Reduction

Bid Bidder A Bidder B Bidder C

1 $15 $12 $13 2 $14 $7 $9 3 $10 $5 $6 4 $8 $3 $4

Price Bidder A Bidder B Bidder C Demand $4.01 4 3 3 10 $5.01 4 2 3 9 $6.01 4 2 2 8 $7.01 4 1 2 7 $8.01 3 1 2 6

Payment $24 $8 $16 $48 Value $39 $12 $22 $73 Gain $15 $4 $6 $25

In the example, the auctioneer opens the bids and determines that at a price of $4.01

for the commodity the total demand is for ten units. This consists of four units for

Bidder A, and three for Bidders B and C.184 At a price of $5.01, the third bid of

Bidder B is eliminated and total demand drops to nine units. This process continues

until a price of $8.01 is reached, at which point the demand for six units of the

commodity matches the available supply. The price of $8.01 is the highest losing bid

184 In the open ascending-price auction equivalent the bidding would start at $4 and the auctioneer would progressively increase the price until six units of demand for the commodity remained

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and thus represents the market-clearing price. At this price, Bidder A secures three

units of the commodity, B secures one unit and C secures two units. If we consider the

position of Bidder A, and ignore decimals, the total price paid for the three units that

have been secured is $24, while the total value placed on these units is $39. The gain

(consumer surplus) to bidder A is $15.

Now consider the same scenario, except that Bidder A shades her bids for the third

and fourth units to $6 each, down from $10 and $8. The results are presented in Table

5.3b. At the price of $4.01, total demand now equals eight units, consisting of two

units from Bidder A and three units for Bidders B and C. At the price of $5.01, Bidder

B drops one unit and the demand of seven exceeds supply by one unit. At the price of

$6.01, the demand of six units matches the available supply, and this becomes the

market-clearing price. Each participant secures two units of the commodity at the

price. The shading of bids three and four by Bidder A has resulted in a decline in the

market-clearing price from $8.01 to $6.01. The gain to Bidder A, ignoring decimals,

increases from $15 to $17, despite the fact that she secures two less items. If we

assume that there are no complementarities across the units for Bidder A, such that

acquiring less units does not impact on the utility derived from the secured units, then

this example shows that demand reduction (by shading bids) on the part of Bidder A

is advantageous.

We can observe from this example how demand reduction creates allocative

inefficiency in the sealed-bid uniform price auction. While six units have been

allocated across three participants, they have not been distributed in a manner that

reflects the true value placed on them by the participants. Specifically, Bidder B has

acquired one unit with a marginal value of $7 (bid 2) which has a marginal value of

$10 (bid 3) in the hands of Bidder A. As a result, demand reduction on the part of

Bidder A has resulted in a social inefficiency. Further, the total revenue in the hands

of the seller of the commodity has declined from $48 to $36 as a result of the shading

of bids by Bidder A.

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Table 5.3b

Uniform Price Auction: Demand Reduction

Bid Bidder A Bidder B Bidder C

1 $15 $12 $13 2 $14 $7 $9 3 $6 $5 $6 4 $6 $3 $4

Price Bidder A Bidder B Bidder C Demand $4.01 2 3 3 8 $5.01 2 2 3 7 $6.01 2 2 2 6

Payment $12 $12 $12 $36 Value $29 $19 $22 $70 Gain $17 $7 $10 $34

Table 5.3c

Uniform Price Auction: Demand Reduction

Bid Bidder A Bidder B Bidder C

1 $15 $12 $13 2 $14 $7 $9 3 $10 $5 $6 4 $3 $4

Price Bidder A Bidder B Bidder C Demand $4.01 3 3 3 9 $5.01 3 2 3 8 $6.01 3 2 2 8 $7.01 3 1 2 6

Payment $21 $7 $14 $42 Value $39 $12 $22 $73 Gain $18 $5 $8 $31

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Finally, consider the case where rather than shading bids, Bidder A reduces her

demand for the commodity by entirely removing her fourth bid.185 The results are

presented in Table 5.3c. At the price of $4.01, total demand now equals nine units,

consisting of three units for each participant. At the price of $7.01, the demand of six

units matches the available supply, and this becomes the market-clearing price. The

withdrawal of the fourth bid by Bidder A has resulted in a lower market-clearing price

of $7.01, from $8.01 without demand reduction. Under this scenario, the gain to

Bidder A, ignoring decimals, increases from $15 to $18, despite the fact that she

secures one less item.186

5.3.2 Incentive-Compatible Multi-Unit Auction Format

We have seen in 5.3.1 that while the uniform price auction format is similar in

concept to the second-price sealed-bid auction, the efficiency results that are

characteristic of the second-price auction are not preserved in the uniform price

auction format when participants can bid for more than one unit of the divisible

commodity. The source of inefficiencies is demand reduction on the part of auction

participants.

These inefficiencies can be eliminated if the pricing rule changes such that the price

paid for each unit equals the value of the bid that it displaces. This means the price

that is charged to each successful bidder for each unit is the opportunity cost of

assigning the unit to that bidder. A participant pays the amount of the highest rejected

bid (other than their own) for the first unit, the second-highest rejected bid (other than

their own) for the second unit, and so on, paying the nth highest rejected bid (other

185 Cramton (1998) cites direct evidence of demand reduction in the Federal Communications Commission narrowband spectrum auctions conducted in the United States between 1994 and 1996. PageNet cut back its bidding on three large licences to two on the basis that continuing to bid for three licences would drive prices on all large licences to disadvantageously-high levels (even though the auction price had not reached PageNet’s incremental marginal value for the third licence). Indirect evidence of demand reduction is indicated by the withdrawal from the MTA Broadband auction of the largest bidders (AT&T, WirelessCo and PrimeCo) on some units at prices that were well below plausible values of the spectrum licences. 186 This again assumes there are no complementarities across the units, such that Bidder A does not need four units to derive value from the three units that have been secured in the auction.

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than their own) for the nth object. This format embodies the fundamental prescription

of the second-price sealed-bid of Vickrey (1961), being that the price paid for a

winning bid is independent of the bid that secured the item. This means there should

be no incentives for misrepresentation by bidders, with the dominant strategy being to

bid for items as their marginal valuation. The result is a social optimum with respect

to the allocation of the units. Further, the incentive for demand reduction should be

eliminated when the payment for an item is decoupled from the bid. We will

demonstrate this shortly.

Let us return to the previous example comprising three participants – bidders A, B and

C – to demonstrate the operation of this auction mechanism. Each participant again

seeks four units of the divisible commodity to be sold, and six units are available for

distribution. Table 5.4 displays the marginal valuations placed on each unit by the

three participants, which are the same as in the previous example. The auction

proceeds as follows. The auctioneer allocates the six units of the commodity to the

highest bids – the result being Bidder A secures three units, B secures one unit and

Table 5.4

‘Vickrey’ Multi-Unit Auction

Bid Bidder A Bidder B Bidder C

1 $15* $12* $13* 2 $14* $7 $9* 3 $10* $5 $6 4 $8 $3 $4

Displaced Bids

Bidder A Bidder B Bidder C $7 $8 $8 $6 $7 $5 Total

Payment ($7+$6+$5) $8 ($8+$7) $41 Value $39 $12 $22 $73 Gain $21 $4 $7 $32

225

Bidder C secures two units. Recall that under the present auction prescription, the

winner pays an amount equal to the externality exerted on competing bidders, which

represents the value competitors could have realised had the winner not participated in

the auction. Referring to Table 5.4, consider the case of Bidder A. If Bidder A had not

been present in the auction, bids two and three of Bidder B ($7 + $5) and bid three of

Bidder C ($6) would have been allocated units in the auction. Consequently Bidder A

is required to pay $18 for the units that have been secured by her in the auction. These

units carried marginal values equal to $39, giving a total gain to Bidder A of $21.187

The participation of Bidder B in the auction displaced one bid, this being bid four of

Bidder A ($8). Consequently Bidder B is required to pay $8 for the unit secured in the

auction, representing a $4 gain on the $12 marginal valuation of this unit. Bidder C

secured two units in the auction, and had this participant not been present in the

auction, bid four of Bidder A ($8) and bid two of Bidder B ($7) would have been

allocated units in the auction. This means the payment required by Bidder C is $15,

representing a $7 gain on the $22 marginal value of the two units secured by C in the

auction.

It can be shown that these results will also be achieved under an open auction with

ascending bids.188 Under this format participants secure units sequentially as bidding

progresses, with the price paid for a unit being that which held at the time the unit was

secured. This process is continued until the supply of units available for distribution is

exhausted. The application of this auction format to the current example is shown in

Table 5.5. Again, six units of the divisible commodity are available for distribution.

At the price of $4.01, Bidder A requires four units, B requires three units and Bidder

C requires three units, giving total demand for ten units. At the price of $5.01, Bidder

B reduces her demand to two units. Now, at this price, the combined demand of

Bidders B and C is five units, and with six units available for distribution, Bidder A is

assured of securing at least one unit.189 The price of $5.01 is thus pivotal for Bidder A

187 Note that this exceeds the gain to Bidder A under the uniform price auction format in the cases of no demand reduction and demand reduction by this bidder. 188 See Ausubel (1998b). 189 The combined demand of Bidders A and C is seven units, so Bidder B is not assured of securing any units at the price of $5.01. Similarly, the combined demand of Bidders A and B is six units, so Bidder C is also not assured of securing any units at this price.

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securing one unit of the commodity, and represents the price to be paid for the said

unit. Next consider the outcome at the price of $6.01. Here Bidder C drops one bid,

leading to a total demand of eight units for the commodity. At this price the combined

demand for Bidders B and C is four units, and again with six units remaining for

distribution, Bidder A is assured of securing a second unit in the auction. As the price

of $6.01 is pivotal to securing the second unit, this is the price that Bidder A will pay

for the unit.190

Table 5.5

Ascending-Bid Multi-Unit Auction

Bidder A Bidder B Bidder C

1 $15 $12 $13 2 $14 $7 $9 3 $10 $5 $6 4 $8 $3 $4

Price Bidder A Bidder B Bidder C Demand Win $4.01 4 3 3 10 $5.01 4 2 3 9 A $6.01 4 2 2 8 A $7.01 4 1 2 7 A + C $8.01 3 1 2 6 B + C

Payment ($5+$6+$7) $8 ($7+$8) $41 Value $39 $12 $22 $73 Gain $21 $4 $7 $32

At the price of $7.01, total demand drops to seven units. The combined demand of

Bidders B and C has dropped by one unit to three units, again assuring Bidder A of

securing a third unit. Also, at this price, the combined demand of Bidders A and B has

dropped from six units to five units, assuring Bidder C of securing one unit.

Consequently the price of $7.01 is pivotal to Bidders A and C securing their third and

first units in the auction, representing the price they will pay for these units.

190 At the $6.01 price, the combined demand of Bidders A and C is six units, so Bidder B is not assured of securing any units in the auction. The combined demand of Bidders A and B is also six units, so Bidder C is also not assured on securing any units in the auction at this price.

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Finally at the price of $8.01, the total demand of six units matches the available

supply. The combined demand of Bidders A and C drops from six to five units,

assuring the Bidder B will secure one unit in the auction. Further, the combined

demand of Bidders A and B at this price drops from five units to four units, assuring

that Bidder C will secure one unit in the auction. This price is pivotal to Bidder B

securing her first unit and Bidder C securing her second unit in the auction. A

comparison of these results with those in Table 5.4 shows that the allocation of units

to the three participants is identical, and the prices paid are identical (ignoring

decimals). Thus the open ascending-bid auction produces the same allocation outcome

as the sealed-bid auction equivalent.

It remains to be shown that the auction formats demonstrated in Tables 5.4 and 5.5 do

not encourage demand reduction on the part of participants. Table 5.6 shows the

outcome for the open ascending-bid auction if Bidder A withdraws her fourth bid.

Having secured three units in the auction the gain to Bidder A is $21, which is the

same outcome for Bidder A without reducing demand. Consequently Bidder A has no

incentive to shade or withdraw bids. This outcome arises, as previously discussed, due

to the separation between the prices bid and the prices paid under the Vickrey and

ascending-bid auction formats. We can conclude that these auction formats are

incentive-compatible (the dominant strategy for participants is to truthfully bid their

marginal valuation for units) and allocative efficient (units are allocated to those

participants who value them the most highly). These auction formats thus meet our

prescription for the allocation of risk capital in the banking firm. It is to this we turn in

the next section of this chapter.

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Table 5.6

Ascending-Bid Multi-Unit Auction: Demand Reduction

Bidder A Bidder B Bidder C

1 $15 $12 $13 2 $14 $7 $9 3 $10 $5 $6 4 $3 $4

Price Bidder A Bidder B Bidder C Demand Win $4.01 3 3 3 9 $5.01 3 2 3 8 A + C $6.01 3 2 2 7 A + B $7.01 3 1 2 6 A + C

Payment ($5+$6+$7) $6 ($5+$7) $36 Value $39 $12 $22 $73 Gain $21 $6 $10 $37

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5.4 AN AUCTION MECHANISM FOR RISK CAPITAL

5.4.1 Overview

It has been demonstrated that the equilibrium strategy for participants in a Vickrey

multi-unit auction is to bid the true marginal value for each unit of the item under

consideration. This equilibrium strategy is a dominant strategy because it is optimal to

reveal the truth even if a bidder assigns a positive probability to other bidders

deviating from their from their equilibrium strategies. Risk-aversion on the part of

participants has no impact on the optimal bidding strategy because bidding more

aggressively, while increasing the probability of winning units in the auction, may

lead to a bidder paying above the marginal valuation for the item if the second highest

bid exceeds this level.191 Finally, the auction mechanism is allocatively efficient

because bidders with the highest marginal valuations will secure the items that are

available for distribution. This remainder of this chapter examines how the Vickrey

multi-unit auction format can be applied to achieve an optimal allocation of risk

capital in a banking firm.

The essential feature in the design of this ‘internal capital market’ is that the

compensation payment function for managers is influenced by the actual capital

utilised on a risky activity and its risk-adjusted return (RORAC). Bonuses are based

on the realised RORAC on a risky activity and the opportunity cost of assigning

capital to the risky activity. A Vickrey multi-unit auction is employed as the

mechanism to distribute capital among competing claims within the bank. This

mechanism induces truthful revelation of managers’ private expectations on current

and expected volatility because the size of their potential bonuses will be independent

of their bids in the auction on expected yields. We describe the workings of this

mechanism, it assumptions and qualifications in the sections that follow.

191 Milgrom (2004) p.122 shows this is not the case in a first-price auction, where a small increase in the bid of a participant slightly increases the probability of winning at the cost of slightly reducing the value of winning.

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5.4.2 Design The first consideration in the design of the auction is whether to employ a sealed-bid

(simultaneous) format or an ascending-bid (sequential) format. The ascending-bid

format provides participants with information (price discovery) through the process of

bidding, whereas the sealed-bid format does not. However, as shown in 5.3.2, both

formats provide truthful revelation of the marginal value of units under auction in a

private-value setting because the price paid for units are independent of a participant’s

individual bids. The sealed-bid format has a slight advantage to the extent that it

avoids the need to bring parties together. Rothkopf et al (1990) argue that a sealed-bid

format may not be desirable because the incentive to reveal true value is lost if this

information is relevant to subsequent transactions. The ascending-bid format reveals

only that the winning bidders are prepared to pay at least the amount bid, with the

upper-portion of the demand curve never being revealed. However this problem can

be overcome in the sealed-bid case if the centre chooses not to reveal the winning bids

across the bank. Our model assumes a sealed-bid format.

The truth-revealing properties of the Vickrey payment rule arise because each

successful bidder is essentially offered a ‘rebate’ that is of sufficient size as to remove

the incentive for bidders to misrepresent their respective valuations. The rebate

corresponds to the difference between the marginal valuation curve of the successful

bidder and the marginal valuation curve of the highest-losing bidder, had the latter

instead secured units in the auction. We use this notion of a ‘rebate’ in the

compensation payment function for managers in the bank model. Specifically, our

framework requires that managers receive a rebate based on the actual profits of the

activities for which they secured capital, the actual risk capital utilised by these

activities over the measurement period, and the highest rejected yield that was pivotal

in securing the capital for these activities in the auction. This requires that specific

positions/transactions/portfolios can be tracked to the bids and the capital allocated

against the bids. We elaborate below.

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The auction mechanism requires that managers in the bank periodically submit to the

centre a downward-sloping demand schedule detailing the required risk capital ‘units’

and the expected return per unit of risk capital. A risk capital ‘unit’ is a fixed dollar

amount, say $1 million.192 The centre subsequently ‘awards’ risk capital against those

bids presenting the highest expected risk-adjusted returns, within constraints set by the

overall operational and strategic plan of the bank.193

The rebate paid to managers in our framework is determined as follows:

pi = � [Kai (r i – ci)]

where pi is the rebate associated with a given activity i, Kai is the actual risk capital

utilised by the activity i, r i is the actual risk-adjusted return on capital (RORAC) on

position i, ci is the yield bid by a losing competitor at which Kai was secured, and ✁ is a

coefficient that determines the proportion of the surplus [Kai (r i – ci)] paid to managers.

The RORAC (r i) is based on capital utilised, rather than capital allocated.

5.4.3 Example

The following example demonstrates the workings of our internal capital market for

risk capital and truth-revealing properties of the allocation mechanism.

Suppose a manager estimates a risk capital requirement of $10 million to support a

$100 million credit portfolio. This risk capital is based on the private information of the

manager on the expected distribution of returns, and is unrelated to the historical

distribution of returns on similar portfolios. We also assume that the risk capital

requirement has been determined using the DSD of the portfolio, as discussed in

192 A downward sloping sealed-bid might comprise the following: Bid 1: 29%, Bid 2: 28% and Bid 3: 27%, where each bid corresponds to $1 million in capital and the yield represents the expected before-tax RORAC. 193 For example, a high-yielding loan portfolio may not be allocated funds because the bank is overexposed to this industry or customer segment, that is, the bank has a concentration of credit risk in this area.

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chapter three of this thesis. The expected earnings on the portfolio over the

measurement period are $2.8 million based on the difference between the interest

income on the portfolio and the interest expense charged to the portfolio, where the

latter is established using a matched-duration funds transfer pricing system.194 This will

be of significance in the model because it implies the expected earnings on the portfolio

are verifiable ex-ante.195 The expected risk-adjusted return on the portfolio is 28%:

RORAC = $2.8 million / $10 million = 28%

In our setup, the manager is required to bid for the risk capital to support the $100

million portfolio under a sealed-bid Vickrey multi-unit auction format, as described

above. The manager will submit a demand curve detailing the volume of capital

required (in specified units) and the expected return on each unit of capital. The

manager is aware that compensation at the end of the measurement period will be based

on the formula:

pi = � [Kai (r i – ci)]

Let us consider three bidding strategies faced by the manager:

1. Understate the expected risk capital requirement

This will act to increase the expected RORAC on the portfolio, thereby increasing

the probability that the manager secures the required funding for the portfolio. Let

us assume that under this option the manager bids for $9.5 million in capital at a

yield of 29.47%, based on expected portfolio earnings of $2.8 million. Recall that

the portfolio earnings are verifiable ex-ante.

194 This ignores complexities related to overhead cost allocations and taxes. The expected earnings thus relate to an interest margin on the credit portfolio. 195 This is because the pricing on the portfolio is established at origination and funds transfer prices are locked-in over the portfolio until the maturity of the assets that comprise the portfolio.

233

2. Bid truthfully on the expected risk capital requirement

Here the manager places a bid for $10 million in capital at a yield of 28%. This

represents the manager’s realistic estimate of the capital requirement for the

portfolio, based on private information on the expected distribution of returns. The

capital requirement is calculated using a DSD risk measure.

3. Overstate the expected risk capital requirement

This will act to reduce the expected RORAC on the portfolio. While shading the bid

in this manner reduces the probability of securing funds in the auction, it does

increase the potential compensation gain to the manager should funding be secured

at this bid. Let us assume that under this bidding strategy the manager bids for

$10.5 million in capital at a yield of 26.67%. This again is based on expected

portfolio earnings of $2.8 million.

Finally, let us assume that the manager estimates that competitors in the auction may

place bids ranging from 30% to 25%, in percentage point increments, with equal

probability. Based on this information, and the compensation payment function, what is

the optimal bidding strategy for the manager?

Table 5.7 shows the auction result and expected bonus to the manager under the first

case where the risk capital requirement is understated and a bid of 29.47% placed.196

First, note that any competing bid in excess of 29.47% means the manager loses the

auction. Next, any bid between 29.47% and 28% results in an expected negative rebate

which will act to reduce the total bonus pool payable to the manager. This arises

because the highest losing bid exceeds the actual yield that the manager can realistically

expect to achieve given her expectation that the actual utilised capital will be $10

196 The calculations in Tables 4.7 to 4.9 assume a coefficient of � = 1 in the bank compensation function. We do not attempt to derive a figure for the coefficient, but refer to the reader to the discussion in section 2.8.2 of chapter two.

234

million and the realised RORAC will be 28%.197 The negative rebate at a competitor

bid of 29% is $100,000, calculated using the compensation function as follows:

pi = � [Kai (r i – ci)] = [$10m (0.28 – 0.29)] = -$0.1m

A competitor bid of 28% results in the auction being won by the manager but the

expected rebate is zero because the actual RORAC that the manager will achieve on the

portfolio should her expectations prove correct matches the losing bid of 28%, this

being the benchmark upon which the rebate is based.

Table 5.7

Understate Expected Risk Capital Requirement

Bid Competitor bid

Auction result

Actual RORAC

Spread Payoff

29.47% 30.00% Lose 28.00%

29.47% 29.00% Win 28.00% -1.00% -$0.1m

29.47% 28.00% Win 28.00% 0.00%

29.47% 27.00% Win 28.00% 1.00% $0.1m

29.47% 26.00% Win 28.00% 2.00% $0.2m

29.47% 25.00% Win 28.00% 3.00% $0.3m

Expected payoff $0.083m

For any competitor bids below 28%, the manager ‘wins’ the auction and secures the

required funding, in concert with a positive rebate should her expectations on

distribution of portfolio returns prove correct. Table 5.7 shows that the expected rebate

payable to the manager where the bidding strategy is to understate the expected risk

capital requirement is $83,000.

197 The realised RORAC is calculated as the earnings of $2.8million on the actual capital utilised of $10 million.

235

Table 5.8 shows the auction result and expected bonus under the second case where the

manager places a bid that matches her truthful expectations on the expected distribution

of portfolio returns. In this case the bid is for $10 million in risk capital at a yield of

28%. Note now that if the expectations of the manager are correct and the capital

utilised over the period equals $10 million, there is no expected negative rebate in the

compensation payment function. Any competitor bid above 28% results in the auction

being lost by the manager, while bids below 28% result in the funds being secured.

Note further that while the lower bid of 28% reduces the probability of the manager

securing funds in the auction when compared to the alternative option of bidding

29.47%, the expected payoff to the manager is larger at $100,000. Consequently

bidding truthfully on expected risk capital requirements dominates the strategy of

understating risk capital requirements, even though the latter option carries a greater

probability of winning the auction.

Table 5.8

Truthful Bid on Expected Risk Capital Requirement

Bid Competitor bid

Auction result

Actual RORAC

Spread Payoff

28.00% 30.00% Lose 28.00%

28.00% 29.00% Lose 28.00%

28.00% 28.00% Tie 28.00% 0.00%

28.00% 27.00% Win 28.00% 1.00% $0.1m

28.00% 26.00% Win 28.00% 2.00% $0.2m

28.00% 25.00% Win 28.00% 3.00% $0.3m

Expected payoff $0.1m

236

Table 5.9 shows the auction result and expected bonus to the manager under the third

case where the risk capital requirement is overstated and a bid of 26.67% placed. The

manager will only secure funds if competing bids are below this level. Participants in

auctions may bid lower than their marginal value for the item under consideration in

order to secure a gain relative to the price of the item, should they be successful in the

auction. Our example shows that it is not in the best interests for the manager to bid

below the true expected yield of 28% because the probability of winning the auction is

reduced without any compensating gain in monetary payoff. This arises because the

rebate paid to managers in our design is based on the opportunity cost of assigning

capital to them, this being the spread between the actual yield on the portfolio and the

yield bid by unsuccessful competitors in the auction. The expected payoff to our

manager under the strategy of overstating the risk capital requirement for the portfolio

is $83,000, as shown in Table 5.9. This is the same expected rebate as in the first

option, which was to understate risk capital requirements in the auction.

Table 5.9

Overstate Expected Risk Capital Requirement

Bid Competitor bid

Auction result

Actual RORAC

Spread Payoff

26.67% 30.00% Lose 28.00%

26.67% 29.00% Lose 28.00%

26.67% 28.00% Lose 28.00%

26.67% 27.00% Lose 28.00%

26.67% 26.00% Win 28.00% 2.00% $0.2m

26.67% 25.00% Win 28.00% 3.00% $0.3m

Expected payoff $0.083m

237

Our example shows that the dominant strategy for the manager is to bid truthfully on

the expected risk capital requirement for the portfolio. Our design has achieved its

objectives. The highest expected rebate ($100,000) is associated with managers

truthfully revealing their expected capital requirements. Notice how the principles of

the Vickrey auction apply in our framework. All three strategies result in a maximum

payoff equivalent to (r i – ci). However the strategy of understating risk capital in order

to improve expected return is dominated because it exposes the bidder to a negative

rebate should the highest losing bid exceed the expected return of the bid in question

(that is, ci > r i). The strategy of bid shading is also dominated because it reduces the

probability of winning the auction without increasing the size of the rebate that would

be secured should the auction be won. Consequently there are no incentives to

misrepresent risk capital requirements. The auction mechanism supports allocative

efficiency because risk capital is allocated to those activities that are expected to

generate the highest RORAC for the bank.

5.5 QUALIFICATIONS

Our model representing an internal capital market for the allocation of risk capital

relies on a number of assumptions. We address each of these in this section and assess

their implications.

Capital utilisation versus capital allocation. The truth-revealing properties of the

auction mechanism require that capital utilised by a risky activity be incorporated into

the calculation of the rebate paid to managers. This requires that a bank has the

capability to mark-to-market its credit portfolios and other risky activities on a regular

basis, in order that the actual distribution of returns for an activity can be derived and

used to assess the capital utilised over the measurement period.

What are the implications for the bank if the capital utilised by a specific activity turns

out not to match the capital allocated in the auction?

238

While our model relies on capital utilised, it could be argued that allocated capital

should form the basis upon which performance is measured because capital is a costly

resource provided by bank owners and remunerating managers based on utilised

capital fails to take into account the capital actually contributed by owners. If, for

example, it turns out that the capital utilised over the period is lower than the capital

secured and invested at auction, then the actual RORAC will be higher for a given

level of earnings, and the rebate paid to managers in our model will be larger. At one

level, this could be justified as a suitable reward for managing or reducing volatility to

a lower level than expected, for a given level of net earnings. At another level,

however, it could be argued that managers should be penalised for utilising less

capital than allocated because the excess capital represents a resource that could have

been invested elsewhere in the bank.198

It is our position that the latter argument is only valid if managers are required to

outline to the centre their capital needs at the beginning of the period, but have no

incentive to truthfully reveal expected risks. In this case if managers have no

responsibility for underutilised capital, they may have an incentive to request more

capital than needed in order to ensure adequate funding for their proposed risky

activities. In this case meeting targets for RORAC at the unit or portfolio level may

not be sufficient at the bank entity level to ensure an adequate return to bank owners.

Further, if the centre adopts a top-down allocation process and RORAC targets are

based on capital utilised, this too can have the effect of eliminating the responsibility

of a manager or business unit to fully exploit the resources assigned to it on behalf of

the owners of the bank. In our model it is not a dominant strategy for a manager to bid

for more capital than needed – which would be the case if the manager expects that

capital utilised will be lower and the rebate larger. We have shown that the dominant

strategy in our model is to bid truthfully on the on the expected risk capital

requirement. Table 5.9 reveals that bidding for a larger amount of capital than

required reduces the probability of winning funds in the auction, and has a lower

expected rebate than bidding truthfully on risk capital.

198 This argument assumes that the capital available for investment capital is a scare resource, with competing demands for its use within the bank.

239

The preceding paragraph suggests that we are putting our faith in the truth-revealing

properties of the integrated auction mechanism and compensation payment function to

ensure that if capital utilised turns out the be less or more than expected, then this

must be due to factors outside the control of managers. It is for this reason that

managers are not be penalised for underutilised capital in our model. There is also a

strong supporting argument for this if we consider that penalising managers for

achieving lower volatility than expected may reduce their incentives to actively

reduce risk in their activities over the measurement period. Somewhat perversely,

managers could in fact be encouraged to take on extra risk in their activities should it

appear that actual volatility was going to be less than anticipated, in order to ensure

the capital utilised did not fall below the capital bid in the auction. This would have

the double impact of reducing the likelihood of a negative rebate and increasing

potential returns on risky activities.

Note that in the case where capital utilised turns out to be greater than capital bid in

the auction, an implicit penalty applies to the extent that the actual RORAC will be

lower when capital utilised is used in the RORAC denominator, and the rebate

consequently smaller. Our figures in Table 5.7 show that the rebate can be negative if

managers deliberately understate capital requirements to ensure funds are secured in

the auction.

Technical capabilities of managers. Our model assumes that managers have the

technical capacity to calculate their risk capital requirements, based on their private

information of expected risks. Specifically, this would require managers to derive the

expected distribution of returns and calculate the expected downside semi-deviation

(DSD) from the distribution of returns, incorporating the risk tolerance level (loss

threshold) set by the bank.199

199 The use of DSD is based on our recommendation of this risk measure in chapter two, given its alignment with the risk preference function of the centre of the bank.

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Expected income is verifiable. Our model assumes that the expected earnings on a

risky activity for which capital is being bid are verifiable and thus cannot be

misrepresented by managers. The mechanism is designed to incentivise managers to

bid accurately with respect to their expected risk capital requirements, and this is

achieved by bidding on the basis of expected yield. If the expected income is not

verifiable, then managers can misrepresent their capital requirements by adjusting the

expected earnings on the activity. The requirement that managers stipulate in their

bids the capital required and the expected yield, they are committing to a given level

of earnings, which can be verified by the centre if required. Put differently, the

verifiable earnings and bid yield commit the manager to an expected risk capital

requirement.

For expected earnings to be verifiable, it is necessary that the interest rate or fee

income on the activity be traceable, and for this the bank must employ a funds transfer

pricing system which acts to lock-in an interest margin on a funded position over its

expected duration using a matched-duration marginal cost of funds. This system

insulates managers against risks that are beyond their control – for example, the

performance of a credit portfolio manager is based on the credit spread on the

portfolio and is insulated from changes in margins associated with interest rate risk.

Thus if a manager writes a $100 million five-year duration loan at an interest rate of

15%, and the matched-duration transfer price is 12%, then the expected income on the

loan is based on the 3% credit spread, irrespective of changes in funding costs. The

same holds if the loan is written at an interest rate that is linked to a short-duration

market benchmark, such as the ninety day bank bill rate. It is through the funds

transfer pricing system that risks can be segregated and expected earnings verified.200

200 See Ford (1998).

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Size of bid for capital. It is a requirement that the risk capital available for allocation

is divisible into common-size bid units so that a demand schedule can be submitted

detailing the number of units required and the yield per unit. The appropriate size of

units will be related to the size of the institution, the nature of its business, and/or

whether capital is allocated on a bottom-up basis (transaction or portfolio level) or a

top-down basis (business unit level). The latter would appear too broad given the need

to incentivise managers to reveal their specialised knowledge regarding expected

opportunities and their risks. Further, the larger the minimum bid unit, the more

difficult it would be for the centre to track capital utilised to capital allocated at

auction.

Frequency of auctions. The risk capital allocation process should be sufficiently

flexible to ensure that business units have the risk capital necessary to support their

day-to-day operations. Without this flexibility it is conceivable that a business unit

would be placed in a situation where it had to reject valuable business opportunities

because it could not be guaranteed of securing risk capital to support the business in

the next ‘scheduled’ auction. From this perspective, the auction mechanism for

allocating risk capital should perhaps only be considered when there are large

competing demands for capital at the beginning of the planning period. While the

mechanism has attractive truth-revealing properties and is proven to be allocatively

efficient, too rigid an application may result in costs for the bank that reduce the

benefits.

Term of capital. Our model assumes a one-year performance measurement period,

and this calibrates with the regulatory solvency standard for determining minimum

bank capital requirements, which is based on a one-year default rate.201 Implicit in this

approach is the assumption that a bank can costlessly recapitalise should actual

unexpected losses on a risky activity turn out to significantly exceed the capital that

has been allocated against the activity. If this assumption can be challenged, then a

case may be established for requiring longer-term positions to carry a larger capital

charge.

201 See Jackson, et al (2002).

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If longer facilities require a larger capital charge than equivalent shorter facilities,

then bidding on the basis of annual yield would not be an appropriate basis for

allocating scarce capital in the auction, without adjustments being made to recognise

that yields on longer activities will be biased downwards. Rather than viewing capital

as an insurance policy subscribed at origination and remaining linked with the facility

until maturity, our approach views capital as insurance against that fraction of risk

that is unavoidable only in the short-run. This implies that capital management is an

ongoing and proactive process, where corrective action can be taken to reduce risks or

limit losses whenever capital utilisation exceeds capital allocated against a risky

activity.

Diversification considerations. One implication of our model is that the allocation

mechanism relies on the notion that some managers/business units will ‘miss out’

when it comes to distributing surplus risk capital. In a practical setting this may not be

desirable, particularly if some activities that do not acquire the required capital in the

auction provide diversification benefits or other diversification benefits across the

banking entity. A mortgage insurance business line, for example, may not generate

high returns relative to other businesses when measured on a stand-alone basis. The

business may, however, provide a gateway to new mortgage business or provide other

benefits across the organisation. A problem arises to the extent that our model only

allows for managers of such business lines to bid for capital on the basis of expected

yields on the stand-alone business. Further, line managers may not be aware that their

businesses provide diversification or other benefits across the organisation, if this

information is only observable at the centre. If expected returns on the stand-alone

business are low, but the business provides synergistic benefits across the

organisation, then the centre itself may need to make ex-post adjustments to bid yields

to ensure these benefits are preserved at the entity level.

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Changes in risk attitude. While the dominant strategy in our model is for managers to

bid truthfully on the expected risk capital requirements, the model cannot prevent

managers changing their risk attitude in relation to ex-post perceptions on

performance relative to the aspiration levels. If a position deteriorates over the

measurement period and managers fear that aspiration levels for bonus will not be

reached, the possibility remains that managers may increase risk-taking in order to

increase the likelihood that performance will achieve target. Additionally, managers

who fear that not securing the required funding in the auction will prevent them from

expanding their businesses may still understate their expected risk capital

requirements (overstate expected yield) in order to increase the probability of securing

funds in the auction. While this is not a dominant strategy in the sense that negative

rebates are likely under this scenario, these managers may perceive that higher ex-post

risk-taking will act to increase potential returns and subvert the potential negative

rebate. The potential for post-contractual moral hazard such as described indicates

that our internal capital market for risk capital does not eliminate the need for

monitoring of the activities of managers at the centre of the bank.

Repeated versus one-shot games. In a game-theoretic setting, the allocation

mechanism developed in this chapter represents an infinitely-repeated game in the

sense that auctions for risk capital are conducted periodically and in line with the

relevant planning period within the bank. Repeated games allow for players to

observe the outcomes of previous games before playing later games, and thus

condition their optimal actions based on what other players have done in the past. In

the current context, however, the outcomes of previous auctions should not result in

the optimal bidding strategy for managers to diverge from that of bidding the truth

with respect to their expected risk capital requirements. This arises because the risk

profile of investment opportunities available to managers within the bank are

independent – that is, while managers are competing against each other internally for

a limited supply of capital from the centre, their investment opportunity sets are not

correlated and in the sense that that they are competing for the same external business.

The private information of managers with respect to the expected risks in their

investment opportunity sets is unique to the particular market (business type or

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region) for which the manager is responsible, and in this sense, the previous bids of

managers in other divisions of the bank should not cause the optimal bidding strategy

to change. To bid otherwise, as demonstrated, reduces the expected payoff to

managers under the compensation system developed in this chapter. Further, we have

argued that a manager’s private information regarding future investment opportunities

and risks should be independent of historical portfolio variance. This means that a

manager’s bid in the current period should be unrelated to his bid in the previous

period, and it is thus unlikely that a reputational effect on the part of the bidding

behaviour of a manager will cause the centre to change its decision rule. The optimal

bidding strategy in the current period should not be influenced by past bidding

behaviour unless it can be demonstrated that future portfolio risks are correlated to

historical portfolio risks.

5.6 CHAPTER SUMMARY

The main findings of this chapter are summarised as follows:

1. When there are competing demands for capital within a banking firm, managers

have an incentive to misrepresent expected risks in order to favourably influence

the capital allocation decisions of the centre – for a given position or activity

managers may understate expected risks or overstate expected profits in order to

inflate the expected RORAC and increase the size of any bonus that may be

linked to this measure. The result is capital may not be deployed to its most

valuable uses within the bank.

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2. An auction mechanism based on the second-price sealed-bid format can be

employed for the allocation of risk capital based on bids from managers. The

attraction of this format in the bank setting is that the dominant strategy is for a

participant to bid her true value, independent of the actions of other participants

in the auction. From the perspective of the centre of the bank, this design allows

for the private information of managers with respect to expected risk-adjusted

returns to be revealed to the centre, allowing for an optimal allocation of capital

based on the ‘true’ expected distribution of portfolio returns.

3. The pricing rule in the auction requires that the price paid for each unit equals

the value of the bid that it displaces. This means the price that is charged to each

successful bidder for each unit is the opportunity cost of assigning the unit to

that bidder. This overcomes the allocative inefficiencies associated with a

uniform pricing rule where participants have incentives to reduce demand on

subsequent bids in order to increase potential gains in the auction.

4. In order that managers truthfully reveal their expectations on future risks, it is a

requirement that the auction mechanism be linked to the compensation payment

function in the bank. The function, in turn, is based on the actual capital utilised

on a risky activity and its risk-adjusted return (RORAC). Bonuses are based on a

rebate linked to the realised RORAC and the opportunity cost of assigning

capital to the risky activity.

5. The internal market mechanism for allocating capital requires that capital

utilisation can be measured, expected earnings are verifiable and managers are

technically capable of calculating risk capital requirements based on their

private expectations on the distribution of returns.

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Chapter Six

Whole of Bank Perspective:

Dynamics of Target Credit Rating, Hurdle Rates and

the Pricing of Bank Assets

“If a car has three wheels and you add a fourth – now that is

synergy. But if you add two more, all you get is an extra

expense”

Harold Geneen, 1997

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6.1 INTRODUCTION

There are essentially two types of capital that banking firms must manage and

optimise: regulatory capital and economic capital. Regulatory capital is the equity

base that a bank must hold in order to satisfy minimum regulatory requirements.

Economic capital is the equity base that a bank should hold in order to meet its

desired external credit rating, which is in turn linked to the target solvency standard of

the bank.202 While regulatory capital is determined according to approaches and

methodologies set by regulatory authorities, economic capital is determined by banks

using their own data and internally-derived models. Economic capital thus should

embody the unique risk and cash flow characteristics of individual banks, and this,

combined with the solvency standard targeted by banks, implies that economic capital

may diverge significantly from the regulatory capital minimum for a bank.

We have seen that internal credit risk models developed by banks are based on the

concepts of ‘expected loss’ and ‘unexpected loss.’ Expected losses represent the

amount a bank expects to lose, on average, over a given time frame, and these are

typically provisioned for as a charge against bank revenues. Unexpected losses are

measured as the volatility of actual losses around the expected loss rate. Economic

capital is calculated as a multiple of unexpected losses, and depends on the loss

distribution and the choice of confidence interval that is consistent with the target

credit rating of the bank on its debt.

This chapter examines how the target credit rating (solvency standard) of a bank

impacts on the pricing of its assets. To achieve this we develop a loan pricing model

for a bank that incorporates the concept that a bank prices its assets in order to return a

minimum return on economic capital. We establish that as a bank increases its

economic capital relative to total assets, two opposing forces act to influence the

minimum rate at which it can price its assets, while maintaining a target hurdle return

202 We argued in chapter two of this thesis that this perspective for measuring capital may not reflect the risk preferences of all bank investors. We also demonstrated that risk measures based on the probability of default, when used internally, may lead to non-optimal portfolio selection on the basis of credit portfolio managers.

248

on equity. A larger equity base increases the after-tax net income that the bank must

earn in order to maintain the hurdle rate on equity. This higher net income results in a

higher lending rate on the bank’s assets, holding other factors constant. Offset against

this is the impact of a higher equity base on the external credit rating on the bank’s

publicly-rated debt securities. A more highly-capitalised bank should achieve a higher

rating on its public debt, all else equal, which in turn should reduce the credit spread

(margin above Treasury bond) that the bank pays on its debt.203 The resulting lower

cost of funds will act to reduce the minimum rate at which the bank must price its

loans in order to achieve the target return on equity.

We take a whole-of-bank perspective in this chapter to show that it is not possible to

separate pricing decisions on credit portfolios from decisions made by the centre of

the bank with respect to target credit rating and minimum hurdle rates. This is

essential to understanding the dynamics of the risk-adjusted performance

measurement framework of a bank, and questions the relevance of bonus-linked

compensation systems in banks when a significant proportion of factors affecting the

performance of credit portfolios are beyond the control of portfolio managers. In

addition to issues of relevance, it has been a consistent theme in this thesis that such

systems may provide considerable unintended outcomes, particularly when managers

have a propensity to increase risk-taking in order to achieve aspiration levels.

We model the opposing forces of changes in credit rating using Standard and Poor’s

credit loss data (expected default frequencies) for borrowers of different credit ratings.

Our base case begins with a bank that has achieved a BBB credit rating on its senior

debt. We then quantify the impact on pricing of loans and the cost of funds for the

bank increasing its target credit rating through the spectrum of A, AA and AAA. We

begin with the assumption of a uniform hurdle rate for the return on target equity of

the bank, and subsequently vary the hurdle rate in line with changes in the leverage of

the bank. We establish that for a bank to gain from a higher credit rating, the

reduction in funding costs associated with a higher credit rating must be greater than

the increase in the price of loans arising from a higher target capital base. This in turn

203 This is supported by empirical data on bank credit spreads, which we present later in this chapter.

249

depends on the proportion of the bank funding book that is sensitive to changes in

credit rating. While debt issued in capital markets may be sensitive to credit rating,

this may not be the case for retail deposits (except in circumstances where changes in

credit rating arise from severe financial distress in the bank). That is, if retail

depositors are insensitive to credit rating, a change in credit rating from, say BBB to

A, is unlikely to result in an increase in new retail deposits into the bank or existing

depositors revising their expected risk premium and accepting a lower interest rate on

their deposits. Indeed, we show that capital market evidence suggests retail depositors

are unaware of the specific credit rating of a bank, or the implications of a change in

rating. This may be driven by (possibly ill-conceived) perceptions that central banks

would intervene and provide support to a bank with a high proportion of retail

deposits, in the event of a temporary liquidity crisis.

We established in chapter four that a uniform hurdle rate was not consistent with a

fixed solvency standard.204 We will see in this chapter that assumptions regarding the

hurdle rate, and in particular, whether it should adjust to reflect changes in leverage,

are critical to determining the optimal credit rating for a bank. This in turn impacts on

pricing decisions and the market value of credit portfolios, and consequently, the

performance of credit portfolio managers.

The chapter proceeds as follows. In section 6.2 we develop our loan pricing model

based on a target return on economic capital. In section 6.3 we explain the loss

distribution used to determine the capital multiplier for the purposes of measuring

economic capital against loan portfolios by risk class. Section 6.4 presents our

hypothesis regarding the trade-off between loan prices and funding costs arising from

the bank’s choice of credit rating. Section 6.5 presents and discusses our results.

Section 6.6 presents limitations and possible extensions to the framework developed

in the chapter. Section 6.7 provides concluding comments and section 6.8 summarises

the main findings of the chapter.

204 Using a Merton model of default, we demonstrated that keeping the probability of default for a bank constant was inconsistent with a uniform hurdle rate when bank asset volatility varied

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6.2 LOAN PRICING MODEL

6.2.1 Overview

Our loan pricing model is built on the assumption that the amount of economic capital

held by banks is a function of their target insolvency rate (the probability that losses

will exceed a certain threshold), which is linked to an implied credit rating. Table 6.1

shows one and ten year historical default probabilities, in basis points, that apply to

various target Standard and Poor’s credit ratings.

Table 6.1

Historical Default Probabilities 1991-2000: All Countries Source: Standard and Poor’s Risk Solutions

Target Standard & Poor’s Credit Rating

One Year Default Probabilities

(basis points)

Ten Year Default Probabilities

(basis points)

AAA 0 13 AA 3 67 A 4 73

BBB 20 325 BB 83 1378 B 606 3045

CCC 2707 5573

The interpretation of the data in the table is as follows. A bank with a BBB credit rating

is deemed to be holding capital sufficient to have a 99.80% probability of not defaulting

on its rated-debt over the ensuing 12 month period (1 – 0.20%).205 Using a ratings-

based credit risk model, Jackson et al (2002) assess the solvency standard for banks

(survival probability) implied by the 1988 Basel Accord, and find that the one-year

confidence interval implicit in the minimum capital requirements of the Accord to be

the equivalent of a survival threshold of between 99.0% and 99.9%. This threshold is

equivalent to the upper end of the BBB rating category. Notably, the credit rating range

implied by the Basel Accord requirement is consistent with requirements under the

proposed new Accord (Basel II), given the declaration by the Basel Committee that the

205 This corresponds to a one in 454 chance of default.

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current minimum level of bank capital in the financial system should be maintained

under any new system (Bank for International Settlements, 2001).

External credit ratings reflect, among other factors, the actual capital held by a bank,

rather than the regulatory minimum. If banks calibrate their internal models to

determine the economic capital that is consistent with a specific credit rating, then the

credit ratings that banks obtain from external agencies provide an indication as to their

internal solvency targets. In Australia, the top four banks maintain sufficient economic

capital to achieve a target credit rating of AA, which is equivalent to a one-year,

99.97% probability of solvency, which is considerably greater than the solvency

standard implicit in the Basel Accord.206 Using a sample of 251 internationally-active

banks, Jackson et al (2002) suggest that banks in their sample are targeting confidence

levels of around 99.96%.

Our model begins with the premise that the lending bank establishes its desired credit

rating, determines its capital requirements in accordance with this rating, and assigns

capital against its loans. The minimum level of economic capital is measured as a

multiple of the standard deviation of expected loss rates for the particular loan class,

net of any diversification benefits attributed to the loan.207 The expected loss rate is

measured as the product of the expected default frequency (EDF), loss in event of

default (LGD) and the potential size of exposure at default. The multiple is in turn a

function of the target credit rating. Matten (2000) estimates that a one-tailed 99.97%

confidence interval for a skewed-distribution credit portfolio lies somewhere between

eight and twelve standard deviations from the mean.208 This means a loan carrying a

$10 million standard deviation around the expected loss rate would require economic

capital of between $80 million and $120 million, subject to the skew of the

distribution of loss rates, for a target credit rating of AA. The required net income on

the loan, after tax, forms the basis of pricing the loan, and this is determined by the

206 Commonwealth Bank of Australia (2005), Annual Report, p.6. 207 Note the economic capital allocated to the loan may also incorporate operational risk and market risk attributed to the position. The dynamics of pricing these risks into a loan portfolio are yet to be fully explored in the literature, although Sundmacher (2004) provides an overview of the issues as they apply to operational risk. 208 Matten (2000), p.202. In section 3 of this paper we estimate the relevant capital multiplier for each asset class using a beta distribution.

252

economic capital allocated to the loan and the minimum hurdle rate on capital

established by the bank.

6.2.2 Model

Let L represent the principal on a loan facility. Our objective is to determine the

minimum interest rate (ra) on this loan that allows the bank to achieve its hurdle rate

on economic capital. The marginal balance sheet of the bank that arises from the

funding of this loan is as follows:

L + S = EC + Dr + Dd

where S are liquid securities deemed necessary to support the loan, EC is the

economic capital allocated to the loan, Dr are retail deposits issued by the bank and Dd

are debt securities issued by the bank. Liquid securities are calculated as a fixed

percentage (� ) of the loan:

S = � L

The economic capital (EC) required to support the loan is the product of the standard

deviation of the expected loss rate (UL), commonly known as the unexpected loss

rate, and a capital multiplier (CM):

EC = UL x CM

The capital multiplier reflects the target risk tolerance level of the bank and the shape

of the distribution of loan losses.209

209 The derivation of the capital multiplier is explained in section 6.3.

253

The expected losses on the loan (EL) are the product of the expected default

frequency (EDF), the loss given default (LGD) and the expected credit exposure at the

time of default (L)210:

EL = EDF x LGD x L

If the expected credit exposure (L) and the LGD are considered fixed factors, then

unexpected losses (UL) can be calculated as follows:211

UL = � = )( ELLGDEL ✁

The key to the pricing model is the bank must price the loan to earn the target profit

(TP) and cover interest expenses, operating costs, expected losses and taxes. The

target before-tax net income for the bank (TP) on the loan facility is the product of the

economic capital allocated to the loan and the target hurdle rate (rh), adjusted for taxes

where t is the tax rate paid on bank profits212:

TP = (rh EC) / (1 – t)

Interest expense on deposits (ID) depends on the proportion of total deposits that are

retail deposits (pr), the marginal cost of retail deposits (rr) and the marginal cost of

debt securities (rd) at the credit rating on the debt securities213:

ID = (L – S – EC) [(pr rr) + ((1 - pr) rd)]

210 We estimate the potential credit exposure at default to equal the face value of the loan. 211 Matten (2000) p.193 states that most credit risk models use this measure, which assumes that default on a loan is a single binomial event. A more sophisticated approach would incorporate volatility in the LGD. 212 The tax adjustment 1/(1 – tc) applies in this case given our one-period pricing assumption. This adjustment is widely used in finance and works for perpetuities and one-period cases. 213 Given the whole-of-bank perspective taken in this chapter, the pricing model does incorporate funds transfer prices for the retail deposits. We assume that the duration of retail deposits and debt securities matches that of the loan facility, implying that there is no interest rate risk margin to be factored-into the price of the loan.

254

Operating costs (OC) for the bank are calculated as a percentage (c) of the face value

of the loan:214

OC = c L

Interest income on the incremental liquid securities (IS) required to support the loan is

based on an earnings yield of rs:

IS = S rs

The required rate of return on the loan (ra), being the interest rate on the loan that

achieves the target hurdle rate on economic capital, is expressed as follows:

ra = [ TP + EL + OC + ID – IS ] / L

Our model says that the interest rate on a loan should be driven by the minimum net

income required on the loan, which is determined by economic capital multiplied by the

return on target equity. In order to derive the interest rate on the loan, we take the

minimum net income and add back taxes, operating costs, provisions for expected

losses, and the interest expense on the debt/deposit component required to fund the

loan. We subtract earnings on incremental liquids and other securities deemed

necessary to support the loan and maintain the target credit rating. The non-equity

funding component is measured as the difference between the assets arising from the

position (loan plus supporting liquid securities) and the economic capital invested in the

assets. We account for the proportion of funding arising from retail deposits and debt

securities issued in the capital market.

214 We do not distinguish between direct and indirect costs and assume the bank apportions its indirect costs using an acceptable activity-based methodology.

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6.3 WHAT CAPITAL MULTIPLIER?

The amount of economic capital needed to achieve the target credit rating of the bank

is the difference between expected losses and unexpected losses at a confidence level

commensurate with the target credit rating. With knowledge of the shape of the loss

distribution on the loans of particular risk class, the capital multiplier (CM) represents

an estimate of the distance between expected losses and the cut-off point (x) of the

distribution, determined by the selected confidence interval, and expressed as a

multiple of the standard deviation of losses.

The choice of the probability distribution for a loan portfolio is fundamental to

estimating the economic capital that is required to support the portfolio, given the

focus on the tail of the distribution. This reflects skewness in the distribution of loan

returns, with upside potential limited to increases in market value associated with

migration to a higher credit rating, and downside incorporating the possibility of large

losses with low probability. The need to capture the potential for extreme losses in the

tail of the loss distribution is critical in determining the relevant capital multiplier for

which the volatility estimate is made. The loss distribution, in turn, is dependent on

the composition of the loan portfolio. Factors that contribute to the fatness of the tail

region of the distribution are the risk ratings of the assets within the portfolio, the size

of the exposures and the covariance across the assets within the portfolio.

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6.3.1 Beta Distribution

We use the beta distribution for the loan loss distribution in order to estimate the

capital multipliers for unexpected losses.215 The density function of the beta

distribution requires two constants, � and ✁

, where � controls the steepness of the

distribution and ✁

controls the fatness of the tail.216 We specify the mean and variance

for each loan portfolio in our model based on empirical data on expected default

frequencies and recovery rates for loans of various credit ratings. These inputs are

used to determine the relevant shape parameters. The steepness of the distribution (�)

is determined as follows:

� = [(µ2 (1 – µ)) / ✂2] - µ

where µ is the portfolio mean and ✂ = the portfolio standard deviation.217 The fatness

of the tail of the distribution is determined as follows:

✁ = [(µ (1 – µ)2 ) / ✂2] + (µ + 1)

The cumulative distribution function of the beta distribution, denoted pbeta (x, �, ✁

),

is the probability that a quantity which follows a beta distribution with parameters �

and ✁

will be less than or equal to x. We use the BETADIST function on EXCEL™ to

determine the value for x that gives the desired cumulative density function. For

example, the 99.97% confidence level that would apply for an AA target credit rating

for the bank requires that

pbeta (xmax, �, ✁

) = 99.97%

215 The model is calibrated by the beta distribution in accordance with empirical works that show the loss given default probability distribution is skewed to the right, viz. Altman and Kishore (1996), Carty and Lieberman (1996), Duffie and Singleton (1996) and Castle, Keisman and Yang (2000). Further, portfolio credit risk models, including KMV Portfolio Manager, Portfolio Risk Tracker and CreditMetrics, assume a beta distribution (Servigny and Renault, 2004). The beta distribution is also amenable to mathematical tractability. 216 See Ong (1999). 217 The expressions for alpha and beta are derived from Ong (1999), p.166-7.

257

The multiple of unexpected losses (CM), being the relevant number of standard

deviations, is calculated as follows:

CM = [(xmax -EL) / �]

Our calculations for capital multipliers for each class of borrower (rated BB, BBB and

A) and at each bank target credit rating (BBB, A, AA and AAA) are presented in

Table 6.2.

Table 6.2

Capital Multipliers for Unexpected Losses (Fit to Beta Distribution)

Borrower credit rating

Asset profile BB BBB A

Expected default frequency (EDF) 0.83% 0.20% 0.04%

Loss given default (LGD) 39.00% 33.00% 24.00%

Expected loss (EL) 0.32% 0.07% 0.01%

Unexpected loss (UL) 3.54% 1.47% 0.48%

Target rating of bank Capital multipliers

BBB 5.462 2.675 0.001

A 11.694 13.928 7.690

AA 12.711 16.573 11.441

AAA 22.801 46.078 95.833

258

Table 6.2 can be interpreted as follows. Consider the case of the exposure of a bank

rated BBB to a borrower that has been assigned an internal credit rating equivalent to

BB. Under the assumption of a beta distribution, the capital multiplier for this loan is

equivalent to 5.462 standard deviations. Given an unexpected loss rate for the loan of

3.54%, the economic capital requirement for the loan is determined as follows:

EC = UL x CM = 0.0354 x 5.462 = 0.1933

This implies that capital equal to 19.33% of the loan principal is required to support

the loan.218 Supporting calculations for the capital multiplier in this example are

provided in Appendix 15.

Note that as the bank increases its target credit rating from BBB to A the capital

multiplier increases to 11.694 standard deviations. This indicates that the economic

capital required to cushion the bank against unexpected outcomes increases

significantly as the confidence level increases from 99.80% to 99.96% in moving

from a credit rating of BBB to A. Further, if the bank desires a credit rating of AA, the

capital multiplier is 12.711 standard deviations, while the multiplier for an AAA-rated

bank against a BB-rated loan is 22.801 standard deviations. The range that applies to

the capital multipliers for a loan to an A-rated borrower is significantly more

pronounced – from 0.001 standard deviations for a BBB-rated bank to 95.833

standard deviations for an AAA-rated bank. While this range is substantially wider,

this needs to be taken within the context that the expected losses and unexpected

losses that apply to an A-rated credit exposure are substantially lower than for the BB-

rated credit exposure, as shown in Table 6.2.

218 This ignores any diversification benefits that the loan may contribute to a portfolio or the bank in general, and this represents a stand-alone capital allocation.

259

6.4 DYNAMICS OF THE TARGET CREDIT RATING

We now turn to our main proposition. If a bank wants to increases its external credit

rating, the economic capital that it must hold, for a given level of asset risk, will

increase. Specifically, a higher external credit rating requires a larger multiple of the

standard deviation of the expected losses on the loan class. Our model will show that,

all else equal, the interest rate on the loan will increase given the larger economic

capital base and a fixed target return on equity. However, all else is not equal. One of

the main objectives in targeting a higher credit rating will be the expectation by the

bank of a lower cost of funds on its debt securities. This means an increase in the

solvency standard for a bank has opposing effects on bank loan prices: a larger

economic capital allocation has an upward effect on loan rates in a given risk class, but

this is offset by the downward impact on loan rates arising from a lower cost of funds

on its rated-debt securities. An additional potential downward impact on the loan price

will arise if there are grounds for the centre to adjust the hurdle rate on economic

capital. This will arise if the centre believes that the increase in economic capital

relative to a constant bank asset risk will result in bank investors accepting a lower

return on their investment for a reduction in bank leverage.

Jackson et al (2002) examine why banks may target a solvency standard that is more

conservative than that implicit in the Basel Accord. They find that a major

determinant of the target credit rating is cost-effective access to unsecured credit

markets, given interbank rates and counterparty credit limits are highly sensitive to

credit rating.219 In particular, they find that banks that engage in significant swap

volumes (relative to balance sheet size) are consistently and significantly more highly

rated that those that do not. Bhasin (1995) reaches a similar conclusion regarding all

US over-the-counter derivatives participants. These studies thus find that target credit

ratings are driven by access to lower-cost debt markets. Interestingly, there is no

indication in these studies that the potential for lower equity funding costs feature in

bank capital structure decisions.

219 Jackson, et al (2002), p.970.

260

A more relevant question in our framework is whether or not a higher credit rating

results in larger inflow of retail funds into a bank, which would act to reduce the overall

cost of funds. We assert that the flow of retail deposits into banks is largely insensitive

to credit rating. Retail depositors typically rank ahead of rated-debt securities in terms

of the distribution of claims in the event of bank insolvency, and this should make retail

depositors less sensitive to the credit rating on debt securities. Further, retail call

deposits are also highly interest rate-insensitive, with depositors accepting low interest

rates in return for high liquidity and the perceived credit strength of the bank. An

upgrade in credit rating may result in the bank securing additional retail funds, but this

would seem unlikely unless retail depositors perceived their existing banks to be in

financial difficulty. It is certainly unrealistic to assume that retail depositors would

accept lower interest rates for a bank that had a higher credit rating.

It is with this in mind that our test of the benefits of an upgrade in a bank’s credit rating

lies with the impact of the upgrade on the cost of retail funds, the return required by

bank owners, the proportion of the funding book that comprises retail funds. We

propose that a bank with a high proportion of retail funds may find that the benefits of a

higher credit rating are minimal, and indeed, the benefits may not outweigh the costs (in

terms of the competitive consequences of the impact of higher loan rates). In the latter

case, the benefits associated with a reduction in funding costs would be more than

offset by the impact of a larger economic capital requirement on the price of the loan.

Higher loan rates may impact on the competitiveness of the bank in the loan market.

Thus the target credit rating of the bank can have a significant influence on the price of

bank loans. This also has implications for assessing the performance of credit portfolio

managers, given the numerator of the RORAC upon which they are measured may

reflect a considerable number of factors that are beyond their control.

Saunders and Lange (1996) suggest that perceptions of central bank protection of retail

deposits may explain the relative insensitivity of retail depositors to bank asset risk.220

They point to the phenomenon whereby mortgage-backed bonds can receive a higher

credit rating than that of the issuing bank, which arises largely as a result of the ability

220 Saunders and Lange (1996), p.458.

261

of a bank to over-collateralise these bonds. By this process, retail depositors effectively

cross-subsidise bond holders by allowing the bank to allocate against the bonds a

portion of mortgages that rightfully act as indirect security against the deposits.

Saunders and Lange (1996) assert that if retail depositors did not believe that they were

protected by the central bank, then they would be likely to demand higher risk

premiums in order to invest in banks. The fact that retail depositors do appear

insensitive to the riskiness of banks assets or the degree of over-collateralisation of

bank assets against other security holders implies that retail depositors do perceive that

their funds are protected by the central bank against bank insolvency.

6.5 RESULTS AND DISCUSSION

6.5.1 Overview and Assumptions

In this section we use our model to measure the extent to which progressive upgrades in

the credit rating of a bank from a base rating of BBB impacts on the price of a loans of

various credit ratings, and the minimum (breakeven) change in the wholesale cost of

funds that would be required to produce a neutral impact on the loan rate (maintain the

loan rate at a constant level regardless of credit rating of the bank). In order to assess

the impact of retail/wholesale funding mix and leverage on the breakeven cost of funds,

we also vary the proportion of retail funding and the return on target equity, and

measure the results. We then compare the minimum required change in the cost of

wholesale funds (to maintain a neutral loan rate) against the credit spreads on bank debt

using Standards and Poor’s data on bank debt securities. We suggest that if an upgrade

in credit rating is to be valuable to a bank, the impact of lower funding costs must be

greater than the increase in the loan rate associated with a larger economic capital

allocation against the loan. We test the proposition by comparing the required fall in

funding costs against empirical data on credit spreads on bank debt.

262

To measure the impact of a change in a bank’s credit rating on the breakeven price of

bank loans, we incorporate the following assumptions:

� Loans are written for one year and the credit rating of the bank is based on a one

year probability of default; � Expected default frequencies (EDF) for each loan rating are based on Standard

and Poor’s data, as presented in Table 6.1; � Estimates of loss given default (LGD) for each credit rating are for private debt

and based on Carey (1998)221; � Unexpected losses are calculated as [EL x (LGD – EL)]1/2. This assumes that the

LGD is a fixed factor.222 � The hurdle rate for the bank under the base case of a BBB credit rating is 15%.

This is adjusted for changes in the leverage of bank as indicated.223 � A retail cost of funds of 4% and a wholesale cost of funds of 8.50% are assumed

for the base case. Operating costs are equal to 2% of the loan size. The retail

cost of funds is invariant to bank leverage. � The base case assumes retail deposits equal 25% of total deposits. This is varied

where indicated. � The bank holds liquid assets equivalent to 3% of the loan exposure, and these

earn a fixed return of 7%. � The corporate tax rate is 30% and taxes are paid when incurred.

The remainder of this section examines our results under various scenarios for target

credit ratings for the bank.

221 Matten (2000) claims that while LGD rates are around 30-40% for most commercial lending portfolios, professional say all-up recovery costs can be around 60-80% when recovery expenses and time value of money factors are taken into consideration. The data used in our simulations, which is based on Carey (1998), finds recovery rates in the region of 24-39% for the borrower ratings selected for this study. 222 Matten (2000), p.193. Refer discussion in 6.2.1. 223 The hurdle rate is adjusted in proportion to the debt/equity ratio of the bank, adjusted for corporate taxes.

263

6.5.2 BB-Rated Exposure: Fixed Hurdle Rate

Our base case is that of a loan that is internally rated at BB. The credit rating on the

senior debt of the bank is BBB. The results are presented in second column of Table

6.3. The BB-rated loan carries an expected loss rate of 32 basis points, based on an

EDF of 83 basis points and a LGD of 39%. The unexpected loss rate is 3.54%. At a

target credit rating of BBB for the bank, this loan requires economic capital equal to

19.33% of the credit exposure. Given a hurdle rate of 15% after-tax and retail funding

of 25% of total deposits, the minimum interest rate on this loan is 12.43%.

Table 6.3

Impact of Increasing Solvency Standard (Fixed hurdle rates and BB-rated asset)

Target bank rating BBB A AA AAA

Borrower data Internal credit rating BB BB BB BB Expected default frequency (EDF) 0.83% 0.83% 0.83% 0.83% Loss given default (LGD) 39.00% 39.00% 39.00% 39.00% Expected loss (EL) 0.32% 0.32% 0.32% 0.32% Unexpected loss (UL) 3.54% 3.54% 3.54% 3.54%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 19.65% 41.70% 45.30% 81.00% � 0.51% 0.51% 0.51% 0.51% ✁ 3.572 3.572 3.572 3.572

Capital multiplier (CM) 5.462 11.694 12.711 22.801

Bank data Proportion of non-rated debt 25.00% 25.00% 25.00% 25.00% Economic capital 19.33% 41.38% 44.98% 80.68% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 15.00% 15.00% 15.00% 15.00% Leverage (debt/equity) 4.33x 1.49x 1.29x 0.28x

Results Asset price 12.43% 15.52% 16.03% 21.05% Breakeven

✂ in cost of rated-debt -6.71% -8.28% -51.50%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence -6.56% -7.87% -50.98%

264

Now consider the impact on the breakeven loan price if the bank changes its capital

structure such as to achieve an A credit rating. In this scenario, the economic capital

required to support the loan, using the beta distribution that applies to the loan, rises to

from 19.33% to 41.38% of the credit exposure. At a fixed internal hurdle rate of 15%,

the minimum interest rate on the loan rises from 12.43% to 15.52%. This increase in

the loan rate could render the loan uncompetitive, unless there is a compensating

reduction in the cost of rated debt in keeping with the higher credit rating on the debt.

In order to for the loan rate to remain unchanged at 12.43%, Table 6.3 shows that the

cost of rated-debt would need to fall by 6.71 percentage points. However, empirical

data on bank debt rated by Standard and Poor’s, depicted in Table 6.4, shows that the

difference in the credit spread between one-year BBB-rated bank debt and one-year

A-rated bank debt is only 15 basis points. The divergence between market spreads and

the change in the cost of debt according to our model and assumptions is 6.56

percentage points. This indicates, ceteris paribus, that an increase in credit rating from

BBB to A for a bank that holds BB-rated loans would not be in the best interests of

the bank, ignoring any potential diversification benefits provided by the loan.

The divergence between the required change in the cost of rated debt in our model and

empirical data on bank credit spreads widens as the bank targets higher credit ratings.

The minimum rate on the loan should the bank move to a target credit rating of AA

rises to 16.03%, as shown in Table 6.3. This is driven by an increase in economic

capital to 44.98% of the credit exposure. In order to maintain the loan rate at the base

rate of 12.43%, the cost of rated debt would need to fall by 8.28 percentage points.

This is also significantly higher than the empirical result on one-year credit spreads

for a bank that moves from BBB to AA, being 41 basis points. The divergence is 7.87

percentage points. The result is magnified if the bank targets an AAA credit rating.

We find that the economic capital required for the loan for an AAA-rated bank is

80.68% of the credit exposure, leading to a breakeven loan rate of 21.05%. Table 6.3

shows that it is not possible for the cost of the bank rated debt to fall sufficiently at the

AAA credit rating to maintain the base lending rate of 12.43% because the required

fall in the cost of debt exceeds the actual interest rate on the debt.

265

Table 6.4

Bank credit spreads, January 2004224 (Basis points above government bond)

Tenor (years) 1 2 3 5 7 10 30 Rating Aaa/AAA 11 13 24 30 47 59 79 Aa1/AA+ 20 28 29 41 57 70 91 Aa2/AA 22 34 36 45 60 72 94 Aa3/AA- 24 37 38 50 64 76 102 A1/A+ 45 50 54 62 77 91 114 A2/A 48 53 56 64 79 93 118 A3/A- 52 56 59 68 82 96 119 Baa1/BBB+ 60 70 78 88 120 142 168 Baa2/BBB 63 78 86 93 125 148 173 Baa3/BBB- 70 83 88 98 130 155 178 Ba1/BB+ 350 360 370 380 400 420 440 Ba2/BB 360 370 380 390 410 430 450 Ba3/BB- 370 380 390 400 420 440 460 B1/B+ 490 500 510 540 580 620 670 B2/B 500 510 520 550 590 630 680 B3/B- 510 520 530 560 600 640 690 Caa/CCC 910 920 930 955 965 975 1005

224 Source: Standard and Poor’s (2004).

266

6.5.3 Hurdle Rate Revisited

Based on our results in 6.5.2, we conclude that at a fixed hurdle rate on economic

capital, there should be no incentive for a BBB-rated bank that predominantly carries

BB-rated loans to seek to increase its solvency standard. The required decline in the

wholesale cost of funds significantly exceeds the change in credit spreads suggested

by empirical data on rated bank debt. We can draw from this that a bank that increases

its economic capital in order to increase its solvency standard, and which at the same

time maintains a fixed hurdle rate for pricing decisions, may find that it is pricing

uncompetitively. We now consider the impact of changing the hurdle rate on

economic capital in line with changes in the leverage of the bank. Should the hurdle

rate for pricing assets be different across banks with different credit ratings?

Using a Merton model of default, we demonstrated in chapter four that keeping the

probability of default for a bank constant was inconsistent with a fixed hurdle rate

when bank asset volatility varied. In the current setting we hold bank asset volatility

constant by pricing loans of a given credit rating, and assess the impact on the optimal

funding equation for the bank as it changes its capital structure in order to a achieve

different credit ratings. It turns out that the question of varying the hurdle rate on

economic capital in line with changes in the leverage of the bank is critical to our

results. Zaik et al (1996) argue that a bank should maintain a fixed corporate-wide

hurdle rate across different business lines, based largely on the difficulty of assessing

betas for individual lines of business and assuming that diversification benefits across

different risky activities are captured in capital allocations.

In the pricing simulations that follow, we hold the hurdle rate constant across different

asset risk classes, but vary the rate in line with changes in the leverage of the bank.

That is, as the bank increases its target credit rating and its economic capital increases

commensurately, we adjust the hurdle rate using a leverage-adjusted beta approach.

We argued in chapter four that the use of a CAPM-based methodology to determine

internal hurdle rates understates the true cost of economic capital to the bank when, in

addition to market risks, bank investors are concerned with bank-specific risks.

267

Specifically, it was put that a fixed hurdle rate for pricing bank assets is not consistent

with a constant probability of default when bank returns are less than perfectly

correlated with the return on the market portfolio, and consequently the internal

hurdle rate should capture the additional costs to investors associated with bank-

specific risks. In the following simulations we assume there are no significant

diversification benefits across loan classes, and allow the hurdle rate to change in line

with changes in leverage. If the contributors of economic capital to the bank perceive

that bank leverage is governed by minimum regulatory requirements, then a case

might be established for a constant hurdle rate. However as banks target higher

solvency standards, and the gap between economic capital and regulatory capital

widens, an alternative view is that the contributors of capital should be willing to

accept a lower required return in response to lower bank leverage. We assess the

impact on the optimal funding equation for the bank under this condition.

6.5.4 BB-Rated Exposure: Leverage-Adjusted Hurdle Rate

In Table 6.5 we show the results when the hurdle rate on economic capital is adjusted in

line with changes in bank leverage. Our base case is unchanged, being the position for a

BBB-rated bank with a loan assessed by the credit portfolio manager to be equivalent to

BB credit rating. The base case hurdle rate for the bank is 15%, and this is adjusted for

changes in bank leverage using a leverage-adjusted factor of [D/E (1 – t)], where D

represents deposit funding, E is economic capital and t is the tax rate on bank profits.

Calculations and assumptions related to the impact on leverage on the hurdle rate are

provided in Appendix 16.

268

Table 6.5

Impact of Increasing Solvency Standard (Leverage-adjusted hurdle rates and BB-rated asset)

Target bank rating BBB A AA AAA

Borrower data Internal credit rating BB BB BB BB Expected default frequency (EDF) 0.83% 0.83% 0.83% 0.83% Loss given default (LGD) 39.00% 39.00% 39.00% 39.00% Expected loss (EL) 0.32% 0.32% 0.32% 0.32% Unexpected loss (UL) 3.54% 3.54% 3.54% 3.54%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 19.65% 41.70% 45.30% 81.00% � 0.51% 0.51% 0.51% 0.51% ✁ 3.572 3.572 3.572 3.572

Capital multiplier (CM) 5.462 11.694 12.711 22.801

Bank data Proportion of non-rated debt 25.00% 25.00% 25.00% 25.00% Economic capital 19.33% 41.38% 44.98% 80.68% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 15.00% 10.97% 10.68% 9.24% Leverage (debt/equity) 4.33x 1.49x 1.29x 0.28x

Results Asset price 12.43% 13.14% 13.26% 14.41% Breakeven

✂ in cost of rated-debt -1.55% -1.91% -11.87%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence -1.40% -1.50% -11.35%

When the hurdle rate is adjusted for bank leverage the results show a considerable

narrowing of the divergence between the required change in the cost of rated-debt and

empirical data on credit spreads for bank-rated debt. Table 6.5 indicates that as the bank

increases its target credit rating from BBB to A the divergence is 1.40 percentage

points. The equivalent divergence under the case where hurdle rates did not adjust for

leverage was 6.56 percentage points. If the bank moves from BBB to AA, the

divergence narrows from 7.87 percentage points to 1.50 percentage points. The

divergence also narrows substantially for a change in the bank credit rating from BBB

to AAA, from 50.98 percentage points to 11.35 percentage points, but remains large

given the high level of capitalisation required to support an AAA credit rating.

269

The data in Table 6.5 shows that in each case the gap between the required change in

the cost of bank debt and empirical data on credit spreads is negative, which indicates

that the fall in the cost of funds associated with higher credit ratings is insufficient to

provide a neutral effect on the lending rate. Nonetheless, the gap is relatively narrow in

our model for a bank with an A or AA credit rating, suggesting that the hurdle rate on

economic capital – and more specifically, whether there should be changes in the hurdle

rate as leverage decreases – is likely to be a major factor in the choice of the optimal

credit rating and funding structure for a bank.

Our analysis assumes that the spreads on bank debt in Table 6.4 are entirely attributed

to credit risk differentials. These spreads, however, may be partially impacted by

factors unrelated to credit risk, such as the volume of securities on issue at each rating,

demand for the securities in capital markets, and market perceptions of the liquidity of

the bank paper. A smaller bank debt issue, for example, could incorporate a larger

liquidity premium than a larger debt issue - despite the identical credit rating – given

investor perceptions on likely future demand for the securities. We return to this point

with the context of limitations of the study in section 6.7.

6.5.5 Varying the Proportion of Retail Funding

In Table 6.6 we show the impact on the pricing equation if the bank cannot raise

incremental retail deposits to fund the loan, and consequently must fund the loan by

issuing rated-debt securities in the capital market. While this may appear an extreme

scenario, the increasing proportion of the savings of the household sector in Australia

being directed to superannuation funds suggests that such a scenario may not be

unrealistic.225

225 The Flow of Funds Matrix for Australia for 2004/05 shows that the net acquisition of currency and deposits by the household sector was $26.7 billion, whereas the net acquisition of insurance and technical reserves (this being investment in superannuation and insurance funds) was $51.5 billion. Household deposits in banks increased by $23.8 billion over the period. See Australian Bureau of Statistics (2005), p.56 and p.68.

270

Table 6.6

Impact of Increasing Solvency Standard 100% funded by rated-debt

(Leverage-adjusted hurdle rates, BB-rated asset) Target bank rating BBB A AA AAA

Borrower data Internal credit rating BB BB BB BB Expected default frequency (EDF) 0.83% 0.83% 0.83% 0.83% Loss given default (LGD) 39.00% 39.00% 39.00% 39.00% Expected loss (EL) 0.32% 0.32% 0.32% 0.32% Unexpected loss (UL) 3.54% 3.54% 3.54% 3.54%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 19.65% 41.70% 45.30% 81.00% ✂ 0.51% 0.51% 0.51% 0.51% ✄ 3.572 3.572 3.572 3.572

Capital multiplier (CM) 5.462 11.694 12.711 22.801

Bank data Proportion of non-rated debt 0% 0% 0% 0% Economic capital 19.33% 41.38% 44.98% 80.68% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 15.00% 10.97% 10.68% 9.24% Leverage (debt/equity) 4.33x 1.49x 1.29x 0.28x

Results Asset price 13.37% 13.83% 13.91% 14.66% Breakeven

☎ in cost of rated-debt -0.76% -0.94% -5.81%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence -0.61% -0.53% -5.29%

Table 6.6 shows that the impact of reducing the retail funding component of deposits

for the bank is to increase the breakeven loan rate and reduce the divergence between

the required change in the cost of funds and the empirical data on bank credit spreads.

The increase in the loan rate occurs because low cost retail deposits are replaced with

higher cost debt securities. In order to cover for the higher cost of funds, the loan rate

must rise – in the base case the loan rate increases from 12.43% to 13.37% when retail

deposits are reduced from 25% of total deposits to 0% of total deposits. The

divergence between the required change in the cost of rated-debt and the empirical

data on credit spreads narrows because the required change in the cost of rated-debt

applies to a larger volume of rated-debt. We see that the divergence falls from 140

basis points to 61 basis points if the bank moves to an A credit rating, and from 150

271

basis points to 53 basis points if the bank moves to a AA credit rating. In the case

where the bank targets an AAA credit rating, the divergence falls from 11.35

percentage points to 5.29 percentage points.

These results support our proposition that banks carrying a high level of retail deposits

in their funding mix and who seek a higher solvency standard are likely to find little

value from doing so. Our results apply to a BB-rated exposure, which typifies the

rating on most middle-market loans on bank books. Our model indicates that the

required fall in the cost of funds, to support an unchanged loan rate, is larger than the

actual change in credit spreads present in empirical data. This holds even when the

bank carries no retail deposits. The results also highlight that bank hurdle rates – and

more specifically, their sensitivity to balance sheet leverage – may also impact

significantly on the potential benefits associated with a bank moving to a higher

solvency standard.

A key question at this point relates to the cause and effect of retail deposits and bank

credit rating. Does a higher credit rating increase the volume of retail deposits into the

bank, or does the volume of retail deposits held by a bank impact on its credit rating?

Ratings agencies assert that a sizeable capital buffer is not in itself a guarantee of a

good credit rating for a bank.226 The argument that retail deposits are positive for a

bank’s credit rating is based on the observation that these are low interest rate funds and

of long duration. That is, while retail deposits tend to be at call or carry short maturities,

and thus expose the bank to high liquidity risk, historically these deposits tend to be the

most stable and insensitive to changes in market interest rates. Partially offsetting these

benefits are the high infrastructure costs associated with attracting and handling a large

volume of retail deposits. If a bank carries some base level of retail deposits, we take

the position that an increase in credit rating is unlikely to result in a significant increase

in new retail funds into the bank because retail depositors are largely invariant to the

credit rating of the bank. The reverse may hold in extreme scenarios where a bank is in

financial distress and has received large downgrades in its credit rating. In our model,

the credit rating on a bank’s debt securities is directly linked to the economic capital

226 De Servigney (2004), p.273.

272

held by the bank, which in turn, is determined by the loss distribution of the bank’s

assets.

A second issue related to the proportion of retail funding is the potential impact on the

operating costs of a bank if it replaces an increasing proportion of its retail deposits

with debt securities. A bank funded by a large proportion of debt securities may carry

lower operating costs, given economies associated with issuing large denomination

debt securities relative to the high costs of a branch infrastructure associated with

large volumes of retail deposits. In the case where the bank has no retail deposits, our

model indicates that while lower operating costs reduce the rate at which the bank

must lend in order to earn the hurdle rate on economic capital, lower operating costs

have no impact on the breakeven cost of funds required to maintain a neutral loan rate

as the bank changes its credit rating. This arises because economic capital is related to

asset risk and is invariant to bank costs. At the target hurdle rate set by the bank, the

minimum required profit is unchanged, so lower costs translate directly into a lower

loan rate. The required fall in the cost of rated-debt securities to maintain a neutral

loan rate across ratings is unchanged because the relationship between target profit

and the wholesale cost of funds is not impacted by changes in the operating costs of

the bank.

6.5.6 BBB-Rated Exposure: Leverage-Adjusted Hurdle Rate

Now we consider the impact on the optimal funding equation for the bank if it writes

higher credit-quality loans. Table 6.7 shows our results if the bank writes loans that

are given an internal credit rating equal to BBB. The top panel of Table 6.7 shows that

these loans carry expected losses of 0.07% and unexpected losses of 1.47%,227 and

this changes the loan loss distribution and capital multipliers relative to the case for

the loan rated BB. We again assume initially that retail deposits comprise 25% of

deposit funding and rated-debt securities comprising 75% of funding, but later assess

the impact of changing this assumption.

227 Based on Standard and Poor’s data in Table 6.1.

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Table 6.7

Impact of Increasing Solvency Standard (Leverage-adjusted hurdle rates and BBB-rated asset)

Target bank rating BBB A AA AAA

Borrower data Internal credit rating BBB BBB BBB BBB Expected default frequency (EDF) 0.20% 0.20% 0.20% 0.20% Loss given default (LGD) 33.00% 33.00% 33.00% 33.00% Expected loss (EL) 0.07% 0.07% 0.07% 0.07% Unexpected loss (UL) 1.47% 1.47% 1.47% 1.47%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 4.00% 20.60% 24.50% 68.00% � 0.13% 0.13% 0.13% 0.13% ✁ 4.033 4.033 4.033 4.033

Capital multiplier (CM) 2.675 13.928 16.573 46.078

Bank data Proportion of non-rated debt 25.00% 25.00% 25.00% 25.00% Economic capital 3.94% 20.53% 24.43% 67.93% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 25.19% 14.56% 13.42% 9.58% Leverage (debt/equity) 11.50x 4.02x 3.22x 0.52x

Results Asset price 11.81%228 12.21% 12.33% 13.74% Breakeven

✂ in cost of rated-debt -0.64% -0.89% -7.35%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence -0.49% -0.48% -6.83%

We observe two significant outcomes when the credit rating on the loan changes from

BB to BBB. First, in the base case where the bank has a credit rating of BBB, the

economic capital required to support a loan of equal credit rating is equal to 3.94% of

the credit exposure. This is below the existing minimum regulatory capital

requirement of 8% under the Basel Accord. This arises because at a bank credit rating

of BBB, the capital multiplier for a BBB-rated loan portfolio falls from 5.462 to 2.675

standard deviations.

228 Price is based on regulatory capital because economic capital is less than regulatory capital at the BBB credit rating for the bank. Refer discussion in this section.

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Should the loan be priced on the basis of the internal economic capital allocation

(3.94%) or the minimum regulatory requirement (8%)? Not only does this have

implications for pricing, but also for loan portfolio selection by credit portfolio

managers because the decision will impact on the RORAC used to measure and

remunerate performance.

If the internal capital allocation is used, an implicit assumption is being made that the

bank has a sufficient volume of exposures where the economic capital requirement

exceeds the minimum regulatory requirement as to offset those exposures where the

reverse holds. Under the ‘Standardised Approach’ for credit risk under the Basel II

revised capital framework (Bank for International Settlements, 2004), regulatory

capital is more aligned with the internal credit rating of the credit exposure, although

the risk weightings are likely to be not as granular as those implicit in bank’s own

internal models. If a bank is permitted to use its internal models for determining

regulatory capital requirements,229 the centre will need to demonstrate to bank

supervisors that its loan portfolio mix is sufficiently stable as to allow for some loans

to carry less economic capital than the regulatory minimum.230

We adopt the most conservative interpretation and assume the centre determines that

the following applies for capital allocation with respect to economic capital (EC) and

regulatory capital (RC):

If EC > RC Assign EC

If EC � RC Assign RC

This assumes that the centre is not sufficiently confident that the bank can maintain

the desired mix of loans by credit rating as to ensure capital ‘cross-subsidisation’ can

be consistently achieved.

229 This would apply under the ‘Internal Ratings-Based Approaches’ of Pillar One of the revised capital requirements. See Bank For International Settlements (2004). 230 If this cannot be demonstrated, a bank may be forced to revert to the ‘Standardised Approach’ for determining capital requirements, which does not provide as much scope for the use banks’ own internal models as ‘Internal Ratings-Based Approaches’.

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As a consequence we price the loan, in the base case, by setting capital at the

regulatory minimum of 8% of the credit exposure. This places a ceiling on the

debt/equity ratio for the bank of 11.5.231 Our model prices the loan at an interest rate

of 11.81%. Notably, the hurdle rate for the bank rises from 15% to 25.19%, reflecting

the higher leverage associated with an 8% capital requirement. While this may appear

high, it ensures consistency with our assumption that the hurdle rate moves in

proportion with changes in bank leverage.

What are the implications for loan portfolio selection on the part of the credit portfolio

manager? If the centre adopts the above allocation rule, managers may face low

incentives to add loans to their portfolios where the economic capital requirement is

lower than the regulatory requirement due to the implicit penalty that would apply in

terms of the RORAC under a regulatory capital floor. Perversely, these would be

predominantly lower risk loans. This again reinforces the difficulties associated with

implementing risk-adjusted performance measures for managers in a bank when much

of their decision-making may not be easily disentangled from bank-wide factors

related to target credit rating, hurdle rate assumptions and diversification across the

entire bank asset book. For the case in point, if the regulatory floor is not applied by

the centre, the performance of managers cannot be judged in isolation because the

resulting RORAC is influenced by portfolio assumptions made at the level of the

centre of the bank.

The second outcome observed in Table 6.7 is a narrowing of the divergence between

the change required in the cost of rated-debt (to maintain a neutral lending rate) and

actual credit spread differentials on bank debt. This indicates that as the risk-rating of

the borrower improves, the scope for gains in targeting higher bank credit ratings

improves. In our model, as the bank moves from the base credit rating of BBB to a

target rating of A, the economic capital requirement for the BBB-rated loan increases

to 20.53% of the credit exposure, and this exceeds the minimum regulatory

requirement. The hurdle rate adjusts to the change in leverage to 14.56%.

231 This assumes that liquid securities held in the bank’s asset portfolio carry a zero-risk weight and thus require no supporting regulatory capital.

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The combination of higher capital charge and lower hurdle rate result in the interest

rate on the loan rising from 11.81% (base case) to 12.21%. For the impact on the loan

rate to be neutral, our model indicates that the cost of rated-debt for the bank must

decrease by 64 basis points in the move from a BBB rating to an A rating for the

bank. While the gap has narrowed, this still exceeds the market differential of 15 basis

points between BBB and A-rated bank debt. The gap is 49 basis points. We find

similar results for the AA target credit rating, with the divergence being 48 basis

points. The gap is larger for the AAA target credit rating, at 6.83 percentage points,

largely reflecting the high capital multiplier that applies to a bank seeking an AAA-

credit rating on its debt securities. These results, when compared to the case for the

BB-rated credit exposure (Table 6.5), indicate that the benefits of a higher solvency

standard are stronger when the credit quality of the loan portfolio of the bank

improves.

Let us consider the outcome when the proportion of retail funding drops to zero. The

results are presented in Table 6.8. Under this scenario the loan interest rate required to

generate the hurdle rate on capital rises from 11.81% to 12.88%, reflecting the interest

rate differential between retail and wholesale funding assumed in our model. We

observe that the divergence between the change required in the cost of rated-debt and

actual credit spread differentials on bank debt narrows to 16 basis points for if the bank

targets an A credit rating, 3 basis points for a AA rating and 3.08 percentage points for

an AAA rating. The divergence is low and suggests that a bank with a low level of

retail deposits may find it advantageous to increase its target credit rating, provided the

credit quality of loans is high and bank investors adjust their required returns in line

with changes in the bank leverage.

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Table 6.8

Impact of Increasing Solvency Standard 100% funded by rated-debt

(Leverage-adjusted hurdle rates, BBB-rated asset) Target bank rating BBB A AA AAA

Borrower data Internal credit rating BBB BBB BBB BBB Expected default frequency (EDF) 0.20% 0.20% 0.20% 0.20% Loss given default (LGD) 33.00% 33.00% 33.00% 33.00% Expected loss (EL) 0.07% 0.07% 0.07% 0.07% Unexpected loss (UL) 1.47% 1.47% 1.47% 1.47%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 4.00% 20.60% 24.50% 68.00% � 0.13% 0.13% 0.13% 0.13% ✁ 4.033 4.033 4.033 4.033

Capital multiplier (CM) 2.675 13.928 16.573 46.078

Bank data Proportion of non-rated debt 0% 0% 0% 0% Economic capital 3.94% 20.53% 24.43% 67.93% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 25.19% 14.56% 13.42% 9.58% Leverage (debt/equity) 11.50x 4.02x 3.22x 0.52x

Results Asset price 12.88%232 13.14% 13.22% 14.14% Breakeven

✂ in cost of rated-debt -0.31% -0.44% -3.60%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence -0.16% -0.03% -3.08%

232 Price is based on regulatory capital because economic capital is less than regulatory capital at the BBB credit rating for the bank.

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6.5.7 A-Rated Exposure: Leverage-Adjusted Hurdle Rate

Finally we consider the outcome if the loan is assigned an internal credit rating of A

by the credit manager. The results are presented in Table 6.8. The top panel of the

table shows that loans with this rating carry expected losses of 0.01% and unexpected

losses of 0.48%.233 Using this data to derive the parameters for the beta distribution,

we obtain a considerable range of values for the capital multiplier – from 0.001 for a

bank target credit rating of BBB through to 96.833 for an AAA bank target credit

rating.234 Our assumption of 25% retail funding applies.

Given the high credit quality of the borrower, our results show that the economic

capital requirement is lower than the regulatory minimum when the bank is rated any

of BBB, A and AA. In the case of a bank rating of AAA, the economic capital

requirement is equal to 45.99% of credit exposure, which exceeds the regulatory

minimum of 8%. If the assumption holds that banks must hold at least the minimum

regulatory capital, the loans are again priced using regulatory capital as the base

where economic capital falls below regulatory capital.

Our model shows that the loan rate required to achieve the hurdle rate on capital at the

BBB target credit for the bank rating is 11.75%, and this remains unchanged up to the

AA target credit rating because the capital level of the bank is unchanged over this

range given the regulatory capital floor. At a bank target credit rating of AAA, the

loan rate rises to 12.98%. The hurdle rate remains unchanged over the range that the

regulatory capital floor applies.235

233 This is based on Standard and Poor’s data in Table 6.1. 234 The explanation for this wide range was provided in 6.3.1. 235 Note however that the hurdle rate on capital exceeds the base case of 15% over the range BBB to AA given the leverage implied by the minimum regulatory capital level.

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Table 6.9

Impact of Increasing Solvency Standard (Leverage-adjusted hurdle rates and A-rated asset)

Target bank rating BBB A AA AAA

Borrower data Internal credit rating A A A A Expected default frequency (EDF) 0.04% 0.04% 0.04% 0.04% Loss given default (LGD) 24.00% 24.00% 24.00% 24.00% Expected loss (EL) 0.01% 0.01% 0.01% 0.01% Unexpected loss (UL) 0.48% 0.48% 0.48% 0.48%

Beta distribution data Cumulative function pbeta (x,�,

✁) 99.80% 99.96% 99.97% 99.99%

Max (x) 0.01% 3.70% 5.50% 46.00% ✂ 0.03% 0.03% 0.03% 0.03% ✄ 5.168 5.168 5.168 5.168

Capital multiplier (CM) 0.001 7.690 11.441 95.833

Bank data Proportion of non-rated debt 25.00% 25.00% 25.00% 25.00% Economic capital 0.01% 3.69% 5.49% 45.99% Regulatory capital minimum 8.00% 8.00% 8.00% 8.00% Hurdle rate 25.19% 25.19% 25.19% 10.61% Leverage (debt/equity) 11.50x 11.50x 11.50x 1.24x

Results Asset price 11.75% 11.75% 11.75% 12.98% Breakeven

☎ in cost of rated-debt 0.00% 0.00% -2.86%

Change in credit spread from BBB (Standard and Poor’s data)

-0.15% -0.41% -0.52%

Divergence 0.15% 0.41% -2.34%

These results are significant because they indicate that a bank can gain by setting a

higher solvency standard when regulatory capital exceeds economic capital. This

arises because excess capacity on economic capital enables the bank to realise a

reduction in wholesale funding costs as its credit rating increases, while the lending

rate remains unchanged. The latter arises because the capital held by the bank over

most rating levels is invariant to credit rating, where regulatory capital establishes a

minimum capital floor on the bank. Our study shows that this result occurs when the

bank makes loans to high credit quality borrowers. Thus by targeting a credit rating up

to the AA level, given our model parameters, a bank can gain provided the economic

capital requirement determined by its internal modelling is lower than the regulatory

minimum. In all other cases we find that the required fall in the wholesale cost of

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funds is larger than the fall in actual credit spreads as bank increases its target credit

rating.

Our analysis has policy implications for banks with respect to the revised capital

guidelines (Basel II) promulgated by the Bank for International Settlements (Bank for

International Settlements, 2004). Under guidelines for minimum capital requirements,

the ‘Standardised Approach’ requires banks to hold regulatory capital equal to at least

8% of risk-weighted assets.236 Under the ‘Internal Ratings-Based Approaches’, banks

can use their own models to determine risk drivers that serve as inputs (EDF, LGD,

potential credit exposure) for the calculation of minimum capital requirements. The

Basel Committee uses these inputs to determine capital requirements for a particular

bank.237 Examination of the internal ratings-based approaches indicates that a

regulatory capital floor is implicit in the these approaches, meaning that if a bank gains

authority to use its own models for determining capital requirements, it still faces a

regulatory floor for capital regardless of the output of its models. This gives relevance

to our finding that a bank can gain by setting a higher solvency standard when

regulatory capital exceeds economic capital.

236 Under the new proposals, the risk weights are more refined for counterparty risk. 237 At the time of writing, these formulae have not been disclosed by the Basel Committee, and it has not indicated if it intends to disclose them in the future. This might be driven by concerns that banks will engage in regulatory capital arbitrage if such formula were disclosed.

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6.6 LIMITATIONS

Our model shows that banks are more likely to gain from a higher solvency standard

as the proportion of retail funding falls and the credit quality of the bank assets

increases, provided the hurdle rate on economic equity adjusts for changes in

leverage. In the section we consider a number of potential limitations to the model,

which represent possible refinements or extensions to the study.

First, our study used actual data on bank credit spreads as a basis for comparison

against the required decline in the cost of rated-debt to determine whether a bank

could gain value by increasing its target credit rating. The actual data was based on

bank debt rated by Standard and Poor’s as at January 2004 in Table 6.4. As earlier

identified, the actual spreads across debt of different credit ratings may be influenced

by factors other than credit risk differentials, such as the volume of securities rated at

each tranche and investor perceptions regarding liquidity. The demand for bank debt

securities may also be influenced by the volume of sovereign paper available in

capital market at any point in time.

Our study measures the required fall in the cost of rated-debt under each bank ratings

scenario on the basis that the loan rate in our model remained neutral. Our assumption

is that the market for commercial loans in which the bank operates is competitive, and

banks price in order to earn the minimum hurdle rate on capital. Any move to a higher

credit rating results in an increase in the interest rate on the credit exposures, and we

assume that this would have the potential to reduce the competitiveness of the bank in

the loan markets in which it operates. There may, however, be capacity for banks to

price their loans to earn above the minimum hurdle rate on capital. A bank may find

that it can price to earn above the minimum hurdle rate where loan markets are

competitive, and indeed, many banks would expect to do so. If this were the case, our

assumption of loan rate neutrality would not necessarily apply. Nonetheless, we

establish this as a useful basis from which to assess whether changes in the solvency

standard of a bank would be beneficial to the bank.

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Our study also examines loan portfolios in isolation when assessing the impact of

changes in the target credit rating of banks. This in itself has provided insight because

it highlights that a bank may realise greater benefits from higher credit ratings when

its loans are highly-rated. This result, however, arises because under our loss

distribution assumptions, economic capital is generally below the minimum

regulatory capital requirement when the bank holds high quality assets in its portfolio.

More specifically, by examining loan portfolios in isolation, our study necessarily

incorporates the assumption that the capital held by a bank to support its loans will be

at the regulatory floor where economic capital is less than the regulatory minimum, or

higher when economic capital exceeds the regulatory floor. A bank that manages its

capital on a portfolio basis is likely to hold a combination of exposures such that some

carry economic capital less than the regulatory minimum and others above the

regulatory minimum. This would improve capital utilisation for the bank. We note,

however, that a major driver of the Basel II revised capital framework is reduction in

the scope for such regulatory arbitrage. We also note that under Basel II a regulatory

capital floor still exists, even for banks that are given authority by supervisors to use

their own internal models for inputs into the calculation of capital.

The hurdle rates in the model were derived using a neoclassical capital asset pricing

model framework. We argued in chapter four that the capital asset pricing model may

not be an appropriate basis for determining the hurdle rate in banks. Economic capital

takes into account a bank’s concern with total risk that makes a bank behave as if it

were risk-averse. If we accept that capital invested by shareholders should earn a

return driven by systematic considerations, the return on economic capital should

capture the additional costs associated with firm-specific risk in the bank’s portfolio.

However, while the assumption of a leverage-adjusted hurdle rate based on the capital

asset pricing model may not be appropriate based on our earlier arguments, the

absolute level of the hurdle rate is not a major determinant of the results in the study,

given the focus on changes in loan prices and wholesale funding costs as banks target

higher credit ratings.

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In one scenario we have a bank rated BBB lending to a borrower that has been

assigned an internal credit rating of A. Although this in an internal credit rating used

by the bank for pricing risk, it could be argued that this borrower would be more

likely to access capital markets directly. A similar, although weaker, argument could

apply to the cases where the bank credit rating matches that of the borrower. This

occurs on two occasions in our study.238

Our study assumed a one-year horizon for the calculation of economic capital. While

this is the regulatory standard, and somewhat convenient from an accounting

perspective, this ignores the longer maturities that are likely to apply to most

commercial bank loan portfolios. It may be more appropriate to consider economic

capital at a horizon that matches the average duration of bank loan portfolios, or

estimates of the time (and cost) that it would take banks to access capital markets for

liquidity or additional equity funds in times of financial distress. Further, it could be

prudent to move beyond the average maturity of the loan portfolio and assess loss

distributions at various time horizons. For the purposes of this study we calibrated the

one-year horizon for the solvency standard of the bank with one-year duration loans

and credit spreads on one-year rated debt. The study could be expanded considerably

to assess the impact of longer-term exposures and their impact on the target credit

rating. In particular, it could be fruitful to assess the impact of a one-year solvency

standard, for the purposes of calculating economic capital, against credit spreads on

bank-rated debt of longer terms.

Linked to the preceding point, our study employed a default mode paradigm for

defining losses, meaning we recognised loss only when the borrowers defaulted on

their legal obligations within the one year time horizon. Consequently the measure of

economic capital in our model largely represents a book value approach. The default

mode approach was considered appropriate for the model given the one-year

investment horizon selected for the asset portfolios. An alternative approach would be

a mark-to-market paradigm, as embodied in the RAPM in chapter three, recognising

gains and losses in the value of the bank’s assets caused by changes in the credit 238 A BBB-rated bank lends to a BBB-rated borrower, and an A-rated bank lends to an A-rated borrower. This occurs in sections 6.5.6 and 6.5.7.

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quality of the borrower over the specified time horizon. This approach would

recognise, for example, deterioration in the credit quality of a loan portfolio during

times of unexpected recession, even though technical default may not arise. The

decline in the value of the portfolio would impact on the market value of economic

equity. The reverse effect would result if the borrower migrated to a higher credit

rating over the life of the loan obligation, resulting in a lower discount rate and a

higher loan value. This would act to increase the market value of economic capital for

the bank. An additional consideration would be the impact on the market value of

economic capital as the bank itself moved to a higher or lower target credit rating.

This would impact on the leverage of the bank and potentially, hurdle rates.

Finally, the model in this chapter incorporated a single hurdle rate on the basis that

diversification across assets within the banking firm cannot influence its external beta

because investors can create similar diversification benefits by spreading their

portfolios across assets with similar risk profiles. Section 4.5.2 of this dissertation

examined the circumstances under which bank-specific risks should factor into hurdle

rates for various bank activities.

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6.7 CONCLUDING COMMENTS

It is generally accepted that an increase in the external credit rating on the debt of an

organisation is positive. Higher credit ratings are, in the main, associated with lower

perceived volatility in the market value of the assets of the entity that has issued debt.

Lower asset volatility implies more stable and sustainable cash flows, and thus a

lower likelihood of default on debt, leading to a lower credit spread on the debt. The

same principles should apply in regard to the rating of debt securities issued by a

bank, but the impact of a change in rating may be more acute on the issuing bank in

the sense that it impacts on both the price of assets and the cost of funds. An increase

in the volatility of expected losses on a bank’s asset portfolio increases the amount of

economic capital that a bank must hold in order to sustain its target credit rating,

where economic capital is measured as a multiple of the standard deviation of

expected losses (unexpected losses). If a bank desires a higher credit rating, all else

equal, the leverage on its balance sheet must decline (the equity capital must rise). If

banks price their assets to realise a target hurdle rate on capital, then a higher credit

rating will result in higher loan rates if the fall in the bank’s cost of capital, associated

with the lower insolvency risk, is insufficient to offset the additional net income that

the loan must be priced to cover. It is through this mechanism that we assert the credit

rating of a bank impacts on the price of its assets.

The main proposition of this chapter is that the proportion of retail funding held by a

bank will be a major determinant of the benefits that accrue to the bank as a result of a

change in its credit rating. If retail deposits are largely insensitive to credit rating, a

bank holding a large proportion of retail deposits in its funding book may find that the

benefits of a higher credit rating, in terms of a reduction in the cost of funds, are

insufficient to offset the increase in net income that must be achieved to earn the

hurdle rate on capital. In this case, the bank may be forced to increase the interest rate

on some loan classes as its target credit rating increases. This may impact on the

competitive position of the bank in the loan markets in which it operates. We provide

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arguments to support the case that retail deposits are largely insensitive to credit rating

on the bank’s debt securities239.

To test our proposition, we constructed a loan pricing model driven by the hurdle rate

on the capital of the bank. We model the case for a bank increasing its target credit

rating from BBB through to AAA, and assess the breakeven wholesale cost of funds

at which the impact of a higher credit rating on loan price is neutral. We examine the

impact on loans rated BB, BBB and A, and use a beta distribution for loan losses to

determine capital multipliers for the calculation of economic capital. We translate the

breakeven wholesale cost of funds into the minimum amount, in basis points, that the

cost of funds must fall to maintain loan price under each credit rating for the bank. We

then compare the required decline in the cost of rated debt, as estimated by our model,

against actual data on bank credit spreads in order to ascertain the extent to which the

increase in credit rating is beneficial to the bank. We find that the required decline in

the cost of funds exceeds actual credit spreads on bank debt, meaning that the

reduction in funding costs is insufficient to offset the increase on loan rates associated

with higher economic capital. As expected, the divergence increases as the proportion

of retail funds increases.

In some cases we find that the economic capital requirement is large relative to the

size of the credit exposure. For example, the required capital for an AA-rated bank

lending to a BBB-rated borrower is equivalent to 25% of the exposure.240 In a

practical setting, this appears high relative to the total capital held by banks, and

indicates the significance of diversification benefits that must exist in banks across

their portfolio of assets. Our model prices loans on a stand-alone basis, making no

assumptions about diversification benefits across the bank. At the very least, our

figures, which use the beta distribution to model the loss distribution of specific loan

classes, indicate that banks must be incorporating some degree of diversification into

their assessments of capital. They also suggest that capital cross-subsidisation must

exist at the bank-wide level, with some assets requiring more capital than the

239 We do not take deposit insurance into consideration in this argument given such a system does not exist in Australia. 240 Refer Table 6.7.

287

regulatory minimum, and others requiring less. Again, this presents complications

when assessing the performance of credit portfolio managers on the basis of their

RORAC, given economic capital forms the basis of the denominator of this

performance measure.

We find that the hurdle rate on economic capital is a distinguishing factor in

determining whether or not increasing its solvency standard is valuable to a bank. If

the hurdle rate remains fixed regardless of the capital structure of the bank, then an

increase in credit rating may have little impact on the value of the bank given our

findings on the large divergence between the required decline in the cost of wholesale

funds and empirical data on bank credit spreads. However, this divergence is

considerably less pronounced if the hurdle rate is varied in direct proportion to the

leverage of the bank. Hurdle rates form the basis upon which banks price their

products and services in order to earn minimum acceptable returns. There is a dearth

of research on the relationship between hurdle rates and the leverage of the bank. If

hurdle rates are linked to returns required by bank owners, then some form of

adjustment to hurdle rates for changes in credit rating appears justified. Banks and

their analysts appear to have focused little on this issue in the past, possibly driven by

constraints on bank leverage inherent in the Basel Accord of 1988. While the notion

of minimum regulatory capital remains under the revised Basel II capital framework,

the combination of an increasing focus on economic capital and target credit rating on

the part of banks and ratings agencies on the one hand, and the greater scope for banks

to use their own models for the determination of capital requirements under the

revised capital framework on the other, suggests that the relationship between bank

capital and hurdle rates on bank capital should remain on the research agenda.

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6.8. CHAPTER SUMMARY

The main findings of this chapter are summarised as follows:

1. It is not possible to separate pricing decisions at the level of credit portfolio

managers from decisions made by the centre of the bank with respect to target

credit ratings for the bank, hurdle rates and assumptions made by the centre with

respect to diversification across bank assets. This places questions over the

relevance of remunerating managers based on RORAC measures, when a large

proportion of factors impacting on both the numerator and denominator of the

RORAC are outside of the control of managers.

2. The decision of a bank to increase its solvency standard increases the price of

bank assets to the extent that assets are priced to achieve a minimum hurdle rate

on economic capital. A higher credit rating, ipso facto, should reduce the cost of

wholesale funds to the bank. The benefits to the bank of increasing its solvency

standard rest with the extent to which the cost of wholesale funds falls relative to

the increase in the price of bank assets, and the degree of insensitivity of retail

deposits to changes in the credit rating on bank debt.

3. There is a considerable divergence between the change required in the cost of

wholesale funds - to maintain a neutral effect on loan prices - when the bank

increases its target credit rating and empirical data on bank credit spreads. This

divergence narrows, however, as the credit quality of the bank loan book

increases and the proportion of retail deposits falls. The divergence also narrows

considerably when the hurdle rate on capital is allowed to adjust to reflect changes

in bank leverage.

289

4. If the centre requires that a regulatory capital floor be applied to the capital

allocation process, managers may face low incentives to add loans to their

portfolios where the economic capital requirement is lower than the regulatory

requirement due to the implicit penalty that would apply in terms of the RORAC

under a regulatory capital floor.

5. A bank can gain from increasing its solvency standard, in terms of the cost of

funds falling more than the increase in loan prices, when the regulatory capital

requirement exceeds the economic capital requirement. This occurs when banks

make loans to high credit-quality borrowers, because capital ‘capacity’ enables

the bank to realise a reduction in funding costs without an offsetting increase in

loan prices. The revised regulatory capital requirements (Basel II) carry an

implied regulatory capital floor, so these opportunities may exist under the new

regulatory capital regime for banks.

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Chapter Seven

Conclusion

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7.1 OVERVIEW

While all corporate entities face risk, the perspective of a banking firm on risk differs

from that of other entities because the fundamental role of a bank is to accept,

transform, price and monitor risk. The centre of a bank is charged with managing a

complex inventory of credit, market and operational risk. This task might be relatively

straightforward if there existed a universal concept of risk, an accompanying risk

measure that perfectly captured this notion of risk, and if the centre had low-cost access

to all the information it needed to estimate this risk. Then, much in the same way as a

single owner-manager firm might operate, all the centre would need do is calculate the

expected risk of each investment opportunity open to it, compare this to the estimated

profit from each investment, and allocate the available capital and other resources to

those investments offering risk-adjusted returns in excess of the bank’s hurdle rate. If

the supply of capital is limited, the task is reduced to investing in those opportunities

offering the highest risk-adjusted returns, subject to achieving the hurdle rate.

Unfortunately for the centre of the bank, and the stakeholders that it represents, this task

is not straightforward because a universal notion of risk does not exist, and even if it

did, information on the risk profile of each investment in the opportunity set becomes

less accessible to the centre as a bank expands its asset base. The classical literature

shows that there has existed a wide range of perceptions of risk. For example, Domar

and Musgrave (1944) view risk as the probability of loss, Savage (1951) perceives risk

in terms making the wrong investment choice, Roy (1952) views risk as the probability

that future income will be below some ‘disaster’ level, and Baumol (1963) views risk in

terms of variability about an expected value. Given this lack of consensus there exists

no one objective measure of risk. Consequently it is often necessary to draw upon

notions of utility functions and risk preferences in order to discriminate upon the risks

in alternative investments.

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Faced with lack of consensus on the definition of risk, the centre might take comfort

from the fact that minimum capital requirements promulgated by bank regulators are

indexed to measures of risk. In other words, bank capital requirements implicitly

embody a notion of risk, as determined by regulatory authorities. In the case of credit

risk, bank capital regulations are in the process of evolution - from a fixed capital

requirement based on risk-weighted assets to a concept of risk that relates bank capital

requirements to a target level of confidence which applies to the bank defaulting on its

senior debt obligations.241 This approach, which identifies risk as the probability that a

loss will fall below some prespecified level, encapsulates the concept of risk used by

most banks to determine their economic capital requirements. It is also aligned with the

ratings agency perspective of risk, where credit ratings apply to default probabilities on

issued debt.

The problem with such an approach is that a fixed solvency standard implies a neutral

attitude towards risk, which may not reflect the risk attitude of bank stakeholders. The

solvency standard approach implies that the risk measure for capital does not capture

the size of potential losses beyond some threshold, nor place any differential penalty on

larger deviations from the threshold relative to smaller deviations from the threshold.242

Viewed from one perspective, the solvency standard approach for determining capital

suggest bank capital is driven more by the views of external ratings agencies, rather

than a disciplined and consistent analysis of risk based on the full distribution of

potential outcomes.

The challenge for the centre, then, is to determine the relevant concept of risk for the

bank, and select or develop a risk measure for use within the bank that is aligned with

this concept of risk. This is particularly important if the bank has decentralised part of

its operations in order to allow managers to gain specialised knowledge on investment

opportunities and their accompanying risk profiles, and if the centre has vested

authority in these managers to select and manage investments from the opportunity set,

241 This was discussed in chapter one within the context of the Basel II revised framework for the international convergence of capital standards and measurement. 242 Average losses may be identical, but we have seen that investors may be more averse to larger losses than smaller losses, even though the former has a lower probability of occurrence.

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because the risk measure is an integral part of a mechanism by which managers can

rank investments in alignment with the preferences of the centre.

The question of achieving congruence between the risk preferences of the centre of a

bank and the portfolio selection decisions of managers has been the focus of this

dissertation. Our first proposition has been that the basis for measuring risk within a

banking firm will necessarily differ from that by which the bank measures its total

economic capital requirements, if the objective is to achieve risk congruence between

the centre and managers.243 The basis of this proposition is the risk preference function

of the centre of a banking firm is unlikely to be one that embodies a neutral attitude

towards risk - which is implicit in the measurement of capital based on a predetermined

solvency standard. If bank stakeholders are risk-averse and take a bank-specific

perspective of risk, then allocating capital and rewarding performance based on risk

measures linked to a minimum solvency standard may lead managers to make

investment decisions that are against the preferences of bank stakeholders. In this

regard, it was shown in chapter three that the use of type 1 risk measures for the

purposes measuring and remunerating risk-adjusted performance leads managers in

many cases to select inefficient portfolios in terms of stochastic dominance

principles.244

In light of this divide, this study set about the design of an internal risk-adjusted

performance measurement framework that provides an incentive-compatible outcome

between the centre and managers. The key objective has been to establish a mechanism

by which managers select the portfolios that the centre itself would select, if

information on the risk-attributes of each investment in the opportunity set was

costlessly available to the centre – much as was the case for the single owner-manager

setting described at the beginning of this chapter.

243 Unless the conceptual basis for measuring risk for determining total economic capital matches the basis for measuring risk internally. This also assumes that the bank meets its minimum regulatory capital requirements: hence economic capital equals or exceeds minimum regulatory capital. 244 The notion of risk measure ‘types’ is defined in chapter two.

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A number of tasks needed to be undertaken in the design of such a mechanism. These

tasks, which formed the basis for each of the chapters in the dissertation, are listed as

follows:

� Determine the risk preference function for the centre of the bank � Determine which risk measure(s) are compatible with this risk preference

function � Assess the relevance of coherency for the internal risk measure(s) � Assess the impact of the structure of the compensation payment function on

portfolio selection and incentive-compatibility � Identify how agency problems impact on the performance measurement

framework � Design a solution to deal with agency problems � Evaluate how bank-wide factors impact on the decisions of managers and the

assessment of their performance.

We started from the position that managers have no incentive to misrepresent their

information on the expected risk profile of their portfolios. They face an opportunity set

of potential investments, and subject to constraints on available funding and capital, are

required to select portfolios for investment that are compatible with the risk and return

objectives of the centre. Information on the risk profile of all the investments in the set

is not available to the centre. In order that managers select portfolios that are congruent

with the risk preferences of the centre, we propose that some part of their remuneration

has to be linked to the risk-adjusted performance (RORAC) of their portfolios. In this

way, the RAPM is the vehicle by which the goals of the centre are transformed into the

actions of managers. Managers simply select the portfolios that offer the highest

expected RORAC, where the risk measure used for the denominator of this equation

provides a risk-ranking of portfolios that is consistent with the risk preference function

of the centre.

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The potential weak link in this framework rests in the requirement that managers

provide information to the centre on the expected risk profile of their portfolios in order

that a risk measure can be assigned for the purposes of determining the expected

RORAC. If managers have no incentive to misrepresent this information to the centre,

then the framework can be considered robust. If, however, managers are self-interested

and opportunistic, as posited in agency theory, then they can be expected to

misrepresent their private information in order to achieve a favourable allocation of

resources and/or increase the probability of achieving high bonuses. The centre, of

course, could avoid this outcome by using historical portfolio volatility measures to

determine the risk value in the denominator of the RORAC equation. This infers that

historical volatility is a good proxy for future volatility. Such an approach, however,

obviates the motivation for decentralising investment decisions to managers – being to

take advantage of the specialised knowledge of investment opportunities and expected

risks that managers can gain under a decentralisation strategy.

It is on this basis that we developed the second proposition of this dissertation. The

solution to these problems is to embed into the bank’s risk measurement process a

truth-revealing mechanism – one by which managers cannot personally gain by

misrepresenting their private information on expected risks. A second-price auction

design is used as the basis for determining this mechanism. We established the notion

of an internal capital market in which managers are required to periodically bid for their

capital needs. The truth-revealing properties of this mechanism allow for the private

information of managers on expected risk-adjusted returns to be revealed to the centre,

allowing for an optimal allocation of capital based on the ‘true’ expected distribution of

portfolio returns.

The third major proposition of this dissertation is that the performance of managers

cannot easily be separated from bank-wide decisions regarding the bank’s target credit

rating, funding mix and the hurdle rate on equity because these factors combine to

influence the price of bank assets, which in turn, impacts on performance metrics upon

which managers are assessed. Our analysis shows that the benefits of changes in credit

rating are contingent upon assumptions regarding changes in the hurdle rate in response

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to changes in leverage. This in turn impacts on pricing decisions and the market value

of credit portfolios, and consequently, the risk-adjusted performance measures of

portfolios under the control of managers.

The remainder of this concluding chapter proceeds as follows. In section 7.2 we

summarise our key findings. In section 7.3 we identify areas considered fruitful for

further research.

7.2 KEY FINDINGS

Table 1.1 in chapter one provides a list of the key research questions set for the study

on a chapter-by-chapter basis. In this section we present our main findings to these

questions.

Chapter two presented the theoretical framework for determining the optimal class of

incentive-compatible risk measures. This required assessment of the risk preference

function of the centre and the selection of an evaluation tool for risk-ordering portfolios

according to this function. The key questions for the chapter were set as follows:

� What is the risk preference function of the centre of the bank? � Is there a methodology that can be used to rank portfolios by risk in

accordance with the risk preferences of the centre? � Is there a risk measure (or measures) that provide a risk-ordering consistent

with the risk preferences of the centre?

The literature on models of the banking firm was reviewed in order to assess the risk

preference function of a bank. This literature provided no firm direction on issue.

Drawing on the literature, it was determined that bank creditors and regulators are

likely to be risk-averse and have a preference for positive skewness in the distribution

of returns on the grounds that their concerns should not only be with the probability of

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the bank defaulting, but also the size of loses in the event of default. This is because the

size of losses in the event of default directly impacts on both creditors and regulators –

the former in terms of losses and the latter in terms of perceptions of banking system

stability.

Assessing the risk attitude of bank owners is more contentious. When limited liability

and the regulatory safety net are taken into consideration, bank shareholders may be

risk-seeking and have a convex risk preference function. If a bank carries significant

franchise value, however, owners may prefer that the bank acts in a risk-averse manner

in order to preserve the associated benefits. In this case the objective function for the

bank would be concave. This also assumes that the owners are concerned with total

bank risk, and not just systemic risk.245 On the assumption that the value of the

franchise to bank owners exceeds the value of the option associated with limited

liability, we make the assumption that bank owners will also be risk-averse.

Consequently it was concluded that the bank risk preference function should possess

the characteristics of non-satiety, risk-aversion and a preference for positive skewness

in the distribution of returns.

Stochastic dominance is used as the methodology to rank portfolios in accordance with

the risk preferences of the centre. The key to using stochastic dominance criteria is that

the methodology allows portfolios to be ranked without having to specify the exact

form of the investor utility function - different orders of stochastic dominance

correspond to different classes of utility function. Third-order stochastic dominance

(TSD) criteria embody non-satiety, risk aversion and a preference for positive skewness

in the distribution of returns. This makes it the most applicable criteria for risk-ordering

portfolios given our conclusions regarding the characteristics of the risk preference

function of the centre.

It was found that the lower partial moment of order n = 2 (LPM2) provides a measure of

risk that is consistent with the risk preference function of the centre. More specifically,

portfolios that dominate by TSD criteria are decreasing in risk according to the LPM2

245 This is examined in chapter three.

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risk measure. The quadratic power function in this measure means large deviations

from the loss threshold receive a greater penalty than smaller deviations in the risk

measure - consistent with a risk-averse attitude to losses and a preference for positive

skewness in the distribution of returns.

Chapter three examined the incentive-compatible properties of a set of risk measure

candidates against five hypothetical credit portfolios. While each portfolio carries the

same expected value, their probability density functions differ. This allowed the

analysis to extend to the impact of the target threshold for measuring gains and losses.

The key questions for the chapter were set as follows:

� Does the internal risk measure need to be coherent in terms of the

structural properties identified by Artzner et al (1999)? � Does the structure of the bank compensation payment function impact on

incentive-compatibility conditions? � Does the choice of target threshold for measuring gains and losses impact

on portfolio selection? � Should the risk attitude of the centre towards the distribution of gains

feature in internal risk measures?

The question as to whether the internal risk measure should also conform to the axioms

coherency, as detailed by Artzner et al (1999), rests on the risk management goals of

the centre. If the performance of managers is assessed on the basis of the risk-adjusted

returns on individual loans under their control, we find it may be less important that the

internal risk measure conform to the axioms of coherency. This is because coherency

axioms tend to relate to portfolio risk management. However, if the centre wishes to

encourage managers to use their specialised knowledge to seek out assets that provide

diversification and other benefits, then performance should be assessed on a portfolio

basis, and in this case, the issue of coherency is relevant.

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Banks typically manage their credit-related business in broad business lines or

portfolios. Although the portfolio delineation used by individual banks can vary greatly,

the common bonds that define a portfolio may be related to the nature of the customer,

the nature of the transaction, or a combination of the two.246 For this reason, the

portfolio perspective was adopted for this study. As a consequence, coherency

considerations for the internal risk measure were deemed relevant.

The LPM2 risk measure, which was the only measure found to provide a risk-ordering

of portfolios consistent with TSD, fails all four axioms of coherence. This arises

because TSD involves the expectation of squared profits and losses, which is

inconsistent with the notion of a coherent risk measure, which does not allow for the

squared profits and losses. This was demonstrated in section 3.4.1. The consequences of

this divide were found to be particularly relevant with respect to the axioms of positive

homogeneity and subadditivity. In the case of positive homogeneity, it was shown that

the LPM2 for two identical loans that are combined in a portfolio will exceed the sum of

the LPM2 of the individual loans. This may lead managers to reject loans that are

valuable to the bank, even though these loans provide no diversification benefits. It was

also shown that the LPM2 measure also fails the subadditivity axiom – the LPM2

measure of a portfolio that has diversification benefits in the left tail can be greater than

the LPM2 measure of the individual loans that make up the portfolio. By not rewarding

diversification, this measure provides no incentive for managers to seek-out and add

loans to their portfolios that provide risk-reducing benefits.

This led to a quandary. The measure that provided a consistent risk-ordering in terms of

TSD failed when tested for coherency. However, investigation revealed the downside

semi-deviation (DSD) risk measure – the square root of the LPM2 – to be both coherent

and consistent with a TSD risk-ordering of portfolios. DSD is an incentive-compatible

risk measure given our requirements. Consequently the DSD was recommended as the

internal risk measure for capital assignment and performance measurement.

246 Bank for International Settlements (2001b), p.5.

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The structural form of the compensation payment function of the bank impacts on

incentive-compatibility conditions where this function is asymmetrical, with losses

driving the risk measure and bonuses linked to the realisation of gains. This means the

distribution of gains in the numerator of the RAPM and the distribution of losses in the

denominator of the RAPM feature in portfolio selection. Despite a consistent ranking of

portfolios by risk, when the distribution of gains is considered, it is not possible to

determine which portfolios managers will select without specific knowledge of their

utility functions.

Further, the basis for measuring gains and losses impacts on portfolio selection

decisions. The analysis incorporated a mark-to-market approach for assessing gains and

losses.247 Different results are achieved depending on whether gains are measured

relative to portfolio expected value or portfolio face value. It is concluded that the

expected value is the most appropriate benchmark given that gains in market value will

have a positive impact on reducing expected losses, provisioned for ex-ante.

On the question of the attitude of the centre towards the distribution of gains, if the

centre is charged with managing both risk and return, as opposed to only managing

downside risk, then the RAPM upon which managers are remunerated should

incorporate risk preferences of the centre to the upside distribution of portfolio returns.

For example, the centre may have a preference for moderate but more consistent gains,

rather than large but less frequent gains. A reward-to-risk ratio, where the numerator

measures upper partial moments in the distribution of returns, allows for portfolios to

be ranked in accordance with the attitude of the centre towards variability in upside

returns. This ratio effectively represents the shadow price of risk, where we assume risk

is measured by the DSD of the portfolio.

247 As opposed to accounting-based measures of performance. Refer to section 3.2.1.

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Chapter four argues that agency problems related to managerial self-interest and

opportunism are likely to lead managers to misrepresent their private information on the

expected distribution of returns, leading to an inefficient allocation of bank resources

and the potential for the bank to be undercapitalised with respect to risk. Agency issues

are also considered within the context of deriving the hurdle rate used by banks for

pricing assets and benchmarking performance. The key questions for the chapter were

set as follows:

� How do agency problems impact on the robustness of the risk-adjusted

performance measurement framework? � Should internal hurdle rates reflect a total-bank risk perspective or a

systematic risk perspective? � Is a fixed hurdle rate consistent with a fixed probability of default

(solvency standard)?

The framework developed in earlier chapters relied on managers freely and truthfully

disclosing their private information on the expected distribution of returns of their

portfolios. The risk measure is the signal by which managers rank portfolios from the

opportunity set in line with the risk preferences of the centre. There was, however,

implicit recognition of agency difficulties in this framework: while managers might be

directed to select portfolios with the lowest risk profiles, it was assumed that this could

not be assured without linking some proportion of their remuneration to the realised

RORAC on their investments. This became a powerful signalling mechanism to

managers.

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In chapter four the assumption of stewardship on the part of managers was removed. It

was argued that if managers can exploit information asymmetries to achieve favourable

outcomes with respect to resource allocation decisions and expected bonus payments,

then the risk-adjusted performance measurement framework is not sufficiently robust to

guarantee incentive-compatibility between the centre and managers. In this sense, while

the DSD risk measure accurately reflects the risk preferences and portfolio management

objectives of the centre, if managers are opportunistic and self-interested, optimal

outcomes for bank stakeholders cannot be assured.

If the existence of information asymmetries between the centre and managers provide

incentives for the management of bank specific risk, it follows that the economic capital

of a bank should embody a total risk perspective. Consequently, the use of a CAPM-

based methodology to determine the internal hurdle rate for measuring performance,

which conforms to the textbook approach for determining required returns, understates

the true cost of economic capital to the bank. RORAC models that relate risk-adjusted

returns based on total risk in the denominator to hurdle rates that require compensation

only for systematic risk implicitly assume that debtholder risks and shareholder risks

are proportional. Using a framework developed by Crouhy (1999), it is shown that this

is not the case. In this regard, a uniform hurdle rate is not consistent with a constant

probability of default when bank returns are less than perfectly correlated with the

return on the market portfolio. We conclude that the internal hurdle rate should capture

the additional costs to investors associated with bank-specific risks.

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Chapter five develops a solution to the agency problems identified in chapter four,

based on the concept of an internal capital market for the allocation of capital. The DSD

is employed as the risk measure for determining the economic capital requirements of

the bank, and the task of the centre is to allocate this capital to its most productive uses.

In order to get managers to truthfully reveal their expectations on portfolio return and

risk, the allocation mechanism links a second-price sealed bid auction design to the

compensation payment function used to remunerate managers. Managers are required

to bid for risk capital on the basis of capital required and the associated RORAC. The

compensation payment function is based on the actual capital utilised by the investment

and its actual RORAC, with bonuses based on a rebate associated with the realised

RORAC and the opportunity cost of assigning capital to the investment.

It is shown that the dominant strategy for a manager is to bid truthfully on expected

capital requirements and the associated RORAC, independent of the actions of other

participants in the auction. From the perspective of the centre of the bank, this design

allows for the private information of managers regarding expected risk-adjusted returns

to be revealed to the centre, allowing for an optimal allocation of capital based on the

‘true’ expected distribution of portfolio returns. The DSD is embedded as the risk

measure upon which capital requirements are based, given its alignment with the risk

preferences of the centre and its coherence properties.

Chapter six examined how bank-wide decisions regarding hurdle rates, target solvency

standard, loan ratings and funding mix impact on the pricing of bank assets. The

chapter identifies difficulties in separating pricing decisions on credit portfolios

entrusted to managers from decisions made by the centre of the bank with respect to

target credit rating and minimum hurdle rates. While managers typically receive some

proportion of their remuneration based on the risk-adjusted performance of portfolios

under their control, a significant proportion of factors affecting the performance of

credit portfolios are beyond the control of managers.

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The key questions for the chapter were set as follows:

� How does the target credit rating of the bank influence portfolio

selection and pricing? � When might a higher solvency standard beneficial to a banking firm? � Should hurdle rates adjust in line with changes in the target credit

rating of a bank?

Assumptions regarding the hurdle rate, and in particular, whether it should adjust to

reflect changes in leverage, are critical to determining the optimal credit rating for a

bank. This in turn impacts on pricing decisions and the market value of credit

portfolios, and consequently, the risk-adjusted performance measures of portfolios

under the control of managers.

A loan pricing model was constructed to test the impact of changes in the target credit

rating of a bank on the pricing of its loans. The decision of a bank to increase its

solvency standard increases the minimum interest rate on its loans in order to achieve

the required hurdle rate on capital assigned to the loans. Offsetting this upward pressure

is the impact of the reduced funding costs arising from the higher credit rating. If retail

deposit rates are insensitive to an upgrade in the credit rating of bank debt securities, we

find that the benefits to a bank from increasing its target credit rating rest with the

extent to which the cost of wholesale funds falls relative to the increase in the price of

bank loans.

A number of scenarios are employed to measure the impact of bank-wide decisions on

target credit rating and funding mix on the pricing of bank loans. As the bank increases

its target credit rating, there is a significant divergence between the change required in

the cost of wholesale funds to maintain unchanged loan rates and empirical data on

bank credit spreads. This divergence narrows, however, as the credit quality of the bank

loan book increases and the proportion of retail deposits falls. The divergence also

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narrows considerably when the hurdle rate on capital is allowed to adjust to reflect

changes in bank leverage.

Further, our model shows that a bank can gain from increasing its solvency standard, in

the sense that the cost of funds falls more than the increase in loan prices, when the

regulatory capital requirement for the loan exceeds the economic capital requirement.

This occurs when banks make loans to high credit-quality borrowers, because capital

‘capacity’ enables the bank to realise a reduction in funding costs without an offsetting

increase in economic capital, and hence an increase in loan interest rates.

Our analysis shows that the benefits of changes in credit rating are contingent upon

assumptions regarding changes in the hurdle rate in response to changes in leverage.

This is also relevant at the level of managers, where performance on portfolios is

measured by the RORAC against the bank hurdle rate. In chapter four we found that a

fixed hurdle rate for pricing bank assets is not consistent with a constant probability of

default when bank returns are less than perfectly correlated with the return on the

market portfolio. We argued that the internal hurdle rate should capture the additional

costs to investors associated with bank-specific risks. If the contributors of economic

capital to the bank perceive that bank leverage is governed by minimum regulatory

requirements, then a case might be established for a constant hurdle rate. However as

banks target higher solvency standards, and the gap between economic capital and

regulatory capital widens, the contributors of capital should be willing to accept a lower

required return in response to lower bank leverage.

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7.3 AREAS FOR FURTHER RESEARCH

One of the reasons for firms decentralising aspects of their operations is to enable

managers to gain specialised knowledge of their local conditions. For credit managers

in a banking firm, this knowledge may take the form of investment opportunities and

the risk profiles of each of these potential investments. In light of principal-agent

problems that arise when information is asymmetrical, the focal point of this

dissertation has been the development of mechanisms that facilitate the free and

accurate disclosure of this information from agents to the principal at the time

investment decisions are being implemented. These mechanisms are necessary because

managers may have strong incentives to misrepresent their private information when

doing so has the potential to impact on the size of their remuneration. This, in turn, has

a direct impact on the ability of the centre to optimally invest the capital of the bank

and accurately price risk into bank investments. Arising from these issues, we close this

study with a consideration of areas fruitful for further research.

An exclusive focus on downside risk in portfolio selection precludes considerations

regarding the distribution of upside returns. If the centre of a bank is charged with

managing both risk and return, then both the right and left tail of the distribution should

be incorporated into the portfolio selection mechanism. In the framework of chapter

three, we started from a position where portfolios had the same expected value, in order

to allow the analysis to focus exclusively on downside risk. Despite portfolios having

the same expected value, we later showed that the distribution of upside returns for

each portfolio does impact on the decisions of managers, subject to their personal utility

functions. This then led to the question of the attitude of the centre towards the upside

distribution of returns, and if it should be of relevance. We developed an evaluation

tool, being the ratio of reward to DSD, where the risk attitude of the centre to upside

volatility was captured in the order of the upper partial moment. This tool reveals that

the risk attitude of the centre does influence the ranking of portfolios. Not addressed

was the question of how to incorporate such an index into the compensation payment

function of managers, in order to incentivise them to select the portfolios desired by the

centre on the basis a risk preference function capturing, potentially, differential risk

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tolerances to gains and losses in portfolio distributions. The integration of this index

into the compensation payment function of a bank could be a useful topic for further

research.

In chapter four it was concluded that a CAPM-based internal hurdle rate understates the

true cost of economic capital to a bank. It was argued that firm-specific risks impose

real costs on a bank, and consequently a hurdle rate based on the CAPM may lead

banks to underprice risk. The task of determining the cost of economic capital that is

over and above the CAPM-based hurdle rate is an area worthy of further research. The

impact of assumptions regarding the internal hurdle rate in achieving incentive-

compatible outcomes has surfaced many times in this study. There is a lack of

convergence if a hurdle rate based on systematic risk factors is used for economic

capital that is defined on a total risk perspective. Despite the importance of the internal

hurdle rate in pricing, performance measurement and incentive alignment, in a practical

setting, there appears to be very little nexus between hurdle rates and economic capital.

The two, however, should be aligned if the aim is a consistent treatment of risk. One

possibility is that commoditised bank products incorporate hurdle rates based on

systematic factors given their greater liquidity potential and scope for securitisation,

while products that are more bank-specific incorporate hurdle rates that reflect total risk

considerations. This could also be an area worthy of further research.

All performance models implicitly assume that the assessor can infer information about

the effort and skills of managers by referring to the RAPM for the positions under their

control. The internal capital market mechanism developed in chapter five required that

capital actually utilised by an investment be measurable, expected earnings on the

investment verifiable, and that managers are technically capable of calculating risk

capital requirements - according to the DSD measure - based on their private

expectations on the distribution of returns. The notion of capital utilisation can be taken

from two perspectives – capital utilised can reflect the capital assigned to a position ex-

ante, or it can be based on the ex-post volatility in returns over the measurement

horizon. The first case is based on the notion that a position absorbs some proportion of

the capital of a bank, to protect against unexpected losses, irrespective of the ensuing

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actual volatility in returns. In the second case, the actual volatility in returns forms the

basis for measuring the capital that has been utilised, which may be greater or smaller

than the capital allocated to a position ex-ante.

Regardless of the approach use to define capital utilisation, it needs to be recognised

that the risk attitude of managers may change over the measurement period subject to

their perceptions of actual or likely performance of positions relative to the target or

aspiration level upon which their performance is judged. The theoretical literature and

empirical studies related to prospect theory, that were reviewed in chapter two, suggest

that individuals become risk-seeking when perceived to be operating below target.

While our framework provides an alignment of incentives and risk congruence between

the centre and managers at the time that investment decisions are made, it is essential

that the mechanism does not tie managers to their initial expectations in such a way that

they are discouraged from revealing new information impacting on the performance of

their portfolios that subsequently comes to light. That is, if positions subsequently

deteriorate, our mechanism may have the potential to encourage risk-seeking behaviour

to the extent that managers are forced to stand by the initial expectations on risk –

whether or not deterioration is due to factors outside of the control of managers. To

avoid managers becoming risk-seeking in order to achieve their performance targets,

and the potential negative consequences for the overall bank of such a change in risk

attitude, the mechanism must embed sufficient flexibility to encourage managers to

reveal new information on risks in their positions as it becomes available, and allow

them to be proactive in managing or restructuring these positions in light of the

information. The private and specialised information of managers should be

incorporated into portfolio management as an ongoing process, and not restricted only

to portfolio selection decisions.

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329

Appendices

330

APPENDIX 1

Table 1

Testing for Prospect Stochastic Dominance and Markowitz Stochastic Dominance

Probability Distributions

Cumulative Probabilities

Sum of Cumulative Probabilities

Differences

Payoff A B A1 B1 A2 B2 (A2 – B2) (B2 – A2)

-5 20% 0% 20% 0% 20% 0% 20% -20%

10 0% 80% 20% 80% 40% 80% -40% 40%

20 80% 0% 100% 80% 140% 160% -20% 20%

35 0% 20% 100% 100% 240% 260% -20% 20%

Mean 15 15 A PSD B B MSD A

Portfolio A dominates B by Prospect Stochastic Dominance where the reference point is a zero payoff. For payoffs that are below the reference point, (A2 – B2)

� 0. This means greater cumulative probability

weight in the domain of losses for Portfolio A relative to B, and corresponds to a risk-seeking preference in the domain of losses. For payoffs that are above the reference point, (A2 – B2)

✁ 0. This

means lower cumulative probability weight in the domain of gains for Portfolio A relative to B, and corresponds to risk-averting preferences in the domain of gains. We conclude that Portfolio A dominates B by PSD. In the case of (B2 – A2), the reverse holds. Under this condition, and given the equality of the means of portfolios A and B, we conclude that Portfolio B dominates A by MSD.

331

APPENDIX 2

VaR, Expected Shortfall and First-order Lower Partial Moment

Consider the following distributions for portfolios A and D:

Probability Distributions

Loss Beyond VaR (-X-VaR)

Portfolio A D A D

Portfolio Value

30 0.4% 60

60 1% 0

90 0.6% 0

97 5% 5%

98 15% 10%

99 30% 40%

100 40% 41%

101 5% 2%

102 3%

103 1%

106 1%

Expected value 98.99 98.99

Value-at-Risk

VaR � = E(V) - Vp

VaR 99% (A) = 98.99 – 60 = 38.99

VaR 99% (D) = 98.99 – 90 = 8.99

Expected Shortfall

ES�(X) = VaR�(X) + (1 - ✁)-1 E {max [–x – VaR�(X), 0]},

ES 99% (A) = 38.99 + [(1 – 0.99)-1 (0 x 1%)] = 38.99

ES 99% (D) = 8.99 + [(1 – 0.99)-1 [(60 x 0.4%) + (0 x 0.6%)]] = 32.99

First-Order Lower Partial Moment

LPM(1)�(X) = (1 - ✁)-1 E {max [t – x, 0]},

LPM1,99% (A) = (1 – 0.99)-1 [(98.99 – 60) x 1%] = 38.99

LPM1,99% (D) = (1 – 0.99)-1 [((98.99 – 30) x 0.4%) + ((98.99 – 90) x 0.6%)] = 32.99

332

APPENDIX 3

Table 1

Mean Preserving Spread: Portfolio A � B

Portfolio A B

Market Value

$0

30

60 1% 1%

90

97 5% +2.5% 7.5%

98

15%

-2.5% -2.5%

10%

99 30% +2.5% 32.5%

100 40% 40%

101 5% 5%

102 3% 3%

103 1% 1%

104

105

106

Expected value $98.99 $98.99

NOTES Portfolio B is created as a mean preserving spread of Portfolio A by removing some of the probability weight from a point in A and distributing it to the tails in such a way as to leave the mean of the portfolio unchanged. Specifically, probability weight of 5 percentage points is moved from the point of portfolio value of $98 and distributed in equal proportions to the points of $97 and $99. Note that the mean-preserving spread does not influence the probability weight at the extreme left tail of the portfolio distributions. This implies that risk measures based on loss thresholds that are not impacted by the mean-preserving spread will fail to capture the larger intermediate losses that are below the loss threshold. This is examined further in the body of the chapter.

333

APPENDIX 3

Table 2

Stochastic Dominance: Analysis of Portfolio B � A

Probability

Distribution

Cumulative Probability

Difference

Sum of Cumulative Probability

Difference

Portfolio A B A1 B1 (B1 – A1) A2 B2 (B2 – A2)

Market Value

0

30

60 1% 1% 1% 1% 1% 1%

90 1% 1% 31% 31%

97 5% 7.5% 6% 8.5% 2.5% 43% 45.5% 2.5%

98 15% 10% 21% 18.5% -2.5% 64% 64%

99 30% 32.5% 51% 51% 115% 115%

100 40% 40% 91% 91% 206% 206%

101 5% 5% 96% 96% 302% 302%

102 3% 3% 99% 99% 401% 401%

103 1% 1% 100% 100% 501% 501%

104 100% 100% 601% 601%

105 100% 100% 701% 701%

106 100% 100% 801% 801%

Expected value 98.99 98.99 No FSD A SSD B

NOTES Portfolio A does not dominate B by first-order stochastic dominance (FSD) because the cumulative probability distribution functions cross. Portfolio A dominates B by second-order stochastic dominance (A SSD B) because the sum of the cumulative distribution functions do not cross, i.e. (B2 – A2)

✁ 0.

334

APPENDIX 4

Table 1

Mean Preserving Spread: Portfolio B � C

Portfolio B C

Market Value

$0

30

60 1% 1%

90 +2.5% 2.5%

97 7.5% -2.5% 5%

98 10% 10%

99 32.5% -2.5% 30%

100 40% 40%

101 5% 5%

102 3% 3%

103 1% 1%

104

105

106 +2.5% 2.5%

Expected value $98.99 $98.99

NOTES Portfolio C is created as a mean preserving spread of Portfolio B by removing some of the probability weight from two points in A and distributing it to the tails in such a way as to leave the mean of the portfolio unchanged. Specifically, probability weights of 2.5 percentage points are moved from the points of $97 and $99 respectively, and distributed to the points of $90 and $106. Note again that the mean-preserving spread does not influence the probability weight at the extreme left tail of the distributions. This implies that risk measures based on loss thresholds that are not impacted by the mean-preserving spread will fail to incorporate the larger intermediate losses that fall below the loss threshold.

335

APPENDIX 4

Table 2

First Order Stochastic Dominance: Analysis of Portfolio C � A, B

Probability Distributions Cumulative Probability Differences

Portfolio A B C A1 B1 C1 (C1 – A1) (C1 – B1)

Market Value

0

30

60 1% 1% 1% 1% 1% 1%

90 2.5% 1% 1% 3.5% 2.5% 2.5%

97 5% 7.5% 5% 6% 8.5% 8.5% 2.5%

98 15% 10% 10% 21% 18.5% 18.5% -2.5%

99 30% 32.5% 30% 51% 51% 48.5% -2.5% -2.5%

100 40% 40% 40% 91% 91% 88.5% -2.5% -2.5%

101 5% 5% 5% 96% 96% 93.5% -2.5% -2.5%

102 3% 3% 3% 99% 99% 96.5% -2.5% -2.5%

103 1% 1% 1% 100% 100% 97.5% -2.5% -2.5%

104 100% 100% 97.5% -2.5% -2.5%

105 100% 100% 97.5% -2.5% -2.5%

106 2.5% 100% 100% 100%

Expected value 98.99 98.99 98.99 No FSD No FSD

NOTES Portfolio C does not dominate either Portfolios A or B by first-order stochastic dominance because the cumulative probability distribution function of C crosses both A and B.

336

APPENDIX 4

Table 3

Second Order Stochastic Dominance: Analysis of Portfolio C � A, B

Probability Distributions Sum of Cumulative Probability

Differences

Portfolio A B C A2 B2 C2 (C2 – A2) (C2 – B2)

Market Value

0

30

60 1% 1% 1% 1% 1% 1%

90 2.5% 31% 31% 33.5% 2.5% 2.5%

97 5% 7.5% 5% 43% 45.5% 63% 20% 17.5%

98 15% 10% 10% 64% 64% 81.5% 17.5% 17.5%

99 30% 32.5% 30% 115% 115% 130% 15% 15%

100 40% 40% 40% 206% 206% 218.5% 12.5% 12.5%

101 5% 5% 5% 302% 302% 312% 10% 10%

102 3% 3% 3% 401% 401% 408.5% 7.5% 7.5%

103 1% 1% 1% 501% 501% 506% 5% 5%

104 601% 601% 603.5% 2.5% 2.5%

105 701% 701% 701%

106 2.5% 801% 801% 801%

Expected value 98.99 98.99 98.99 A SSD C B SSD C

NOTES Portfolio A dominates C by second-order stochastic dominance (A SSD C) because the sum of the cumulative distribution functions do not cross, i.e. (C2 – A2)

✁ 0.

Portfolio B dominates C by second-order stochastic dominance (B SSD C) because the sum of the cumulative distribution functions do not cross, i.e. (C2 – B2)

✁ 0.

337

APPENDIX 4

Table 4

Third Order Stochastic Dominance: Analysis of Portfolio C � A, B

Probability Distributions

Sum of Cumulative Probabilities

Differences

Portfolio A B C A3 B3 C3 (B3 – A3) (C3 – A3) (C3 – B3)

Market

Value

0

30

60 1% 1% 1% 1% 1% 1% 0% 0% 0%

90 2.5% 496% 496% 498.5% 0% 2.5% 2.5%

97 5% 7.5% 5% 746% 748.5% 836% 2.5% 90% 87.5%

98 15% 10% 10% 810% 812.5% 917.5% 2.5% 107.5% 105%

99 30% 32.5% 30% 925% 927.5% 1047.5% 2.5% 122.5% 120%

100 40% 40% 40% 1131% 1133.5% 1266% 2.5% 135% 132.5%

101 5% 5% 5% 1433% 1435.5% 1578% 2.5% 145% 142.5%

102 3% 3% 3% 1834% 1836.5% 1986.5% 2.5% 152.5% 150%

103 1% 1% 1% 2335% 2337.5% 2492.5% 2.5% 157.5% 155%

104 2936% 2938.5% 3096% 2.5% 160% 157.5%

105 3637% 3639.5% 3797% 2.5% 160% 157.5%

106 2.5% 4438% 4440.5% 4598% 2.5% 160% 157.5%

Expected

Value

98.99 98.99 98.99 A TSD B A TSD C B TSD C

NOTES

Portfolio A dominates B by third-order stochastic dominance (A TSD B) because (B3 – A3) ✁

0.

Portfolio A dominates C by third-order stochastic dominance (A TSD C) because (C3 – A3) ✁

0.

Portfolio B dominates C by third-order stochastic dominance (B TSD C) because (C3 – B3) ✁

0.

338

APPENDIX 5

Table 1

First Order Stochastic Dominance: Analysis of Portfolio D � A, B, C

Probability Distributions Cumulative Probability Differences

Portfolio A B C D A1 B1 C1 D1 (D1 – A1) (D1 – B1) (D1 – C1)

Market

Value

0

30 0.4% 0.4% 0.4% 0.4% 0.4%

60 1% 1% 1% 1% 1% 1% 0.4% -0.6% -0.6% -0.6%

90 2.5% 0.6% 1% 1% 3.5% 1% 0% 0% -2.5%

97 5% 7.5% 5% 5% 6% 8.5% 8.5% 6% 0% -2.5% -2.5%

98 15% 10% 10% 10% 21% 18.5% 18.5% 16% -5% -2.5% -2.5%

99 30% 32.5% 30% 40% 51% 51% 48.5% 56% 5% 5% 7.5%

100 40% 40% 40% 41% 91% 91% 88.5% 97% 6% 6% 8.5%

101 5% 5% 5% 2% 96% 96% 93.5% 99% 3% 3% 5.5%

102 3% 3% 3% 99% 99% 96.5% 99% 0% 0% 2.5%

103 1% 1% 1% 100% 100% 97.5% 99% -1% -1% 1.5%

104 100% 100% 97.5% 99% -1% -1% 1.5%

105 100% 100% 97.5% 99% -1% -1% 1.5%

106 2.5% 1% 100% 100% 100% 100% 0% 0% 0%

Expected

Value

98.99 98.99 98.99 98.99 No FSD No FSD No FSD

NOTES Portfolios A, B and C do not dominate portfolio D by first -order stochastic dominance because the cumulative distribution function of portfolio D crosses A, B and C.

339

APPENDIX 5

Table 2

Second Order Stochastic Dominance: Analysis of Portfolio D � A, B, C

Probability Distributions Sum of Cumulative Probability Differences

Portfolio A B C D A2 B2 C2 D2 (D2 – A2) (D2 – B2) (D2 – C2)

Market

Value

0

30 0.4% 0.4% 0.4% 0.4% 0.4%

60 1% 1% 1% 1% 1% 1% 12.4% 11.4% 11.4% 11.4%

90 2.5% 0.6% 31% 31% 33.5% 25% -6% -6% -8.5%

97 5% 7.5% 5% 5% 43% 45.5% 63% 37% -6% -8.5% -26%

98 15% 10% 10% 10% 64% 64% 81.5% 53% -11% -11% -28.5%

99 30% 32.5% 30% 40% 115% 115% 130% 109% -6% -6% -21%

100 40% 40% 40% 41% 206% 206% 218.5% 206% 0% 0% -12.5%

101 5% 5% 5% 2% 302% 302% 312% 305% 3% 3% -7%

102 3% 3% 3% 401% 401% 408.5% 404% 3% 3% 4.5%

103 1% 1% 1% 501% 501% 506% 503% 2% 2% -3%

104 601% 601% 603.5% 602% 1% 1% -1.5%

105 701% 701% 701% 701% 0% 0% 0%

106 2.5% 1% 801% 801% 801% 801% 0% 0% 0%

Expected

Value

98.99 98.99 98.99 98.99 No SSD No SSD No SSD

NOTES Portfolios A, B and C do not dominate D by second-order stochastic dominance because the sum of each of the cumulative distribution functions cross.

340

APPENDIX 5

Table 3

Third Order Stochastic Dominance: Analysis of Portfolio D � A, B, C

Probability Distributions Sum of Cumulative Probabilities Differences

Portfolio A B C D A3 B3 C3 D3 (D3 – A3) (D3 – B3) (D3 – C3)

Market

Value

0

30 0.4% 0.4% 0.4% 0.4% 0.4%

60 1% 1% 1% 1% 1% 1% 198.4% 197.4% 197.4% 197.4%

90 2.5% 0.6% 496% 496% 498.5% 757% 261% 261% 258.5%

97 5% 7.5% 5% 5% 746% 748.5% 836% 965% 219% 216.5% 129%

98 15% 10% 10% 10% 810% 812.5% 917.5% 1018% 208% 205.5% 100.5%

99 30% 32.5% 30% 40% 925% 927.5% 1047.5% 1127% 202% 199.5% 79.5%

100 40% 40% 40% 41% 1131% 1133.5% 1266% 1333% 202% 199.5% 67%

101 5% 5% 5% 2% 1433% 1435.5% 1578% 1638% 205% 202.5% 60%

102 3% 3% 3% 1834% 1836.5% 1986.5% 2042% 208% 205.5% 55.5%

103 1% 1% 1% 2335% 2337.5% 2492.5% 2545% 210% 207.5% 52.5%

104 2936% 2938.5% 3096% 3147% 211% 208.5% 51%

105 3637% 3639.5% 3797% 3848% 211% 208.5% 51%

106 2.5% 1% 4438% 4440.5% 4598% 4649% 211% 208.5% 51%

Expected

Value

98.99 98.99 98.99 98.99 A TSD D B TSD D C TSD D

NOTES Portfolio A dominates D by third-order stochastic dominance (A TSD D) because (D3 – A3)

✁ 0.

Portfolio B dominates D by third-order stochastic dominance (B TSD D) because (D3 – B3) ✁

0.

Portfolio C dominates D by third-order stochastic dominance (C TSD D) because (D3 – C3) ✁

0.

341

APPENDIX 6

Table 1

First Order Stochastic Dominance: Analysis of Portfolio E � A, B, C, D

Probability Distributions Cumulative Probability

Portfolio A B C D E A1 B1 C1 D1 E1

Market

Value

0

30 0.4% 0.5% 0.4% 0.5%

60 1% 1% 1% 1% 1% 1% 0.4% 0.5%

90 2.5% 0.6% 0.5% 1% 1% 3.5% 1% 1%

97 5% 7.5% 5% 5% 10% 6% 8.5% 8.5% 6% 11%

98 15% 10% 10% 10% 6% 21% 18.5% 18.5% 16% 17%

99 30% 32.5% 30% 40% 25% 51% 51% 48.5% 56% 42%

100 40% 40% 40% 41% 53% 91% 91% 88.5% 97% 95%

101 5% 5% 5% 2% 4% 96% 96% 93.5% 99% 99%

102 3% 3% 3% 1% 99% 99% 96.5% 99% 100%

103 1% 1% 1% 100% 100% 97.5% 99% 100%

104 100% 100% 97.5% 99% 100%

105 100% 100% 97.5% 99% 100%

106 2.5% 1% 100% 100% 100% 100% 100%

Expected

Value

98.99 98.99 98.99 98.99 98.99

Differences

Portfolio (E1 – A1) (E1 – B1) (E1 – C1) (E1 – D1)

Market

Value

0

30 0.5% 0.5% 0.5% 0.1%

60 -0.5% -0.5% -0.5% 0.1%

90 0% 0% -2.5% 0%

97 5% 2.5% 2.5% 5%

98 -4% -1.5% -1.5% 1%

99 -9% -9% -6.5% -14%

100 4% 4% 6.5% -2%

101 3% 3% 5.5% 0%

102 1% 1% 3.5% 1%

103 0% 0% 2.5% 1%

104 0% 0% 2.5% 1%

105 0% 0% 2.5% 1%

106 0% 0% 0% 0%

No FSD No FSD No FSD No FSD

342

APPENDIX 6

Table 2

Second Order Stochastic Dominance: Analysis of Portfolio E � A, B, C, D

Probability Distributions Sum of Cumulative Probability

Portfolio A B C D E A2 B2 C2 D2 E2

Market

Value

0

30 0.4% 0.5% 0.4% 0.5%

60 1% 1% 1% 1% 1% 1% 12.4% 15.5%

90 2.5% 0.6% 0.5% 31% 31% 33.5% 25% 31%

97 5% 7.5% 5% 5% 10% 43% 45.5% 63% 37% 48%

98 15% 10% 10% 10% 6% 64% 64% 81.5% 53% 65%

99 30% 32.5% 30% 40% 25% 115% 115% 130% 109% 107%

100 40% 40% 40% 41% 53% 206% 206% 218.5% 206% 202%

101 5% 5% 5% 2% 4% 302% 302% 312% 305% 301%

102 3% 3% 3% 1% 401% 401% 408.5% 404% 401%

103 1% 1% 1% 501% 501% 506% 503% 501%

104 601% 601% 603.5% 602% 601%

105 701% 701% 701% 701% 701%

106 2.5% 1% 801% 801% 801% 801% 801%

Expected

Value

98.99 98.99 98.99 98.99 98.99

Differences

Portfolio (E2 – A2) (E2 – B2) (E2 – C2) (E2 – D2)

Market

Value

0

30 0.5% 0.5% 0.5% 0.1% Loss

60 14.5% 14.5% 14.5% 3.1% Domain

90 0% 0% 2.5% 6%

97 5% 2.5% -15% 11%

98 1% 1% -16.5% 12%

99 -8% -8% -23% -2%

100 -4% -4% -16.5% -4% Gain 101 -1% -1% -11% -4% Domain 102 0% 0% -7.5% -3%

103 0% 0% -5% -2%

104 0% 0% -2.5% -1%

105 0% 0% 0% 0%

106 0% 0% 0% 0%

No SSD No SSD No SSD No SSD

A MSD E B MSD E No MSD D MSD E

E PSD A E PSD B No PSD E PSD D

343

APPENDIX 6

Table 3

Third Order Stochastic Dominance: Analysis of Portfolio E � A, B, C, D

Probability Distributions Sum of Cumulative Probabilities

Portfolio A B C D E A3 B3 C3 D3 E3

Market

Value

0

30 0.4% 0.5% 0% 0% 0% 0.4% 0.5%

60 1% 1% 1% 1% 1% 1% 198.4% 248%

90 2.5% 0.6% 0.5% 496% 496% 498.5% 757% 946%

97 5% 7.5% 5% 5% 10% 746% 748.5% 836% 965% 1201%

98 15% 10% 10% 10% 6% 810% 812.5% 917.5% 1018% 1266%

99 30% 32.5% 30% 40% 25% 925% 927.5% 1047.5% 1127% 1373%

100 40% 40% 40% 41% 53% 1131% 1133.5% 1266% 1333% 1575%

101 5% 5% 5% 2% 4% 1433% 1435.5% 1578% 1638% 1876%

102 3% 3% 3% 1% 1834% 1836.5% 1986.5% 2042% 2277%

103 1% 1% 1% 2335% 2337.5% 2492.5% 2545% 2778%

104 2936% 2938.5% 3096% 3147% 3379%

105 3637% 3639.5% 3797% 3848% 4080%

106 2.5% 1% 4438% 4440.5% 4598% 4649% 4881%

Expected

Value

98.99 98.99 98.99 98.99 98.99

Differences

Portfolio (E3 – A3) (E3 – B3) (E3 – C3) (E3 – D3)

Market

Value

0

30 0.5% 0.5% 0.5% 0.1%

60 247% 247% 247% 49.6%

90 450% 450% 447.5% 189%

97 455% 452.5% 365% 236%

98 456% 453.5% 348.5% 248%

99 448% 445.5% 325.5% 246%

100 444% 441.5% 309% 242%

101 443% 440.5% 298% 238%

102 443% 440.5% 290.5% 235%

103 443% 440.5% 285.5% 233%

104 443% 440.5% 283% 232%

105 443% 440.5% 283% 232%

106 443% 440.5% 283% 232%

A TSD E B TSD E C TSD E D TSD E

344

APPENDIX 7

Value-at-Risk

VaR � = E(V) - Vp

VaR 99% (A) = 98.99 – 60 = 38.99

VaR 99% (B) = 98.99 – 60 = 38.99

VaR 99% (C) = 98.99 – 60 = 38.99

VaR 99% (D) = 98.99 – 90 = 8.99

VaR 99% (E) = 98.99 – 90 = 8.99

Expected Shortfall

ES�(X) = VaR�(X) + (1 - ✁)-1 E {max [–x – VaR�(X), 0]},

ES 99% (A) = 38.99 + [(1 – 0.99)-1 (0 x 1%)] = 38.99

ES 99% (B) = 38.99 + [(1 – 0.99)-1 (0 x 1%)] = 38.99

ES 99% (C) = 38.99 + [(1 – 0.99)-1 (0 x 1%)] = 38.99

ES 99% (D) = 8.99 + [(1 – 0.99)-1 [(60 x 0.4%) + (0 x 0.6%)]] = 32.99

ES 99% (E) = 8.99 + [(1 – 0.99)-1 [(60 x 0.5%) + (0 x 0.5%)]] = 38.99

345

APPENDIX 8

Risk Attitude Implicit in Order of Lower Partial Moments

Loss Portfolio U Probability

Portfolio V Probability

1 10% 0%

5 0% 42%

10 20% 0%

Expected loss 2.1 2.1

LPM order LPM: U LPM V Selection

0.0 0.300 0.420 U

0.5 0.732 0.939 U

0.9 1.689 1.788 U

1.0 2.100 2.100 Indifferent

1.5 6.425 4.696 V

2.0 20.100 10.500 V

NOTES Portfolios U and V have the same expected loss, but Portfolio U has a larger tail loss. Portfolio U has the lowest value for the LPM when the order n is such that: (0

� n < 1). An investor wishing to

maximise the return on the risk measure would thus select portfolio U, despite its larger tail risk. This indicates that the LPM of order (0

� n < 1) carries an implicit risk-seeking attitude. The LPM of order 1

does not discriminate between the portfolios because the average losses are the same. This indicates that LPM of order 1 carries an implicit risk-neutral attitude in losses. Portfolio V has the lowest value for the LPM when the order is greater than 1. An investor wishing to maximise return on the risk measure would select portfolio V, which carries the lower tail risk of the portfolios. This indicates that LPM of order n > 1 carries an implicit risk-averse attitude to losses.

346

APPENDIX 9

Spectral Measure of Risk: Portfolios A – E

Probability Distribution Loss deviations

Market value

Losses

A

B

C

Order n

(DevA)n

(DevB)n

(DevC)n

30 68.99 2.5

60 38.99 1% 1% 1% 2 15.2022 15.2022 15.2022

90 8.99 2.5% 1.5 0.6739

97 1.99 5% 7.5% 5% 1.2 0.1142 0.1713 0.1142

98 0.99 15% 10% 10% 1 0.1485 0.0990 0.0990

98.99 Threshold

100

106

Spectral risk measure 15.4649 15.4725 16.0893

Probability Distribution Loss Deviations

Market value

Losses

D

E

Order n

(DevD)n

(DevE)n

30 68.99 0.4% 0.5% 2.5 158.134 197.6675

60 38.99 2

90 8.99 0.6% 0.5% 1.5 0.1617 0.1348

97 1.99 5% 10% 1.2 0.1142 0.2284

98 0.99 10% 6% 1 0.0990 0.0594

98.99 Threshold

100

106

Spectral risk measure 158.5089 198.0901

NOTES Losses are measured as deviations from the loss threshold of $98.99, which is the expected value of each portfolio. The expected loss for each portfolio is $1.01, being the difference between the face value of each portfolio of $100 and the expected value.

347

APPENDIX 10

Wang Transform (WT) Risk Measures: Portfolios A - E

Portfolio A

Loss (xi) P(xi) F(Xi) F*(xi) F*(xi) – F*(xi-1) Specifications

0 79.0% 79.0% 6.4% 6.4% � = 99%

0.99 15.0% 94.0% 22.0% 15.6% ✁ = ✂-1(0.99) = 2.326

1.99 5.0% 99.0% 50.0% 28.0%

38.99 1.0% 100.0% 100.0% 50.0%

WT (99%) 20.206

Portfolio B

Loss (xi) P(xi) F(Xi) F*(xi) F*(xi) – F*(xi-1) Specifications

0 81.5% 81.5% 7.6% 7.6% � = 99%

0.99 10.0% 91.5% 17.0% 9.4% ✁ = ✂-1(0.99) = 2.326

1.99 7.5% 99.0% 50.0% 33.0%

38.99 1.0% 100.0% 100.0% 50.0%

WT (99%) 20.244

Portfolio C

Loss (xi) P(xi) F(Xi) F*(xi) F*(xi) – F*(xi-1) Specifications

0 81.5% 81.5% 7.6% 7.6% � = 99%

0.99 10.0% 91.5% 17.0% 9.4% ✁ = ✂-1(0.99) = 2.326

1.99 5.0% 96.5% 30.4% 13.4%

8.99 2.5% 99.0% 50.0% 19.6%

38.99 1.0% 100.0% 100.0% 50.0%

WT (99%) 21.619

Portfolio D

Loss (xi) P(xi) F(Xi) F*(xi) F*(xi) – F*(xi-1) Specifications

0 84.0% 84.0% 9.1% 9.1% � = 99%

0.99 10.0% 94.0% 22.0% 12.9% ✁ = ✂-1(0.99) = 2.326

1.99 5.0% 99.0% 50.0% 28.0%

8.99 0.6% 99.6% 62.7% 12.7%

68.99 0.4% 100.0% 100.0% 37.3%

WT (99%) 27.557

348

Appendix 10 (Continued)

Portfolio E

Loss (xi) P(xi) F(Xi) F*(xi) F*(xi) – F*(xi-1) Specifications

0 83.0% 83.0% 8.5% 8.5% � = 99%

0.99 6.0% 89.0% 13.6% 5.1% ✁ = ✂-1(0.99) = 2.326

1.99 10.0% 99.0% 50.0% 36.4%

8.99 0.5% 99.5% 60.0% 9.9%

68.99 0.5% 100.0% 100.0% 40.1%

WT (99%) 29.347

NOTES The transformation function takes the original cumulative probability function F(Xi) and transforms this through a standard normal inverse transformation to obtain

✂-1(F(x)). The risk-aversion parameter ✁ is

then subtracted and the resulting expression again transformed through a standard normal transformation to achieve the distorted cumulative probability function F*(xi). For a loss variable X with discrete distribution F, the WT risk measure is derived as follows (refer Wang 2002, p.6):

1. For confidence level �, let ✁ = ✂-1(�).

2. Apply the Wang transform: F* (x) = ✂

[✂-1(F(x)) –

✁]

3. Calculate the risk measure as the expected value under F*.

349

APPENDIX 11

Positive Homogeneity of LPMn Risk Measures

The following calculations compare the LPMn of a portfolio that represents the doubling of an investment in portfolio A (a scalar of 2) with the sum of the LPMn of two individual investments in portfolio A. Positive homogeneity (PH) for the risk measure holds if LPMn + LPMn = 2LPMn.

LPM Order LPM0 LPM0.5 LPM1 LPM2

LPM Individual Investment 1 0.210 0.282 0.638 15.547

LPM Individual investment 2 0.210 0.282 0.638 15.547

LPM Sum of individual investments

0.420 0.564 1.276 31.094

LPM Portfolio (scalar = 2) 0.210 0.399 1.276 62.189

Comments Not PH Not PH PH holds Not PH

NOTES The results show that the only measure that is positive homogenous is LPM1. The risk measures for shortfall probability and LPM0.5 are lower for the portfolio that represents a doubling of the investment in the individual portfolios, while the risk measure for LPM2 is larger for the portfolio that represents a doubling of the investment in the individual portfolios. The only LPM measure that is positive homogenous is LPM1.

350

APPENDIX 12

Positive Homogeneity of the DSD Risk Measure

The following calculations compare the DSD of a portfolio that represents the doubling of an investment in portfolio A (a scalar of 2) with the sum of the DSD of two individual investments in portfolio A. Positive homogeneity (PH) for the risk measure holds if DSDA + DSDA = 2DSDA.

DSD

Individual Investment 1 (Port A) 3.943

Individual investment 2 (Port A) 3.943

DSD Sum of individual investments 7.886

DSD Portfolio (scalar = 2) 7.886

Comments DSD is PH

351

APPENDIX 13

Table 1

First-Order Stochastic Dominance by RAPM (Gain/DSD): Analysis of Portfolios A to E

Probability Distributions Cumulative Probability

Portfolio A B C D E A1 B1 C1 D1 E1

RAPM

0% 91% 91% 88.5% 97% 95% 91% 91% 88.5% 97% 95%

20.1% 4% 91% 91% 88.5% 97% 99%

22.5% 2% 91% 91% 88.5% 99% 99%

23.9% 5% 91% 91% 93.5% 99% 99%

25.3% 5% 91% 96% 93.5% 99% 99%

25.4% 5% 96% 96% 93.5% 99% 99%

40.3% 1% 96% 96% 93.5% 99% 100%

47.8% 3% 96% 96% 96.5% 99% 100%

50.6% 3% 96% 99% 96.5% 99% 100%

50.7% 3% 99% 99% 96.5% 99% 100%

71.7% 1% 99% 99% 97.5% 99% 100%

76.0% 1% 99% 100% 97.5% 99% 100%

76.1% 1% 100% 100% 97.5% 99% 100%

134.8% 1% 100% 100% 97.5% 100% 100%

143.6% 2.5% 100% 100% 100% 100% 100%

Differences

Portfolio B1-A1 C1-A1 D1-A1 E1-A1 C1-B1 D1-B1 E1-B1 D1-C1 E1-C1 E1-D1

RAPM

0% 0% -2.5% 6% 4% -2.5% 6% 4% 8.5% 6.5% -2%

20.1% 0% -2.5% 6% 8% -2.5% 6% 8% 8.5% 10.5% 2%

22.5% 0% -2.5% 8% 8% -2.5% 8% 8% 10.5% 10.5% 0%

23.9% 0% 2.5% 8% 8% 2.5% 8% 8% 5.5% 5.5% 0%

25.3% 5% 2.5% 8% 8% -2.5% 3% 3% 5.5% 5.5% 0%

25.4% 0% -2.5% 3% 3% -2.5% 3% 3% 5.5% 5.5% 0%

40.3% 0% -2.5% 3% 4% -2.5% 3% 4% 5.5% 6.5% 1%

47.8% 0% 0.5% 3% 4% 0.5% 3% 4% 2.5% 3.5% 1%

50.6% 3% 0.5% 3% 4% -2.5% 0% 1% 2.5% 3.5% 1%

50.7% 0% -2.5% 0% 1% -2.5% 0% 1% 2.5% 3.5% 1%

71.7% 0% -1.5% 0% 1% -1.5% 0% 1% 1.5% 2.5% 1%

76.0% 1% -1.5% 0% 1% -2.5% -1% 0% 1.5% 2.5% 1%

76.1% 0% -2.5% -1% 0% -2.5% -1% 0% 1.5% 2.5% 1%

134.8% 0% -2.5% 0% 0% -2.5% 0% 0% 2.5% 2.5% 0%

143.6% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

A � B No FSD No FSD A � E No FSD No FSD B � E C � D C � E No FSD

352

APPENDIX 13

Table 2

Second-Order Stochastic Dominance by RAPM (Gain/DSD): Analysis of Portfolios A to E

Probability Distributions Sum of Cumulative Probability

Portfolio A B C D E A2 B2 C2 D2 E2

RAPM

0% 91% 91% 88.5% 97.0% 95% 91% 91% 88.5% 97% 95%

20.1% 4% 1729% 17295 1681.5% 1843% 1809%

22.5% 2% 2002% 2002% 1947% 2136% 2106%

23.9% 5% 2093% 2093% 2040.5% 2235% 2205%

25.3% 5% 2184% 2189% 2134% 2334% 2304%

25.4% 5% 2280% 2285% 2227.5% 2433% 2403%

40.3% 1% 3432% 3437% 3349.5% 3621% 3592%

47.8% 3% 4200% 4205% 4100.5% 4413% 4392%

50.6% 3% 4296% 4304% 4197% 4512% 4492%

50.7% 3% 4395% 4403% 4293.5% 4611% 4592%

71.7% 1% 6276% 6284% 6128% 6492% 6492%

76.0% 1% 6573% 6582% 6420.5% 6789% 6792%

76.1% 1% 6673% 6682% 6518% 6888% 6892%

134.8% 1% 12073% 12082% 11783% 12235% 12292%

143.6% 2.5% 12773% 12782% 12468% 12935% 12992%

Differences

Portfolio B2-A2 C2-A2 D2-A2 E2-A2 C2-B2 D2-B2 E2-B2 D2-C2 E2-C2 E2-D2

RAPM

0% 0% -2.5% 6% 4% -2.5% 6% 4% 8.5% 6.5% -2%

20.1% 0% -47.5% 114% 80% -47.5% 114% 80% 161.5% 127.5% -34%

22.5% 0% -55% 134% 104% -55% 134% 104% 189% 159% -30%

23.9% 0% -52.5% 142% 112% -52.5% 142% 112% 194.5% 164.5% -30%

25.3% 5% -50% 150% 120% -55% 145% 115% 200% 170% -30%

25.4% 5% -52.5% 153% 123% -57.5% 148% 118% 205.5% 175.5% -30%

40.3% 5% -82.5% 189% 160% -87.5% 184% 155% 271.5% 242.5% -29%

47.8% 5% -99.5% 213% 192% -104.5% 208% 187% 312.5% 288% -22%

50.6% 8% -99% 216% 196% -107% 208% 188% 315% 291.5% -21%

50.7% 8% -101.5% 216% 197% -109.5% 208% 189% 317.5% 295% -20%

71.7% 8% -148% 216% 216% -156% 208% 208% 364% 364% 0%

76.0% 9% -152.5% 216% 219% -161.5% 207% 210% 368.5% 371.5% 3%

76.1% 9% -155% 215% 219% -164% 206% 210% 370% 374% 4%

134.8% 9% -290% 162% 219% -299% 153% 210% 452% 509% 57%

143.6% 9% -305% 162% 219% -314% 153% 210% 467% 524% 57%

A � B C � A A � D A � E C � B B � D B � E C � D C � E No SSD

353

APPENDIX 14

Table 1

First-Order Stochastic Dominance by RAPM (Gain/DSD): Analysis of Portfolios A to E

Probability Distributions Cumulative Probability

Portfolio A B C D E A1 B1 C1 D1 E1

RAPM

0% 21% 18.5% 18.5% 16% 17% 21% 18.5% 18.5% 16% 17%

0.2% 30% 32.5% 30% 40% 25% 51% 51% 48.5% 56% 42%

20.3% 53% 51% 51% 48.5% 56% 95%

22.7% 41% 51% 51% 48.5% 97% 95%

24.1% 40% 51% 51% 88.5% 97% 95%

25.6% 40% 40% 91% 91% 88.5% 97% 95%

40.5% 4% 91% 91% 88.5% 97% 99%

45.1% 2% 91% 91% 88.5% 99% 99%

48.0% 5% 91% 91% 93.5% 99% 99%

50.9% 5% 91% 96% 93.5% 99% 99%

51.0% 5% 96% 96% 93.5% 99% 99%

60.6% 1% 96% 96% 93.5% 99% 100%

71.9% 3% 96% 96% 96.5% 99% 100%

76.2% 3% 96% 99% 96.5% 99% 100%

76.3% 3% 99% 99% 96.5% 99% 100%

95.8% 1% 99% 99% 97.5% 99% 100%

101.5% 1% 99% 100% 97.5% 99% 100%

101.7% 1% 100% 100% 97.5% 99% 100%

157.5% 1% 100% 100% 97.5% 100% 100%

167.5% 2.5% 100% 100% 100% 100% 100%

Differences

Portfolio B1-A1 C1-A1 D1-A1 E1-A1 C1-B1 D1-B1 E1-B1 D1-C1 E1-C1 E1-D1

RAPM

0% -2.5% -2.5% -5 -4% 0% -2.5% -1.5% -1.5% -2.5% 1%

0.2% 0% -2.5% 5% -8% -2.5% 5% -9% 7.5% -6.5% -14%

20.3% 0% -2.5% 5% 44% -2.5% 5% 44% 7.5% 46.5% 39%

22.7% 0% -2.5% 46% 44% -2.5% 46% 44% 48.5% 46.5% -2%

24.1% 0% 37.5% 46% 44% 37.5% 46% 44% 8.5% 6.5% -2%

25.6% 0% -2.5% 6% 4% -2.5% 6% 4% 8.5% 6.5% -2%

40.5% 0% -2.5% 6% 8% -2.5% 6% 8% 8.5% 10.5% 2%

45.1% 0% -2.5% 8% 8% -2.5% 8% 8% 10.5% 10.5% 0%

48.0% 0% 2.5% 8% 8% 2.5% 8% 8% 5.5% 5.5% 0%

50.9% 5% -2.5% 8% 8% -2.5% 3% 3% 5.5% 5.5% 0%

51.0% 0% -2.5% 3% 3% -2.5% 3% 3% 5.5% 5.5% 0%

60.6% 0% 0.5% 3% 4% -2.5% 3% 3% 5.5% 6.5% 1%

No FSD No FSD No FSD No FSD No FSD No FSD No FSD No FSD No FSD No FSD

354

APPENDIX 14

Table 2

Second-Order Stochastic Dominance by RAPM (Gain/DSD): Analysis of Portfolios A to E

Probability Distributions Sum of Cumulative Probability

Portfolio A B C D E A2 B2 C2 D2 E2

RAPM

0% 21% 18.5% 18.5% 16% 17% 21% 18.5% 18.5% 16% 17%

0.2% 30% 32.5% 30% 40% 25% 93% 88% 85.5% 88% 76%

20.3% 53% 10344% 10399% 9834% 11344% 8571%

22.7% 41% 11568% 11563% 10998% 12729% 10851%

24.1% 40% 12282% 12277% 11717% 14087% 12181%

25.6% 40% 40% 13087% 13082% 13044.5% 15542% 13606%

40.5% 4% 26646% 26641% 26231% 29995% 27765%

45.1% 2% 30832% 30827% 30302% 34459% 32319%

48.0% 5% 33471% 33466% 32873.5% 37330% 35190%

50.9% 5% 36110% 36110% 39585% 40201% 38061%

51.0% 5% 36206% 36206% 35678.5% 40300% 38160%

60.6% 1% 45422% 45422% 44654.4% 49804% 47665%

71.9% 3% 56270% 56270% 55223% 60991% 58965%

76.2% 3% 60398% 60401% 59372.5% 65248% 63265%

76.3% 3% 60497% 60500% 59469% 65347% 63365%

95.8% 1% 79802% 79805% 78287.5% 84652% 82865%

101.5% 1% 85445% 85449% 83845% 90295% 88565%

101.7% 1% 85644% 85649% 84040% 90493% 88765%

157.5% 1% 141444% 141449% 138445% 145736% 144565%

167.5% 2.5% 151444% 151449% 148198% 155736% 154565%

355

Table 2

(Continued)

Differences

Portfolio B2-A2 C2-A2 D2-A2 E2-A2 C2-B2 D2-B2 E2-B2 D2-C2 E2-C2 E2-D2

RAPM

0% -2.5% -2.5% -5% -4% 0% -2.5% -1.5% -2.5% -1.5% 1%

0.2% -5% -7.5% -5% -17% -2.5% 0% -12% 2.5% -9.5% 12%

20.3% -5% -510% 1000% -1773% -505% 1005% -1768% 1510% -1263% -2773%

22.7% -5% -570% 1161% -717% -565% 1166% -712% 1731% -147% -1878%

24.1% -5% -565% 1805% -101% -560% 1810% -96% 2370% 464% -1906%

25.6% -5% -42.5% 2455% 519% -37.5% 2460% 524% 2497.5% 561.5% -1936%

40.5% -5% -415% 3349% 1119% -410% 3354% 1124% 3764% 1534% -2230%

45.1% -5% -530% 3627% 1487% -525% 3632% 1492% 4157% 2017% -2140%

48.0% -5% -597.5% 3859% 1719% -592.5% 3864% 1724% 4456.5% 2316.5% -2140%

50.9% 0% -525% 4091% 1951% -525% 4091% 1951% 4616% 2476% -2140%

51.0% 0% -527.5% 4094% 1954% -527.5% 4094% 1954% 4621.5% 2481.5% -2140%

60.6% 0% -767.5% 4382% 2243% -767.5% 4382% 2243% 5149.5% 3010.5% -2139%

71.9% 0% -1047% 4721% 2695% -1047% 4721% 2695% 5768% 3742% -2026%

76.2% 3% -1025.5% 4850% 2867% -1028.5% 4849% 2864% 5875.5% 3892.5% -1983%

76.3% 3% -1028% 4850% 2868% -1031% 4849% 2865% 5878% 3896% -1982%

95.8% 3% -1514.5% 4850% 3063% -1517.5% 4849% 3060% 6364.5% 4577.5% -1787%

101.5% 4% -1600% 4850% 3120% -1604% 4846% 3116% 6450% 4720% -1730%

101.7% 5% -1604% 4849% 3121% -1609% 4844% 3116% 6453% 4725% -1728%

157.5% 5% -2999% 4292% 3121% -3004% 4287% 3116% 7291% 6120% -1171%

167.5% 5% -3247% 4292% 3121% -3251% 4287% 3116% 7538% 6367% -1171%

No SSD C � A No SSD No SSD C � B No SSD No SSD No SSD No SSD No SSD

356

APPENDIX 15

Calculating the Capital Multiplier

This appendix shows the workings for the capital multiplier for a BBB-rated bank and a BB-rated

borrower from chapter five. It uses data from Table 6.2.

In order to calculate the capital multiplier under this example, we first calculate unexpected losses:

UL = � = )( ELLGDEL ✁ = 0.0354

where expected losses are 0.32% and the loss given default is 39%, as shown in Table 6.2. The shape

parameters for the beta distribution are determined as follows:

✂ = [(µ2 (1 – µ)) / �2] – µ = 0.0051

✄ = [(µ (1 – µ)2 ) / �2] + (µ + 1) = 3.572

where µ is 0.32% and � is 3.54%. The 99.80% confidence level that would apply to a BBB target credit

rating for the bank requires that

pbeta (xmax, ✂, ✄

) = 99.80%

Using the BETADIST function on EXCEL™ we determine the value for xmax that gives the desired

cumulative density function of 99.8% is 19.65%. Using our data, the capital multiplier is calculated as

follows:

CM = [(xmax -EL) / �] = (0.1965 – 0.0032)/0.0354 = 5.462

The capital multiplier is 5.462.

357

APPENDIX 16

Leverage-Adjusted Hurdle Rate Calculation

This appendix shows workings and assumptions for the leverage-adjusted hurdle rate calculations in

Table 6.5.

We assume an asset beta for the bank of 0.29, a risk-free rate of 6.82% and a market risk premium of

7%. The base case for the bank requires economic capital equal to 19.33% of the credit exposure,

equivalent to a D/E ratio for the bank of 4.33. Adjusting the asset beta for leverage provides a beta for

the bank of 1.17:

�L =

�U [1+ ((D/E)(1 – t))] = 0.29 [1+ (4.33 x (1 – 0.3))] = 1.17

Using the capital asset pricing model, the hurdle rate for the base case is 15%:

rh = r f + (

�L x market risk premium) = 6.82% + (1.17 x 7% ) = 15%

In the case of the bank moving to an A credit rating, the D/E for the bank changes to 1.49 (refer Table

5.5), and the beta changes to 0.59:

�L =

�U [1+ ((D/E)(1 – t))] = 0.29 [1+ (1.49 x (1 – 0.3))] = 0.59

The hurdle rate changes to 10.97%:

rh = r f + (

�L x market risk premium) = 6.82% + (0.59 x 7% ) = 10.97%

The same approach applies for each change in the target credit rating of the bank.