Post on 04-May-2023
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
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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.
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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.
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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
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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
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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
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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
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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
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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
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Chapter Seven Conclusion 290 7.1 Overview 291
7.2 Key Findings 296 7.3 Areas for Further Research 306 Bibliography 309 Appendices 328
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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
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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
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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.
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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.
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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)
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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
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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.
106
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.
108
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
110
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
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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
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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.
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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
151
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.
152
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).
154
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]
158
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
160
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.
232
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.
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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.
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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.
251
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
280
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
286
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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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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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.