Robust optimal decisions with imprecise forecasts

Computational Statistics & Data Analysis 51 (2007) 3595–3611www.elsevier.com/locate/csda

Robust optimal decisions with imprecise forecasts

Nalan Gülpınara,∗, Berc Rustemb

aWarwick Business School, The University of Warwick, Coventry CV4 7AL, UKbDepartment of Computing, Imperial College London, South Kensington Campus, SW7 2AZ, UK

Available online 15 December 2006

Abstract

A robust minimax approach for optimal investment decisions with imprecise return forecasts and risk estimations in financialportfolio management is considered. Single-period and multi-period mean–variance optimization models are extended to worst-casedesign with multiple rival risk estimations and return forecasts. In multi-period stochastic formulation of classical mean–varianceportfolio optimization problem, an investor makes an investment decision based on expectations and/or scenarios up to someintermediate times prior to the horizon and, consequently, rebalances or restructures the portfolio. Multi-period portfolio optimizationentails the construction of a scenario tree representing a discretized estimate of uncertainties and associated probabilities in futurestages. It is well known that return forecasts and risk estimations are inherently inaccurate and there are different rival estimates,or scenario trees. Robust optimization models are presented and imprecise nature of moment forecasts to reduce the risk of makinga decision based on the wrong scenario is addressed. The worst-case performance is guaranteed in view of all rival risk and returnscenarios and will only improve when any scenario other than the worst-case is realized. The ex-ante performance of minimaxmodels is tested using historical data and backtesting results are presented.© 2006 Elsevier B.V. All rights reserved.

Keywords: Rival scenarios; Robust optimization; Minimax

1. Introduction

A stochastic decision model under uncertainty consists of a problem of estimation and managing risk associatedwith random parameter. Risk management procedures are developed to reduce or prevent high losses incurred frommaking an incorrect decision. In particular, in finance, risk management techniques combine both rigorous and eleganttheoretical results and practical effectiveness. Different loss functions and risk metrics are defined in terms of a decisionvector and a stochastic vector which describes the uncertainty in the model (Khan and Ahmed, 2006). In particular, arisk measure introduces the ordering relationships for risks, so that one is able to discriminate “less risky” decisionsfrom “more risky” ones. In general, the choice of a risk measure is dictated by the nature of uncertainties and risks inthe underlying problem.

One of the fundamental investment problem is to allocate a capital over a number of available assets to maximizethe return on the investment while minimizing the risk. Portfolio selection problem was first formulated by Markowitz(1952). This classical Markowitz model requires the knowledge of the first and the second moments of the distributionof returns and capture errors in the data. The inherent inaccuracy in forecasting the expected value and estimating

∗ Corresponding author. Tel.: +44 24 7652 4491; fax: +44 24 7652 4539.E-mail addresses: [email protected] (N. Gülpınar), [email protected] (B. Rustem).

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3596 N. Gülpınar, B. Rustem / Computational Statistics & Data Analysis 51 (2007) 3595–3611

covariance matrices is well known. Typically, one would regress against historical data for estimation of the randomvariables (Schmidt, 1993). As with many estimation techniques based on historical data, the quality of the estimatesor forecasts are therefore critically dependent on the applicability of the given historical data for forecasting over thegiven period, which is difficult to determine. Even though it can be claimed that a good set of historical data is chosen,the estimation techniques will be bound to further uncertainty. Therefore, the imprecise nature of the moment forecastsneeds to be tackled to reduce the risk of decision-making on the wrong scenario. The misforecasting problem has beeninvestigated through forecast pooling, for instance, see Granger and Newbold (1977) and Lawrence et al. (1986). Thepooling has been often achieved using stochastic programming approaches where the probability of each scenario isconsidered and their weighted average is evaluated (e.g. see Kall and Wallace, 1994; Lawrence et al., 1986; Makridakisand Winkler, 1983). Robust pooling using min–max has been introduced in Becker et al. (2000), Rustem (1994) andRustem and Howe (2002). A min–max algorithm for stochastic programs based on a bundle method is discussed inBreton and El Hachem (1995).

A fundamental weakness of the mean–variance framework is that the model is based on a single forecast of returnand risk and places a low weight to the worst-case effect of the uncertainty. There are certain situations where theworst-case scenarios might have a disastrous impact on the portfolio should they materialize. Sometimes one mightnot even know which scenario is the worst-case. Therefore, the decisions based on a single forecast often need to bejustified in view of worst-case scenarios. One technique to assess the worst-case effect is min–max. In the presence ofthe uncertainty, a robust approach is to compute expectations using the worst-case probability distribution. Aoki (1967)provides a detailed discussion of this approach within the context of general dynamical systems.

The significance of worst-case robust strategies is increasingly recognized as attitudes towards risk evolved ineconomics, financial markets and engineering. Rustem and Howe provide an extensive coverage on min–max andworst-case analysis in Rustem and Howe (2002). In economics, Chow (1969) and Becker et al. (1986) consider rivalmacroeconomics models of the same economy. A worst-case estimator based on min–max approach for econometricmodels with unobservable components has been proposed by Esteban-Bravo and Vidal-Sanz (2007). An application ofworst-case decisions to portfolio optimization in finance have been investigated by a number of researchers, for examplesee, Dert and Oldenkamp (1997), Gulpinar and Rustem (2006), Howe and Rustem (1997) and Howe et al. (1994).Ben-Tal and Nemirovski (1997) discuss truss topology design under rival load scenarios in engineering. Electricityspot market when relevant empirical information is incomplete is modelled using min–max approach (Esteban-Bravoand Rustem, 2006).

In this paper, we present robust mean–variance portfolio optimization models with multiple rival risk estimationsand return forecasts. Rival return scenarios/scenario trees represent rival views of the future. The portfolio risk isrepresented by variance of returns and the estimated covariance matrices are considered as rival risk scenarios. Weassume that all rival scenarios are reasonably likely and that it is not possible to rule out any of these using statisticalanalysis. Wise decision making would therefore entail the incorporation of all scenarios to provide an integrated andconsistent framework. We propose worst-case robust optimal strategies which yield guaranteed performance. In riskmanagement terms, such a robust strategy would ensure that performance is optimal in the worst-case and this willfurther improve if the worst-case scenarios do not materialize. We note that there may well be more than one scenariocorresponding to the worst-case. This non-inferiority is the guaranteed performance character of min–max. Min–maxoptimization is more robust to the realization of worst-case scenarios than considering a single scenario or an arbitrarypooling of scenarios. It is suitable for situations which need protection against risk of adopting the investment strategybased on the wrong scenario since the min–max mean–variance optimization model constructs an optimal portfoliosimultaneously with the worst-case scenario.

The rest of this paper is organized as follows. In Section 2, we describe min–max approach and discuss some issuesthat arise from the approach. Section 3 focuses on applications of worst-case analysis to single-period and multi-periodfinancial portfolio management. Scenario generation is discussed in Section 4. Section 5 presents the computationalresults and Section 6 summarizes our conclusions.

2. The min–max approach

The worst-case approach is to minimize the objective function with respect to the worst possible outcome of theuncertain variables. Let F : Rn+m → R be a function of a decision variable x and a stochastic vector y standing foruncertainties in the model. Due to unknown values of vector y we are unable to find the actual values of F(x, y). To

N. Gülpınar, B. Rustem / Computational Statistics & Data Analysis 51 (2007) 3595–3611 3597

be on the conservative side, the decision x is required to be optimal with respect to each observation of y. Therefore,x is chosen to minimize the objective function, where nature chooses y to maximize it. This leads to the following thegeneral min–max optimization problem:

minx

maxy

F (x, y)

s.t.

x ∈ X ⊂ Rn,

y ∈ Y ⊂ Rm. (1)

Therefore, the risk of generating incorrect decision in an uncertain environment is managed using the min–max ap-proach. In general, the uncertainties are characterized either in terms of a number of rival scenarios or ranges in whichthe uncertain parameters or exogenous effects may vary. Such characterization of uncertainty leads to discrete and con-tinuous min–max models (Rustem and Howe, 2002). The discrete min–max approach determines the optimal strategyin view of all specified rival scenarios simultaneously, rather than any single scenario. In this case, Y is a finite set, and(1) becomes the discrete min–max problem which is formulated as follows:

minx

maxy∈Y

F (x, y)

s.t.

Y = {y1, y2, . . . , yk}. (2)

It can be shown that min–max problem (2) is equivalent to the following optimization problem:

minx,z

z

s.t.

F(x, yj )�z ∀j = 1, . . . , k. (3)

The continuous min–max strategy provides a guaranteed optimal performance in view of continuum of scenariosvarying between upper and lower bounds. Thus, there are an infinite number of future scenarios in the continuousmin–max framework. Assume that Y has an infinite number of elements. Then (1) is called continuous min–max whichcan be stated as the following semi-infinite optimization problem:

minx,z

z

s.t.

F(x, y)�z ∀y ∈ Y .

Notice that there is an infinite number of constraints corresponding to the elements in Y.Similarly, using the fact that the maximum over a set of scalars is equal to the maximum over their convex hull, (2)

can be equivalently expressed as the continuous min–max problem

minx

max�

k∑j=1

�jF (x, yj )

s.t.k∑

j=1

�j = 1,

�j �0 ∀j = 1, . . . , k.

2.1. Robustness of the min–max approach

Robustness arises from the non-inferiority of the worst-case optimal strategy. Let x∗, y∗ be an optimal solution ofgeneral min–max problem (1). Then, the following inequality is valid for all feasible y

F(x∗, y∗)�F(x∗, y).


This inequality simply states the optimality of y∗ for the corresponding problem (1). It also signifies the robustness ofmin–max in that performance is assured to improve if the worst-case y∗ does not happen. This means that expectedperformance is guaranteed to be at the level corresponding to the worst-case and will improve if any scenario otherthan the worst-case is realized. Similarly, under the same assumptions, for all feasible x, we have

F(x∗, y∗)�F(x, y∗).

3. Financial portfolio management

In financial portfolio management, maximization of return for a level of risk is the accepted approach to decisionmaking. A fundamental example is the single-stage Markowitz model in which expected portfolio return is maximizedand risk measured by the variance of portfolio return is minimized (Markowitz, 1952).

3.1. Single-stage mean–variance optimization

3.1.1. Markowitz modelThe single-stage model of Markowitz considers a portfolio of n assets defined in terms of a set of weights wi for

i = 1, . . . , n, which sums to unity. In other words, the initial budget is normalized to 1 as

1′w = 1. (4)

Given an expected rate of return r̄ , the optimal portfolio is defined in terms of the solution of the following quadraticprogramming problem:

minw

{〈w, �w〉|w′r = r̄ , 1′w = 1, w�0},

where � is the covariance matrix of asset returns. The quadratic program yields the minimum variance portfolio. Manytraditional portfolio analysis models seek only to maximize expected return. This objective can be achieved with aclassical stochastic linear programming formulation which incorporates the mean term. This approach is risk neutraland does not take risk-attitudes into account.

If the investor currently has holdings of assets 1, . . . , n, then vector h (scaled so that 1′h = 1) represents his currentposition. If the investor currently has no holdings (wishing to buy), then h = 0. Let b and s define decision variablesfor transactions of buying and selling, respectively. Then the allocation of the initial budget is represented with thefollowing constraints:

h + b − s = w. (5)

Transaction costs, �, incurred by moving to strategy w from current position h, subject to costs cb, cs can be incorporatedinto the mean–variance optimization model. The transaction cost of the purchase or sale is formulated as

c′bb + c′

ss = �. (6)

Let w be benchmark portfolio. The trade-off between expected return and risk is achieved by solving the followingquadratic programming problem:

min �(w − w)′�(w − w) − [(w − w)′r − �]s.t.

Constraints (4).(6),

w, b, s�0,

where the scaling constant � determines the level of risk-aversion optimized for. By sliding from � = 0 (total risk-seeking) to �=∞ (total risk-aversion), the entire range of efficient investment strategies is obtained. These investmentstrategies basically define the efficient frontier.


3.1.2. Single-stage min–max portfoliosLet J and I be the number of rival return and risk scenarios, respectively. Let S denote the number of benchmarks

provided. A compact representation of the min–max portfolio allocation problem is as follows:

minw

{� max

i,k{(w − wk)

′�i (w − wk)} − minj,k

{(w − wk)′rj − t (w)}

}, (7)

where i = 1, . . . , I , j = 1, . . . , J and k = 1, . . . , S. Function t (w) represents the transaction costs incurred by movingto strategy w from current position h, subject to costs cb, cs. The min–max problem (7) can be reformulated as aquadratically constrained mathematical program:

S—MMX

min �� − �

s.t.


(w − wk)′�i (w − wk)��, i = 1, . . . , I, k = 1, . . . , S,

(w − wk)′rj − ��, j = 1, . . . , J, k = 1, . . . , S,

w, b, s�0.

If I = 0 (no risk scenarios provided), then quadratic constraints are omitted and the objective function becomesminw −� ≡ maxw �, which is a purely linear problem. If J =0 (no return scenarios provided), then linear performanceconstraints are omitted and the objective function becomes simply minw �. When S = 1, then only one benchmarkportfolio is considered. The reader is referred to Rustem et al. (2000) and Rustem and Settergren (2002) for moredetails on single-stage min–max portfolios.

3.2. Multi-stage mean–variance optimization

The single-stage asset allocation problem can be extended to a multi-stage framework using stochastic programming.In the multi-stage case, after the initial investment, we can rebalance our portfolio (subject to any desired bounds) tomaximize profit at the investment horizon and minimize the risk at discrete stages and redeem at the end of the stage.The future is seen in terms of scenarios that are essentially a discrete set of realization of uncertainties. In the multi-stagecase, each scenario evolves into a set of scenarios in the next stage, with associated probabilities (Gulpinar et al., 2004).Instead of risk and return forecasts over a single-stage, the forecast takes the shape of a scenario tree, which dividesthe investment horizon into discrete time intervals, at which the portfolio may be rebalanced.

In this section, we first briefly summarize multi-stage mean–variance optimization model; see Gulpinar et al. (2003)for more details. Then we extend it to mean–variance min–max model with multiple rival risk and rival return sce-narios/scenario trees. The optimization model integrates benchmark relative computations in view of scalable (notfixed) transaction costs. Min–max mean–variance optimization constructs an optimal portfolio simultaneously with theworst-case scenario.

3.2.1. Multi-stage mean–variance modelWe consider n risky assets and construct a portfolio over an investment horizon T. After the initial investment

(t = 0), the portfolio may be restructured at discrete times t = 1, . . . , T − 1 in terms of both return and risk, andredeemed at the end of the investment horizon (t = T ). Multi-stage portfolio optimization entails the construction ofa scenario tree representing a discretized estimate of uncertainties and associated probabilities in future stages. Themulti-stage stochastic mean–variance approach takes account of the approximate nature of the discrete set of scenariosby considering a variance term around the return scenarios. Hence, uncertainty on return values of instruments isrepresented by a discrete approximation of a multivariate continuous distribution as well as the variability due to thediscrete approximation.

A scenario is defined as a possible realization of the stochastic variables {�1, . . . , �T }. Hence, the set of scenarioscorresponds to the set of leaves of the scenario tree, NT , and nodes of the tree at level t �1 (the set Nt ) correspond topossible realizations of �t . A set of interior nodes of the scenario tree, excluding the root node and leaves, is denotedas NI . A node of the tree (or event) is denoted by e = (s, t), where s is a scenario (path from root to leaf), and stage t


Fig. 1. Scenario tree for multi-stage mean–variance.

specifies a particular node on that path. The root of the tree is 0 = (s, 0). The ancestor (parent) of event e = (s, t) isdenoted a(e) = (s, t − 1). pe is the conditional probability of event e, given its parent event a(e). The path to event e iscalled a partial scenario with probability Pe = ∏

i=1,...,tp(s,i). The approaches to generate scenario tree are describedin Gulpinar et al. (2004).

Let re be stochastic realization of rt in event e: re ∼ N(rt (�t ), �). Expectation of rt (�

t ) for event e is conditional on�t−1 and denoted by r̂e ≡ E(re(�t |�t−1)). we, be, se define decision variables at node e ∈ Nt for asset allocations andtransactions of buying and selling, respectively. The decision at a given stage does not depend on the future realizationof the random events. Each node at a level t corresponds to a decision {wt , bt , st } which must be determined at time t,and depends in general on �t , the initial wealth w0 and past decisions {wj , bj , sj }, j = 1, . . . , t − 1. This process isadapted to �t as wt , bt , st cannot depend on future events �t+1, . . . , �T which are not yet realized. Hence wt = wt (�

t ),bt = bt (�

t ), and st = st (�t ). However, for simplicity, we shall use the terms wt , bt and st , and assume their implicit

dependence on �t . Notice that �t can take only finitely many values. Fig. 1 displays an example of scenario tree wherep∗ and r∗ are the branching probability and return vector indexed with corresponding node label. WT is the finalexpected wealth as defined in (14).The dynamic structure of the model is captured by investment strategy which isdefined by asset weights at each interior node of the scenario tree as follows:

r̂e ◦ wa(e) + (1 − cb) ◦ be − (1 + cs) ◦ se = we, e ∈ NI . (8)

Note that vector multiplication for u, v ∈ Rn is u ◦ v = (u1v1, u2v2, . . . , unvn)′. Since transactions of buying and

selling at the last stage are not allowed, asset weights at t = T are computed as

r̂e ◦ wa(e) = we, e ∈ NT . (9)

We require subsequent transactions (buy = bt , sell = st ) not to alter the wealth within the stage t. Hence, we have thecondition

1′be − 1′se = 0, e ∈ NI . (10)

The allocation of the initial budget of 1 can be represented with the following constraints:

h + (1 − cb)b0 − (1 + cs)s0 = w0, (11)

1′b0 − 1′s0 = 1 − 1′h, (12)

where h represents the investor’s current position. If the investor has no holdings and is wishing to buy in at time t = 0,then h = 0. The following box constraints are defined on we, be, se for each event e ∈ NI to prevent the short-sale and


enforce any restriction imposed by the investor:

wLe �we �wU

e ,

0�be �bUe ,

0�se �sUe . (13)

The expected wealth at time t, arising from period [t − 1, t] is E[rt (�t )′wt−1]. Given the possible events e ∈ Nt (the

discretization of �t ), the expected wealth at time t relative to benchmark wt , is given by

Wt = E[rt (�t |�t−1)′(wt−1 − wt−1)] = E

⎡⎣∑

e∈Nt

Pe(r′e (wa(e) − wa(e)))

⎤⎦

=∑e∈Nt

Pe(r̂′e(wa(e) − wa(e))),

where r̂e and we (e ∈ Nt ) are the various realizations of stochastic quantities rt (�t |�t−1) and wt−1(�t−1). The expected

terminal wealth, WT , is computed as

WT =∑

e∈NT

Pe((wa(e) − wa(e))′r̂e). (14)

Risk, for any realization of �t , is measured as the variance of the portfolio return relative to the benchmark w. Giventhe covariance matrix �t and expected portfolio return Wt , we can compute the portfolio risk at t as follows:

Var[rt (�t |�t−1)′(wt−1 − wt−1)]

= E[(rt (�t |�t−1)′(wt−1 − wt−1))2] − (E[rt (�t |�t−1)′(wt−1 − wt−1)])2

= E[(wt−1 − wt−1)′(�t + rt (�

t |�t−1)rt (�t |�t−1)′)(wt−1 − wt−1)]

− (E[rt (�t |�t−1)′(wt−1 − wt−1)])2

=∑e∈Nt

Pe((wa(e) − wa(e))′(�t + r̂er̂′

e)(wa(e) − wa(e))) − (Wt )2, (15)

where Pe is the probability of event e.Let �e be covariance matrix at each node e ∈ Nt . The multi-stage portfolio reallocation problem can be expressed

as the minimization of the trade-off between risk and expected wealth subject to constraints which describe the growthof the portfolio along all the various scenarios and bounds on the decision variables as follows:

minw,b,s

�T∑

t=1

�t

∑e∈Nt

Pe(wa(e) − wa(e))′(�e + (r̂e)

′r̂e)(wa(e) − wa(e)) −∑

e∈NT

Pe(wa(e) − wa(e))′r̂e

s.t.


where �t for each time period t = 1, . . . , T reflects a discount for the risk in the future when �t = t for ∈ (0, 1).When �t = 1, then risk at each time period is weighted equally.

The level of risk-aversion optimized for is determined by the scaling constant �. When � = 0, the pure risk-seekinginvestment strategy (at the highest end of the efficient frontier) is obtained by solving a linear programming problem.When � = ∞, completely risk-averse strategy (at the lowest end of the efficient frontier) is obtained by solving thequadratic programming problem (by ignoring the expected portfolio return).

3.2.2. Multi-stage min–max portfolios with single scenario treeMulti-stage mean–variance optimization framework can be extended to worst-case design with multiple rival return

and risk scenarios. The optimal portfolio is constructed (relative to benchmark) simultaneously with the worst-case totake account of all rival scenarios. The portfolio is balanced at each stage incorporating scalable (not fixed) transaction


cost and its relative performance is measured in terms of returns and the volatility of returns. Robustness arises fromthe non-inferiority of the worst-case optimal strategy.

Assume that a number of risk scenarios are considered at each event of the future realizations. Therefore, we havethe same number of covariance matrices, Ie, at each node of the scenario tree, e ∈ Nt for t = 1, . . . , T . The covariancematrices and a scenario tree are an input to the min–max model. Rival return scenarios are determined by the scenariotree as number of events at the first stage since investors wish to survive at the first stage. Let K denote the number ofrival return scenarios.

Given the rival risk and return scenarios (or a scenario tree), the general min–max model for multi-stage assetallocation problem is formulated as follows:

minw,b,s

⎧⎨⎩�

T∑t=1

�t

∑e∈Nt

maxi

[Pe(wa(e) − wa(e))′(�i + (r̂e)

′r̂e)(wa(e) − wa(e))]⎫⎬⎭

+ minw,b,s

⎧⎪⎨⎪⎩− min

k

⎡⎢⎣ ∑

e∈NkT

Pe(wa(e) − wa(e))′r̂e

⎤⎥⎦

⎫⎪⎬⎪⎭ ≡ min

w,b,s

⎧⎨⎩�

T∑t=1

�t

∑e∈Nt

maxi

[J ie (w)] − min

k[Wk

T (w)]⎫⎬⎭ ,

where

J ie (w) = Pe(wa(e) − wa(e))

′(�i + (r̂e)′r̂e)(wa(e) − wa(e)),

and

WkT (w) =

∑e∈Nk

T

Pe(wa(e) − wa(e))′r̂e,

for i ∈ Ie, k = 1, . . . , K, t = 1, . . . , T and e ∈ Nt . Let �e and � be the worst-case risk at node e ∈ Nt and the worst-case return, respectively. The min–max problem above can be reformulated as a quadratically constrained mathematicalprogram:

M—MMXsingle

minw,b,s

�T∑

t=1

�t

∑e∈Nt

�e − �

s.t.

Constraints (8).(13),∑e∈Nk

T

Pe(wa(e) − wa(e))′r̂e ��, k = 1, . . . , K ,

Pe(wa(e) − wa(e))′(�i + (r̂e)

′r̂e)(wa(e) − wa(e))��e, i ∈ Ie, e ∈ Nt, t = 1, . . . , T .

The worst-case risk �e for each event e ∈ Nt is calculated as the maximum risk value, which is computed by im-plementing the min–max strategy on specific rival risk scenarios and selecting maximum one among �i , i ∈ Ie. Theworst-case return � is obtained as the minimum expected return at the last stage among W 1

T , . . . , WKT corresponding

to each sub-scenario tree (see Fig. 2).Alternative min–max formulations of the multi-period mean–variance optimization problem with single scenario

tree are introduced in Gulpinar and Rustem (2006). The difference between these formulations is characterized by theway that rival risk scenarios are integrated.

3.3. Multi-stage min–max portfolios with multi scenario trees

In multi-period mean–variance framework, the discretization using scenario trees approximates future expectations.Yet, each such approximation has an associated error. The inaccuracy in forecasting scenario trees can be addressed


r1r0 r2

r10

p00 p10 p22p01 p20 p21

r01r00 r20r21 r22

WT WTWT

2 31

V2

V22V21V20V10

V0 V1

V01V00

Fig. 2. Multi-stage worst-case mean–variance on a scenario tree.

WT

r0 r2

r21r10r00

1 r11 1 r0 r2

2 r12 2 r0 r2

3 r13 3

111

1 WT2

WT3

1

11 r01 r20r21r10r002

2222r01 r20

r21r10r003

3333r01 r20

Ji2

Ji3

Ji

Fig. 3. Multi-stage worst-case mean–variance with rival scenario trees.

through the specification of rival scenario trees using min–max. In other words, we can consider worst-case analysiswith different scenario trees. Therefore, the rival scenario trees are treated as rival views of the future.

Let L denote the number of scenario trees with the same topology in terms of time periods and branching at eachintermediate time periods. The case of L = 3 is presented in Fig. 3 where J l

i and WlT denote the worst-case portfolio

risk and expected wealth at T for scenario trees, l =1, 2, 3. We consider I number of rival risk scenarios associated witheach scenario tree. The min–max problem for multi-period mean–variance model with multiple scenario trees can be


formulated as follows:

minw,b,s

⎧⎨⎩� max

i,l

T∑t=1

�t

∑e∈Nt

[P le(wa(e) − wa(e))

′(�li + (r̂l

e)′r̂l

e)(wa(e) − wa(e))]⎫⎬⎭

+ minw,b,s

⎧⎨⎩− min

l

⎡⎣ ∑

e∈NT

P le(wa(e) − wa(e))

′r̂le

⎤⎦

⎫⎬⎭ ≡ min

w,b,s

{� max

i,l[J l

i (w)] − minl

[WlT (w)]

},

where

J li (w) =

T∑t=1

�t

∑e∈Nt


′(�li + (r̂l

e)′r̂l

e)(wa(e) − wa(e))

and

WlT (w) =

∑e∈NT


′r̂le,

for i = 1, . . . , I and l = 1, . . . , L. Let � and � be the worst-case risk and the worst-case return values over the scenariotrees, respectively.

Robust minimum variance portfolio is to minimize the worst-case variance of portfolio subject to worst-case portfolioexpected return to be at least preferred level as follows:

minw,b,s

⎧⎨⎩max

i,l

T∑t=1

�t

∑e∈Nt


′(�li + (r̂l

e)′r̂l

e)(wa(e) − wa(e))]⎫⎬⎭

s.t.


minl=1,...,L

⎡⎣ ∑

e∈NT


′r̂le

⎤⎦ ��,

r̂le ◦ wa(e) + (1 − cb) ◦ be − (1 + cs) ◦ se = we, e ∈ NI , l = 1, . . . , L,

r̂le ◦ wa(e) = we, e ∈ NT , l = 1, . . . , L.

Robust maximum return portfolio is to maximize the worst-case portfolio return subject to constraint on worst-casevariance as

maxw,b,s

⎧⎨⎩min

l

⎡⎣ ∑

e∈NT


′r̂le

⎤⎦

⎫⎬⎭

s.t.


maxi=1,...,I, l=1,...,L

T∑t=1

�t

∑e∈Nt


′(�li + (r̂l

e)′r̂l

e)(wa(e) − wa(e))]��,


r̂le ◦ wa(e) = we, e ∈ NT , l = 1, . . . , L.


The general min–max mean–variance portfolio allocation problem with multi rival scenario trees is reformulated as aquadratic programming problem:

M—MMXmulti

minw,b,s

�� − �

s.t.

Constraints (10).(13),∑e∈NT


′r̂le ��, l = 1, . . . , L,

T∑t=1

�t

∑e∈Nt


′(�li + (r̂l

e)′r̂l

e)(wa(e) − wa(e))��, l = 1, . . . , L, i = 1, . . . , I ,


r̂le ◦ wa(e) = we, e ∈ NT , l = 1, . . . , L.

where � and � are the worst-case portfolio risk and expected portfolio return with multi scenario trees l = 1, . . . , L.

4. Scenario tree generation

Multi-period stochastic programming requires a coherent representation of uncertainty that is expressed in termsof multivariate continuous distributions. Hence, a decision model is generated with internal sampling or a discreteapproximation of the underlying continuous distribution. In multi-period mean–variance framework, return uncertaintyis expressed by a multivariate continuous distribution which is represented by finitely many discrete approximations.Therefore, it is assumed that �t+1 at time period t + 1 can take only finitely many values, given the event history upto time t. A scenario tree represents this branching process. Finding a set of discrete outcomes of the random variabledepends on the scenario tree generation method. The root node of a scenario tree corresponds to the decision made“today” and the nodes further down represent the conditional decisions at later stages. The arcs linking the nodesrepresent various realizations of the uncertain variables. The dynamic of decision making is captured as the decisionsare adjusted according to realizations of uncertainty. The discrete approximations of the underlying distribution andprobability space must take into account all extreme events of the problem under consideration so that the future isadequately represented.

For the multi-period worst-case portfolio problem, we generate scenario trees by an approach based on probabilisticsimulation. A discussion of the relative merits of this approach is beyond the scope of this paper. Therefore, themethod is briefly described here; for a detailed description of this and alternative approaches, the reader is referredto Gulpinar et al. (2004). In this approach, the return scenarios at each time period are generated as the centroids ofsimulations generated in parallel or sequentially. Given the topology of the tree to generate, mean growth rates, and acovariance matrix obtained by fitting the history to the expected mean growth, one time period of growth from “today”to “tomorrow” in each scenario is simulated. Large numbers of randomly generated simulations are then clustered intoa number of nodes defined by the branching at that time period. The distance measure to determine which seed eachscenario is the closest to can be chosen with great flexibility as well as the way the simulations are generated. In thisway, the centre point of each clustering is chosen as the best representative return scenario.

5. Computational experiments

5.1. Design of experiments

The single-period and multi-period min–max optimization models explained in the previous section are imple-mented and integrated with a software package called robust. Robust is written in C + + and uses the interior pointlinear/quadratic solver BPMPD to optimize the linear and quadratic programming problems (Meszaros, 1997). Allcomputational experiments are carried out on a 1.7 GHz Pentium M, running Windows XP with 1.5 GB RAM.


Table 1Problem statistics and CPU time taken to solve a quadratic program

Problem statistics M—MMXsingle M—MMXmulti

Variables 160 147Linear constraints 84 225Quadratic constraints 14 3

CPU time (s) 10.92 64.09

In the computational experiments, the statistical parameters are measured from historical data which consists ofmonthly prices of randomly selected five and 10 FTSE 100 stocks from 1988 to mid-2000. The historical data is fittedto an exponential growth curve. The obtained monthly growth rates are annualized, and used to simulate future growth.A covariance matrix is measured from the residuals of the exponential fit, and used in the simulation-based scenariogeneration method.

The specification of rival covariance matrices can be realized by observing the data during different periods inthe past. Dividing a given observation period into a number of subperiods and measuring volatility in each one isan effective way of estimating risk scenarios. It is well known that the estimates corresponding to each subperiodcan be substantially different, and employing the worst-case scenario arising from this consideration yields a robuststrategy. Other covariance estimation methods such as ARCH-GARCH models and bootstrapping are also clearlyuseful Audrino (2006). We consider three rival risk scenarios which are estimated from the historical data. The firstcovariance matrix is obtained by an exponential fit of 150 monthly price data. For the second and the third rival riskscenarios, we divide the past price data into two parts. The covariance matrices (corresponding to the second andthird rival risk scenarios) are measured from the residuals of the exponential fit of the first part and the second part ofthe data.

A scenario tree is input to robust and can have arbitrary branching structure and depth (limited only by com-puter memory). The size of an optimization problem depends on the topology of the scenario tree and numberof assets. We illustrate the computational effectiveness of models using three rival scenario trees with three timeperiods and 2–2–2 branching topology and one rival risk scenario. The problem statistics of M—MMXsingle andM—MMXmulti models are given in Table 1 in terms of the number of variables and constraints. Table 1 alsopresents the CPU time (in seconds) taken to solve a quadratic program for obtaining an efficient min–maxportfolio.

The results show that the CPU time taken to solve the most general min–max problem depends on the number ofrival scenario trees and their branching topology.

5.2. Results

The performance of min–max optimization models is measured in terms of worst-case risk-return efficient frontiersand backtesting. For multi-period min–max model with single scenario tree, rival return scenarios are determined bysub-scenario trees rooted at the first time period of the scenario tree since the investor wishes to survive at the first timeperiod. The scenario tree with three periods and 4–2–2 branching topology consists of four rival return scenarios definedby number of events realized at the first time period. They are denoted by R1, R2, R3, R4. Risk-return efficient frontierspresented in Figs. 4 and 5 are obtained by solving M—MMXsingle model where risk scenarios are associated at eachnode of scenario tree (Gulpinar and Rustem, 2006). In Fig. 4, four efficient frontiers above the min–max risk-returnfrontier (based on all rival risk and return scenarios) are the min–max strategies obtained by solving M—MMXsinglewith a single rival return scenario Ri for i = 1, 2, 3, 4 and three rival risk scenarios. This shows that the min–maxinvestment strategy leads to a guaranteed portfolio performance.

The benefit of using min–max optimization is displayed in Fig. 5. The top four curves in Fig. 5 represent the efficientfrontiers obtained by optimizing with each of the four return scenarios alone and three rival risk scenarios. The middlecurve is the min–max strategy obtained by optimization models with rival four return and three risk scenarios. The fourcurves at the bottom show what actually would happen if the worst-case of the four return scenarios with respect tooptimized portfolio was actually realized.


Fig. 4. Robust portfolios obtained by multi-period min–max model with single scenario tree.

Fig. 5. Worst-case analysis of multi-period min–max model with single scenario tree.

We also investigate the performance of multi-period min–max model with multi scenario trees. The scenario treesare generated using simulation-based approach with antithetic variates (ANT), sequential method (SEQ) and Sobollow-discrepancy random generator with 5000 (SOBOL) and 10,000 (SOBOL1) seeds. Four rival scenario trees are inputto M—MMXmulti model. The same rival risk scenarios (three covariance matrices) are associated with each scenariotree. Fig. 6 illustrates the min–max risk-return efficient frontier (at the bottom) obtained by solving M—MMXmultioptimization problem with four rival scenario trees and three rival risk scenarios. Four efficient frontiers (at the top)are obtained by solving M—MMXsingle optimization problem with each rival scenario tree and the same single riskscenario. Since the min–max model constructs an optimal portfolio simultaneously with the worst-case scenario, theoptimal investment strategy has the best lower bound performance which can only be improved when any scenarioother than the worst-case is realized.

Non-inferiority of the min–max strategy (obtained by solving M—MMXmulti with all rival risk scenarios and rivalscenario trees) is presented in Fig. 7. The min–max efficient frontier is the one at the bottom. The other curves above


Fig. 6. Comparison of min–max strategies obtained by solving multi-stage worst-case mean–variance optimization models with single and multirival scenario trees.

Fig. 7. Robustness of the min–max strategy.

the min–max strategy are obtained by implementing the min–max strategy on single rival scenario tree. This confirmsa consequent feature of min–max optimization: if the worst-case does not occur, then the performance of min–maxportfolio can only be better. This ensures the robustness of the min–max strategy.

The min–max implementations of optimization models presented in this paper are also backtested by generatingscenario trees with three time periods and 2–2–2 and 4–2–2 branching topologies. The historical data consists ofmonthly price time series of 10 arbitrarily selected FTSE 100 stocks through the 1990s. Since M—MMXmulti is amore general model, we only present its backtesting results in Figs. 8 and 9. See Gulpinar et al. (2004), for more specificdetails of the backtesting procedures, but a summary follows.

At any particular “present” time, the previous 10 time periods of past history are fitted to exponential growth curvesthat model the expected current “growth” rate in the market. Scenario trees are generated by the simulation-basedscenario generation method. The min–max optimization software robust is used to find an optimal robust investmentstrategy with single or multi scenario trees. The resulting investment strategy is implemented at the “present” prices,


2

4

6

8

10

12

0 20 40 60 80 100 120 140

Po

rtfo

lio V

alu

e

Time (months)

Index

ANT

SEQ

SOBOL

SOBOL1

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140

Po

rtfo

lio V

alu

e

Time (months)

Index

ANT

SEQ

SOBOL

SOBOL1

Fig. 8. Backtesting results obtained by multi-stage min–max model with different scenario trees (three periods and 2–2–2 branching topology) at50% (top) and 75% (bottom) risk levels.

and portfolio value is updated according to “tomorrow” prices. Then the present is moved forward one time period,and the process is repeated.

For any given scenario tree, the optimizer can yield the entire range of efficient strategies, from risk-seeking torisk-averse (in other words from the highest expected return to the lowest risk). So, the desired risk level is anotherparameter that needs to be specified during a backtesting experiment. Risk levels are measured as follows: 100%risk corresponds to the total risk-seeking investment strategy and 0% risk level corresponds to the strategy that ex-clusively minimizes risk with no consideration of performance. In our computational experiments, we also consider50%, 75% risk levels for illustrative purposes. The results are plotted against an equally weighted index of the as-sets involved. The backtesting results presented in Figs. 8 and 9 show that the forecasting technique as well as thetopology of the scenario trees play an important role on the performance of the multi-period min–max optimizationmodels.


2

4

6

8

10

12

0 20 40 60 80 100 120 140

Po

rtfo

lio V

alu

e

Time (months)

IndexANTSEQ

SOBOLSOBOL1

2

4

6

8

10

12

0 20 40 60 80 100 120 140

Po

rtfo

lio V

alu

e

Time (months)

IndexANTSEQ

SOBOLSOBOL1

Fig. 9. Backtesting results obtained by multi-stage min–max model with different scenario trees (three periods and 4–2–2 branching topology) at50% (top) and 75% (bottom) risk levels.

6. Conclusion

In this paper, we present a min–max approach for single-period and multi-period mean–variance optimizationproblems with multiple rival risk and rival return scenarios. The advantage of worst-case analysis is to providea guaranteed performance that will improve if any scenario other than the worst-case occurs. The issueof inaccuracy in asset return forecasting and risk estimation is addressed using a robust min–maxstrategy.

Our computational experiments illustrate that using rival risk and return forecasts within a robust framework isdesirable. The approach also provides some flexibility in the modelling of rival risk scenarios in a multi-stage pro-cess. The min–max is thus an alternative approach which yields a cautious strategy to provide cover and avoidlosses.


Acknowledgements

This research was supported by EPSRC Grant GR/T02560/01. We would like to thank anonymous referees and theeditor of this journal for helpful comments and suggestions.

References

Aoki, M., 1967. Optimization of Stochastic Systems. Academic Press, London, New York.Audrino, F., 2006. The impact of general non-parametric volatility functions in multivariate GARCH models. Comput. Statist. Data Anal. 50 (11),

3032–3052.Becker, R., Dwolatzky, B., Karakitsos, E., Rustem, B., 1986. Simultaneous use of rival models in policy optimization. Econ. J. 96, 89–101.Becker, R.G., Hall, S., Rustem, B., 2000. Robust optimal decisions with stochastic nonlinear economic systems. J. Econom. Dyn. Control 18,

125–148.Ben-Tal, A., Nemirovski, A., 1997. Robust truss topology design via semidefinite programming. SIAM J. Optim. 7 (4), 991–1016.Breton, M., El Hachem, S., 1995. Algorithms for the solution of stochastic dynamic minimax problems. Comput. Optim. Appl. 4, 317–345.Chow, G.C., 1969. Effective use of econometric models in macroeconomics policy formulation. Optimal Control Econom. Models, 31–39.Dert, C., Oldenkamp, B., 1997. Optimal guaranteed return portfolios and the casino effect. Research Report, Department DFS, Free University of

Amsterdam.Esteban-Bravo, M., Rustem, B., 2006. Worst-case modelling for management decisions in electricity markets with incomplete information. Research

Paper, Imperial College, London.Esteban-Bravo, M., Vidal-Sanz, J.M., 2007. Worst-case estimation for econometric models with unobservable components. Comput. Statist. Data

Anal., in press.Granger, C., Newbold, P., 1977. Forecasting Economic Time Series. Academic Press, New York.Gulpinar, N., Rustem, B., 2006. Worst-case optimal robust decisions for multi-period mean–variance portfolio optimization. Eur. J. Oper. Res.,

in press.Gulpinar, N., Rustem, B., Settergren, R., 2003. Multistage stochastic mean–variance portfolio analysis with transaction cost. Innov. Financ. Econom.

Networks 3, 46–63.Gulpinar, N., Rustem, B., Settergren, R., 2004. Optimization and simulation approaches to scenario tree generation. J. Econom. Dyn. Control 28

(7), 1291–1315.Howe, M.A., Rustem, B., 1997. A robust hedging algorithm. J. Econom. Dyn. Control 21, 1065–1092.Howe, M.A., Rustem, B., Selby, M., 1994. Minimax hedging strategy. Comput. Econom. 7 (4), 245–275.Kall, P., Wallace, S.W., 1994. Stochastic Programming. Wiley, New York.Khan, B.U., Ahmed, S.E., 2006. Comparisons of improved risk estimators of the multivariate mean vector. Comput. Statist. Data Anal. 50 (2),

402–421.Lawrence, M.J., Edmunson, R.H., O’Connor, M.J., 1986. The accuracy of combining judgemental and statistical forecasts. Manag. Sci. 32,

1521–1532.Makridakis, S., Winkler, R., 1983. Average of forecasts: some empirical results. Manag. Sci. 29, 987–996.Markowitz, H., 1952. Portfolio selection. J. Finance 7, 77–91.Meszaros, C., 1997. BPMPD User’s Manual Version 2.2. Department of Computing Research Report, Imperial College London.Rustem, B., 1994. Stochastic and robust control of nonlinear economic systems. Eur. J. Oper. Res. 73, 304–318.Rustem, B., Howe, M., 2002. Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, London,

New Jersey.Rustem, B., Settergren, R., 2002. Scenario specification for robust portfolio analysis. In: Computational Methods in Decision Making, Economics

and Finance: Optimization Models. Kluwer Academic Publishers, Dordrecht, Boston.Rustem, B., Becker, R., Marty, W., 2000. Robust min–max portfolio strategies for rival forecast and risk scenarios. J. Econom. Dyn. Control 24,

1591–1623.Schmidt, K., 1993. On the performance of minimax estimators in linear regression. Comput. Statist. Data Anal. 16 (4), 455–468.

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