European Journal of Operational Research 212 (2011) 417–428

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Interfaces with Other Disciplines

Robust optimization and portfolio selection: The cost of robustness

Christine Gregory a,⇑, Ken Darby-Dowman b, Gautam Mitra a

a CARISMA: The Centre for the Analysis of Risk and Optimisation Modelling, Brunel University, Uxbridge, Middlesex UB8 3PH, UKb Brunel University, Uxbridge, Middlesex UB8 3PH, UK

a r t i c l e i n f o

Article history:Received 26 September 2008Accepted 9 February 2011Available online 19 February 2011

Keywords:Uncertainty modellingRobust optimizationPortfolio selection

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.02.015

⇑ Corresponding author.E-mail address: christine.gregory@brunel.ac.uk (C.

a b s t r a c t

Robust optimization is a tractable alternative to stochastic programming particularly suited for problemsin which parameter values are unknown, variable and their distributions are uncertain. We evaluate thecost of robustness for the robust counterpart to the maximum return portfolio optimization problem. Theuncertainty of asset returns is modelled by polyhedral uncertainty sets as opposed to the earlier proposedellipsoidal sets. We derive the robust model from a min-regret perspective and examine the properties ofrobust models with respect to portfolio composition. We investigate the effect of different definitions ofthe bounds on the uncertainty sets and show that robust models yield well diversified portfolios, in termsof the number of assets and asset weights.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Robust optimization is a min-regret modelling methodologythat seeks to minimise the negative impact of future events whenthe values of model parameters are unknown, variable and theirdistributions are uncertain. Variability, commonly expressed as astatistical measure (e.g., standard deviation, variance), is the ‘‘nat-urally occurring, unpredictable change’’ (Burgman, 2005) of aparameter and is not reducible by the acquisition of more knowl-edge (Vose, 2000). That is, no matter how much information is ac-quired about the parameter it is not going to change its behaviour.Uncertainty, however, reflects a lack of knowledge about a futureevent and can be reduced (but not necessarily eliminated) by gath-ering more information (Vose, 2000); for example, by collectingmore data, parameter distributions can be more precisely esti-mated. To illustrate the difference between variability and uncer-tainty, consider the random walk of a stock price, which depictsits ‘‘naturally occurring, unpredictable change’’. It is not possibleto change the random walk, i.e. make it less volatile, no matterhow much data is collected or knowledge acquired regarding itspast behaviour – this is variability. However, if we estimate the dis-tribution of a random walk, acquiring more information about itspast behaviour will increase the precision of that estimate – thisis uncertainty.

Model robustness can be defined in many different ways. Thispaper refers to a specific branch of optimization under uncertaintyknown as robust optimization, whose roots can be found in the fieldof robust control and in the work of Soyster (1973) as well as laterworks by Ben-Tal and Nemirovski (1997, 1998) and independently

ll rights reserved.

Gregory).

by El Ghaoui and Lebret (1997) and El Ghaoui et al. (1998). Conse-quently, we define a model as robust if it guarantees, with highprobability, that the optimal objective will be achieved or exceededand that the solution will be feasible for all possible realisations ofeach unknown parameter, contained within the bounds of anuncertainty set, even if the assumed distributions and estimatesof the parameters are imprecise. Although parameter values are un-known, historical data (if available) may be used to estimate theuncertainty set, which does not need to encapsulate every possiblerealisation of the parameter, but only the ‘‘most likely’’ values, thespecification of which is partially a subjective decision. The twomost common ways of defining the geometry of uncertainty setsare polyhedral and ellipsoidal sets (see Section 2).

The purpose of this paper is to analyze the behaviour, robust-ness and cost of the robust counterpart formulation of the portfoliooptimization problem purposed by Bertsimas and Sim (2004), inwhich unknown parameters were modelled by polyhedral sets.The majority of previous robust portfolio optimization researchhas stemmed from the work of Ben-Tal and Nemirovski (1999)who model unknown parameters by ellipsoidal sets (discussed fur-ther in Section 2). It has been suggested that polyhedral sets are acrude model of uncertainty and the resulting robust counterpart istoo simplistic (Ben-Tal and Nemirovski, 1998). However, it isworthwhile investigating whether or not the robust portfolio opti-mization model in question produces quality solutions, eventhough it is a more simplistic formulation. To our knowledge, anempirical investigation of this model, its properties (i.e. portfoliocomposition), an analysis of the cost of robustness and the effectsof changing the size of the uncertainty set has not been carried outbefore.

In Section 2, we discuss the motivation behind robust optimiza-tion and provide insight into the uncertainty set U, particularly

418 C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428

with respect to the definition and estimation of the parameterswhich define the scale of U. In Section 3, we highlight the literaturesupporting robust portfolio optimization. Section 4 shows the der-ivation of the linear robust counterpart purposed by Bertsimas andSim (2004) and an interpretation of the robust counterpart. In Sec-tion 5, we discuss two models for portfolio selection introduced byBertsimas and Sim (2004) and the composition of robust portfolios.In addition, we discuss an alternative to their ‘correlated’ model. InSection 6, we introduce measures of robustness and the cost ofrobustness and evaluate robust portfolios based on these mea-sures. Lastly, in Section 7, we present our conclusions.

2. Robust decisions of uncertain mathematical programs

The robust counterpart of an uncertain mathematical programis a deterministic worst case formulation in which model parame-ters are assumed to be uncertain, but symmetrically distributedover a bounded interval known as an uncertainty set, U. The struc-ture and scale of U is specified by the modeller, typically based onstatistical estimates. Structure refers to the geometry or shape ofthe constraint set U, such as ellipsoidal or polyhedral. Scale refersto the magnitude of the deviations of the uncertain parametersfrom their nominal values; it can be thought of as the size of thestructure defining U. A general form of the robust counterpart toan uncertain LP is given as

Max ½MinðcT xÞ�:Subject to Ax 6 b; 8ðA; b; cÞ 2 U:

ð1Þ

As defined by Ben-Tal and Nemirovski (1998), feasible solutionsto (1) are robust feasible solutions and the optimal solution to (1) isa robust optimal solution. Bertsimas and Sim (2004) introduced theconcept ‘‘price of robustness’’ which considers how ‘‘heavily’’ theobjective function value is penalised when we are guarded againstobjective underperformance and/or constraint violation. Implicitly,this is the difference between the robust optimal solution and theobjective function value of the nominal problem. In Section 6, weexplicitly define a similar measure called the cost of robustness.

Soyster (1973) was the first to show that uncertain linear pro-grams could be formulated as robust convex linear programs suchthat feasibility was preserved for all possible values of the uncer-tain parameters defined within a set. Consider an uncertain linearprogram in which the uncertainty is in the constraints. In Soyster’smodel, the worst case solution is guaranteed to be feasible for all(A, b, c) e U, where the uncertainty set U includes every possiblerealisation of the uncertain parameters such that the probabilityof violating the ith constraint is zero. In a less conservative modelintroduced by Ben-Tal and Nemirovski (1998), it is possible to scaledown the uncertainty set U such that it only includes the ‘‘mostlikely’’ values of the uncertain parameters, which are determinedby the user, based upon statistical estimates. Consequently, thetrue value of any uncertain parameter may be outside the boundsof U, for which the worst case solution has not accounted; there-fore, feasibility is no longer 100% guaranteed. Ben-Tal and Nemi-rovski prove that the probability of violating the ith constraint,given by a confidence term c, is greater than 0 and bounded aboveby an exponential term that is a function of the scale of U.

Bertsimas and Sim (2004) introduced an alternative robustcounterpart with budgeted uncertainty, which is referred to as abudgeted robust counterpart. They relaxed the condition that thesolution must be feasible for all (A, b, c) e U under the assumptionthat not every parameter will take its worst case value. Thus, thesolution must be feasible only for some (A, b, c) e U. The numericalvalue of ‘for some’ is represented by a user defined parameter C,which can take any real value between 1 and |J|, where J is set ofuncertain parameters, hence |J| is the cardinality of J. The value

of C affects the structure of U and thus, the guaranteed robustnessof the solution. Clearly, when C = 0 none of the uncertain parame-ters take their worst case value; thus, the budgeted robust counter-part is similar to the non-robust nominal problem. When C = |J|, allof the uncertain parameters take their worst case value; thus, thebudgeted robust counterpart is similar to (2) – the solution mustbe feasible for all (A, b, c) e U. The budgeted robust counterpart ofBertsimas and Sim (2004) builds on similar principles to those ofBen-Tal and Nemirovski’s model. In their model, it is also possibleto adjust U such that only ‘‘most likely’’ values and feasibility is not100% guaranteed. In addition, the probability of constraint viola-tion is bounded above by an exponential term, however, the expo-nential term is not a function of the scale of U, but the structure ofU.

In the robust optimization framework, the true value ai, of anuncertain parameter, is given by the following equation:

ai ¼ �ai þ aigi; 8i; ð2Þ

where �ai is a statistical estimate of the expected value of ai (com-monly referred to as a point estimate), ai is a statistical estimateof the maximum distance that ai is ‘‘likely’’ to deviate from the pointestimate �ai and gi is a random variable which is bounded by andsymmetrically distributed within the interval [�1, 1]. The natureof the uncertain parameters will determine how gi is distributedover this interval (for example, gi may be stochastic in nature, uni-formly distributed, etc.). For both models, consider ai e A, 8i, as un-known and symmetrically distributed with respect to �ai on theinterval ½�ai � ai; �ai þ ai�.

Ellipsoidal uncertainty sets (Ben-Tal and Nemirovski, 1998) aregiven by

UX ¼ a 2 Rn :X ðai � �aiÞ2

a2i

6 X2; kgk1 6 1

( ); ð3Þ

where X is a user defined parameter and adjusts the trade-off be-tween robustness and optimality. As X increases, the area of theellipsoid defining the uncertainty set also increases. Hence, theupper bound on the probability of constraint violation, e�X2=2, de-creases (i.e. the model is more robust). Therefore, the scale of theuncertainty sets (or ellipsoids) and the probability of constraint vio-lation are determined by the parameter X, which is dependentupon the user’s risk preference.

Bertsimas and Sim (2004) modelled uncertainty by budgetedpolyhedral uncertainty sets,

U ¼ a : jai � �aij 6 ai; kgk1 6 1; kgk1 6 C� �

: ð4Þ

Recall that C is a user defined parameter that adjusts the robustnessof the model and interpreted as the maximum number of uncertainparameters allowed to take their worst case value. We can see from(4) that changing C will change the number of bounds, which definethe polyhedron, thus changing the structure of U and adjusting therobustness of the model. Bertsimas and Sim prove that the probabil-ity of constraint violation is bounded above by e�C2=2jJj. Thus, as Cincreases, more protection is given and the solution is more robust.In contrast to the parameter X, introduced by Ben-Tal and Nemirov-ski (1997), C does not affect the size of the uncertainty set, shownin (3) and (4), respectively.

As mentioned previously, there are two main aspects of uncer-tainty sets: structure and scale. We discussed the most commonstructures, ellipsoidal and polyhedral. We now address thequestion of scale. For the purpose of clarity, we redefine theuncertain parameter ai, which was given by (2), earlier in thissection. Introducing a scaling factor c, we redefine ai by the follow-ing equation:

ai ¼ �ai þ caigi; 8i: ð5Þ

C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428 419

In this form we have factored out the coefficient c from ai to dis-tinguish between the deviation measure ai from its scaling factor c.Intuitively, ai tells us how ai deviates from the point estimate �ai,while c tells us by how much. Therefore, ai lies on the interval½�ai � cai; �ai þ cai�. For example, if we determine that the distribu-tion of ai is best represented by defining ai as the standard devia-tion of ai, then c would represent the number of standarddeviations ai can deviate from �ai. Thus, ai lies on the interval½�ai � cai; �ai þ cai�.

Consequently, when we speak of scale, we are concerned withhow �ai and ai are defined, how their values are estimated andhow ai is scaled (the value of c). There is very little research pub-lished that addresses these questions. Almost all of the literaturein robust optimization only mentions the structure, but not thescale of U. One work which discusses scale is by Tütüncü and Koe-nig (2004), in which �ai and ai are defined as the 50th percentile andestimated using a bootstrapped sample. The deviation measure ai

was scaled by c = 47.5%. Thus, ai lies between 2.5 and 97.5 percen-tile values estimated from the bootstrap sample. Moreover, theauthors conclude that the scale of U should be dependent uponthe risk preferences of the decision-maker. Our empirical investi-gation of cost and robustness (see Section 6) suggests that definingboth �ai and ai as the 50th percentile of distribution of returns yieldsportfolios which are counterintuitive with respect to the relation-ship between c and the trade-off between cost and robustness;such portfolios increased in cost and decreased in robustness whenthe scale of uncertainty set U was increased.

The most recent work we are aware of, which investigates thescale of an uncertainty set, considers how ai are defined (Chenand Tan, 2009) by letting aþi and a�i to be the upside and downsidedeviations of ai, respectively. They compare their results with thoseof two other robust models: (1) with symmetric interval randomuncertainty which follows a normal distribution and (2) what theauthors term ‘interval uncertainty’. For the interval uncertaintysets, percentiles obtained from the distribution of returns wereused to estimate �ai and ai (similar to Tütüncü and Koenig, 2004).

In response to the lack of literature regarding the scale of U, weinvestigate different definitions of �ai and ai, estimated from a his-torical dataset, combined with different values of c (see Section 6).We evaluate the corresponding uncertainty sets by measuring thecost and robustness of a budgeted robust counterpart of the port-folio selection problem, where the structure of U is polyhedral. Ourresults suggest that for this problem, the most appropriate defini-tions of �ai and ai are measures of central tendency and spread,respectively, while c depends on the risk preferences of theinvestor.

3. Robust optimization and portfolio selection

In the portfolio selection problem an investor chooses the pro-portion of capital to be invested in each of N assets such that a de-sired set of goals is achieved; thus, this problem is essentiallymulti-objective. The first notable work to consider risk in portfoliooptimization was published in 1952, when Markowitz presentedthe well-known Expected value–Variance (E–V) model for portfoliooptimization, in which a portfolio that achieves a specified ex-pected return at minimum risk (defined by portfolio variance) isdetermined.

An underlying assumption of Markowitz’s model is that preciseestimates of li and rij have been obtained. Consequently, li and rij

are treated as known constants; however, asset returns are vari-able. It is reasonable to conclude that a model which treats returnsas known constants will produce a portfolio whose realised returnis different from the optimal portfolio return given by the objectivefunction value. In particular, when the realised asset returns are

less than the estimates used to optimise the model, the realisedportfolio return will be less than the optimal portfolio return givenby the objective. Therefore, it is worthwhile exploring alternativeframeworks, such as robust optimization, for application to theportfolio selection problem.

Although the distributions of asset returns are uncertain, in therobust optimization framework, we may assert that l or r, or both,belong to an uncertainty set, the bounds of which we can define.Most robust portfolio models describe asset returns by ellipsoidaluncertainty sets, based on the methodology of Ben-Tal and Nemi-rovski (1998) and El Ghaoui and Lebret (1997), in which the userdefined parameter X adjusts the guaranteed and achieved robust-ness of the portfolio. Previously, robustness has been evaluatedbased upon performance, particularly worst case performance,then compared to the worst case performance of a non-robustmodel such as the E–V model. In addition to worst case perfor-mance, we suggest that it is also important to evaluate robustnessbased upon whether a model yields portfolios that achieve theirguaranteed robustness in practice (see Section 6).

In 2000, Lobo and Boyd presented several different methods formodelling the uncertainty sets for the expected returns vector andcovariance matrices, such as box or ellipsoidal sets. Each robustmodel was a semi-definite program solved via interior point meth-ods. Their results focused on the performance of the solution meth-od rather than on the robustness of the optimal portfolios.

Goldfarb and Iyengar (2003) defined asset returns by robust fac-tor models in which uncertainty was modelled by ellipsoidal sets.Robustness was evaluated based on performance, particularly inworst case scenarios, and compared to the E–V portfolio model. Re-sults showed the worst case performance of the robust model wasapproximately 200% better than the non-robust model; thus, theyconcluded that robust portfolios were more apt to withstand noisydata.

El Ghaoui et al. (2003), introduced and evaluated the robustnessof a worst case VaR model in which the uncertainty (of both l andr) was modelled by ellipsoidal sets. Results showed that the non-robust model ‘wins’ if there is no uncertainty, but the robust model‘wins’ in the worst case scenario, which is what one would expect.Most recently, Huang et al. (2010), presented a relative robust con-ditional value-at-risk (RCVaR) portfolio selection problem, compar-ing its performance with that of the CVaR model of Rockafellar andUryasev (2000) and the worst case robust CVaR (WCVaR) model ofZhu and Fukushima (2009), for both hedge fund and equity portfo-lio management examples. In both examples, the RCVaR modeltypically resulted in better returns in the worst case.

Tütüncü and Koenig (2004) compare the performance of theirrobust portfolios to that of E–V portfolios. Results showed that ro-bust portfolios are only ‘‘marginally inefficient’’ when returns taketheir expected value but E–V portfolios are ‘‘severely inefficient’’when returns take their worst case values (as defined by theiruncertainty set). Ceria and Stubbs (2006) presented a model whichminimised the difference between the estimated and actual effi-cient frontiers while maximising portfolio return. Typically thetrue frontier lies between the estimated and actual frontiers,hence, minimising their distance will bring them closer to the truefrontier. Results showed that the robust model yielded greater‘‘realized returns’’ in most cases. Kim and Boyd (2007) analysedthe basic properties of a worst case efficient frontier representingthe optimal trade-off between worst case risk-return pairs, inwhich the uncertainty in l and r are independent, in order to con-struct several models of uncertainty, which the authors claim arecomputationally tractable.

Pflug and Wozabal (2007) took a slightly different perspectiveon modelling uncertainty by considering the probability model ofan asset to be unknown. They constructed a robust portfolio selec-tion model in which a ‘confidence set’ described the probability

420 C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428

distribution of asset returns. In addition, they evaluated the trade-off between risk, robustness and portfolio return. Their resultsshowed that as robustness increased, risk and portfolio return de-creased and portfolios were more diversified.

Robust optimization techniques have been criticized for givingequal weight to all possible values of the uncertain parameters,specified within their respective uncertainty sets, which may notbe a realistic assumption (Bienstock, 2007). Bienstock (2007) ad-dressed this criticism by defining two types of uncertainty sets thatgave higher weight to more significant data realizations. Similarly,as mentioned in Section 2, Chen and Tan (2009) consider intervalrandom uncertainty sets to capture the asymmetry of upside anddownside deviations. Their results suggested that their intervalrandom uncertainty model improved the worst case performanceof portfolios.

A multi-period portfolio optimization model given by Bertsimasand Pachamanova (2008) builds upon the approach of Ben-Talet al. (1999), but with polyhedral, instead of ellipsoidal, uncer-tainty sets. They compared the computational performance of theirlinear robust models with a single period mean–variance modelusing simulations of future returns of three assets. Results sug-gested that a robust multi-period approach should be consideredas an alternative to single period E–V models.

As is evident from the literature, nearly all robust portfoliomodels construct uncertainty sets as ellipsoids, based on the workof Ben-Tal and Nemirovski (1998), El Ghaoui and Lebret (1997) andEl Ghaoui et al. (1998). Typically, solution robustness is evaluatedby comparing the worst case performance of the robust modelwith that of a non-robust model. In addition, there is not an explicitevaluation of the cost of robustness. In this paper, we consider therobust portfolio model of Bertsimas and Sim (2004), which con-structs polyhedral uncertainty sets; we investigate the guaranteedand achieved robustness of the solution as well as the cost ofrobustness. Furthermore, by altering model parameters, we evalu-ate the robustness of the model itself, that is, as model parameterschange, how much do our decisions change?

4. Linear robust counterpart to portfolio optimization

In this Section, we discuss the formulation of the linear robustcounterpart to the portfolio optimization problem, first from aduality perspective (introduced by Bertsimas and Sim, 2004) andwe then explain the rationale for the model.

4.1. Robust counterpart by duality

The basic portfolio optimization problem is defined as follows:

MaxX

i

riwi;

S:t:X

i

wi ¼ 1;ð6Þ

where wi P 0 for all i. Asset returns, ri, are uncertain parameterswith unknown distributions defined as bounded and symmetricwith respect to a point estimate �ri:

ri 2 ½�ri � ri;�ri þ ri�: ð7Þ

Even though the true distribution of ri is unknown, historicaldata can be used to estimate the mean log return of asset i, andsubstituted for the point estimate �ri. A new stochastic variable gi

(Bertsimas and Thiele, 2006) measures the deviation of parameterri from �ri and takes values in [�1, 1], gi ¼ ðri � �riÞ=ri. By rearrangingthis equation, ri can be expressed as:

ri ¼ �ri þ rigi: ð8Þ

Let |J| be the number of parameters, ri, that are uncertain; then forSoyster’s and Ben-Tal and Nemirovski’s model

Xi

jri � �rijri

¼ jJj orX

i

jgij ¼ jJj:

Bertsimas and Sim (2004) relaxed this condition by defining a newparameter C (the budget of uncertainty) as the number of uncertainparameters that take their worst case value �ri � ri. Therefore,P

ijgij 6 C, such that C e [0, |J|]. Rewriting the initial portfolio opti-mization problem using (8) for ri:

Maxwi

Pi

�riwiþMingi

Pi

rigiwi

� �: � Max

wi

Pi

�riwi�Maxgi

Pi

rigiwi

� �:

S:t:P

iwi¼1; S:t:

Pi

wi¼1;Pijgij6C;

Pigi6C;

wi P 0;�16gi61; 8i: wi P 0;06gi61; 8i:

By duality (Bertsimas and Sim, 2004), the inner maximization prob-lem subject to the stochastic constraints becomes:

Minp;qi

CpþX

i

qi:

S:t: pþ qi P riwi; 8i;

p P 0;qi P 0; 8i:

Substituting this result, we obtain the following robust counterpart:

Maxwi

Pi

�riwi �Minp;qi

ðCpþP

iqiÞ

� �� Max

Pi

�riwi � Cp�P

iqi:

S:t:P

iwi ¼ 1; S:t:

Pi

wi ¼ 1;

pþ qi P riwi; 8i; pþ qi P riwi; 8i;

p P 0; p P 0;wi; qi P 0; 8i: wi; qi P 0; 8i:

4.2. Interpretation of robust counterpart

The robust counterpart of an uncertain optimization problem isa max–min or min–max model; the objective is to optimise theworst case performance. Soyster’s and Ben-Tal and Nemirovski’smodels stipulate that every constraint be feasible for every uncer-tain parameter defined within a bounded symmetric set (an uncer-tainty set). That is, their models were optimized for every uncertainparameter taking its worst case value.

Bertsimas and Sim (2004) introduced a model that assumes atmost C uncertain parameters will take their worst case values,not every parameter. Applying this concept, consider the basicportfolio model stated in Section 4.1, but using the definition ofri in (8).

We desire the portfolio with the best worst case return giventhat C asset returns take their worst case values, �ri � ri. Therefore,

Max MinðP

i

�riwi�Pt2T

rtwt� �

: � MaxP

i

�riwi�Min MaxPt2T

rtwt

� �:

S:t:P

iwi¼1; S:t:

Pi

wi¼1;

ð9Þ

where I ¼ fij1 6 i 6 Ng; T # I; jTj ¼ C, that is, T is the subset of C as-sets that take their worst case values, �rt � rt ; t 2 T. The min–maxterm in the objective seeks to minimise the worst case. The innermaximisation term seeks to choose C assets with the largest riwi

as the subset T whilst the outer minimisation term seeks to makethe sum as small as possible with respect to wi. For the moment,

C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428 421

assume that the quantity rtwt is the same for all t, and denoted by p.Then the term

Pt2T rtwt becomes Cp.

Now consider the possibility that pt – p for some t, wherept ¼ rtwt for all t. Clearly, at optimality, the term

Pt2T rtwt will be

greater than or equal to zero, hence, we will only consider the casewhen pt P p. Therefore, the difference pt � p, for all t, needs to beadded onto Cp and the quantity Min(Max

Pt2T rtwt) becomes:

Min½CpþX

t

ðpt � pÞ�; 8t ¼ ftjðpt � pÞP 0g: ð10Þ

We can restrict the difference pt � p to be greater than or equal tozero if we introduce a new variable qt given by the equation:

qt ¼maxð0;pt � pÞ: ð11Þ

The question now is: Which pt is chosen as our p value? Recallthat the min–max term in the objective seeks to maximize thequantity

Pt2T rtwt by choosing the C largest riwi as the subset T

and that the quantityP

tqt is greater than zero only when pt > p.Thus, p is chosen as the smallest rtwt , over all t, which means itis the Cth largest riwi, over all i.

The final portfolio optimization model becomes:

MaxX

i

�riwi � Cp�X

i

qi:

S:t:X

i

wi 6 1:

qi ¼ Max 0;X

i

riwi

! ð12Þ

Remark.P

i2Iqi can be substituted forP

t2T qt because every riwi,where i R T, will be less than p. Therefore, pi � p will be less thanzero and the corresponding qi will be zero.

Fig. 1. Number of assets selected for the portfolio at each C, when N = 30.

5. Robust portfolio models

5.1. The ‘Uncorrelated’ robust portfolio selection model

Bertsimas and Sim (2004) reformulated a maximum expectedreturn portfolio model as a linear robust optimization problem, gi-ven by (12), the derivation and interpretation of which was dis-cussed in Section 4. This model is termed ‘uncorrelated’ becauseasset correlations are not explicitly incorporated in the construc-tion of the uncertainty sets. This does not mean that the assetsthemselves are not correlated. Bertsimas and Sim (2004) also pre-sented a ‘correlated model’ which is discussed further inSection 5.2.

MaximizeXn

i¼1

�riwi � pC�Xn

i¼1

qi ð13Þ

Subject toXn

i¼1

wi ¼ 1;

pþ qi P criwi; 8i;

where �ri is the point estimate for the log return on asset i (e.g. themedian or mean log return), ri is the true log return of asset i, wi P 0is the proportion of total wealth invested in asset i and variables qi

and p are nonnegative. The true log return of asset i, ri, belongs tothe interval ½�ri � cri;�ri þ cri�, where ri is chosen by the user anddetermines how the uncertainty set defines ri, and c 2 Rþ definesthe magnitude of the range of the set U, as discussed in Section 2.For example, if ri is the standard deviation of asset i, then c woulddetermine the width of the interval in terms of the number of stan-dard deviations. Alternatively, if ri ¼ �ri, where �ri is the mean log re-turn, then c would determine the width of the interval in terms ofthe percentage of �ri that the true log return deviates from �ri. The

user defined parameter C is given a value between 0 and n. As C in-creases, the probability of underperforming the robust optimalobjective decreases. At optimality, p is the Cth largest criwi andqi ¼maxð0; criwi � pÞ for each asset i. The focus of Bertsimas andSim’s paper was to present their robust approach and not portfoliooptimization per se; their experimental results were for a set of 150stocks with �ri and ri generated by arithmetic progressions.

5.1.1. Portfolio composition5.1.1.1. Data set. The data set used to construct Figs. 1–3, consist ofthe monthly returns of 30 stocks selected at random from the FTSE100 index beginning 1 January 1992 through to 1 December 2002.Other investigations were completed using larger asset pools: 169stocks selected from the FTSE250, 248 stocks selected from theFTSE350, and 441 assets selected from the S&P500, beginning 1October 1998 through to 1 September 2008. In this investigation,the properties of the robust model which include diversification,the selection of assets and the weighting of assets were observedto be the same for N = 30 and for these larger asset pools. We choseto illustrate these properties using the results from N = 30 to sim-plify illustrations.

5.1.1.2. Diversification. Consider N + 1 consecutive portfolios corre-sponding to integer values of C from 0 to N. When C = 0, the port-folio consisted of 1 asset; this is simply the maximum returnproblem with no robustness. As C increased, the number of assetsincreased until a maximum number of assets was reached, whichin most cases was N. From this point, the composition of portfoliosfor successive values of C remained constant until all but 1 assetwere dropped, corresponding to C = Cdrop. Results show that whenC = Cdrop � 1, the resulting portfolio is the most robust (greatestprobability of optimality) diversified portfolio consisting of at leastas many assets as each portfolio corresponding to all other valuesof C. Lastly, for C P Cdrop the optimal portfolio consisted of the as-set with the largest risk-adjusted return, �ri � cri. This behaviour isshown in Fig. 1. Our results are quite different from those ofTütüncü and Koenig (2004) who found their robust portfolios tobe much less diversified, as they tended to concentrate on a smallset of asset classes, having chosen mostly large capital value stocks.

Fig. 2. Assets in descending order by �ri. An example of how assets are selected andhow weights change as more assets are included in the portfolio.

Fig. 3. Assets in ascending order by ri . An example of how robust models weightassets.

Fig. 4. Plot of p (riwi ¼ p; 8i) withP

iqi for C = 1. . .N.

422 C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428

5.1.1.3. Selection and weights. In Fig. 2, the assets along the x-axisare in descending order by �ri (mean log return); asset 1 has thelargest �ri and asset 30 has the smallest �ri. Fig. 2 has two purposes:Firstly, to show the assets held in each portfolio when C = 0through to C = 5 and secondly, to show how the weight of each as-set changed when C increased to C + 1. Again, consider N + 1 con-secutive optimal portfolios corresponding to integer values of Cfrom 0 to N. As C increased, the number of assets held also in-creased until a maximum number of assets (in this case N) wasreached; those with a larger �ri were selected first. In addition, aninteresting relationship exists between the weights of each assetheld in successive portfolios, i.e. from C to C + 1 (Fig. 2). Let NumAC

be the number of assets selected in a portfolio, for C = 0. . .N. IfNumAC+1 P NumAC then every asset held at C decreased by the

same percentage in order to include additional assets at C + 1(Fig. 2). Lastly, we consider how weight is distributed amongstthe chosen assets. In Fig. 3, the assets along the x-axis are inascending order by ri (standard deviation); asset 1 has the smallestri and asset 30 has the largest ri. Observe that once selected, the as-sets with the smallest ri were given the most weight. Asset weightsare only shown for values of C from one to five because at C = 0only one asset is held, at C = 6 through to C = Cdrop � 1 the portfo-lio composition and weights are the same as at C = 5 and atC = Cdrop through to C = N only the asset with the largest �ri � cri

is held.

5.1.1.4. Note on p and qi’s. Recall from Section 4.2 we deduced thatp is chosen as the Cth largest riwi, over all i. In our empirical re-sults, not only was p the Cth largest riwi, but riwi ¼ p for all i, giventhat wi > 0 and C 6 Cdrop � 1. Consequently, qi = 0 for all i. WhenC P Cdrop, each portfolio held only 1 asset, thus, p = 0 andqi ¼ riwi (which is 0 for all but one i). The relationship between pand qi is illustrated in Fig. 4. Only p and

Piqi are plotted for all C

instead of plotting pi (riwi) and qi for all i because riwi ¼ p (for alli, if wi > 0) and when

Piqi – 0, qi for the asset with the largest

risk-adjusted return is the only qi that does not equal zero; thus,Piqi equals the riwi of the asset with the largest worst case return.

5.2. The ‘correlated’ robust portfolio selection model

In addition to (13), Bertsimas and Sim (2004) proposed a robustportfolio model which they termed a ‘correlated model’. In ourview, this is not necessarily a robust portfolio selection model withasset correlations, but a type of robust factor model.

In their ‘correlated model’, there are K sources of data uncer-tainty (which are henceforth referred to as factors). The impactof each k e K on every asset i is given by the parameter gki, however,they do not state how this impact is measured; here we assumethat gki represents the magnitude of the impact and is thus non-negative. The true return of asset i is given by the following:

ri ¼ �ri þXk2K

gkgki; 8i: ð14Þ

C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428 423

Where ri and �ri are the true return and point estimate of asset i,respectively, and gk e [�1, 1]. Thus, the robust portfolio model is gi-ven by the following maximisation problem:

MaximizeX

i

�riwi � pC�X

k

qk

Subject toX

i

wi ¼ 1;

pþ qk PX

i

gkiwi; 8k;

ð15Þ

where wi, qi and p are nonnegative. The correlated model can beinterpreted similarly to the uncorrelated model as discussed in Sec-tion 4.2. The objective function in (15) can be expressed in a similarform to (9):

MaxX

i

�riwi �MinðMaxXt2T

Xi

gtiwiÞ; ð16Þ

where I = {i|1 6 i 6 N}, K is the set of factors k and t e T, where T # Kand |T| = C, that is, T is the subset of C factors that have the greatestimpact over all the assets. The min–max term in (16) seeks to min-imise the worst case. The inner maximisation term seeks to chooseC factors with the largest

Pigkiwi as the subset T whilst the outer

minimisation term seeks to make the sum as small as possible withrespect to wi. As with the uncorrelated model, assume that thequantity

Pigtiwi is the same for all t, and denoted by p. Then the

termP

teTP

igtiwi becomes Cp.Now consider the possibility that pt – p for some t, where

pt =P

igtiwi for all t. Clearly, at optimality, the termP

teTP

igtiwi,will be greater than or equal to zero, hence, only the case whenpt P p will be considered. Therefore, the difference pt � p, for allt, needs to be added onto Cp and the quantity Min(MaxP

teTP

igtiwi)becomes:

Min½CpþX

t

ðpt � pÞ�; 8t ¼ ftjðpt � pÞP 0g ð17Þ

As the difference pt � p is restricted to be greater than or equal tozero, a new variable qt, given by (11), is introduced:

qt ¼maxð0;pt � pÞ: ð18Þ

To determine which pt is chosen as the p value, recall that the min–max term in the objective seeks to maximize the quantityP

teTP

igtiwi by choosing the C factors, k, for whichP

igkiwi is thegreatest, as the subset T and that the quantity

Ptqt is greater than

zero only when pt > p. Thus, p is chosen as the smallestP

igtiwi, overall t, which means it is the Cth largest

Pigkiwi, over all k.

The final portfolio optimization model becomes:

MaxX

i

�riwi � Cp�X

k

qk: ð19Þ

S:t:X

i

wi 6 1;

qk ¼ Maxð0;X

i

gkiwiÞ; 8k:

Remark. As with the uncorrelated model,P

keKqk can be substi-tuted for

PteTqt because every

Pigkiwi where k R T, will be less than

p, thus pk � p will be less than zero and the corresponding qk willbe zero.

Applying what was learned about the properties of optimal ro-bust portfolios given by (13), we can have an understanding of thecomposition of robust portfolios resulting from (15). The MaxP

i�riwi term in (16) tells us that assets with the largest �ri will be

selected first, whereas the Min (MaxP

teTP

igtiwi)term indicatesthat of the assets selected, those which are least impacted by theU factors (i.e.

PteTgtiwi is the smallest) will be given more weight.

Note that in this model, we are no longer considering a set of Uassets taking their worst case value, but all assets. In addition,

unlike the ‘uncorrelated’ model, the bounds of the uncertaintyset (and hence the worst case return of each asset) are determinedat optimization and dependent upon the value of U specified by theuser. Similar to (7), the true return of asset i, ri, belongs to abounded interval centred on the point estimate �ri; ½�ri �

Pkgki;

�ri þP

kgki�. At optimality, U factors have been selected, thus, gk

has been given a value of �1 (selected) or 0 (not selected) for eachk (recall the definition of ri from (14)). We emphasize that gk is in-dexed by k and not i, therefore, this formulation does not considerthe set of k factors which have the greatest impact for each asset,but one set of U factors is chosen to impact all assets.

5.3. An alternate ‘correlated’ robust portfolio selection model

The correlated model presented by Bertsimas and Sim (2004)has several potential disadvantages. Firstly, as mentioned at theconclusion of Section 5.2, because gk is indexed by k and not i, theirformulation considers one set of U factors (decided at optimality)to impact all assets – it does not allow the user to identify the indi-vidual factors that have the greatest impact on each individual as-set. Secondly, the model is no longer seeking to maximise theworst case portfolio return, but is essentially minimising the worstcase impact of a set of factors K on the optimal portfolio return.Thirdly, it seems that defining the uncertainty set in this mannerimposes extra work on the optimization stage that could be deter-mined before hand by the user, in particular, the selection of fac-tors that effect each asset. Lastly, with this definition ofuncertainty, we can no longer distinguish between the followingthree aspects of scale discussed in Section 2, namely the definitionand estimation of ri, and the scaling of ri by the factor c.

To overcome these disadvantages, we have modified the defini-tion of ri in (14):

ri ¼ �ri þ gi

Xk2K

gki; 8i: ð20Þ

Where gki > 0 if factor k has a statistically significant impact on asseti, otherwise gki = 0. In this way, from the set of factors K, we candetermine the factors that have the greatest impact on each asseti. Thus, the uncertainty set can be constructed similarly to theuncorrelated model: ri ¼

Pkgki and the true return of asset i belongs

to the interval ½�ri �P

kgki;�ri þP

kgki�. Consequently, the ‘uncorre-lated’ model from (13) can be applied. In this way, the bounds ofthe uncertainty set are defined by the impact of the set K on eachasset, whilst the objective seeks to maximise the worst case portfo-lio return. In Section 6, the set of factors K is the set of N assets andgki is the contribution of asset k to the variance of asset i; we thentook the square root of the sum

Pkgki to obtain an estimate of each

assets standard deviation. That is, the interval for ri is½�ri �

ffiffiffiffiffiffiffiffiffiffiffiffiffiPkgki

p;�ri þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiPkgki

p�. In our investigation, principal compo-

nent analysis was used to determine the set of factors, K, whichhad the greatest impact on each asset i.

5.4. Computational platform

We chose to optimise this particular linear robust portfoliomodel using the solver CPLEX version 10.1, a common computa-tional platform, within the modelling language AMPL. More specia-lised modelling languages, such as CVX which is implemented inMatlab (Grant et al., 2008), may be chosen for solving such convexproblems.

6. The cost of robustness

Robustness, viewed as a performance guarantee, comes at acost. In the case of portfolio optimization, it is the probability guar-antee that the portfolio return will be at least equal to that of the

424 C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428

optimal robust solution. One would expect that in order to achieverobustness, a sacrifice, in terms of optimal objective value, will oc-cur. But how much does this sacrifice cost? And is it worth it?There is a two-fold motivation to the investigation detailed in thissection. Firstly, to provide a measure for the cost of robustness anddetermine if the robust methodology in Section 4 is, in reality, ro-bust. Secondly, to examine how the cost and robustness (bothguaranteed and achieved robustness) are affected when the follow-ing are changed: (i) the point estimate of asset i (�ri), the deviationparameter (ri), which indicates the maximum amount the truereturn of asset i may deviate from its point estimate, and scalingfactor c defining the uncertainty set U, of ri, given by ri 2 ½�ri�cri;�ri þ cri�; and/or (ii) the size of the set of historical data usedto estimate �ri and ri.

Recall that there are two aspects of an uncertainty set U: struc-ture and scale. Throughout this chapter, the structure remains con-stant in that it is polyhedral. However, because we allow thepossibility for C to be different, the dimension of the polyhedralset can change. With respect to the scale of U, different definitionsof �ri and ri and/or different values of the scaling factor c were usedto define each model; �ri and ri were estimated from the historicaldata set for each model (see Table 1).

The data set used in this investigation consists of 120 monthlylog returns of 68 assets from the FTSE 100 starting 1 February 1996through to 1 January 2007. Robust models, Rj, where j = 1 . . . 13(Table 1), each have a different uncertainty set defining ri, the truelog return of asset i, which is unknown and variable. In order to ob-tain a distribution for both measures of cost and both measures ofrobustness (detailed in Sections 6.1 and 6.2), three sets of 100 trialswere generated, one for each m e M, where M = {20, 40, 60}. That is,for each set of 100 trials, a set of m months was randomly selectedfor each trial, t, and considered as the set of available historicaldata used to estimate model parameters. One portfolio (i.e. oneC) was selected as the optimal portfolio for each of the 13 robustmodels for each t and m. The C value selected yielded the most ro-bust diversified portfolio (i.e. the portfolio with the smallest prob-ability of underperformance and consisting of more than oneasset), which corresponded to C�1

drop. Thus, a total of 1300 optimalportfolios were considered for each set of m months. Lastly, theset of m months was also used for in-sample back-testing in theevaluation of cost and robustness.

The value of C�1drop varied for all 1300 portfolios, for each set m,

hence, the probability of underperformance, given by (21), wasalso varied; however, as there were only N = 68 possible valuesof C�1

drop, many of the portfolios did have the same probability ofunderperformance. Bertsimas and Thiele (2006) define the proba-bility of underperformance as follows:

PrðZtruej;t;m;l 6 ZOpt

j;t;mÞ 6 1�UððC� 1Þ=ffiffiffiffiNpÞ ð21Þ

where each robust optimal objective function value for C, ZOptj;t;m, was

obtained using (13) from Section 5.1, Ztruej;t;m;l was the realised portfo-

lio return for model Rj, during trial t, given the set of m months,evaluated in-sample at month l, such that l = 1. . .m and N is the totalnumber of assets.

A note on portfolio composition: Before discussing the analysisof cost and robustness, it is important to briefly mention thecomposition of portfolios over all trials for each model and each

Table 1Summary of robust models.

Model Rj �ri ri c

j = 1, 2, 3 Mean log return Standard deviation 1, 2, 3j = 4, 5, 6, 7 Mean log return Mean log return 0.90, 0.95, 0.98, 1j = 8, 9, 10 Median log return Standard deviation 1, 2, 3j = 11, 12, 13 Mean log return Asset correlations 1, 2, 3

set of m months, a summary of which is available as additionalonline material. As will be evident in Sections 6.1 and 6.2, cost islargely dependent upon total portfolio return, which is a functionof Wj,t (the vector of optimal asset weights for model j at trial t),whilst robustness is largely dependent upon Udrop � 1. Thus, ifWj,t is the same for corresponding t between any Rj, then the costwill be the same. If this occurs for a large number of correspondingt, then the distributions will be similar. In contrast, if Wj,t is thesame for corresponding t between any Rj, the robustness will notnecessarily be affected; the guaranteed robustness of Rj, irrespec-tive of t, will only be the same if Udrop � 1 is the same. In addition,actual (or achieved) robustness is largely dependent upon m and cas well as Udrop � 1, hence, even if Wj,t is the same for correspond-ing t between any Rj (in which �ri and ri are defined in the sameway, respectively) results may show that the model with a largerc will be more robust. Lastly, models Rj, for j = 11, 12, 13, showedunique behaviour. As m increased from 20 to 60, not only werethe number of corresponding t for which Wj,t was the same be-tween j increasing, but this vector was actually the same for mostt, within and between each model. For example, when m = 60,W11,t = W12,t = W13,t for 93 trials and W12,t = W13,t for all trials. Thisis very important to keep in mind when evaluating the remainderof Section 6.

6.1. Measures of cost

For each trial t and set m, robust model Rj yields a n-vector ofoptimal asset weights, w�j;t;m. Therefore, the total portfolio returnof Rj for each t and m, is given by PTotal

j;t;m in (22):

PTotalj;t;m ¼

Xn

i¼1

�ri;t;mw�i;j;t;m; 8j; t;m ð22Þ

where �ri;t;m is the mean log return of asset i over a set of m monthsfor trial t. We introduce two measures for the cost of robustness. LetrMMax

t;m denote the return of the asset with the largest mean log returnfor trial t and set of m months. Cost1 and Cost2 measure the cost ofthe robust optimal portfolio, PTotal

j;t;m , with respect to rMMaxt;m . Cost1 mea-

sures the deviation between the value of the non-robust solution(i.e. with just a single asset) and the value of the robust solutions,whereas Cost2 measures the deviation as a ratio with respect torMMax

t;m . In other words, Cost2 can be thought of as a cost-to-maximumpotential reward ratio.

Cost1j;t;m ¼ rMMaxt;m � PTotal

j;t;m ; 8j; t;m: ð23Þ

Cost2j;t;m ¼ rMMaxt;m � PTotal

j;t;m

=rMMax

t;m ; 8j; t;m: ð24Þ

Bertsimas and Sim (2004) introduced the concept ‘‘price of robust-ness’’ as the difference between the robust optimal objective func-tion value and that of the nominal problem. The measures of costgiven by (23) and (24) differ from the price of robustness in thatthey measure the difference between the optimal objective of thenominal problem and the objective function value of the nominalproblem evaluated at the robust optimal solutions.

6.1.1. Measures of cost: resultsThe motivation for the cost analysis is to (1) determine the dis-

tributions of the cost of PTotalj;t;m for each model and (2) observe how

changing the definitions of �ri; ri and the scaling factor c and/orthe size of the data set m e M, where M = {20, 40, 60}, affects cost.We will only present the results for Cost2; the results for Cost1are available as additional online material.

The distribution of Cost2 for each model, Rj, is shown in Fig. 5(m = 20 months and m = 60 months). The minimum, maximum,median and mean are plotted for each model; each set of piecewiselinear functions correspond to a different scaling factor c, for fixed

C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428 425

definitions of �ri and ri. Observe the effect of changing c on each sta-tistic and distribution (Fig. 5). As with Cost1, an increase in ctended to correspond to an increase in Cost2 and each modeltended to cost less when m = 60 than when m = 20, although onlyslightly.

One distinction from Cost1 is that models R4 to R7 tended to costless than the other six models. Observed by their distributions andhistograms, R4 to R7 had a much smaller spread and were closer tobeing normally distributed, with means close to the other six mod-els, but without any outliers. The histograms of Cost2 (m = 20, 60)for each Rj, j = 1 . . . 3, 8 . . . 10, 11 . . . 13 were close to Normal, butwith slightly higher peaks and positively skewed. Models R1 to R3

had 1 or 2 outliers which were maximum whereas R8 had an out-lier that was a minimum, but only for m = 20 months. From ananalysis of Fig. 5, we conclude that one can expect models R1–R3,R8–R10 and R11–R13 to cost approximately 75–83% on average, andmodels R4–R7 to cost approximately 70–85% on average, whenthe number of months in the historical data set is between 20and 60 months. It is possible that further increasing the numberof months in the historical data set would result in decreased costs,however, further investigations not presented here suggest thatthe mean of Cost2 would not decrease significantly.

To summarise, the distribution of the loss in portfolio returnwith respect to rMMax

t;m for all 13 models was very similar, with nonehaving a significant advantage over the others, particularly with re-spect to the mean and median of both Cost1 and Cost2. The meanpercentage loss for each model, regardless of �ri; ri, m or c, wasapproximately 70–85%. However, the distributions of the percent-age loss in portfolio return with respect to rMMax

t;m showed that mod-els R4–R7 had a more consistent percentage loss and were closer toa Normal distribution without outliers. In addition, results suggestthat increasing the scaling factor c, i.e. increasing the scale (or size)of the structure of the uncertainty set U defining ri, tended to in-crease costs whereas increasing the number of months in the setof historical data tended to decrease costs.

6.2. Measures of robustness

Robustness is measured by (1) the probability of underperfor-mance, which is dependent upon C, and (2) the proportion of

Fig. 5. Distributions of Cost2 for each mode

evaluated portfolios that underperform the robust optimal objec-tive (PLO: Proportion of portfolios Less than Objective), given by(25) and (26) respectively.

PLOMaxj;t;m ¼ ð1� PrðZtrue

j;t;m;l P ZOptj;t;mÞÞ ¼ ð1�UððC� 1Þ=

ffiffiffiffiNpÞÞ; 8j; t;m:

ð25ÞPLOEval

j;t;m ¼X

l

dj;t;m;l=m; 8j; t;m; ð26Þ

where dj,t,m,l is 1 if Ztruej;t;m;l < ZOpt

j;t;m and 0 otherwise. Because both (25)and (26) are measures of underperformance, as they decrease,robustness increases and as they increase, robustness decreases.Comparing the distributions of PLOMax

j;t;m and PLOEvalj;t;m;8j;m, we can

evaluate the robustness of this methodology for the stated defini-tions of ri, c and m.

6.2.1. Measures of robustness: resultsThe motivation of the robust analysis is to determine (1) if the

robustness guaranteed by each model is actually achieved and (2)how changing the definition of �ri; ri, the scaling factor c and/or thesize of the data set m e M, affects the robustness of the solution(guaranteed and achieved).

The distribution of the guaranteed robustness, PLOMaxj;t;m, is shown

in Fig. 6 (m = 20 months and m = 60 months). For each model, theminimum, maximum, median and mean probability of underper-formance is plotted; each set of piecewise linear functions corre-sponds to different values of c for fixed values of �ri and ri. Forboth 20 months and 60 months, an increase in c corresponded toa decrease in the probability of underperformance; the distribu-tions became tighter and means and medians were closer to 0. Inaddition, the maximum for each model when m = 20 was muchhigher than the maximum when m = 60 (Fig. 6) which suggeststhat an increase in m also tended to result in a decrease in theprobability of underperformance, thus, greater guaranteed robust-ness. For example, R1 had a max close to 0.30 when m = 20 but amax of about 0.02 when m = 60. In addition, when m = 60 months,the distribution of each model had a much smaller spread and amean and median closer to 0. Recall that the probability of under-performance, PLOMax

j;t;m, is dependent upon Cdrop � 1, as largerCdrop � 1 yield a smaller PLOMax

j;t;m. Therefore, a larger data set of

l Rj, when m = 20 and m = 60 months.

Fig. 6. Distributions of PLOMaxj;t;m , for each model Rj, when m = 20 and m = 60 months.

426 C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428

m months yielded diversified portfolios for larger values of C (i.e.portfolios with greater guaranteed robustness). Lastly, althoughthere is not one type of model that guaranteed significantly morerobustness, those which define �ri and ri as the mean log return ofasset i (R4–R7), were characterised by distributions with slightlymore spread and higher maximum values, suggesting that theremaining nine models, j = 1 . . . 3, 8 . . . 13, are slightly moreadvantageous.

The distribution of the proportion of portfolios that actuallyunderperform the robust optimal objective (PLOEval

j;t;m), is shown formodel Rj, for m = 20 and m = 60 months in Fig. 7. For each Rj, theminimum, maximum, median and mean underperformance areplotted; each set of piecewise linear functions correspond to a dif-ferent scaling factor c for fixed definitions of �ri and ri. When �ri wasdefined as the mean or median log return and ri as the standarddeviation or asset correlations of asset i, increasing c also increased

Fig. 7. Distribution of PLOEvalj;t;m , for each mod

robustness. However, when both �ri and ri were defined as the meanlog return (R4–R7), increasing c tended to decrease actual robust-ness (Fig. 7). Recall that an increase in c also resulted in highercosts for these particular definitions of �ri and ri. Thus, as their costincreased, so did their actual underperformance (i.e. robustnessdecreased).

Table 2 shows the number of trials in which the proportion ofportfolios that underperformed the optimal objective was less thanthe probability of underperformance (i.e. the number of instancesin which PLOEval

j;t;m was less than PLOMaxj;t;m). Models R3, R10 and R13, in

which c = 3, had the largest percentage of trials that achieved or ex-ceeded the guaranteed level of robustness, i.e. in over 60% of thetrials the percentage of portfolios that underperformed the robustoptimal objective was less than the probability of underperfor-mance. Conversely, for models R4–R7, not one trial achieved or ex-ceeded the guaranteed level of robustness for any m.

el Rj, when m = 20 and m = 60 months.

Table 2The number of trials (out of 100) in which the guaranteed level of robustness wasachieved or exceeded, for each model Rj for a given set of m months.

How often is the guaranteed robustness achieved?

m R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13

20 1 29 63 0 0 0 0 2 27 63 30 47 9640 1 11 73 0 0 0 0 1 9 67 4 26 9660 0 9 64 0 0 0 0 0 8 62 0 17 98

C. Gregory et al. / European Journal of Operational Research 212 (2011) 417–428 427

Lastly, increasing m tended to decrease the proportion of port-folios that actually underperformed, shown by decreased max val-ues and tighter distributions for the same model (Fig. 7). However,this did not necessarily correspond to an increase in the number oftrials that achieved or exceeded the guaranteed robustness (Ta-ble 2). In addition, an increase in m also decreased the probabilityof underperformance (Fig. 6), which in most cases was less than 1%.In order for the actual percentage of portfolios that underperformto be less than 1%, for any given trial, the portfolio return had tobe greater than the robust optimal objective for every monthl(l = 1 . . . m) – not one out of the m months can underperform.For models R2, R3, R9, R10, R12 and R13, when m = 40 or 60 months,many trials did not achieve guaranteed robustness because onlyone portfolio, out of the 40 or 60, underperformed the robust opti-mal objective.

In summary, the distributions of the probability of underperfor-mance suggest that increasing the scale of the structure of uncer-tainty set U, defining ri, decreases the probability that the actualportfolio return will underperform the robust optimal objective(PLOMax

j;t;m). Likewise, when �ri was the mean or median log returnand ri was the standard deviation of asset i, the actual proportionof portfolios that underperform (PLOEval

j;t;m) also decreases, i.e. theywere more robust. However, when both �ri and ri were the meanlog return of asset i, the actual proportion of portfolios that under-perform tended to increase, which means portfolios were less ro-bust than they were guaranteed to be; these models were alsomuch less robust than the other 9 models.

6.3. Discussion of cost and robustness results

Given that the mean log return of asset i is uncertain and lies onthe interval ½�ri � cri;�ri þ cri�, and applying the robust portfoliomodel given in (5.1), we have provided measures to assess the costof robustness and examined how the cost and robustness (bothguaranteed and achieved robustness) are affected when the follow-ing are changed: (i) �ri; ri and scaling factor c, which define theuncertainty set of ri, and/or (ii) the size of the set of historical dataused to estimate �ri and ri. When �ri and ri were both specified as themean log return of asset i (R4–R7), portfolios were slightly lesscostly, with respect to Cost1 (the difference between the non-ro-bust and robust total portfolio return) and Cost2 (cost-to-maxi-mum potential reward ratio), but also less robust than the othermodels, particularly with respect to achieved robustness. In addi-tion, when �ri and ri were both specified as the mean log return ofasset i, an increase in the scale of uncertainty set U not only in-creased cost but decreased actual robustness, which is counterin-tuitive. One would expect that in exchange for a sacrifice inportfolio return there would be an increase in achieved robustness,i.e. fewer portfolios would underperform the optimal objectivefunction value. For the other 9 models, if increasing the range ofthe uncertainty set also increased cost, it was in exchange forincreased robustness. The results suggest that models whichdefine �ri as the mean or median log return and ri as the standarddeviation of asset i have the most desirable trade-off between costand achieved robustness as well as guaranteed and achievedrobustness. Recall from Section 2, that both Ceria and Stubbs

(2006) and Chen and Tan (2009) evaluate robust models whereboth �ri and ri are percentiles of the distribution of asset i. Ourresults suggest that their results may be improved by using ameasure of spread for ri.

7. Summary and conclusions

In this paper we interpreted the robust optimization methodol-ogy of Bertsimas and Sim (2004), both their ‘uncorrelated’ and ‘cor-related’ models from a min-regret point of view; we also presentedan alternate correlated model. Robust optimization is best appliedin situations where parameter values are unknown, variable, andtheir distributions are uncertain. In the case when distributionscan be precisely estimated one should consider other methodolo-gies such as stochastic programming. We have shown that thecomposition of robust portfolios are intuitive in nature because as-sets are first selected for the portfolio in descending order by meanlog return and then those with smaller ri are given more weight. Inaddition, robust portfolios are diversified both in terms of the num-ber of assets and in weight, which is an advantageous feature,especially because portfolios are more diversified for larger valuesof C (through to Udrop � 1) which corresponds to greater guaran-teed robustness.

The results of the investigation reported in this paper show thatrobust models in which �ri is the mean or median log return and ri isthe standard deviation of asset i, yield the most robust and costeffective portfolios. Furthermore, for these models, uncertaintysets with a larger range tend to result in higher costs, but increasedrobustness. Results suggest that a value of c P 2 is preferred, i.e. ri

is defined as being within at least two standard deviations of itsmean or median log return. The risk preferences of the investordetermine the value of c chosen. In exchange for increased guaran-teed and achieved robustness, the 99% confidence interval for themean of each model suggests that one can expect to sacrificeapproximately 68–90%, on average, in optimality with respect tothe optimal objective value of the maximum return problem,which simply chooses the asset with the largest mean log return.We do recognise that this is an empirical investigation and thatalthough individual cases of cost and robustness for each modelare stable, particularly models R1–R3 and R8–R10, these results aredependent on one particular set of data (FTSE 100 over the period01 January 1992 through to 01 December 2007). Furthermore, it isimportant to note that every model, robust or not, comes at a cost.In future work we will compare the cost of Expected value–Variance (E–V) portfolios to that of the robust portfolios.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.ejor.2011.02.015.

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