CORC Technical Report TR-2002-03

Robust portfolio selection problems

D. Goldfarb G. Iyengar

Submitted: Dec. 26, 2001.Revised: May 25th, 2002, July 24th, 2002

Abstract

In this paper we show how to formulate and solve robust portfolio selection problems. Theobjective of these robust formulations is to systematically combat the sensitivity of the optimalportfolio to statistical and modeling errors in the estimates of the relevant market parameters.We introduce “uncertainty structures” for the market parameters and show that the robustportfolio selection problems corresponding to these uncertainty structures can be reformulatedas second-order cone programs and, therefore, the computational effort required to solve themis comparable to that required for solving convex quadratic programs. Moreover, we show thatthese uncertainty structures correspond to confidence regions associated with the statisticalprocedures employed to estimate the market parameters. Finally, we demonstrate a simplerecipe for efficiently computing robust portfolios given raw market data and a desired level ofconfidence.

Keywords: Robust optimization, Mean-variance portfolio selection, Value-at-risk portfolio selec-tion, Second-order cone programming, Linear regression.

1 Introduction

Portfolio selection is the problem of allocating capital over a number of available assets in orderto maximize the “return” on the investment while minimizing the “risk”. Although the benefits ofdiversification in reducing risk have been appreciated since the inception of financial markets, thefirst mathematical model for portfolio selection was formulated by Markowitz (1952, 1959). In theMarkowitz portfolio selection model, the “return” on a portfolio is measured by the expected valueof the random portfolio return, and the associated “risk” is quantified by the variance of the portfolioreturn. Markowitz showed that, given either an upper bound on the risk that the investor is willingto take or a lower bound on the return the investor is willing to accept, the optimal portfolio canbe obtained by solving a convex quadratic programming problem. This mean-variance model hashad a profound impact on the economic modeling of financial markets and the pricing of assets –the Capital Asset Pricing Model (CAPM) developed primarily by Sharpe (1964), Lintner (1965)and Mossin (1966) was an immediate logical consequence of the Markowitz theory. In 1990, Sharpeand Markowitz shared the Nobel Memorial Prize in Economic Sciences for their work on portfolioallocation and asset pricing.

1

In spite of the theoretical success of the mean-variance model, practitioners have shied awayfrom this model. The following quote from Michaud (1998) summarizes the problem: “AlthoughMarkowitz efficiency is a convenient and useful theoretical framework for portfolio optimality,in practice it is an error prone procedure that often results in error-maximized and investment-irrelevant portfolios.” This behavior is a reflection of the fact that solutions of optimization prob-lems are often very sensitive to perturbations in the parameters of the problem; since the estimatesof the market parameters are subject to statistical errors, the results of the subsequent optimiza-tion are not very reliable. Various aspects of this phenomenon have been extensively studied inthe literature on portfolio selection. Chopra and Ziemba (1993) studies the cash-equivalent lossfrom the use of estimated parameters instead of the true parameters. Broadie (1993) investigatesthe influence of errors on the efficient frontier and Chopra (1993) investigates the turnover in thecomposition of the optimal portfolio as a function of the estimation error (see also Part II of Ziembaand Mulvey (1998) for a summary of this research).

Several techniques have been suggested to reduce the sensitivity of the Markowitz-optimal portfoliosto input uncertainty: Chopra (1993) and Frost and Savarino (1988) propose constraining portfolioweights, Chopra et al. (1993) proposes using a James-Stein estimator (see Huber (1981) for detailson Stein estimation) for the means, while Klein and Bawa (1976), Frost and Savarino (1986), andBlack and Litterman (1990) suggest Bayesian estimation of means and covariances. Although thesetechniques reduce the sensitivity of the portfolio composition to the parameter estimates, they arenot able to provide any guarantees on the risk-return performance of the portfolio. Michaud (1998)suggests resampling the mean returns µ and the covariance matrix Σ of the assets from a confidenceregion around a nominal set of parameters, and then aggregating the portfolios obtained by solvinga Markowitz problem for each sample. Recently scenario-based stochastic programming modelshave also been proposed for handling the uncertainty in parameters (see Part V of Ziemba andMulvey (1998) for a survey of this research). Both the sampling-based and the scenario-basedapproaches do not provide any hard guarantees on the portfolio performance and become veryinefficient as the number of assets grows.

In this paper we propose alternative deterministic models that are robust to parameter uncertaintyand estimation errors. In this framework, the perturbations in the market parameters are modeledas unknown, but bounded, and optimization problems are solved assuming worst case behaviorof these perturbations. This robust optimization framework was introduced in Ben-Tal and Ne-mirovski (1999) for linear programming and in Ben-Tal and Nemirovski (1998) for general convexprogramming (see also Ben-Tal and Nemirovski (2001)). There is also a parallel literature on robustformulations of optimization problems originating from robust control (see El Ghaoui and Lebret(1997), El Ghaoui et al. (1998), and El Ghaoui and Niculescu (1999)).

Our contributions in this paper are as follows:

(a) We develop a robust factor model for the asset returns. In this model the vector of randomasset returns r ∈ Rn is given by

r = µ+VT f + ε,

where µ ∈ Rn is the vector of mean returns, f ∈ Rm is the vector of random returns of them(< n) factors that drive the market, V ∈ Rm×n is the factor loading matrix and ε is thevector of residual returns. The mean return vector µ, the factor loading matrix V, and thecovariance matrices of the factor return vector f and the residual error vector ε are knownto lie within suitably defined uncertainty sets. For this market model, we formulate robustanalogs of classical mean-variance and value-at-risk portfolio selection problems.

2

(b) We show that the natural uncertainty sets for the market parameters are defined by thestatistical procedures employed to estimate these parameters from market return data. Thisclass of uncertainty sets is completely parametrized by the market data and a parameter ωthat controls the confidence level; thereby allowing one to provide probabilistic guarantees onthe performance of the robust portfolios. In all previous work on robust optimization (Ben-Taland Nemirovski, 1998, 1999; El Ghaoui and Lebret, 1997; El Ghaoui et al., 1998; Halldorssonand Tutuncu, 2000), the uncertainty sets for the parameters are assumed to have a certainstructure without any explicit justification. Also, there is no discussion of how these sets areparametrized from raw data.

(c) We show that the robust optimization problems corresponding to the natural class of uncer-tainty sets (defined by the estimation procedures) can be reformulated as second-order coneprograms (SOCPs). SOCPs can be solved very efficiently using interior point algorithms (Nes-terov and Nemirovski, 1993; Lobo et al., 1998; Sturm, 1999). In fact, both the worst case andpractical computational effort required to solve an SOCP is comparable to that for solvinga convex quadratic program of similar size and structure, i.e. in practice the computationaleffort required to solve these robust portfolio selection problems is comparable to that requiredto solve the classical Markowitz mean-variance portfolio selection problems.

In a recent related paper, Halldorsson and Tutuncu (2000) show that if the uncertain mean returnvector µ and the uncertain covariance matrix Σ of the asset returns r belong to the component-wiseuncertainty sets Sm = {µ : µL ≤ µ ≤ µU} and Sv = {Σ : Σ º 0,ΣL ≤ Σ ≤ ΣU} respectively, therobust problem reduces to a nonlinear saddle-point problem that involves semidefinite constraints.Here A º 0 (resp. Â 0) denotes that the matrix A is symmetric and positive semidefinite (resp.definite). This approach has several shortcomings when applied to practical problems - the model isnot a factor model (in applied work factor models are popular because of the econometric relevanceof the factors), no procedure is provided for specifying the extreme values (µL,µU ) and (ΣL,ΣU )defining the uncertainty structure and, moreover, the solution algorithm, although polynomial,is not practicable when the number of assets is large. A multi-period robust model, where theuncertainty sets are finite sets, was proposed in Ben-Tal et al. (2000).

The organization of the paper is as follows. In Section 2 we introduce the robust factor model anduncertainty sets for the mean return vector, the factor loading matrix, and the covariance matrix ofthe residual return. We also formulate robust counterparts of the mean-variance optimal portfolioselection problem, the maximum Sharpe ratio portfolio selection problem and the value-at-risk(VaR) portfolio selection problem. The uncertainty sets introduced in this section are ellipsoidal(intervals in the one dimensional case), and may appear quite arbitrary. Before demonstrating thatthese sets are, indeed, natural, we first establish in Section 3 that the robust mean-variance portfolioselection problem in markets where the factor loading and mean returns are uncertain, but the factorcovariance is known and fixed, can be reformulated as an SOCP. An SOCP formulation of the robustmaximum Sharpe ratio problem in such markets follows as a corollary. In Section 4 we developan SOCP reformulation for the robust VaR portfolio selection problem. In Section 5 we justifythe uncertainty sets introduced in Section 2 by relating them to linear regression. Specifically, weshow that the uncertainty sets correspond to confidence regions around the least-squares estimateof the market parameters and can be constructed to reflect any desired confidence level. In thissection, we collect the results from the preceding sections and present a recipe for robust portfolioallocation that closely parallels the classical one. In Section 6 we improve the factor model byallowing uncertainty in the factor covariance matrix and show that, for natural classes of uncertaintysets, all robust portfolio allocation problems continue to be SOCPs. We also show that these

3

natural classes of uncertainty sets correspond to the confidence regions associated with maximumlikelihood estimation of the covariance matrix. In Section 7 we present results of some of preliminarycomputational experiments with our robust portfolio allocation framework.

2 Market model and robust investment problems

We assume that the market opens for trading at discrete instants in time and has n traded assets.The vector of asset returns over a single market period is denoted by r ∈ Rn, with the interpretationthat asset i returns (1 + ri) dollars for every dollar invested in it. The returns on the assets indifferent market periods are assumed to be independent. The single period return r is assumed tobe a random variable given by

r = µ+VT f + ε, (1)

where µ ∈ Rn is the vector of mean returns, f ∼ N (0,F) ∈ Rm is the vector of returns of thefactors that drive the market, V ∈ Rm×n is the matrix of factor loadings of the n assets andε ∼ N (0,D) is the vector of residual returns. Here x ∼ N (µ,Σ) denotes that x is a multivariateNormal random variable with mean vector µ and covariance matrix Σ.

In addition, we assume that the vector of residual returns ε is independent of the vector of factorreturns f , the covariance matrix F Â 0 and the covariance matrix D = diag(d) º 0, i.e. di ≥ 0,i = 1, . . . , n. Thus, the vector of asset returns r ∼ N (µ,VTFV +D). Although not required forthe mathematical development in this paper, the eigenvalues of the residual covariance matrix Dare typically much smaller than those of the covariance matrix VTFV implied by the factors, i.e.the VTFV is a good low rank approximation of the covariance to the asset returns.

Except in Section 6, the covariance matrix F of the factor returns f is assumed to be stable andknown exactly. The individual diagonal elements di of the covariance matrix D are assumed to liein an interval [di, di], i.e. the uncertainty set Sd for the matrix D is given by

Sd = {D : D = diag(d), di ∈ [di, di], i = 1, . . . , n}. (2)

The columns of the matrix V, i.e. the factor loadings of the individual assets, are also assumed tobe known approximately. In particular, V belongs to the elliptical uncertainty set Sv given by

Sv ={

V : V = V0 +W, ‖Wi‖g ≤ ρi, i = 1, . . . , n}

, (3)

where Wi is the i-th column of W and ‖w‖g =√wTGw denotes the elliptic norm of w with

respect to a symmetric, positive definite matrix G.

The mean returns vector µ is assumed to lie in the uncertainty set set Sm given by

Sm = {µ : µ = µ0 + ξ, |ξi| ≤ γi, i = 1, . . . , n} ; (4)

i.e. each component of µ is assumed to lie within a certain interval. The choice of the uncertaintysets is motivated by the fact that the factor loadings and the mean returns of assets are estimatedby linear regression. The justification of the uncertainty structures and suitable choices for thematrix G, and the bounds ρi, γi, di, di, i = 1, . . . , n, are discussed in Section 5.

An investor’s position in this market is described by a portfolio φ ∈ Rn, where the i-th componentφi represents the fraction of total wealth invested in asset i. The return rφ on the portfolio φ is

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given byrφ = rTφ = µTφ+ fTVφ+ εTφ ∼ N (φTµ,φT (VTFV +D)φ).

The objective of the investor is to choose a portfolio that maximizes the “return” on the investmentsubject to some constraints on the “risk” of the investment. The mathematical model for portfolioselection proposed by Markowitz (1952, 1959) assumes that the expected value E[r] of the assetreturns and the covariance Var

[

r]

are known with certainty. In this model investment “return” isthe expected value E[rφ] of the portfolio return and the associated “risk” is the variance Var

[

rφ]

.The objective of the investor is to choose a portfolio φ∗ that has the minimum variance amongstthose that have expected return at least α, i.e. φ∗ is the optimal solution of the convex quadraticoptimization problem

minimize Var[

rφ]

subject to E[rφ] ≥ α,1Tφ = 1.

(5)

As was pointed out in the introduction, the primary criticism leveled against the Markowitz modelis that the optimal portfolio φ∗ is extremely sensitive to the market parameters (E[r],Var

[

r]

) –since these parameters are estimated from noisy data, φ∗ often amplifies noise. By introducing“measures of uncertainty” in the market models, we are attempting to correct this sensitivity toperturbations. The uncertainty sets Sm, Sv and Sd represent the uncertainty of our limited (inexact)information of the market parameters and we wish to select portfolios that performs well for allparameter values that are consistent with this limited information. Such portfolios are solutions ofappropriately defined min-max optimization problems called robust portfolio selection problems.

The robust analog of the Markowitz mean-variance optimization problem (5) is given by

minimize max{V∈Sv ,D∈Sd}Var[

rφ]

subject to min{µ∈Sm}E[rφ] ≥ α,

1Tφ = 1.

(6)

The objective of the robust minimum variance portfolio selection problem (6) is to minimize theworst case variance of the portfolio subject to the constraint that the worst case expected return onthe portfolio is at least α. We expect that the sensitivity of the optimal solution of this mathematicalprogram to parameter fluctuations will be significantly smaller than it would be for its classicalcounterpart (5).

A closely related problem, the robust maximum return problem, is the dual of (6). In this problem,the objective is to maximize the worst case expected return subject to a constraint on the worstcase variance, i.e. to solve the mathematical program

maximize min{µ∈Sm}E[rφ]subject to max{V∈Sv ,D∈Sd}Var

[

rφ]

≤ λ,

1Tφ = 1.

(7)

Another variant of the robust optimization problem (6) is the robust maximum Sharpe ratio problem.Here the objective is to choose a portfolio that maximizes the worst case ratio of the expected excessreturn on the portfolio, i.e. the return in excess of the risk-free rate rf , to the standard deviationof the return. The corresponding max-min problem is given by,

maximize{φ:1Tφ=1} min{µ∈Sm,V∈Sv ,D∈Sd}

{

E[rφ]−rf√

Var[

rφ

]

}

(8)

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All these variants are studied in Section 3. We show that for the uncertainty sets Sd, Sm and Sv,defined in (2)-(4) above, all of these problems reduce to SOCPs. Although the factor model (1)is crucial for the SOCP reduction, the assumption that factor and residual returns are Normallydistributed can be relaxed. From the results in Bertsimas and Sethuraman (2000) it follows thatall of our results continue to hold with slight modifications if one assumes that the distributionsare unknown but second moments of the factor and residual returns lie in the sets Sv and Sdrespectively. We leave this extension to the reader.

In Section 4 we study robust portfolio selection with value-at-risk (VaR) constraints where theobjective is to maximize the worst case expected return of the portfolio subject to the constraintthat the probability of the return rφ falling below a threshold α is less than a prescribed limit, i.e.the objective is to solve the following mathematical program.

maximize min{µ∈Sm}E[rφ]subject to max{V∈Sv ,µ∈Sm,D∈Sd}P(rφ ≤ α) ≤ β.

(9)

VaR was introduced as a performance analysis tool in the context of risk management. Re-cently there has been a growing interest in imposing VaR-type constraints while optimizing creditrisk (Kast et al., 1998; Mausser and Rosen, 1999; Andersson et al., 2001). We show that, for theuncertainty sets defined in (2)-(4), this problem can also be reduced to an SOCP and, hence, canbe solved very efficiently. Our technique can be extended to another closely related performancemeasure called the conditional VaR by using the results in Mausser and Rosen (1999). El Ghaouiet al. (2002) presents an alternative robust approach to VaR problems that results in semidefiniteprograms.

The optimization problems of interest here fall in the general class of robust convex optimizationproblems. Ben-Tal and Nemirovski (1998, 2001) propose the following structure for generic robustoptimization problem.

minimize cTxsubject to F (x, ξ) ∈ K ⊂ Rm, ∀ξ ∈ U , (10)

where ξ are the uncertain parameters in the problem, x ∈ Rn is the decision vector, K is a convexcone and, for fixed ξ ∈ U , the function F (x, ξ) is K-concave.The essential ideas leading to this formalism were developed in robust control (see Zhou et al. (1996)and references therein). Robustness was introduced to mathematical programming by Ben-Tal andNemirovski (1998, 1999, 2001). They established that for suitably defined uncertainty sets U ,the robust counterparts of linear programs, quadratic programs, and general convex programs arethemselves tractable optimization problems. Robust least squares problems and robust semidefiniteprograms were independently studied by El Ghaoui and his collaborators (El Ghaoui and Lebret,1997; El Ghaoui et al., 1998). Ben-Tal et al. (2000) have studied robust modeling of multi-stageportfolio problems. Halldorsson and Tutuncu (2000) study robust investment problems that reduceto saddle point problems.

3 Robust mean-variance portfolio selection

This section begins with a detailed analysis of the robust minimum variance problem (6). It isshown that for uncertainty sets defined in (2)-(4) this problem reduces to an efficiently solvable

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SOCP. This result is subsequently extended to the maximum return problem (7) and the robustmaximum Sharpe ratio problem (8).

3.1 Robust minimum variance problem

Since the return rφ ∼ N (µTφ,φT (VTFV+D)φ), the robust minimum variance portfolio selectionproblem (6) is given by

minimize max{V∈Sv}{

φTVTFVφ}

+max{D∈Sd}{

φTDφ}

subject to min{µ∈Sm} µTφ ≥ α,

1Tφ = 1.

(11)

The bounds di ≤ di ≤ di imply that φTDφ ≤ φTDφ, where D = diag(d). Also, since thecovariance matrix F of the factor f is assumed to be strictly positive definite, the function ‖x‖f :

x 7→√xTFx defines a norm on Rm. Thus, (11) is equivalent to the robust augmented least squares

problemminimize max{V∈Sv} ‖Vφ‖2f + φTDφsubject to min{µ∈Sm} µ

Tφ ≥ α,

1Tφ = 1.

(12)

By introducing auxiliary variables ν and δ, the robust problem (12) can be reformulated as

minimize ν + δ

subject to max{V∈Sv} ‖Vφ‖2f ≤ ν,

φTDφ ≤ δ,min{µ∈Sm} µ

Tφ ≥ α,

1Tφ = 1.

(13)

If the uncertainty sets Sv and Sm are finite, i.e. Sv = {V1, . . . ,Vs} and Sm = {µ1, . . . ,µr}, then(13) reduces to the convex quadratically constrained problem

minimize λ+ δ

subject to ‖Vkφ‖2f ≤ λ, for all k = 1, . . . , s,

φTDφ ≤ δ,

φTµk ≥ α, for all k = 1, . . . , r,1Tφ = 1.

(14)

This problem can be easily converted to an SOCP (see Lobo et al. (1998) or Nesterov and Nemirovski(1993) for details). In El Ghaoui and Lebret (1997) it was shown that for

Sv ={

V : V = V0 +W, ‖W‖ =√

Tr(WTW) ≤ ρ}

,

the problem can still be reformulated as an SOCP. Methodologically speaking, our results can beviewed as an extension of the results in El Ghaoui and Lebret (1997) to other classes of uncertaintysets more suited to the application at hand.

When the uncertainty in V and µ is specified by (3) and (4) respectively, the worst case meanreturn of a fixed portfolio φ is given by,

min{µ∈Sm}

µTφ = µT0 φ− γT |φ| ; (15)

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and the worst case variance is given by,

maximize ‖(V0 +W)φ‖2fsubject to ‖Wi‖g ≤ ρi, i = 1, . . . , n.

(16)

Since the constraints ‖Wi‖g ≤ ρi, i = 1, . . . , n imply the bound,

‖Wφ‖g =∥

∥

∥

n∑

i=1

φiWi

∥

∥

∥

g≤

n∑

i=1

|φi| ‖Wi‖g ≤n∑

i=1

ρi |φi| , (17)

the optimization problem,maximize ‖V0φ+w‖2fsubject to ‖w‖g ≤ r,

(18)

where r = ρT |φ| =∑ni=1 ρi |φi|, is a relaxation of (16), i.e. the optimal value of (18) is at least as

large as that of (16).

The objective function in (18) is convex; therefore, the optimal solution w∗ lies on the boundaryof the feasible set, i.e. ‖w∗‖g = r. For i = 1, . . . , n, define

W∗i =

{

φi|φi|

ρir w

∗, φi 6= 0,ρir w

∗, otherwise.(19)

Then ‖W∗i ‖g = ρi, for all i = 1, . . . , n, i.e. W∗ is feasible for (16), and W∗φ =

∑ni=1 φiW

∗i = w∗.

Therefore, the optimal value of (16) and (18) are, in fact, the same.

Thus, for a fixed portfolio φ, the worst-case variance is less than ν if, and only if,

max{y:‖y‖g≤r}

‖y0 + y‖2f ≤ ν, (20)

where y0 = V0φ and r = ρT |φ|.The following lemma reformulates this constraint as a collection of linear equalities, linear inequal-ities and restricted hyperbolic constraints (i.e. constraints of the form: zT z ≤ xy, z ∈ Rn, x, y ∈ Rand x, y ≥ 0).

Lemma 1 Let r, ν > 0, y0,y ∈ Rm and F,G ∈ Rm×m be positive definite matrices. Then theconstraint

max{y:‖y‖g≤r}

‖y0 + y‖2f ≤ ν (21)

is equivalent to either of the following:

(i) there exist τ, σ ≥ 0, and t ∈ Rm+ that satisfy

ν ≥ τ + 1T tσ ≤ 1

λmax(H) ,

r2 ≤ στ,w2i ≤ (1− σλi)ti, i = 1, . . . ,m,

(22)

where QΛQT is the spectral decomposition of H = G− 12FG−

12 , Λ = diag(λi), and w =

QTH12G

12y0;

8

(ii) there exist τ ≥ 0 and s ∈ Rm+ that satisfy

r2 ≤ τ(ν − 1T s),u2i ≤ (1− τθi)si, i = 1, . . . ,m,τ ≤ 1

λmax(K) ,(23)

where PΘPT is the spectral decomposition of K = F12G−1F

12 , Θ = diag(θi), and u =

PTF12y0.

Proof: By setting y = ry, we have that (21) is equivalent to

(ν − yT0 Fy0)− 2ryT0 Fy − r2yTFy ≥ 0, (24)

for all y such that 1− yTGy ≥ 0. Before proceeding further, we need the following:

Lemma 2 (S-procedure) Let Fi(x) = xTAix+2bTi x+ ci, i = 0, . . . , p be quadratic functions ofx ∈ Rn. Then F0(x) ≥ 0 for all x such that Fi(x) ≥ 0, i = 1, . . . , p, if there exist τi ≥ 0 such that

[

c0 bT0b0 A0

]

−p∑

i=1

τi

[

ci bTibi Ai

]

º 0.

Moreover, if p = 1 then the converse holds if there exists x0 such that F1(x0) > 0.

For a discussion of the S-procedure and its applications, see Boyd et al. (1994). Since y = 0 isstrictly feasible for 1− yTGy ≥ 0, the S-procedure implies that (24) holds for all 1− yTGy ≥ 0 ifand only if there exists a τ ≥ 0 such that

M =

[

ν − τ − yT0 Fy0 −ryT0 F−rFy0 τG− r2F

]

º 0. (25)

Let the spectral decomposition of H = G− 12FG−

12 be QΛQT , where Λ = diag(λ), and define

w = QTH12G

12y0 = Λ

12QTG

12y0. Observing that yT0 Fy0 = wTw, we have that the matrix M º 0

if and only if

M =

[

1 0T

0 QTG−12

]

M

[

1 0T

0 G−12Q

]

=

[

ν − τ −wTw −rwTΛ12

−rΛ 12w τI− r2Λ

]

º 0.

The matrix M º 0 if and only if τ ≥ r2λi, for all i = 1, . . . ,m (i.e. τ ≥ r2λmax(H)), wi = 0 forall i such that τ = r2λi, and the Schur complement of the nonzero rows and columns of τI− r2Λ

ν − τ −wTw − r2

∑

i:τ 6=r2λi

λiw2i

τ − r2λi

= ν − τ −∑

i:σλi 6=1

w2i1− σλi

≥ 0,

where σ = r2

τ . It follows that

max{y:‖y‖g≤r}

‖y0 + ry‖2f ≤ ν,

9

if and only if there exists τ, σ ≥ 0 and t ∈ Rm+ satisfying,

ν ≥ τ + 1T t,r2 = στ,w2i = (1− σλi)ti, i = 1, . . . ,m,σ ≤ 1

λmax(H) .

(26)

It is easy to establish there exist τ, σ ≥ 0, and t ∈ Rm+ that satisfy (26) if and only if there

exist τ, σ ≥ 0, and t ∈ Rm+ that satisfy (26) with the equalities replaced by inequalities, i.e. that

satisfy (22).

To establish the second representation, note that (21) holds if and only if

yTGy ≥ r2, (27)

for all y such that ‖y0 + y‖2f ≥ ν. Since ‖y0 + y‖2f > ν for all sufficiently large y, the S-procedureimplies that (27) holds for all ‖y0 + y‖2f ≥ ν if and only if there exists a τ ≥ 0 such that

M =

[

−r2 − τ(yT0 Fy0 − ν) −τyT0 F−τFy0 G− τF

]

º 0. (28)

Let PΘPT be the spectral decomposition of K = F12G−1F

12 , where Θ = diag(θ), and u =

PTF12y0. Then M º 0 if and only if

M =

[

1 0T

0 PTF−12

]

M

[

1 0T

0 F−12P

]

=

[

−r2 − τ(uTu− ν) −τuT−τu Θ−1 − τI

]

º 0.

But M º 0 if and only if τ ≤ 1θi, for all i = 1, . . . ,m (i.e. τ ≤ 1/λmax(K)), ui = 0 for all i such

that τθi = 1, and the Schur complement of the nonzero row and columns of Θ−1 − τI

r2 − τ(uTu− ν)− τ 2

∑

i:τθi 6=1

u2iθ−1i − τ

= −r2 + τ

ν −∑

i:τθi 6=1

u2i1− τθi

≥ 0.

Hence,max

{y:‖y‖g≤r}‖y0 + ry‖2f ≤ ν,

if and only if there exists τ ≥ 0 and s ∈ Rm+ satisfying,

r2 ≤ τ(ν − 1T s),u2i = (1− τθi)si, i = 1, . . . ,m,τ ≤ 1

λmax(K) .(29)

Completely analogous to the proof of Part (i), we have that there exists τ ≥ 0 and s ∈ Rm+

satisfying (29) if and only if there exist τ ≥ 0 and s ∈ Rm+ satisfying (23). This proves the second

result.

To illustrate the equivalence of parts (i) and (ii) of Lemma 1, we note that if F = κG thenK = H = κI and w = u. Hence, if (τ, s) satisfies (23) then (σ, τ , t) = (τ, ν − 1T s, s) satisfies (22).

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The restricted hyperbolic constraints, zT z ≤ xy, x, y ≥ 0, can be reformulated as second-order coneconstraints as follows (see Nesterov and Nemirovski, 1993, Section 6.2.3)

zTz ≤ xy ⇔ 4zTz ≤ (x+ y)2 − (x− y)2 ⇔∥

∥

∥

∥

[

2zx− y

]∥

∥

∥

∥

≤ x+ y. (30)

The above result and Lemma 1 motivate the following definition.

Definition 1 Given V0 ∈ Rm×n, and F,G ∈ Rm×m positive definite, define H(V0,F,G) to bethe set of all vectors (r; ν;φ) ∈ R×R×Rn such that (r, ν,y0 = V0φ) satisfy (22), i.e. there existσ, τ ≥ 0 and t ∈ Rm

+ that satisfy

τ + 1T t ≤ ν,σ ≤ 1

λmax(H) ,∥

∥

∥

∥

[

2rσ − τ

]∥

∥

∥

∥

≤ σ + τ,∥

∥

∥

∥

[

2wi1− σλi − ti

]∥

∥

∥

∥

≤ 1− σλi + ti, i = 1, . . . ,m,

whereQΛQT is the spectral decomposition ofH = G− 12FG−

12 , Λ = diag(λi) and w = QTH

12G

12V0φ.

From (15), (20), and Definition 1, it follows that (13) can be reformulated as

minimize ν + δ,

subject to

∥

∥

∥

∥

∥

[

2D12φ

1− δ

]∥

∥

∥

∥

∥

≤ 1 + δ,

µT0 φ− γT |φ| ≥ α,ρT |φ| ≤ r,1Tφ = 1,

(r; ν;φ) ∈ H(V0,F,G).

(31)

where the equality r = ρT |φ| has been relaxed by recognizing that the relaxed constraint willalways be tight at an optimal solution. Although (31) is a convex optimization problem, it is notan SOCP. However, replacing |φ| by a new variable ψ ∈ Rn and adding the 2n linear constraints,ψi ≥ |φi|, i = 1, . . . , n, leads to the following SOCP formulation for the robust minimum varianceportfolio selection problem (11)

minimize ν + δ,

subject to

∥

∥

∥

∥

∥

[

2D12φ

1− δ

]∥

∥

∥

∥

∥

≤ 1 + δ,

µT0 φ− γTψ ≥ α,ψi ≥ φi, i = 1, . . . , n,ψi ≥ −φi, i = 1, . . . , n,

1Tφ = 1,(ρTψ; ν;φ) ∈ H(V0,F,G).

(32)

Another possible transformation leading to an SOCP is to replace φ by φ = φ+−φ−, φ+,φ− ∈ Rn+.

An alternative SOCP formulation is obtained if one employs Part (ii) of Lemma to characterize theworst case variance. The derivation of these results is left to the reader.

11

To keep the exposition simple, in the rest of this paper it will assumed that short sales are notallowed, i.e. φ ≥ 0. In each case the result can be extended to general φ by employing the abovetransformations.

3.2 Robust maximum return problem

The robust maximum return problem is given by

maximize min{µ∈Sm} µTφ,

subject to max{V∈Sv ,D∈Sd}φT (VTFV +D)φ ≤ λ,

1Tφ = 1,φ ≥ 0,

or equivalently, from (15), by

maximize (µ− γ)Tφ,subject to max{V∈Sv}φ

TVTFVφ ≤ λ− δ,φTDφ ≤ δ,1Tφ = 1,φ ≥ 0.

(33)

From Definition 1, it follows that the robust maximum return problem is equivalent to the SOCP

maximize (µ− γ)Tφ,

subject to

∥

∥

∥

∥

∥

[

2D12φ

1− δ

]∥

∥

∥

∥

∥

≤ 1 + δ,

1Tφ = 1,φ ≥ 0,

(ρTφ;λ− δ;φ) ∈ H(V0,F,G).

(34)

3.3 Robust maximum Sharpe ratio problem

The robust maximum Sharpe ratio problem is given by,

max{φ:φ≥0,1Tφ=1}

min{V∈Sv ,µ∈Sm,D∈Sd}

µTφ− rf√

Var[

rφ]

, (35)

where rf is the risk-free rate of return. We will assume that the optimal value of this max-minproblem is strictly positive, i.e. there exists a portfolio with finite worst case variance whose worstcase return is strictly greater than the risk-free rate rf . In practice the worst case variance of everyasset is bounded, therefore this constraint qualification reduces to the requirement that there is atleast one asset with worst case return greater than rf .

Since the components of the portfolio vector φ add up to 1, the objective of the max-min prob-lem (35),

µTφ− rf√

Var[

rφ]

=(µ− rf1)Tφ√

Var[

rφ]

,

12

is a homogeneous function of the portfolio φ. This implies that the normalization condition 1Tφ = 1can be dropped and the constraint min{µ∈Sm}(µ− rf1)

Tφ = 1 added to (35) without any loss ofgenerality. With this transformation, (35) reduces to minimizing the worst case variance; i.e. (35)is equivalent to

minimize max{V∈Sv}φTVTFVφ+ φTDφ

subject to (µ0 − γ − rf1)Tφ ≥ 1,φ ≥ 0,

(36)

where the constraint min{µ∈Sm}(µ− rf1)Tφ = 1 has been relaxed by recognizing that the relaxedconstraint will always be tight at an optimal solution. Consequently, the robust maximum Sharperatio problem is equivalent to a robust minimum variance problem with α = 1, µ replaced byµ− rf1, and φ no longer normalized. Exploiting this relationship, we have that (35) is equivalentto the SOCP

minimize ν + δ,

subject to

∥

∥

∥

∥

∥

[

2D12φ

1− δ

]∥

∥

∥

∥

∥

≤ 1 + δ,

(µ0 − γ − rf1)Tφ ≥ 1,(ρTφ; ν;φ) ∈ H(V0,F,G).

(37)

The crucial step in the reduction of (35) to (37) is the realization that the objective function in(35) is homogeneous in φ, and therefore, its numerator can be restricted to 1 without any loss ofgenerality. This homogenization goes through even when there are additional inequality constraintson the portfolio choices.

Suppose the portfolio φ is constrained to satisfy Aφ ≥ b (φ ≥ 0 is assumed to be subsumed inAφ ≥ b). Then, the robust maximum Sharpe ratio problem,

max{φ:Aφ≥b,1Tφ=1}

min{V∈Sv ,µ∈Sm,D∈Sd}

µTφ− rf√

Var[

φ]

, (38)

is equivalent to the robust minimum variance problem

minimize max{V∈Sv}φTVTFVφ+ φTDφ

subject to (µ0 − γ − rf1)Tφ ≥ 1,Aφ ≥ ζb,1Tφ = ζ,

ζ ≥ 0,

(39)

where ζ is an auxiliary variable that has been introduced to homogenize the constraints Aφ ≥ band 1Tφ = 1. The problem (39) can easily be converted into a second-order cone problem by usingthe techniques developed above.

4 Robust Value-At-Risk (VaR) portfolio selection

The robust VaR portfolio selection problem is given by

maximize minµ∈Sm E[rφ]subject to max{µ∈Sm,V∈Sv ,D∈Sd}P(rφ ≤ α) ≤ β,

1Tφ = 1,φ ≥ 0.

(40)

13

This optimization problem maximizes the expected return subject to the constraint that the short-fall probability is less than β. Note that for ease of exposition we again assume that no shortsales are allowed, i.e. φ ≥ 0. The solution in the general case can be obtained by employing atransformation identical to the one detailed at the end of Section 3.1.

For fixed (µ,V,D), the return vector rφ = N (µTφ,φT (VTFV +D)φ). Therefore

P(rφ ≤ α) ≤ β ⇔ P

(

µTφ+ Z√

φT (VTFV +D)φ ≤ α

)

≤ β,

⇔ P

Z ≤ α− µTφ√

φT (VTFV +D)φ

≤ β,

⇔ α− µTφ√

φT (VTFV +D)φ≤ F−1Z (β), (41)

where Z ∼ N (0, 1) is the standard Normal random variable and FZ(·) is its cumulative densityfunction. In typical VaR applications β << 1, therefore F−1Z (β) < 0. Thus, the probabilityconstraint for fixed (µ,V,D) is equivalent to the second-order cone constraint

−F−1Z (β)

√

∥

∥F12Vφ

∥

∥

2+∥

∥D12φ∥

∥

2 ≤ µTφ− α. (42)

On incorporating (42), the problem (40) can be rewritten as

maximize minµ∈Sm µTφ

subject to max{µ∈Sm,V∈Sv ,D∈Sd}

{

−F−1Z (β)

√

∥

∥F12Vφ

∥

∥

2+∥

∥D12φ∥

∥

2 − µTφ}

≤ α,

1Tφ = 1,φ ≥ 0.

(43)

Assuming that the uncertainty sets Sd, Sv and Sm are given by (2)-(4), (43) reduces to

maximize (µ0 − γ)Tφsubject to −F−1Z (β)

∥

∥

∥

∥

[

νδ

]∥

∥

∥

∥

≤ (µ0 − γ)Tφ− α,∥

∥

∥D12φ∥

∥

∥ ≤ δ,

max{V∈Sv}∥

∥

∥F12Vφ

∥

∥

∥ ≤ ν,

1Tφ = 1,φ ≥ 0.

(44)

From Part (ii) of Lemma 1, it follows that

max{V∈Sv}

∥

∥F12Vφ

∥

∥ ≤ ν,

if and only if there exist τ ≥ 0 and s ∈ Rm+ such that,

r2 ≤ τ(ν2 − 1T s),u2i ≤ (1− τ θi)si, i = 1, . . . ,m,

λmax(K)τ ≤ 1,(45)

14

where K = PΘPT is the spectral decomposition of K = F12G−1F

12 , Θ = diag(θi), and u =

PTF12V0φ. On introducing the change of variables, τ = ντ and s = 1

ν s, (45) becomes

r2 ≤ τ(ν − 1T s),u2i ≤ (ν − τθi)si, i = 1, . . . ,m,

λmax(K)τ ≤ ν.(46)

Therefore, the robust VaR portfolio selection problem (40) is equivalent to the SOCP

maximize (µ0 − γ)Tφ,subject to u = PTF

12VT

0 φ,

−F−1Z (β)

∥

∥

∥

∥

[

νδ

]∥

∥

∥

∥

≤ (µ0 − γ)Tφ− α,∥

∥

∥D

12φ∥

∥

∥≤ δ,

∥

∥

∥

∥

[

2ρTφ(τ − ν + 1T s)

]∥

∥

∥

∥

≤ (τ + ν − 1T s),∥

∥

∥

∥

[

2ui(ν − τθi − si)

]∥

∥

∥

∥

≤ (ν − τθi + si), i = 1, . . . ,m,

ν − τλmax(K) ≥ 0,1Tφ = 1,φ ≥ 0,τ ≥ 0.

(47)

As in the case of the minimum variance problem, an alternative SOCP formulation for the robustVaR problem follows from Part (i) of Lemma 1.

5 Multivariate regression and norm selection

In this section results from the statistical theory of multivariate linear regression are used justifythe uncertainty structures Sd, Sv and Sm proposed in Section 2 and motivate natural choices forthe matrix G defining the elliptic norm ‖·‖g and the bounds ρi, γi, di, i = 1, . . . , n.

In Section 2, the return vector r is assumed to be given by the linear model,

r = µ+VT f + ε, (48)

where ε is the residual return. In practice, the parameters (µ,V) of this linear model are estimatedby linear regression. Given market data consisting of samples of asset return vectors and thecorresponding factor returns, the linear regression procedure computes the least squares estimates(µ0,V0) of (µ,V). In addition, the procedure also gives multi-dimensional confidence regionsaround these estimates with the property that the true value of the parameters lie in these regionswith a prescribed confidence level. The structure of the uncertainty sets introduced in Section 2is motivated by these confidence regions. The rest of this section describes the steps involved inparameterizing the uncertainty structures, i.e. computing µ0, V0, G, ρi, γi, di, i = 1, . . . , n.

Suppose the market data consists of asset returns, {rt : t = 1, . . . , p}, for p periods and thecorresponding factor returns {f t : t = 1, . . . , p}. Then the linear model (48) implies that

rti = µi +n∑

j=1

Vjiftj + εti, i = 1, . . . , n, t = 1, . . . , p.

15

In linear regression analysis, typically, in addition to assuming that the vector of residual returns εt

in period t is composed of independent normals, it is assumed that the residual returns of differentmarket periods are independent. Thus, {εti : i = 1, . . . , n, t = 1, . . . , p} are all independent normalrandom variables and εti ∼ N (0, σ2i ), for all t = 1, . . . , p, i.e. the variance of the residual return ofthe i-th asset is σ2i . The independence assumption can be relaxed to ARMA models by replacinglinear regression by Kalman filters (see Hansen and Sargent, 2001).

Let S = [r1, r2, . . . , rp] ∈ Rn×p be the matrix of asset returns and B = [f 1, f2, . . . , fp] ∈ Rm×p bethe matrix of factor returns. Collecting together terms corresponding to a particular asset i overall the periods t = 1, . . . , p, we get the following linear model for the returns {rti : t = 1, . . . , p}

yi = Axi + εi,

whereyi =

[

r1i r2i . . . rpi

]T, A =

[

1 BT]

, xi =[

µi V1i V2i . . . Vmi]T,

and εi = [e1i , . . . , epi ]T is the vector of residual returns corresponding to asset i.

The least squares estimate xi of the true parameter xi is given by the solution of the normalequations

ATAxi = ATyi;

i.e.xi = (ATA)−1ATyi, (49)

if rank(A) = m+ 1. Substituting yi = Axi + εi, we get

xi − xi = (ATA)−1AT εi = N (0,Σ),

where Σ = σ2i (ATA)−1. Hence

X =1

σ2i(xi − xi)

T (ATA)(xi − xi) ∼ χ2m+1, (50)

is a χ2 random variable with (m + 1) degrees of freedom. Since the true variance σ2i is unknown,(50) is not of much practical value. However, a standard result in regression theory states that ifσ2i in the quadratic form (50) is replaced by (m + 1)s2i , where s

2i is the unbiased estimate of σ2i

given by

s2i =‖yi −Axi‖2p−m− 1

, (51)

then the resulting random variable

Y =1

(m+ 1)s2i· (xi − xi)

T (ATA)(xi − xi) (52)

is distributed according to the F -distribution with (m + 1) degrees of freedom in the numeratorand (p−m− 1) degrees of freedom in the denominator (Anderson, 1984; Greene, 1990).

Let 0 < ω < 1, FJ denote the cumulative distribution function of the F -distribution with J degreesof freedom in the numerator and p −m − 1 degrees of freedom in the denominator and cJ(ω) bethe ω-critical value, i.e. the solution of the equation FJ(cJ(ω)) = ω.

16

Then the probability Y ≤ cm+1(ω) is ω, or equivalently,

P(

(xi − xi)T (ATA)(xi − xi) ≤ (m+ 1)cm+1(ω)s

2i

)

= ω. (53)

DefineSi(ω) =

{

xi : (xi − xi)T (ATA)(xi − xi) ≤ (m+ 1)cm+1(ω)s

2i

}

. (54)

Then, (53) implies that Si(ω) is a ω-confidence set for the parameter vector xi corresponding toasset i. And since the residual errors {εi : i = 1, . . . , n} are assumed to be independent, it followsthat

S(ω) = S1(ω)× S2(ω)× . . .× Sn(ω), (55)

is a ωn-confidence set for (µ,V).

Let Sm(ω) denote the projection of S(ω) along the vector µ, i.e.

Sm(ω) = {µ : µ = µ0 + ν, |νi| ≤ γi, i = 1, . . . , n},

where

µ0,i = µi, γi =

√

(m+ 1)(ATA)−111 cm+1(ω)s2i , i = 1, . . . , n. (56)

Then (55) implies that Sm(ω) is a ωn-confidence set for the mean vector µ. Note that the uncer-

tainty structure (4) for the mean µ assumed in Section 2 is identical to Sm(ω).

Let Q = [e2, e3, . . . , em+1]T ∈ Rm×(m+1) be projection matrix that projects xi along Vi. Define

the projection Sv(ω) of S(ω) along V as follows

Sv(ω) ={

V : V = V0 +W, ‖Wi‖g ≤ ρi, i = 1, . . . , n}

,

whereV0 = [V1 . . . Vn],

G = (Q(ATA)−1QT )−1 = BBT − 1p(B1)(B1)

T ,

ρi =√

(m+ 1)cm+1(ω)s2i , i = 1, . . . , n.

(57)

Then Sv(ω) is a ωn-confidence set for the factor loading matrix V. As in the case of Sm(ω), theuncertainty structure (3) for the factor covariance V assumed in Section 2 has precisely the samestructure as Sv(ω) defined above.

The construction of the confidence regions can be done in the reverse direction as well, i.e. individualconfidence regions can be suitably combined to yield joint confidence regions. Let Sm(ω) ⊂ Rn

and Sv(ω) ⊂ Rm×m be any ω-confidence regions for µ and V respectively. Then

P(

(µ,V) ∈ Sm(ω)× Sv(ω))

= 1−P(

(µ,V) 6∈ Sm(ω)× Sv(ω))

,

≥ 1−P(

µ 6∈ Sm(ω))

−P(

V 6∈ Sv(ω))

,

= 2ω − 1, (58)

i.e. Cartesian products of individual confidence regions are joint confidence regions. This leads toan alternative means of constructing joint confidence regions for market parameters (µ,V).

Let Q ∈ RJ×m. Then

Y =1

Js2i(Qxi −Qxi)

T(

Q(ATA)−1QT)−1

(Qxi −Qxi), (59)

17

is distributed according to the F -distribution with J degrees of freedom in the numerator andp−m−1 degrees of freedom in the denominator, i.e. the probability Y ≤ cJ(ω) is ω, or equivalently,

P(

(Qxi −Qxi)T (Q(ATA)−1QT )−1(Qxi −Qxi) ≤ JcJ(ω)s

2i

)

= ω. (60)

Set Q = eT1 . Then, Qxi = µi, the least squares estimate of the mean return of asset i and Qxi = µi,the true mean return of asset i. Therefore, (60) implies that

P

(

|µi − µi| ≤√

(ATA)−111 c1(ω)s2i

)

= ω. (61)

Since the residual errors εi are assumed to be independent, it follows that

Sm(ω) = {µ : µ = µ0 + ν, |νi| ≤ γi, i = 1, . . . , n},

where

µ0,i = µi, γi =

√

(ATA)−111 c1(ω)s2i , i = 1, . . . , n, (62)

is a ωn-confidence set for the mean vector µ.

Next setQ = [e2, e3, . . . , em+1]T ∈ Rm×(m+1). Then, the projectionQxi = [V1i V2i . . . Vim]

T = Vi,the least squares estimate of the true factor loading Qxi = Vi, and (60) implies that

P(

(Vi −Vi)T (Q(ATA)−1QT )−1(Vi −Vi) ≤ mcm(ω)s

2i

)

= ω.

As in the case of the mean vector µ, it follows that

Sv(ω) ={

V : V = V0 +W, ‖Wi‖g ≤ ρi, i = 1, . . . , n}

,

V0 = V, G = (Q(ATA)−1QT )−1 = BBT − 1p(B1)(B1)

T ,

ρi =√

mcm(ω)s2i , i = 1, . . . , n,(63)

is ωn-confidence set for the factor loading matrixV. From (58), it follows that S(ω) = Sm(ω)×Sv(ω)is a joint confidence region with (2ωn − 1) confidence.

The two families of uncertainty sets defined by (56)–(57) and (62)–(63) are not entirely comparable.Since the critical value cJ,m(ω) is an increasing function of J , it follows that, for a fixed ω ∈ (0, 1),S(ω) ⊂ S(ω). On the other hand, if ω is chosen such that 2ωn − 1 = ωn, the direction of inclusiondepends on the dimension m.

After determining the uncertainty sets for µ and V, all that remains to be established is the upperbound di on the variance of regression error σ2i . From the analysis above, it follows that the naturalchoices for the bounds are given by the confidence interval around the least squares estimate s2i ofthe error variance σ2i . Unfortunately, the regression procedure yields only a single unbiased samples2i of the error variance. One possible solution is to construct bootstrap confidence intervals (seeEfron and Tibshirani, 1993). Since the bootstrapping step can often be computationally expensiveand since the robust optimization problems only require an estimate of the worst case error variance,any reasonable estimate of the worst case error variance could alternatively be used as the bound di.

Incorporating the results developed in this section, we have the following recipe for solving therobust portfolio selection problem:

18

1. Collect data on the returns r of the assets and the returns f of the factors.

2. Using (49) one asset at a time, evaluate the least squares estimates µ0 and V0 of the meanµ and the factor loading matrix V, respectively.

3. Choose a confidence threshold ω.

(a) Construct a bootstrap ω-confidence interval around σ2i . Alternatively use any estimateof the worst case error variance.

(b) Define Sm and Sv using (56) and (57) respectively (or (62) and (63) respectively).

4. Solve the robust problem of interest.

A word of caution about the choice of ω. If ω is chosen very high, the uncertainty sets willbe very large, i.e. implicitly one is demanding robustness with respect to a very large set ofparameter values. The resulting portfolio will be very conservative, and therefore its performancefor a particular set of parameters (µ0,V0) will be significantly worse than the portfolio designedfor that set of parameters. On the other hand, if ω is chosen too low, the portfolio choice willnot be robust enough. The typical choices of ω lie in the range 0.95-0.99. See Section 7 for moredetails on the implications of the choice of ω and Appendix A for a discussion of the probabilisticguarantees on the performance of the optimal robust portfolio.

Although we propose a strictly data driven approach in this section, this analysis extends toBayesian estimation methods such as those in Black and Litterman (1990), as well as Empiri-cal Bayes estimation. In the Bayesian setting the confidence regions are given by the posteriordistribution.

Recall that the maximum likelihood estimate Fml of covariance matrix F of the factors is given by,

Fml =1

p− 1

[

BBT − 1

p(B1)(B1)T

]

, (64)

i.e. G = (p − 1)Fml. Suppose the true covariance matrix F is approximated by the maximumlikelihood estimate Fml. Then F = κG, where κ = 1/(p− 1). Recall that the worst case variance

maxV∈Sv

{

φTVTFVφ}

≤ ν2, (65)

if and only ifmax

{y:‖y‖g≤r}‖V0φ+ y‖f ≤ ν,

where r = ρTφ. Since F = κG, the above norm constraint is equivalent to

max{u:‖u‖≤r√κ}

∥

∥

∥F12V0φ+ u

∥

∥

∥ ≤ ν, (66)

where ‖·‖ is now the usual Euclidean norm. It is easy to see that the maximum in (66) is attainedat

u =r√κ

‖F 12V0φ‖

· F 12V0φ

19

and the corresponding maximum value is∥

∥

∥F

12V0φ

∥

∥

∥+r√κ. Thus, the worst case variance constraint

(65) is equivalent to the second-order cone constraint

∥

∥

∥F

12V0φ

∥

∥

∥≤ ν −√κρTφ. (67)

From the fact that ν2 ≤ τ is equivalent to a second-order cone constraint, it follows that in thespecial case F = κG the robust minimum variance reduces to the following simple SOCP

minimize τ + δ,subject to (µ0 − γ)Tφ ≥ α,

1Tφ = 1,∥

∥

∥F12V0φ

∥

∥

∥ ≤ ν −√κρTφ,∥

∥

∥

∥

∥

[

2D12φ

1− δ

]∥

∥

∥

∥

∥

≤ 1 + δ,

∥

∥

∥

∥

[

2ν1− τ

]∥

∥

∥

∥

≤ 1 + τ,

φ ≥ 0.

(68)

The robust maximum return problem, the robust maximum Sharpe ratio, as well as the robustVaR problem can all be simplified in a similar manner.

In practice, however, the covariance matrix F is assumed to be stable, and is, typically, estimatedfrom a much larger data set and by taking several extraneous macroeconomic indicators into ac-count (see Ledoit, 1996). If this is the case, then the above simplification cannot be used and onewould have to revert back to the formulation in (31).

6 Robust portfolio allocation with uncertain covariance matrices

The market model introduced in Section 2 assumes that the covariance matrix F of the factors iscompletely known and stable. While this is a good first approximation, a more complete marketmodel is one that allows some uncertainty in the covariance matrix and optimizes over it. In orderfor any uncertainty structure for covariance matrices to be useful it must be flexible enough to modela variety of perturbations, while at the same time restricted enough to admit fast parameterizationand efficient optimization. Our goal in this section is to develop such an uncertainty structure forcovariance matrices. We assume the market structure is the one introduced in Section 2, exceptthat the covariance matrix is no longer fixed.

6.1 Uncertainty structure for covariance inverse

Consider the following uncertainty structure for the factor covariance matrix F:

Sf−1 ={

F : F−1 = F−10 +∆ º 0,∆ =∆T ,∥

∥F120∆F

120

∥

∥ ≤ η}

, (69)

where F0 Â 0. The norm ‖A‖ in (69) is given by ‖A‖ = maxi |λi(A)|, where {λi(A)} are theeigenvalues of A.

20

The uncertainty set Sf−1 restricts the perturbations ∆ of the covariance matrix to be symmetric,

bounded in norm relative to a nominal covariance matrix F0 and subject to F−1 = F−10 +∆ º 0.Clearly this uncertainty structure is not the most general – one could, for example, allow for eachelement of the covariance matrix to lie in some uncertainty interval subject to the constraint thatthe matrix is positive semidefinite. However, the robust optimization problem corresponding tothis uncertainty structure is not very tractable (Halldorsson and Tutuncu, 2000). On the otherhand, if the covariance uncertainty is given by (69), all the robust portfolio allocation problemsintroduced in Section 2 can be reformulated as SOCPs. Moreover, as in the case of the uncertaintystructures for µ and V, the uncertainty structure Sf−1 for F corresponds to the confidence regionassociated with statistical procedure used to estimate F. In particular, we show that maximumlikelihood estimation of the covariance matrix F provides a confidence region of the form Sf−1 andyields a value of η that reflects any desired confidence level.

All the robust portfolio selection problems introduced in Section 2 can be formulated for this marketas well. For example, the robust Sharpe ratio problem in this market model is given by,

maximize{φ:1Tφ=1} min{µ∈Sm,V∈Sv ,D∈Sd,F∈Sf−1}

{

µTφ−rf√

φT(VTFV+D)φ

}

.

Lemma 3, below, establishes that the worst case variance for a fixed portfolio is given by a collectionof linear and restricted hyperbolic constraints – the critical step in reformulating robust portfolioselection problems as SOCPs.

Lemma 3 Fix a portfolio φ and let Sf−1 be given by (69). Then the following results hold.

(i) If the bound η ≥ 1, the worst-case variance is unbounded, i.e.

max{V∈Sv ,F∈Sf−1}

φTVTFVφ =∞.

(ii) If η < 1, the worst-case variance max{V∈Sv ,F∈Sf−1}φTVTFVφ ≤ ν if and only if (ρTφ; (1−

η)ν;φ) ∈ H(V0,F0,G), where H(·, ·, ·) is defined in Definition 1.

Proof: Define ∆ = F120∆F

120 . Then

Sf−1 ={

F : F−1 = F− 1

20 (I+ ∆)F

− 12

0 º 0, ∆ = ∆T, ‖∆‖ ≤ η

}

.

Fix V ∈ Sv and define x = VTφ. Then

supF∈S

f−1

{xTFx} = sup{

(

F120 x)T

(I+ ∆)−1(

F120 x)

: ∆ = ∆T, ‖∆‖ ≤ η, I+ ∆ º 0

}

.

First consider the case η ≥ 1. Choose ∆ = −βI where 0 ≤ β ≤ 1. Then ‖∆‖ = β ≤ η andI+ ∆ = (1− β)I º 0. Thus,

supF∈S

f−1

{

xTFx}

≥ sup{β:0≤β≤1}

1

1− β(

xTF0x)

=∞.

21

Next, suppose η < 1. Since η < 1 implies that I+ ∆ º 0 for all ‖∆‖ ≤ η,

Sf−1 ={

F : F−1 = F− 1

20 (I+ ∆)F

− 12

0 , ∆ = ∆T, ‖∆‖ ≤ η

}

.

Therefore

supF∈S

f−1

{xTFx} = sup

{

(F120 x)(I+ ∆)−1(F

120 x) : ∆ = ∆

T, ‖∆‖ ≤ η

}

=

m∑

i=1

(

qTi (F120 x)

)2

1 + λi(∆),

where {qi, i = 1, . . . ,m} is the set of eigenvectors of ∆. Since |λi(∆)| ≤ η, it follows that

supF∈S

f−1

{xTFx} ≤ 1

1− η ·m∑

i=1

(

qTi (F120 x)

)2=

1

1− η · xTF0x.

Moreover, the supremum is achieved by ∆ = −ηI. Thus,

max{V∈Sv ,F∈Sf−1}

φT (VTFV)φ =1

1− η · maxV∈Sv

φTVTF0Vφ.

From Part (i) of Lemma 1, we have that

1

1− η · maxV∈Sv

φTVTF0Vφ ≤ ν

if and only if (ρTφ; (1− η)ν;φ) ∈ H(V0,F0,G).

From Lemma 3 it follows that if η < 1 then all robust portfolio problems introduced in Section 2can be reformulated as SOCPs.

The next step is to justify the uncertainty structure Sf−1 and show how it can be parametrizedfrom market data. In Section 5 we show that the natural uncertainty structures for (µ,V) and theirparameterizations are implied by the confidence regions associate with the statistical proceduresused to compute the point estimates. Here, we show that the uncertainty structure Sf−1 and itsparametrization, i.e. the choice of the nominal matrix F0 and the bound η, arise naturally frommaximum-likelihood estimation typically used to compute the point estimate of F.

Suppose the covariance matrix of the factor returns is estimated from the data B = [f 1, f2, . . . , fp]over p market periods. Then the maximum likelihood estimate Fml of the factor covariance matrixF is given by (64).

Suppose one is in a Bayesian setup and assumes a non-informative conjugate prior distribution forthe true covariance F. Then the posterior distribution for F conditioned on the data B is given by

F|B ∼W−1p−1(

(p− 1)Fml)

, (70)

where W−1q (A) denotes an inverse Wishart distribution with q degrees of freedom and parameter

A (Anderson, 1984; Schafer, 1997), i.e. the density f(F|B) is given by

f(F|B) =

{

c|Fml|(p−m−2)/2|F−1|(p−1)/2e−(

(p−1)2

Tr(F−1Fml))

, F Â 0,0, otherwise,

(71)

22

40 60 80 100 120 140 160 180 200

0.985

0.99

0.995

1

PSfrag replacements

ωmax

p

ωm

ax

Figure 1: ωmax vs p (m = 40)

where |A| = det(A) and c is a normalization constant. This density can be rewritten as follows.

f(F|B) =

{

c|F−1ml |(m−1)/2|F12mlF

−1F12ml|(p−1)/2e

−(

(p−1)2

Tr(F12ml

F−1F12ml))

, F Â 0,0, otherwise.

(72)

From (72) it follows that the natural choice for the nominal covariance matrix F0 in the definitionof Sf−1 is F0 = Fml.

Let λ ∈ Rm be the vector of eigenvalues of F120F

−1F120 . Then (72) implies that the density of λ is

given by

f(λ|B) =

{

cλ∏m

i=1 λ(p−1)/2i e−

(p−1)2

λi , λ ≥ 0,0, otherwise,

(73)

where cλ is a normalization constant, i.e. the eigenvalues λi, i = 1, . . . ,m, are IID Γ(

p+12 , p−12

)

distributed random variables.

Let ∆ = F−1 − F−10 be the deviation of the F−1 from the nominal inverse F−10 and let ∆ =

F120∆F

120 = F

120F

−1F120 − I. Then ‖∆‖ ≤ η if and only if 1− η ≤ λi ≤ 1 + η, i.e.

P(‖∆‖ ≤ η) =(

P(1− η ≤ Gp ≤ 1 + η))m

=(

FΓp(1 + η)−FΓp(1− η))m

, (74)

where Gp ∼ Γ(

p+12 , p−12

)

and FΓp is the corresponding CDF.

If η < 1, F−1 = F−10 +∆ Â 0 for all ‖∆‖ ≤ η, and therefore (74) implies that

P(F ∈ Sf−1) = (FΓp(1 + η)−FΓp(1− η))m. (75)

Thus, in order to satisfy a desired confidence level ωm, the parameter η must be set equal to theunique solution of

FΓp(1 + η)−FΓp(1− η) = ω. (76)

23

Since the problem is unbounded for η ≥ 1, (76) implicitly states that the confidence level ωm that

can be supported by p return samples is at most ωmax(p) =(

FΓp(2))m

. Figure 1 plots ωmax as

a function of the data length p for m = 40 (the plot begins at p = m + 1 since at least m + 1observations are needed to estimate the covariance matrix). From the plot, one can observe thatfor p ≥ 50, ωmax(p) ≥ 0.995. Thus, the restriction ωm ≤ ωmax(p) is not likely to be restrictive.

This methodology can be modified to accommodate prior information about the structure of F.Suppose the prior distribution on F is the informative conjugate priorW−1

k ((k−1)F), k ≥ 1. Then,the posterior distribution F|B ∼W−1

k+p((p−1)Fml+(k−1)F). In this case, the nominal covariance

matrix F0 =1

k+p

(

(p− 1)Fml + (k − 1)F)

, and η is given by

FΓk+p(1 + η)−FΓk+p(1− η) = ω,

where FΓk+p denotes the CDF of a Γ(

k+p+22 , k+p2

)

random variable.

From the results in this section, it follows that, as far as the portfolio selection problems areconcerned, all that the uncertainty in the covariance matrix does is “shrink” the MLE F−10 to(1− η)F−10 , where the shrinkage factor (1− η) is a function of the desired confidence level ω. Thus,this procedure has the flavor of robust statistics (see Huber, 1981).

6.2 Uncertainty structure for the covariance

Another possible uncertainty structure for the factor covariance matrix is given by

Sf ={

F : F = F0 +∆ º 0,∆ =∆T ,∥

∥N−12∆N−

12

∥

∥ ≤ ζ}

, (77)

where F0 º 0. The norm ‖A‖ in (77) is either ‖A‖ = maxi |λi(A)|, or ‖A‖ =√

∑

i λ2i (A), where

{λi(A)} are the eigenvalues of A.

In order for (77) to be a viable uncertainty structure, the corresponding robust problem shouldbe efficiently solvable and the structure ought to be easily parameterizable from raw market data.The first requirement is established by the following lemma.

Lemma 4 Fix a portfolio φ ∈ Rn+ and let Sf be given by (77). Then max{V∈Sv ,F∈Sf}φ

T (VTFV)φ ≤ν if and only if (ρTφ; ν;φ) ∈ H(V0, (F0 + ζN),G), where H(·, ·, ·) is defined in Definition 1.

Proof: Define ∆ = N−12∆N−

12 . Then

Sf ={

F : F = F0 +N12 ∆N

12 º 0, ∆ = ∆

T,∥

∥∆∥

∥ ≤ ζ}

.

Fix V ∈ Sv and define x = VTφ. Then

maxF∈Sf

{xTFx} = max{

xTF0x+ (N12x)T ∆(N

12x) : ∆ = ∆

T, ‖∆‖ ≤ ζ,F0 +N

12 ∆N

12 º 0

}

,

≤{

xTF0x+ (N12x)T ∆(N

12x) : ∆ = ∆

T, ‖∆‖ ≤ ζ

}

, (78)

≤ xTF0x+ ζ(N12x)T (N

12x), (79)

24

where (79) follows from the properties of the matrix norm.

Since ‖∆‖ = max{|λi(∆)|} or√

∑

i λ2i (∆) and N Â 0, the bound (79) is achieved by

∆∗= ζ · (N

12x)(N

12x)T

‖N 12x‖2

,

unless x = 0. Thus, the right hand side of (78) is given by xT (F0 + ζN)x and is achieved by

∆∗= ζ · NxxTN

xTNx, unless x = 0. Since F0 +N

12 ∆

∗N

12 º 0, it follows that the inequality (78) is, in

fact, an equality, i.e.maxF∈Sf

{

xTFx}

= xT (F0 + ζN)x.

Thus,max

{V∈Sv ,F∈Sf}φT (VTFV)φ = max

V∈SvφTVT (F0 + ζN)Vφ.

From Lemma 1, we have that

maxV∈Sv

φTVT (F0 + ζN)Vφ ≤ ν

if and only if (ρTφ; ν;φ) ∈ H(V0, (F0 + ζN),G) where H(·, ·, ·) is defined in Definition 1.

Notice that, unlike in the case Sf−1 , here we allow N 6= F0 and do not require ζ < 1.

The next step is to devise a method for parameterizing this uncertainty structure. The parametriza-tion will once again be implied by the statistical procedure used to estimate F.

Let F be the true (unknown) covariance matrix of the factor returns and let Fml be the MLE of Fcomputed from the return data B. Then it is well known (Anderson, 1984) that if p > m (i.e. thenumber of observations is larger than the dimension of the matrix) then Fml  0 with probability 1and

Fml ∼Wp−1(

(p− 1)F)

, (80)

where Wq(A) denotes a Wishart distribution with q degrees of freedom centered at the matrix A,i.e. the density f(Fml | F) is given by

f(Fml|F) ={

c|(F)−1|(p−1)/2|Fml|(p−m−2)/2e−(

(p−1)2

Tr(F−1Fml))

, Fml  0,0, otherwise.

(81)

An argument very similar to the one in the previous section establishes that Fml belongs to the set

S ={

F : F = F+∆ Â 0,∆ =∆T , ‖F− 12∆F−

12 ‖ ≤ β

}

, (82)

with probability ωm if the bound β is set equal to the solution of

FΓp(1 + β)−FΓp(1− β) = ω, (83)

where FΓp denotes the CDF of a Γ((p+ 1)/2, (p− 1)/2) random variable. Note that this equationis identical to (76), i.e. the solution β of (83) is equal to the solution η of (76).

25

Suppose the bound β < 1, or equivalently ωm < ωmax(p) = (FΓp(2))m. Then we have that

∥

∥F− 1

2ml (F− Fml)F

− 12

ml

∥

∥ =∥

∥F− 1

2ml F

12F

12 (F− Fml)F

− 12F

12F

− 12

ml

∥

∥,

≤∥

∥F− 1

2ml F

12

∥

∥

∥

∥F−12 (F− Fml)F

− 12

∥

∥

∥

∥F12F

− 12

ml

∥

∥,

≤ β

1− β , (84)

where (84) follows from the fact that Fml ∈ S defined in (82). From (84) we have that

S =

{

F : Fml +∆ º 0,∆ =∆T ,∥

∥F− 1

2ml ∆F

− 12

ml

∥

∥ ≤ β

1− β

}

, (85)

contains a ωm-confidence set for the (true) covariance matrix F, i.e. a natural parametrization ofthe uncertainty structure in (77) is F0 = N = Fml and ζ = β

1−β . Thus, the uncertainty structure(77) is completely parametrized by considering the confidence regions around the MLE.

Note that for N = F0 and ζ = β1−β , Lemma 4 implies that

max{V∈Sv ,F∈Sf}

φT (VTFV)φ ≤ ν

if and only if

(ρTφ; ν;φ) ∈ H(

V0, (1 + β/(1− β))F0,G)

= H(

V0, 1/(1− β)F0,G)

.

As noted above, the solution β of (83) is equal to the solution η of (76), and from Definition 1 itfollows that

(

ρTφ; ν;φ)

∈ H(

V0, 1/(1− β)F0,G)

⇔(

ρTφ; (1− η)ν;φ)

∈ H(

V0,F0,G)

.

Thus, max{V∈Sv ,F∈Sf}φT (VTFV)φ ≤ ν if and only if (ρTφ; (1 − η)ν;φ) ∈ H

(

V0,F0,G)

. Since

this is the precisely the condition in Part (ii) of Lemma 3, it follows that, although the two un-certainty sets (69) and (77) are not the same, they imply the same worst-case variance constraint.However, unlike the parametrization of Sf−1 , the parametrization of Sf cannot be biased to reflectprior knowledge about F.

7 Computational results

In this section we report the results of our preliminary computational experiments with the robustportfolio selection framework proposed in paper. The objective of these computational experimentswas to contrast the performance of the classical portfolio selection strategies with that of the robustportfolio selection strategies. We conducted two types of computational tests. The first set of testscompared performance on simulated data and the second set compared sample path performanceon real market data. In these experiments our intent was to focus on the benefit accrued fromrobustness; therefore, we wanted to avoid any user defined variables, such as the minimum return αin the robust minimum variance problem or the probability threshold β in the robust VaR problem.Thus, in our tests both the classical and robust portfolios were selected by solving the correspondingmaximum Sharpe ratio problem. We expect the qualitative aspects of the computational resultswill carry over to the other portfolio selection problems. All the computations were performed usingSeDuMi V1.03 (Sturm, 1999) within Matlab6.1R12 on a Dell Precision 340 workstation runningRedHat Linux 7.1. The details of the experimental procedure and the results are given below.

26

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.2

1.4

1.6

1.8PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical worst case Sharpe ratios

ω

Figure 2: Performance as a function of ω (Run 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11

1.2

1.4

1.6

1.8

2PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical worst case Sharpe ratios

ω

Figure 3: Performance as a function of ω (Run 2)

27

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7

0.75

0.8

0.85

0.9

0.95

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.6

1.8

2

2.2

2.4

2.6PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical worst case Sharpe ratios

ω

Figure 4: Performance as a function of ω (Run 3)

7.1 Computational results for simulated data

For our computational tests on simulated data, we fixed the number of assets n = 500 and thenumber of factors m = 40. A symmetric positive definite factor covariance matrix F was randomlygenerated except that we ensured that the condition number of F, i.e. λmax(F)/λmin(F), was atmost 20 by adding a suitable multiple of the identity. This factor covariance matrix was assumedto be known and fixed. The nominal factor loading matrix V was also randomly generated. Thecovariance matrix D of the residual returns ε was assumed to be certain (i.e. D = D = D) andset to D = 0.1diag(VTFV), i.e. it was assumed that the linear model explains 90% of the assetvariance.

The risk-free rate rf was set to 3 and the nominal asset returns µi were chosen independentlyaccording to a uniform distribution on [rf − 2, rf + 2]. Next, we generated a sequence of asset andfactor return vectors according to the market model (1) in Section 2 for an investment period oflength p = 90 and used equations (62) and (63) in Section 5 to set the parameters V0, µ0, G, ρ andγ. (Note that we did not estimate D from data.) Next, the robust and the classical portfolios φrand φm were computed by solving the robust maximum Sharpe ratio problem (8) and its classicalcounterpart, respectively. Although the precise numbers used in these simulation experiments werearbitrary, we expect that the qualitative aspects of the results are not dependent on the precisevalues.

In the first set of simulation experiments we compared the performance of the robust and classicalportfolios as the confidence threshold ω (see Section 5 for details) was increased from 0.01 to 0.95.The experimental results for three independent runs are shown in Figure 2 - Figure 4. In each ofthe three figures, the top plot is the ratio of the mean Sharpe ratio of the robust portfolio to thatof the classical portfolio. The mean Sharpe ratio of any portfolio φ is given by

(µ0 − rf1)Tφ√

φT (VT0 FV0)φ

.

The classical portfolio φm maximizes the mean Sharpe ratio. The bottom plot in Figure 2 - Figure 4

28

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.95

1

1.05

1.1

1.15

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

2

3

4

5

6

7PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical worst case Sharpe ratios

σ2

Figure 5: Performance as a function of σ (Run 1)

is the ratio of the worst case Sharpe ratio of the robust portfolio to that of the classical portfolio.The worst case Sharpe ratio of a portfolio φ is given by

min{V∈Sv ,µ∈Sm,D∈Sd}

(µ− rf1)Tφ√

φT (VFV +D)φ,

where Sd, Sv and Sm are given by (2)-(4). The robust portfolio φr maximizes the worst case Sharperatio. The performance on the three runs is almost identical – the ratio of the mean Sharpe ratiosdrops from approximately 1 to approximately 0.8 as w increases from 0.01 to 0.95 while the ratioof the worst case Sharpe ratios increases from approximately 1 to approximately 2. Thus, at amodest 20% reduction in the mean performance the robust framework delivers an impressive 200%increase in the worst case performance. Notice that the drop in mean performance in run 3 (seeFigure 4) is close to 25% but the corresponding improvement in the worst case performance is closeto 260%.

The second set of simulation experiments compared the performance of robust and classical port-folios as a function of the upper bound on D. For this set of experiments, we set ω = 0.95 andD = σ2 diag(VT

0 FV0), where σ2 increased from 0.01 to 1. Since the performance of both the

robust and classical portfolio was very sensitive to the sample path – particularly for large values ofσ2 – the results shown in Figure 5 - Figure 7 were averaged over 10 runs for every value of σ2. Asbefore, the top plot is the ratio of the mean performances of the robust and classical portfolios andthe bottom plot is the ratio of the worst case performances. (The three plots truncate at differentvalues of σ2 because we only compare performance for values of σ2 for which the worst case Sharperatio of the classical portfolio is non-negative). Again the essential features of the performance wereindependent of the particular run – the mean performance of the robust portfolio did not degradesignificantly with the increase in noise variance, if it did so at all; and the worst case performance ofthe robust portfolio was significantly superior to that of the classical portfolio as the data becamenoisy. This is not unexpected since the robust portfolios were designed to combat noisy data.

Figure 8 compares the CPU time needed to solve the robust and classical maximum Sharpe ratioproblem as a function of the number of assets. For this comparison the number of factors m =

29

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.9

0.92

0.94

0.96

0.98

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

1.5

2

2.5PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical worst case Sharpe ratios

σ2

Figure 6: Performance as a function of σ (Run 2)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.96

0.97

0.98

0.99

1

1.01

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

2

3

4

5

6PSfrag replacements

Robust/Classical mean Sharpe ratios

Robust/Classical of worst case Sharpe ratios

σ2

Figure 7: Performance as a function of σ (Run 3)

30

0 200 400 600 800 1000 1200 1400 1600 1800 20000

100

200

300

400

500

600RobustClassical

PSfrag replacements

CPU

tim

e(s

econds)

Number of assets n (m = d0.1ne)

Figure 8: Complexity of Robust and Classical strategies

d0.1ne, the risk free rate rf was set to 3, µ and F were generated as described above, the noisecovarianceD was set to 0.1diag(VTFV), the confidence level ω was set to 0.95, and the parameters(µ0,V0,G,ρ,γ) were computed from randomly generated factor and asset return vectors for aninvestment period p = 2m. These experiments were conducted using SeDuMi V1.03 (Sturm, 1999)within Matlab6.1R12 on a Dell Precision 340 machine. The running times were averaged over100 randomly generated instances for each problem size n. It is clear from Figure 8 that thecomputation time for the classical and robust strategies is almost identical and grows quadraticallywith the problem size. We should note, however, that algorithms based on the active set methodcan be used to solve the classical problem and these may be more efficient in some cases.

7.2 Computational results for real market data

In this section we compare the performance of our robust approach with the classical approachon real market data. The universe of assets that were chosen for investment were those currentlyranked at the top of each of 10 industry categories by Dow Jones in August, 2000. In total therewere n = 43 assets in this set (see Table 1). The base set of factors were five major market indices(see Table 2), to which we added the eigenvectors corresponding to the 5 largest eigenvalues of thecovariance matrix of the asset returns, i.e. the total number of factors used was m = 10. Thischoice of factors was made to ensure that the linear model (1) would have good predictability,i.e. small values for the residual return variances. (Selecting appropriate factors to explain thecovariance structure of the returns is a sophisticated industry, and our choice of factors is by nomeans claimed to be the most appropriate.) The data sequence consisted of daily asset returns fromJanuary 2, 1997 through December 29, 2000. Given that that assets were selected in August 2000,the data sequence suffers from the survivorship bias, i.e. we apriori knew that the companies wewere considering in our universe were the major stocks in their industry category in August 2000.It is expected that this bias would affect both strategies in a similar manner; therefore, relativeresults are still meaningful.

A complete description of the experimental procedure is as follows. The entire data sequence wasdivided into investment periods of length p = 90 days. For each investment period t, we first

31

Aerospace industry TelecommunicationAIR AAR corporation T AT&TBA Boeing Corp. LU Lucent TechnologiesLMT Lockheed Martin NOK NokiaUTX United Technologies MOT MotorolaSemiconductor Computer SoftwareAMD Applied Materials ARBA AribaINTC Intel Corp. CMRC Commerce One Inc.HIT Hitachi MSFT MicrosoftTXN Texas Instruments ORCL OracleComputer Hardware Internet and OnlineDELL Dell Computer Corp. AKAM AkamaiPALM Palm Inc. AOL AOL Corp.HWP Hewlett Packard CSCO Cisco SystemsIBM IBM Corp. NT Northern TelecomSUNW Sun Microsystems PSIX PsiNet Inc.Biotech and Pharmaceutical UtilitiesBMV Bristol-Myers-Squibb ENE Enron CorporationCRA Applera Corp.- Celera DUK Duke Energy CompanyCHIR Chiron Corp. EXC Exelon Corp.LLV Eli Lilly and Co. PNW Pinnacle WestMRK Merck and Co.Chemicals Industrial goodsAVY Avery Denison Corp. FMC FMC Corp.DD Du Pont GE General ElectricDOW Dow Chemical HON HoneywellEMN Eastman Chemical Co. IR Ingersoll Rand

Table 1: Assets

DJA Dow Jones Composite Average

NDX Nasdaq 100

SPC Standard and Poor’s 500 Index (S&P500)

RUT Russell 2000

TYX 30-year bond

Table 2: Base factors

32

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

PSfrag replacements

Relative performance of robust vs classical portfolios

Period

Rela

tive

perc

enta

ge

perf

orm

ance

Figure 9: Evolution of relative wealth for ω = 0.95

estimated the covariance matrix ΣR of the asset returns based on the market data of the previousp trading days and extracted the eigenvectors corresponding to the 5 largest eigenvalues (if anasset did not exist during the entire period it was removed from consideration). These eigenvectorstogether with the base market indices defined the factors for a particular period, and their returnswere used to estimate the factor covariance matrix F. Next, the equations (62) and (63) in Section 5were used to set V0, µ0, G, ρ and γ. The bound di on the variance of the residual return wasset to di = s2i , where s

2i is given by (51), and the risk-free rate rf was set to zero. Once all the

parameters were set, the robust portfolio φtr (resp. classical portfolio φtm) was computed by solving

the robust (resp. classical) maximum Sharpe ratio problem. The portfolio φtr and φtm were held

constant for the period t and then rebalanced to the portfolio φ(t+1)r and φ

(t+1)m in period t+ 1.

Let wtr (resp. wt

m) denote the wealth at the end of period t of an investor with initial wealth w0and employing the robust (resp. classical) portfolio strategy. Then

w(t+1)r =

[

(

∏

tp<k≤(t+1)p(1+ rk)T)

φtr

]

· wtr,

w(t+1)m =

[

(

∏

tp<k≤(t+1)p(1+ rk)T)

φtm

]

· wtm.

(86)

Since both the robust and classical strategies require a block of data of length p = 90 to estimatethe parameters, the first investment period labeled t = 1 starts from the time instant p + 1. Thetime period Jan 2, 1997-Dec 29, 2000 contains 11 periods of length p = 90, i.e. in all there are 10investment periods.

Figure 9 is a plot of the relative performance of robust strategy with respect to the classical strategy,

i.e. it plots 100( wtrwtm− 1)

as a function of t, for a confidence threshold of ω = 0.95. At the end of 11

periods the wealth generated by the robust investment strategy is 40% greater than that generatedby the classical investment strategy. Notice that at t = 7 there is a precipitous drop in the relativeperformance of the robust strategy. This drop is probably because the model estimate in periodt = 6 is remarkably close to the realization in period t = 7, i.e. the robust strategy is beingunnecessarily conservative. Therefore, one would expect that this problem would be somewhat

33

0 1 2 3 4 5 6 7 8 9 10−20

−10

0

10

20

30

40

PSfrag replacements

Relative performance of robust vs classical portfolios

Period

Rela

tive

perc

enta

ge

perf

orm

ance

Figure 10: Evolution of relative wealth for ω = 0.7

0 1 2 3 4 5 6 7 8 9 10−40

−20

0

20

40

60

80

PSfrag replacements

Relative performance of robust vs classical portfolios

Period

Rela

tive

perc

enta

ge

perf

orm

ance

ω = 0.2

ω = 0.4

ω = 0.6

ω = 0.8

ω = 0.95

Figure 11: Evolution of relative wealth as a function of ω

34

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−20

−10

0

10

20

30

40

50

60

PSfrag replacements

Relative performance of robust vs classical portfolios

ω

Fin

alre

lati

ve

wealt

h

Figure 12: Final relative wealth as a function of ω

mitigated if the threshold ω were reduced. Figure 10 plots the relative performance of the robustand classical investment strategies for a confidence threshold of ω = 0.7. It is obvious that the dropin period 7 is reduced; however, the final wealth generated by the robust strategy is, in fact, 9% lessthan that generated by the classical portfolio ! Thus, it is not guaranteed that the robust strategywill always outperform the classical portfolio. Figure 11 plots the time evolution of the relativeperformance for various values of ω. As one would expect for small values of ω the performanceof the robust and classical strategies is quite close. For intermediate values, the dip at t = 7 isreduced but unimpressive performance over the other periods drags the overall relative performanceof the robust strategy down. The performance improves as ω ↑ 1 (for ω = 0.99 the robust strategygenerates a final wealth that is 50% larger than that generated by the classical strategy). Figure12 plots the final wealth ratio as a function of ω (results were obtained for ω = 0.1, 0.2, . . ., 0.9,0.95, 0.99). It is clear that the performance is not monotonic in ω.

An important aspect of any investment strategy is the cost of implementing it. Since we areinterested in comparing the costs of implementing the robust strategy with that of implementingthe classical strategy we will quantify the transaction costs by

∥

∥φt − φ(t−1)∥

∥

1. In Figure 13 we

plot the ratio of the costs, i.e.∥

∥φtr − φ(t−1)r

∥

∥

1

/∥

∥φtm − φ(t−1)m

∥

∥

1, for a confidence threshold of

ω = 0.95. The average cost is 0.9623, i.e. the transaction costs incurred by the robust strategywere approximately 4% less that that incurred by the classical strategy. Figure 14 plots the samequantity for ω = 0.7 and now the average cost is 1.0057, i.e. as the robust strategy become lessconservative it pays more in transaction costs. Figure 15 shows the average cost as a function ofω. The average cost remains almost constant at a value slightly greater than 1 until ω > 0.9 andthen it decreases monotonically.

7.3 Summary of computation results

The summary of our simulation experiments and the experiments with real market data is asfollows:

35

2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

PSfrag replacements

Relative cost of robust vs classical portfolios

Period

Average cost = 0.9623

Rela

tive

cost

Figure 13: Relative cost per period for ω = 0.95

2 3 4 5 6 7 8 9 100.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

PSfrag replacements

Relative cost of robust vs classical portfolios

Period

Average cost = 1.0057

Rela

tive

cost

Figure 14: Relative cost per period for ω = 0.7

36

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

PSfrag replacements

Average relative cost of robust vs classical portfolios

ω

Rela

tive

cost

Figure 15: Average cost vs ω

(a) The mean performance of the robust portfolios does not significantly degrade as the confidencelevel ω is increased. Even at ω = 0.95 the relative loss of the robust portfolios is only about20%. On the other hand the worst case performance of the robust portfolios is about 200%better.

(b) Robust portfolios are able to withstand noisy data considerably better than classical portfo-lios.

(c) Summarizing the performance of robust strategies on the real market data sequence is not easy.Based on the simulation data one would expect a monotonic improvement in performance asthe threshold ω is increased. However, the experimental results do not uphold this hypothesis.For the particular data sequence used in our experiments, the robust strategy was clearlysuperior; i.e. it generated a larger wealth at a smaller cost, when ω was sufficiently large;whereas for small ω there was no discernible trend.

These computational experiments, particularly those with the real data sequence, are by no meanscomprehensive. For one the problem size – (m = 40, n = 500) for the simulation experimentsand (m = 10, n = 43) for market data experiments – was small. For the robust strategy to beacceptable one would have to ensure that the complexity of the robust optimization problems isnot significantly higher than that of the classical problems. A simple comparison of run timessuggests that this is indeed the case.

The experimental results on the real data suggests (see Figure 11) that if one wants the robuststrategy to consistently outperform the classical strategy one would want to choose ω ≈ 1. But sucha choice of ω makes the robust strategy extremely conservative which would hamper its performanceif the noise in the model was low. Since the noise is not known a priori, the correct choice of ωremains a vexing problem. Our preliminary experiments suggest that the factor model we employedwas probably noisy. More extensive experiments have to be conducted before one can assert thatthis is almost always the case (e.g., using “better” factors may significantly change the results),and therefore, setting ω ≈ 1 is a good choice. We would expect that in practice one would have

37

to adjust ω dynamically by comparing the robust strategy with the classical benchmark. Also, wehave not tested robust portfolios based on our model in Section 6 which incorporates uncertaintyin the factor covariance matrix. These and other experimental studies are planned.

A Probabilistic interpretation

In this section, we interpret the choice of the uncertainty sets Sm, Sv and Sf in terms of the impliedprobabilistic guarantees on the risk-return performance.

First consider the case where the factor covariance F is fixed. Define Sm(ω) and Sv(ω) for aconfidence level ω using (56) and (57). Let φ∗ be the optimal solution of the robust maximumSharpe ratio problem (37) and let s∗(ω) be the corresponding Sharpe ratio.

Since µTφ∗ ≥ 1 for all µ ∈ Sm(ω), it follows that µTφ∗ ≥ 1 for (µ,V) in the joint uncertainty setS(ω) defined in (55). Similarly, we have that√

(φ∗)TVTFVφ∗ ≤ 1

s∗(ω), ∀ V ∈ Sv(ω) ⇒

√

(φ∗)TVTFVφ∗ ≤ 1

s∗(ω), ∀(µ,V) ∈ S(ω).

Therefore, we have that Sharpe ratio of the portfolio φ∗

µTφ∗√

(φ∗)TVTFVφ∗≥ s∗(ω), ∀(µ,V) ∈ S(ω),

i.e. the Sharpe ratio is at least as large as s∗(ω) with confidence ωn. Similar probabilistic guaranteescan be provided for the performance of the optimal solutions of robust minimum variance, maximumreturn and VaR problems. Computing probabilistic guarantees on the performance of the robustportfolio when the uncertainty sets are defined by (62)–(63) is left to the reader.

Thus, in contrast to the classical Markowitz portfolio selection, the robust formulation allows oneto impose confidence thresholds on the value of the optimization problem. Setting a low value forω results in a high value for the corresponding robust Sharpe ratio s∗(ω) but the confidence thatthe corresponding optimal portfolio φ∗ would indeed achieve the Sharpe ratio is low. On the otherhand, a high value for ω would imply a lower Sharpe ratio s∗(ω) but it would ensure that thecorresponding optimal portfolio will achieve the Sharpe ration with a higher confidence. Thus, theparameter ω can be viewed as a surrogate for risk aversion.

Next, consider the case where the factor covariance matrix F is uncertain. Suppose one assumesthe following Bayesian setup. The market parameters (µ,V) and F are assumed to be a prioriindependent and distributed according to a non-informative conjugate prior (see Greene, 1990;Anderson, 1984, for appropriate choices for the conjugate prior). Suppose also that the residualreturn ε is independent of µ, V and F. Then the market model (1) implies that the conditionaldistribution f(S,R | µ,V,F) of the asset returns S = [r1, r2, . . . , rp] and factor returns B =[f1, f2, . . . , fp] can be factored as follows

f(S,R | µ,V,F) = f(B | F)f(S | µ,V,B). (87)

The posterior distribution f(µ,V,F | S,B) is, therefore, given by

f(µ,V,F | S,B) =f(B | F)f(F)f(S | µ,V,B)f(µ,V)

f(S,R),

=(

c1f(B | F)f(F))

·(

c2f(S | µ,V,B)f(µ,V))

, (88)

38

where c1, c2 are suitable normalizing constants. From (88) it follows that F and (µ,V) are a poste-riori independent. Therefore, the uncertainty sets for (µ,V,F) can be represented as a Cartesianproduct S × Sf−1 .

Suppose the confidence threshold is set to ω ≤ (FΓp(2))m, where FΓp is the CDF of a Γ(

p+12 , p−12

)

random variable (for a discussion of the upper bound on the achievable confidence see Section 6).Set F0 = Fml and set η by solving (76). Then F ∈ Sf−1(η) with confidence ωm. As before, letS(ω) be the ωn confidence set for (µ,V). Let φ∗ be the optimal solution of the robust Sharpe ratioproblem with uncertain covariance and let s∗(ω) be the corresponding value. Then the confidencethat (µ,V,F) ∈ S(ω)×Sf−1(η) is ω(m+n), i.e. with a confidence level ω(m+n), the realized Sharperatio of φ∗ is at least as large as s∗(ω).

The latter development relied on the fact that an independent Bayesian prior ensures that the jointconfidence set for (µ, V, F) is the Cartesian product of the individual confidence sets for (µ,V)and F. Such a partition is not immediately obvious in the non-Bayesian setup, and, consequently,it is not clear how the uncertainty set Sf given by (77) could lead to probabilistic guarantees.

Acknowledgments

D. Goldfarb’s research was partially supported by DOE grant GE-FG01-92ER-25126, and NSFgrants DMS-94-14438, CDA-97-26385 and DMS-01-04282. G. Iyengar’s research partially supportedby NSF grants CCR-00-09972 and DMS-01-04282.

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D. Goldfarb: IEOR Department, Columbia University, e-mail: gold@ieor.columbia.eduG. Iyengar: IEOR Department, Columbia University, e-mail: garud@ieor.columbia.edu.

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