Robust Portfolio Selection Problems

8/4/2019 Robust Portfolio Selection Problems

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MATHEMATICS OF OPERATIONS RESEARCH

Vol. 28, No. 1, February 2003, pp. 138

Printed in U.S.A.

ROBUST PORTFOLIO SELECTION PROBLEMS

D. GOLDFARB and G. IYENGAR

In this paper we show how to formulate and solve robust portfolio selection problems. The

objective of these robust formulations is to systematically combat the sensitivity of the optimal

portfolio 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 robust portfolio

selection problems corresponding to these uncertainty structures can be reformulated as second-

order cone programs and, therefore, the computational effort required to solve them is comparable

to that required for solving convex quadratic programs. Moreover, we show that these uncertainty

structures correspond to confidence regions associated with the statistical procedures employed to

estimate the market parameters. Finally, we demonstrate a simple recipe for efficiently computing

robust portfolios given raw market data and a desired level of confidence.

1. Introduction. Portfolio selection is the problem of allocating capital over a number

of available assets in order to maximize the return on the investment while minimizing

the risk. Although the benefits of diversification in reducing risk have been appreciated

since the inception of financial markets, the first mathematical model for portfolio selection

was formulated by Markowitz (1952, 1959). In the Markowitz portfolio selection model, the

return on a portfolio is measured by the expected value of the random portfolio return,

and the associated risk is quantified by the variance of the portfolio return. Markowitz

showed that, given either an upper bound on the risk that the investor is willing to take or

a lower bound on the return the investor is willing to accept, the optimal portfolio can be

obtained by solving a convex quadratic programming problem. This mean-variance model

has had a profound impact on the economic modeling of financial markets and the pricing

of assetsthe 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, Sharpe and Markowitz shared the Nobel Memorial Prize in Economic

Sciences for their work on portfolio allocation and asset pricing.Despite the theoretical success of the mean-variance model, practitioners have shied

away from this model. The following quote from Michaud (1998) summarizes the problem:

Although Markowitz efficiency is a convenient and useful theoretical framework for portfo-

lio 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 problems are often very sensitive to perturbations in the parameters of the

problem; since the estimates of the market parameters are subject to statistical errors, the

results of the subsequent optimization are not very reliable. Various aspects of this phe-

nomenon have been extensively studied in the literature on portfolio selection. Chopra and

Ziemba (1993) studies the cash-equivalent loss from the use of estimated parameters instead

of the true parameters. Broadie (1993) investigates the influence of errors on the efficient

frontier, and Chopra (1993) investigates the turnover in the composition of the optimal

Received January 1, 2002; revised May 25, 2002, and July 24, 2002.

MSC 2000 subject classification. Primary: 91B28; secondary: 90C20, 90C22.

OR/MS subject classification. Primary: Finance/portfolio; secondary: programming/quadratic.

Key words. Robust optimization, mean-variance portfolio selection, value-at-risk portfolio selection, second-order

cone programming, linear regression.

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0364-765X/03/2801/0001/$05.001526-5471 electronic ISSN, 2003, INFORMS


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2 D. GOLDFARB AND G. IYENGAR

portfolio as a function of the estimation error (see also Part II of Ziemba and Mulvey 1998for a summary of this research).

Several techniques have been suggested to reduce the sensitivity of the Markowitz-optimalportfolios to input uncertainty: Chopra (1993) and Frost and Savarino (1988) propose con-

straining portfolio weights, Chopra et al. (1993) proposes using a James-Stein estimator(see Huber 1981 for details on Stein estimation) for the means, while Klein and Bawa

(1976), Frost and Savarino (1986), and Black and Litterman (1990) suggest Bayesian esti-mation of means and covariances. Although these techniques reduce the sensitivity of the

portfolio composition to the parameter estimates, they are not able to provide any guaran-tees on the risk-return performance of the portfolio. Michaud (1998) suggests resampling

the mean returns and the covariance matrix of the assets from a confidence regionaround a nominal set of parameters, and then aggregating the portfolios obtained by solv-

ing a Markowitz problem for each sample. Recently scenario-based stochastic programmingmodels have also been proposed for handling the uncertainty in parameters (see Part V of

Ziemba and Mulvey 1998 for a survey of this research). Both the sampling-based and thescenario-based approaches do not provide any hard guarantees on the portfolio performance

and become very inefficient as the number of assets grows.

In this paper we propose alternative deterministic models that are robust to parame-ter uncertainty and estimation errors. In this framework, the perturbations in the marketparameters are modeled as unknown, but bounded, and optimization problems are solved

assuming worst case behavior of these perturbations. This robust optimization frameworkwas introduced in Ben-Tal and Nemirovski (1999) for linear programming and in Ben-Tal

and Nemirovski (1998) for general convex programming (see also Ben-Tal and Nemirovski2001). There is also a parallel literature on robust formulations of optimization problems

originating from robust control (see El Ghaoui and Lebret 1997, El Ghaoui et al. 1998, andEl 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

random asset returns r Rn is given by

r = + VTf+

where Rn is the vector of mean returns, f Rm is the vector of random returns of them< n factors that drive the market, V Rmn is the factor loading matrix and is thevector of residual returns. The mean return vector , the factor loading matrix V, and the

covariance matrices of the factor return vector f and the residual error vector are knownto lie within suitably defined uncertainty sets. For this market model, we formulate robust

analogs of classical mean-variance and value-at-risk portfolio selection problems.(b) We show that the natural uncertainty sets for the market parameters are defined by

the statistical procedures used to estimate these parameters from market return data. Thisclass of uncertainty sets is completely parametrized by the market data and a parameter

that controls the confidence level, thereby allowing one to provide probabilistic guaranteeson the performance of the robust portfolios. In all previous work on robust optimization

(Ben-Tal and Nemirovski 1998, 1999; El Ghaoui and Lebret 1997; El Ghaoui et al. 1998;Halldrsson and Ttnc 2000), the uncertainty sets for the parameters are assumed to have

a certain structure without any explicit justification. Also, there is no discussion of howthese sets are parametrized from raw data.

(c) We show that the robust optimization problems corresponding to the natural class ofuncertainty sets (defined by the estimation procedures) can be reformulated as second-order

cone programs (SOCPs). SOCPs can be solved very efficiently using interior point algo-rithms (Nesterov and Nemirovski 1993, Lobo et al. 1998, Strm 1999) In fact, both the

worst case and practical computational effort required to solve an SOCP is comparable to


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ROBUST PORTFOLIO SELECTION PROBLEMS 3

that for solving a convex quadratic program of similar size and structure; i.e., in practice,the computational effort required to solve these robust portfolio selection problems is com-

parable to that required to solve the classical Markowitz mean-variance portfolio selectionproblems.

In a recent related paper, Halldrsson and Ttnc (2000) show that if the uncertainmean return vector and the uncertain covariance matrix of the asset returns r belongto the component-wise uncertainty sets Sm = L U and Sv = 0L U, respectively, the robust problem reduces to a nonlinear saddle-point problemthat involves semidefinite constraints. Here A 0 (resp. 0) denotes that the matrix A issymmetric and positive semidefinite (resp. definite). This approach has several shortcomingswhen applied to practical problemsthe model is not a factor model (in applied work factor

models are popular because of the econometric relevance of the factors), no procedure isprovided for specifying the extreme values LU and LU defining the uncertaintystructure and, moreover, the solution algorithm, although polynomial, is not practical when

the number of assets is large. A multiperiod robust model, where the uncertainty sets arefinite sets, was proposed in Ben-Tal et al. (2000).

The organization of the paper is as follows. In 2 we introduce the robust factor model

and uncertainty sets for the mean return vector, the factor loading matrix, and the covariancematrix of the residual return. We also formulate robust counterparts of the mean-varianceoptimal portfolio selection problem, the maximum Sharpe ratio portfolio selection problem,

and the value-at-risk (VaR) portfolio selection problem. The uncertainty sets introduced inthis section are ellipsoidal (intervals in the one-dimensional case) and may appear quitearbitrary. Before demonstrating that these sets are, indeed, natural, we first establish in

3 that the robust mean-variance portfolio selection problem in markets where the factorloading and mean returns are uncertain, but the factor covariance is known and fixed, can

be reformulated as an SOCP. An SOCP formulation of the robust maximum Sharpe ratioproblem in such markets follows as a corollary. In 4 we develop an SOCP reformulation for

the robust VaR portfolio selection problem. In 5 we justify the uncertainty sets introducedin 2 by relating them to linear regression. Specifically, we show that the uncertainty sets

correspond to confidence regions around the least-squares estimate of the market parametersand can be constructed to reflect any desired confidence level. In this section, we collectthe results from the preceding sections and present a recipe for robust portfolio allocation

that closely parallels the classical one. In 6 we improve the factor model by allowinguncertainty in the factor covariance matrix and show that, for natural classes of uncertainty

sets, all robust portfolio allocation problems continue to be SOCPs. We also show that thesenatural classes of uncertainty sets correspond to the confidence regions associated with

maximum likelihood estimation of the covariance matrix. In 7 we present results of someof preliminary computational experiments with our robust portfolio allocation framework.

2. Market model and robust investment problems. We assume that the market opensfor 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 interpretation that asset i returns1 + ri dollars for every dollar invested in it. The returns on the assets in different marketperiods are assumed to be independent. The single period return r is assumed to be arandom variable given by

r = +VTf+(1)

where Rn is the vector of mean returns, f 0 F Rm is the vector of returns ofthe factors that drive the market, V Rmn is the matrix of factor loadings of the n assets,and 0 D is the vector of residual returns. Here x denotes that x is amultivariate normal random variable with mean vector and covariance matrix .


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In addition, we assume that the vector of residual returns is independent of the vectorof factor returns f, the covariance matrix F 0 and the covariance matrix D = diagd 0,i.e. di 0, i = 1 n. Thus, the vector of asset returns r VTFV +D. Althoughnot required for the mathematical development in this paper, the eigenvalues of the residual

covariance matrix D are typically much smaller than those of the covariance matrix VTFVimplied by the factors; i.e., the VTFV is a good low-rank approximation of the covariance

to the asset returns.Except in 6, the covariance matrix F of the factor returns f is assumed to be stable and

known exactly. The individual diagonal elements di of the covariance matrix D are assumed

to lie in an interval di di, i.e., the uncertainty set Sd for the matrix D is given by

Sd = D D = diagd di di di i = 1 n(2)

The columns of the matrix V, i.e., the factor loadings of the individual assets, are alsoassumed to be known approximately. In particular, V belongs to the elliptical uncertaintyset Sv given by

Sv = V V = V0 +W Wig i i = 1 n (3)where Wi is the ith column of W and wg =

wTGw denotes the elliptic norm of w with

respect to a symmetric, positive definite matrix G.The mean returns vector is assumed to lie in the uncertainty set Sm given by

Sm = = 0 + i i i = 1 n(4)

i.e., each component of is assumed to lie within a certain interval. The choice of the

uncertainty sets is motivated by the fact that the factor loadings and the mean returns ofassets are estimated by linear regression. The justification of the uncertainty structures andsuitable choices for the matrix G, and the bounds i, i, di, di, i = 1 n, are discussedin 5.

An investors position in this market is described by a portfolio Rn

, where the ithcomponent i represents the fraction of total wealth invested in asset i. The return r on

the portfolio is given by

r = rT= T+ fTV+T TTVTFV +D

The objective of the investor is to choose a portfolio that maximizes the return on theinvestment subject to some constraints on the risk of the investment. The mathematicalmodel for portfolio selection proposed by Markowitz (1952, 1959) assumes that the expected

value Er of the asset returns and the covariance Varr are known with certainty. Inthis model investment return is the expected value Er of the portfolio return andthe associated risk is the variance Varr. The objective of the investor is to choose a

portfolio that has the minimum variance among those that have expected return at least

i.e., is the optimal solution of the convex quadratic optimization problemminimize Varr

subject to Er 1T= 1

(5)

As was pointed out in the introduction, the primary criticism leveled against theMarkowitz model is that the optimal portfolio is extremely sensitive to the marketparameters Er Varrsince these parameters are estimated from noisy data, oftenamplifies noise. By introducing measures of uncertainty in the market models, we are


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attempting to correct this sensitivity to perturbations. The uncertainty sets Sm, Sv and Sdrepresent the uncertainty of our limited (inexact) information of the market parameters, andwe wish to select portfolios that perform well for all parameter values that are consistentwith this limited information. Such portfolios are solutions of appropriately 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 maxVSv DSd

Varr

subject to minSm

Er 1T= 1

(6)

The objective of the robust minimum variance portfolio selection problem (6) is to minimizethe worst case variance of the portfolio subject to the constraint that the worst case expectedreturn on the portfolio is at least . We expect that the sensitivity of the optimal solutionof this mathematical program to parameter fluctuations will be significantly smaller than itwould be for its classical counterpart (5).

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

maximize minSm

Er

subject to maxVSv DSd

Varr 1T= 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 ratioof the expected excess return on the portfolio, i.e., the return in excess of the risk-free raterf, to the standard deviation of the return. The corresponding max-min problem is given by

maximize 1T=1

minSm VSv DSd

Er rf

Varr

(8)

All these variants are studied in 3. We show that for the uncertainty sets Sd, Sm, andSv, defined in (2)(4) above, all of these problems reduce to SOCPs. Although the factormodel (1) is crucial for the SOCP reduction, the assumption that factor and residual returnsare normally distributed can be relaxed. From the results in Bertsimas and Sethuraman(2000) it follows that all of our results continue to hold with slight modifications if oneassumes that the distributions are unknown but second moments of the factor and residualreturns lie in the sets Sv and Sd, respectively. We leave this extension to the reader.

In 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

constraint that the probability of the return r falling below a threshold is less than aprescribed limit; i.e., the objective is to solve the following mathematical program.

maximize minSm

Er

subject to maxVSvSm DSd

Pr (9)

VaR was introduced as a performance analysis tool in the context of risk management.Recently there has been a growing interest in imposing VaR-type constraints while opti-mizing credit risk (Kast et al. 1998, Mausser and Rosen 1999, Andersson et al. 2001). We


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show that, for the uncertainty sets defined in (2)(4), this problem can also be reduced to an

SOCP and, hence, can be solved very efficiently. Our technique can be extended to another

closely related performance measure called the conditional VaR by using the results in

Mausser and Rosen (1999). El Ghaoui et al. (2002) presents an alternative robust approach

to VaR problems that results in semidefinite programs.The optimization problems of interest here fall in the general class of robust convex opti-

mization problems. Ben-Tal and Nemirovski (1998, 2001) propose the following structure

for generic robust optimization problem.

minimize cTx

subject to F x Rm (10)

where are the uncertain parameters in the problem, x Rn is the decision vector, is aconvex cone and, for fixed , the function F x is -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 and Nemirovski (1998, 1999, 2001). They established that for suit-

ably defined uncertainty sets , the robust counterparts of linear programs, quadratic pro-grams, and general convex programs are themselves tractable optimization problems. Robust

least-squares problems and robust semidefinite programs 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 multistage portfolio problems. Halldrsson

and Ttnc (2000) study robust investment problems that reduce to saddle point problems.

3. Robust mean-variance portfolio selection. This section begins with a detailed anal-

ysis of the robust minimum variance problem (6). It is shown that for uncertainty sets

defined in (2)(4) this problem reduces to an efficiently solvable SOCP. This result is sub-

sequently extended to the maximum return problem (7) and the robust maximum Sharpe

ratio problem (8).

3.1. Robust minimum variance problem. Since the return

r TTVTFV +D

the robust minimum variance portfolio selection problem (6) is given by

minimize maxVSv

TVTFV

+ maxDSd

TD

subject to minSm

T 1T= 1

(11)

The bounds di di di imply that T

D T

D, where D = diagd. Also, since thecovariance matrix F of the factor f is assumed to be strictly positive definite, the function

xf x

xTFx defines a norm on Rm. Thus, (11) is equivalent to the robust augmented

least-squares problem

minimize maxVSv

V2f +TD

subject to minSm

T

1T= 1

(12)


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

minimize + subject to max

VSv V

2f

TD min

SmT 1T = 1

(13)

If the uncertainty sets Sv and Sm are finite, i.e., Sv = V1 Vs and Sm = 1 r,then (13) reduces to the convex quadratically constrained problem

minimize + subject to Vk2f for all k = 1 s

TD Tk for all k = 1 r

1T = 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 =

TrWTW the problem can still be reformulated as an SOCP. Methodologically speaking, our results

can be viewed as an extension of the results in El Ghaoui and Lebret (1997) to other classes

of uncertainty sets more suited to the application at hand.

When the uncertainty in V and is specified by (3) and (4), respectively, the worst case

mean return of a fixed portfolio is given by,

minSm

T=T0T (15)

and the worst case variance is given by,

maximize V0 +W2fsubject to Wig i i = 1 n

(16)

Since the constraints Wig i, i = 1 n imply the bound,

Wg = n

i=1 iWi

g n

i=1 i Wig n

i=1 i i (17)

the optimization problem,

maximize V0+ w2f subject to wg r

(18)

where r= T =ni=1 i i is a relaxation of (16), i.e., the optimal value of (18) is atleast as large as that of (16).


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

Wi =

i

ii

rw

i =0

ir

w otherwise(19)

Then Wi g = i, for all i = 1 n; i.e., W is feasible for (16), and W =ni=1 iW

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

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

maxy ygr

y0 +y2f (20)

where y0 = V0 and r= T .The following lemma reformulates this constraint as a collection of linear equalities, linear

inequalities and restricted hyperbolic constraints (i.e. constraints of the form: zT

z xy,z Rn, x y R and x y 0).Lemma 1. Letr > 0, y0 y Rm andF G Rmm be positive definite matrices. Then

the constraint

maxyyg r

y0 +y2f (21)

is equivalent to either of the following:

(i) there exist 0, and t Rm+ that satisfy

+ 1Tt

1

maxH

r2 w2i 1 iti i = 1 m

(22)

where QQT is the spectral decomposition of H = G1/2FG1/2, = diagi, and w =QTH1/2G1/2y0;

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

r2 1Tsu2i 1 isi i = 1 m

1

maxK

(23)

where PPT is the spectral decomposition of K = F1/2G1F1/2, = diagi, and u =PTF1/2y0.

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

yT0 Fy0 2ryT0 Fy r2yTFy 0(24)

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


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Lemma 2 (-procedure). Let Fix = xTAix + 2bTi x + ci, i = 0 p be quadraticfunctions ofx Rn. Then F0x 0 for all x such thatFix 0, i = 1 p, if there existi 0 such that

c0 bT0

b0 A0

p

i=1

i ci bTi

bi Ai 0

Moreover, ifp = 1 then the converse holds if there exists x0 such that F1x0 > 0.For a discussion of the -procedure and its applications, see Boyd et al. (1994). Since

y = 0 is strictly feasible for 1 yTGy 0, the -procedure implies that (24) holds for all1 yTGy 0 if and only if there exists a 0 such that

M =

yT0 Fy0 ryT0 FrFy0 G r2F

0(25)

Let the spectral decomposition of H = G1/2FG1/2 be QQT, where = diag, anddefine w = QTH1/2G1/2y0 =1/2QTG1/2y0. Observing that yT0 Fy0 = wTw, we have that thematrix M 0 if and only if

M =

1 0T

0 QTG1/2

M

1 0T

0 G1/2Q

=

wTw rwT1/2r1/2w I r2

0

The matrix M 0 if and only if r2i, for all i = 1 m (i.e., r2maxH), wi = 0for all i such that = r2i, and the Schur complement of the nonzero rows and columnsof I r2,

wTw r2

i=r2i

iw2i

r2i

=

ii=1

w2i1 i

0

where = r2/. It follows that

maxy ygr

y0 +

ry

2

f

if and only if there exists 0 and t Rm+ satisfying,

+ 1Ttr2 =

w2i = 1 iti i = 1 m 1

maxH

(26)

It is easy to establish that there exist 0 and t Rm+ that satisfy (26) if and only ifthere exist 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 +y2f . Since y0 +y2f > for all sufficiently large y, the-procedure implies that (27) holds for all y0 +y2f if and only if there exists a 0 such that

M =r2 yT0 Fy0 yT0 F

Fy0 G F

0(28)


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Let PPT be the spectral decomposition of K = F1/2G1F1/2, where = diag, andu = PTF1/2y0. Then M 0 if and only if

M

= 1 0T

0 PTF1/2M1 0

T

0 F1/2P=

r2 uTu uT

u 1 I 0However, M 0 if and only if 1/i, for all i = 1 m (i.e., 1/maxK), ui = 0for all i such that i = 1, and the Schur complement of the nonzero row and columns of1 I,

r2 uTu 2

ii=1

u2i1i

= r2 +

ii=1

u2i1 i

0

Hence,

maxyygr

y0 + ry2f

if and only if there exists 0 and s Rm+ satisfying,

r2 1Tsu2i = 1 isi i = 1 m 1

maxK

(29)

Completely analogous to the proof of part (i), we have that there exists 0 and s Rm+satisfying (29) if and only if there exist 0 and s Rm+ satisfying (23). This proves thesecond result.

To illustrate the equivalence of parts (i) and (ii) of Lemma 1, we note that if F = G,then K = H = I and w = u. Hence, if s satisfies (23), then t = 1Ts ssatisfies (22).

The restricted hyperbolic constraints, zTz xy, x y 0, can be reformulated as second-order cone constraints as follows (see Nesterov and Nemirovski 1993, 6.2.3):

zTz xy 4zTz x + y2 x y2

2z

x y x + y(30)

The above result and Lemma 1 motivate the following definition.

Definition 1. Given V0 Rmn, and F G Rmm positive definite, define V0 F Gto be the set of all vectors r R R Rn such that r y0 = V0 satisfy (22);i.e., there exist 0 and t Rm+ that satisfy

+ 1Tt

1maxH

2r

+

2wi

1 i ti

1 i + ti i = 1 mwhere QQT is the spectral decomposition of H = G1/2FG1/2, = diagi and w =QTH1/2G1/2V0.


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

minimize +

subject to2D1/2

1 1+ T0T

T r1T = 1

r V0 F G

(31)

where the equality r = T has been relaxed by recognizing that the relaxed constraintwill always be tight at an optimal solution. Although (31) is a convex optimization problem,

it is not an SOCP. However, replacing by a new variable Rn and adding the 2nlinear constraints, i i, i = 1 n, leads to the following SOCP formulation for therobust minimum variance portfolio selection problem (11):

minimize +subject to

2D1/21

1 +T0T

i i i = 1 ni i i = 1 n

1T = 1T V0 F G

(32)

Another possible transformation leading to an SOCP is to replace by = + ,+ Rn+. An alternative SOCP formulation is obtained if one uses part (ii) of thelemma to characterize the worst case variance. The derivation of these results is left to thereader.

To keep the exposition simple, in the rest of this paper it will assumed that short sales are

not allowed; i.e., 0. In each case the result can be extended to general by employingthe above transformations.

3.2. Robust maximum return problem. The robust maximum return problem is given

bymaximize min

SmT

subject to maxVSv DSd

TVTFV + D 1T= 1 0

or equivalently, from (15), by

maximize Tsubject to max

VSv TVTFV

TD 1T = 1 0

(33)


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

maximize T

subject to2D1/21 1 +

1T = 1 0

T V0 F G

(34)

3.3. Robust maximum Sharpe ratio problem. The robust maximum Sharpe ratioproblem is given by,

max0 1T=1

minVSv Sm DSd

T rf

Varr

(35)

where rf is the risk-free rate of return. We will assume that the optimal value of this max-min problem is strictly positive; i.e., there exists a portfolio with finite worst-case variance

whose worst-case return is strictly greater than the risk-free rate rf. In practice the worst-case variance of every asset is bounded; therefore, this constraint qualification reduces tothe requirement that there is at least one asset with worst-case return greater than rf.

Since the components of the portfolio vector add up to 1, the objective of the max-minproblem (35),

T rfVarr

= rf1T

Varr

is a homogeneous function of the portfolio . This implies that the normalization condition

1T= 1 can be dropped and the constraint min Smrf1T= 1 added to (35) withoutany loss of generality. With this transformation, (35) reduces to minimizing the worst casevariance; i.e., (35) is equivalent to

minimize maxVSv

TVTFV+TD

subject to 0 rf1T 1 0

(36)

where the constraint minSm rf1T = 1 has been relaxed by recognizing that therelaxed constraint will always be tight at an optimal solution. Consequently, the robustmaximum Sharpe ratio problem is equivalent to a robust minimum variance problem with

= 1, replaced by rf1, and no longer normalized. Exploiting this relationship, wehave that (35) is equivalent to the SOCP

minimize +

subject to

2D1/21

1 +0 rf1T 1

T V0 F G

(37)

The crucial step in the reduction of (35) to (37) is the realization that the objective functionin (35) is homogeneous in and, therefore, its numerator can be restricted to 1 withoutany loss of generality. This homogenization goes through even when there are additional

inequality constraints on the portfolio choices.


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Suppose the portfolio is constrained to satisfy A b ( 0 is assumed to besubsumed in A b). Then, the robust maximum Sharpe ratio problem,

max Ab 1T=1 minVSv Sm DSd T rfVar (38)

is equivalent to the robust minimum variance problem

minimize maxVSv

TVTFV+TDsubject to 0 rf1T 1

A b1T =

0

(39)

where is an auxiliary variable that has been introduced to homogenize the constraintsA b and 1T= 1. The problem (39) can easily be converted into a second-order coneproblem by using the techniques developed above.

4. Robust Value-at-Risk (VaR) portfolio selection. The robust VaR portfolio selection

problem is given by

maximize minSm

Er

subject to maxSm VSv DSd

Pr

1T= 1

0

(40)

This optimization problem maximizes the expected return subject to the constraint that the

shortfall probability is less than . Note that for ease of exposition we again assume that

no short sales are allowed; i.e., 0. The solution in the general case can be obtained byusing a transformation identical to the one detailed at the end of 3.1.

For fixed V D, the return vector r =TTVTFV + D. Therefore,

Pr PT+

TVTFV +D

(41)

P

T

TVTFV+ D

TTVTFV +D

1

where 0 1 is the standard normal random variable and is its cumulativedensity function. In typical VaR applications 1; therefore 1 < 0. Thus, the prob-ability constraint for fixed V D is equivalent to the second-order cone constraint

1 F1/2V2 +D1/22 T(42)


14/39


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

maximize minSm

T

subject to maxSm VSv DSd

1

F1/2V2 +D1/22 T 1T= 1 0

(43)

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

maximize 0 T

subject to 1

0 T

D1/2 max

VSv F1/2V

1T = 1 0

(44)

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

maxVSv

F1/2V if and only if there exist 0 and s Rm+ such that,

r2 2 1Tsu2i 1 isi i = 1 m

maxK 1(45)

where K = PPT is the spectral decomposition of K = F1/2G1F1/2, = diagi, andu = PTF1/2V0. On introducing the change of variables, = and s = 1/s, (45)becomes

r2 1Tsu2i isi i = 1 m

maxK (46)

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

maximize 0 Tsubject to u = PTF1/2VT0

1

0 T D1/2

(47)


15/39


2T

+ 1Ts

+ 1Ts

2ui

i

si

i

+si i

=1 m

maxK 01T = 1 0 0

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

5. Multivariate regression and norm selection. In this section results from the statis-tical theory of multivariate linear regression are used justify the uncertainty structures Sd,Sv and Sm proposed in 2 and motivate natural choices for the matrix G defining the ellipticnorm

g and the bounds i, i,

di, i=

1 n.

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

r = +VTf+(48)

where is the residual return. In practice, the parameters V of this linear model areestimated by linear regression. Given market data consisting of samples of asset returnvectors and the corresponding factor returns, the linear regression procedure computes theleast squares estimates 0 V0 of V. In addition, the procedure also gives multi-dimensional confidence regions around these estimates with the property that the true valueof the parameters lie in these regions with a prescribed confidence level. The structure ofthe uncertainty sets introduced in 2 is motivated by these confidence regions. The rest ofthis section describes the steps involved in parameterizing the uncertainty structures, i.e.,computing 0, V0, G, i, i, di, i = 1 n.

Suppose the market data consists of asset returns, rt

t = 1 p, for p periods andthe corresponding factor returns ft t = 1 p. Then the linear model (48) implies that

rti = i +n

j=1Vji f

tj + ti i = 1 n t = 1 p

In linear regression analysis, typically, in addition to assuming that the vector of residualreturns t in period t is composed of independent normals, it is assumed that the residualreturns of different market periods are independent. Thus, ti i = 1 nt = 1 pare all independent normal random variables and ti 0 2i , for all t = 1 p; i.e.,the variance of the residual return of the ith asset is 2i . The independence assumption canbe relaxed to ARMA models by replacing linear regression by Kalman filters (see Hansenand Sargent 2001).

Let S = r1

r

2

r

p

Rn

p

be the matrix of asset returns and B = f1

f

2

f

p

Rmp be the matrix of factor returns. Collecting together terms corresponding to a particularasset i over all the periods t = 1 p, we get the following linear model for the returnsrti t = 1 p,

yi = Axi +iwhere

yi =

r1i r2i r

pi

T A = 1 BT xi = i V1i V2i VmiT

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


16/39


The least-squares estimate xi of the true parameter xi is given by the solution of thenormal equations

ATAxi = ATyi

i.e.,

xi = ATA1ATyi(49)

if rankA = m +1. Substituting yi = Axi +i, we get

xi xi = ATA1ATi = 0

where = 2i ATA1. Hence

= 12i

xi xiTATAxi xi 2m+1(50)

is a 2 random variable with m

+1 degrees of freedom. Since the true variance 2i is

unknown, (50) is not of much practical value. However, a standard result in regressiontheory states that if 2i in the quadratic form (50) is replaced by m +1s2i , where s2i is theunbiased estimate of 2i given by

s2i =yi Axi2p m 1 (51)

then the resulting random variable

= 1m +1s2i

xi xiTATAxi xi(52)

is distributed according to the F-distribution with m+1 degrees of freedom in the numer-ator and p m 1 degrees of freedom in the denominator (Anderson 1984, Greene1990).

Let 0 < < 1, J denote the cumulative distribution function of the F-distribution with J

degrees of freedom in the numerator and p m1 degrees of freedom in the denominatorand let cJ be the -critical value, i.e., the solution of the equation JcJ = .

Then the probability cm+1 is , or equivalently,

P

xi xiTATAxi xi m +1cm+1s2i= (53)

Define

Si =

xi xi xiTATAxi xi m +1cm+1s2i

(54)

Then, (53) implies that Si is a -confidence set for the parameter vector xi correspondingto asset i. Since the residual errors i i = 1 n are assumed to be independent, itfollows that

S = S1 S2 Sn(55)

is a n-confidence set for V.

Let Sm denote the projection of S along the vector ; i.e.,

Sm = = 0 + i i i = 1 n


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where

0i = i i =

m +1ATA111 cm+1s2i i = 1 n(56)

Then (55) implies that Sm is an n

-confidence set for the mean vector . Note that theuncertainty structure (4) for the mean assumed in 2 is identical to Sm.

Let Q = e2 e3 em+1T Rmm+1 be projection matrix that projects xi along Vi.Define the projection Sv of S along V as follows:

Sv =

V V = V0 + W Wig i i = 1 n

where

V0 = V1 VnG = QATA1QT1 = BBT 1

pB1B1T

i=m +1cm+1s

2i i

=1 n

(57)

Then Sv is an n-confidence set for the factor loading matrix V. As in the case of

Sm, the uncertainty structure (3) for the factor covariance V assumed in 2 has preciselythe same structure as Sv defined above.

The construction of the confidence regions can be done in the reverse direction aswell; i.e., individual confidence regions can be suitably combined to yield joint confidenceregions. Let Sm Rn and Sv Rmm be any -confidence regions for and Vrespectively. Then

P

V Sm Sv = 1 P V Sm Sv(58)

1 P SmPV Sv=

2

1

i.e., Cartesian products of individual confidence regions are joint confidence regions. This

leads to an alternative means of constructing joint confidence regions for market parameters V.

Let Q RJm. Then

= 1Js2i

Qxi QxiT

QATA1QT1

Qxi Qxi(59)

is distributed according to the F-distribution with J degrees of freedom in the numeratorand p m1 degrees of freedom in the denominator, i.e., the probability cJ is ,or equivalently,

P Qxi

Q

xi

TQATA1QT1Qxi

Qx

i

JcJ

s2

i= (60)

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

P

i i

ATA111 c1s

2i

= (61)

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

Sm = = 0 + i i i = 1 n


18/39


19/39


for a discussion of the probabilistic guarantees on the performance of the optimal robustportfolio.

Although we propose a strictly data-driven approach in this section, this analysis extendsto Bayesian estimation methods such as those in Black and Litterman (1990), as well as

empirical Bayes estimation. In the Bayesian setting, the confidence regions are given by theposterior distribution.

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

Fml =1

p 1

BBT 1

pB1B1T

(64)

i.e., G = p1Fml. Suppose the true covariance matrix F is approximated by the maximumlikelihood estimate Fml. Then F = G, where = 1/p 1. Recall that the worst casevariance

maxVSv

TVTFV

2(65)

if and only if maxy ygr

V0+yf

where r= T. Since F = G, the above norm constraint is equivalent tomax

u ur

F1/2V0+u (66)where is now the usual Euclidean norm. It is easy to see that the maximum in (66) isattained at

u = r

F1/2V0 F1/2V0

and the corresponding maximum value is

F1/2V0+r. Thus, the worst-case variance

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

F1/2V0 T(67)From the fact that 2 is equivalent to a second-order cone constraint, it follows thatin the special case F = G, the robust minimum variance reduces to the following simpleSOCP

minimize + subject to 0 T

1T = 1F1/2V0 T

2D1/21

1 +

21 1 + 0

(68)

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

In practice, however, the covariance matrix F is assumed to be stable and is, typically,estimated from a much larger data set and by taking several extraneous macroeconomicindicators into account (see Ledoit 1996). If this is the case, then the above simplificationcannot be used, and one would have to revert back to the formulation in (31).


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6. Robust portfolio allocation with uncertain covariance matrices. The marketmodel introduced in 2 assumes that the covariance matrix F of the factors is completely

known and stable. While this is a good first approximation, a more complete market modelis one that allows some uncertainty in the covariance matrix and optimizes over it. For

any uncertainty structure for covariance matrices to be useful, it must be flexible enoughto model a variety of perturbations while at the same time restricted enough to admit fastparameterization and efficient optimization. Our goal in this section is to develop such an

uncertainty structure for covariance matrices. We assume the market structure is the one

introduced in 2, except that the covariance matrix is no longer fixed.

6.1. Uncertainty structure for covariance inverse. Consider the following uncertaintystructure for the factor covariance matrix F:

Sf1 =

F F1 = F10 + 0=TF1/20 F1/20 (69)

where F0 0. The norm A in (69) is given by A = maxi iA, where iA arethe eigenvalues of A.

The uncertainty set Sf1 restricts the perturbations of the covariance matrix to besymmetric, bounded in norm relative to a nominal covariance matrix F0 and subject to

F1 = F10 + 0. Clearly, this uncertainty structure is not the most generalone could,for example, allow for each element of the covariance matrix to lie in some uncertaintyinterval subject to the constraint that the matrix is positive semidefinite. However, the

robust optimization problem corresponding to this uncertainty structure is not very tractable

(Halldrsson and Ttnc 2000). On the other hand, if the covariance uncertainty is givenby (69), all of the robust portfolio allocation problems introduced in 2 can be reformu-

lated as SOCPs. Moreover, as in the case of the uncertainty structures for and V, theuncertainty structure Sf1 for F corresponds to the confidence region associated with the

statistical procedure used to estimate F. In particular, we show that maximum likelihood

estimation of the covariance matrix F provides a confidence region of the form Sf1 and

yields a value of that reflects any desired confidence level.All the robust portfolio selection problems introduced in 2 can be formulated for this

market as well. For example, the robust Sharpe ratio problem in this market model is givenby

maximize 1T=1

minSm VSv DSd FSf1

T rf

TVTFV +D

Lemma 3, below, establishes that the worst-case variance for a fixed portfolio is given by a

collection of linear and restricted hyperbolic constraintsthe critical step in reformulatingrobust portfolio selection problems as SOCPs.

Lemma 3. Fix a portfolio and let Sf1 be given by (69). Then the following resultshold.

(i) If the bound

1, the worst-case variance is unbounded; i.e.,

maxVSv FSf1

TVTFV=

(ii) If < 1, the worst-case variance maxVSv FSf1 TVTFV if and only if

T 1 V0 F0 G, where is defined in Definition 1.Proof. Define = F1/20 F1/20 . Then

Sf1 =

F F1 = F1/20 I + F1/20 0 = T


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Fix V Sv and define x = VT. Then

supFSf1

xTFx = sup

F1/20 x

TI + 1F1/20 x = T I + 0

First consider the case 1. Choose = I where 0 1. Then = andI + = 1 I 0. Thus,

supFSf1

xTFx

sup 01

1

1

xTF0x=

Next, suppose < 1. Since < 1 implies that I + 0 for all ,

Sf1 =

F F1 = F1/20 I + F1/20 = T

Therefore

supF

Sf

1

xTFx = supF1/20 xI + 1F1/20 x = T

=m

i=1

qTi F

1/20 x

21 +i

where qi i = 1 m is the set of eigenvectors of . Since i , it follows that

supFSf1

xTFx 11

mi=1

qTi F

1/20 x

2= 1

1 xTF0x

Moreover, the supremum is achieved by = I. Thus,

maxVSv FSf1

TVTFV= 11 maxVSv

TVTF0V

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

1

1 maxVSv TVTF0V

if and only if T 1 V0 F0 G. From Lemma 3 it follows that if < 1, then all robust portfolio problems introduced in

2 can be reformulated as SOCPs.The next step is to justify the uncertainty structure Sf1 and show how it can be param-

eterized from market data. In 5 we show that the natural uncertainty structures for Vand their parameterizations are implied by the confidence regions associate with the statis-tical procedures used to compute the point estimates. Here, we show that the uncertaintystructure Sf1 and its parameterization, i.e., the choice of the nominal matrix F0 and the

bound , arise naturally from maximum-likelihood estimation typically used to computethe point estimate of F.

Suppose the covariance matrix of the factor returns is estimated from the data B =f1 f2 fp over p market periods. Then the maximum likelihood estimate Fml of thefactor covariance matrix F is given by (64).

Suppose one is in a Bayesian setup and assumes a noninformative conjugate prior distri-bution for the true covariance F. Then the posterior distribution for F conditioned on thedata B is given by

FB W1p1

p 1Fml

(70)


22/39


where W1q A denotes an inverse Wishart distribution with q degrees of freedom andparameter A (Anderson 1984, Schafer 1997); i.e., the density f FB is given by

f FB

= c

Fml

pm2/2

F1

p1/2ep1/2 TrF

1Fml F

0

0 otherwise(71)

where A = detA and c is a normalization constant. This density can be rewritten asfollows:

f FB =

cF1ml m1/2F1/2ml F1F1/2ml p1/2ep1/2 TrF1/2ml F

1F1/2ml F 00 otherwise

(72)

From (72) it follows that the natural choice for the nominal covariance matrix F0 in the

definition of Sf1 is F0 = Fml.Let Rm be the vector of eigenvalues ofF1/20 F1F1/20 . Then (72) implies that the density

of is given by

f B =

c

mi=1

p1/2i e

p1/2i 0

0 otherwise

(73)

where c is a normalization constant, i.e., the eigenvalues i, i = 1 m, are IIDp + 1/2 p 1/2 distributed random variables.

Let = F1 F10 be the deviation of the F1 from the nominal inverse F10 and let= F1/20 F1/20 = F1/20 F1F1/20 I. Then if and only if 1 i 1 + ; i.e.,

P = P1 p 1 +m = p 1 + p 1 m(74)where p p +1/2 p 1/2 and p is the corresponding CDF.

If < 1, F1

= F1

0 + 0 for all , and therefore (74) implies thatPF Sf1 = p 1 + p 1 m(75)

Thus, in order to satisfy a desired confidence level m, the parameter must be set equalto the unique solution of

p 1 + p 1 = (76)

Since the problem is unbounded for 1, (76) implicitly states that the confidence levelm that can be supported by p return samples is at most maxp = p 2m. Figure 1plots max as a function of the data length p for m = 40 (the plot begins at p = m+1 sinceat least m + 1 observations are needed to estimate the covariance matrix). From the plot,one can observe that for p 50, maxp 0995. Thus, the restriction

m

maxp isnot 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 prior W1k k 1F,k 1. Then, the posterior distribution FB W1k+pp 1Fml + k 1F. In this case, thenominal covariance matrix F0 = 1/k + pp 1Fml + k 1F, and is given by

k+p 1 + k+p 1 =

where k+p denotes the CDF of a k +p +2/2 k +p/2 random variable.


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40 60 80 100 120 140 160 180 200

0.985

0.99

0.995

1

p

max

Figure 1. max vs p (m = 40).

From the results in this section, it follows that, as far as the portfolio selection problemsare concerned, all that the uncertainty in the covariance matrix does is shrink the MLEF10 to 1F10 , where the shrinkage factor 1 is a function of the desired confidencelevel . Thus, this procedure has the flavor of robust statistics (see Huber 1981).

6.2. Uncertainty structure for the covariance. Another possible uncertainty structurefor the factor covariance matrix is given by

Sf = F F = F0 + 0=TN

1/2N1/2 (77)where F0 0. The norm A in (77) is either A = maxi iA, or A =

i

2i A,

where iA are the eigenvalues of A.For (77) to be a viable uncertainty structure, the corresponding robust problem should be

efficiently solvable, and the structure ought to be easily parameterizable from raw marketdata. The first requirement is established by the following lemma.

Lemma 4. Fix a portfolio Rn+ and let Sf be given by Equation (77). ThenmaxVSv FSf

TVTFV if and only if T V0 F0 + N G, where is defined in Definition 1.

Proof. Define = N1/2N1/2. Then

Sf = F F = F0 +N1/2N1/2 0 = T

Fix V Sv and define x = VT. Then

maxFSf

xTFx = max

xTF0x + N1/2xTN1/2x = T F0 + N1/2N1/2 0

xTF0x + N1/2xTN1/2x = T

(78)

xTF0x +N1/2xTN1/2x(79)

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


24/39


25/39


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

=

F Fml + 0=T

F1/2ml F

1/2ml

1

(85)

contains an m-confidence set for the (true) covariance matrix F; i.e., a natural parame-terization of the uncertainty structure in (77) is F0 = N = Fml and = /1 . Thus,the uncertainty structure (77) is completely parameterized by considering the confidenceregions around the MLE.

Note that for N = F0 and = /1 , Lemma 4 implies that

maxVSv FSf

TVTFV

if and only if

T V0 1 +/1 F0 G= V0 1/1 F0 GAs noted above, the solution of (83) is equal to the solution of (76), and from Defini-tion 1 it follows that

T V0 1/1 F0 G T 1 V0 F0 G

Thus, maxVSv FSfTVTFV if and only if T 1 V0 F0 G.

Since this is precisely the condition in part (ii) of Lemma 3, it follows that, although the twouncertainty sets (69) and (77) are not the same, they imply the same worst-case varianceconstraint. However, unlike the parameterization of Sf1 , the parameterization of Sf cannotbe biased to reflect prior knowledge about F.

7. Computational results. In this section we report the results of our preliminary com-putational experiments with the robust portfolio selection framework proposed in this paper.The objective of these computational experiments was to contrast the performance of theclassical portfolio selection strategies with that of the robust portfolio selection strategies.

We conducted two types of computational tests. The first set of tests compared performanceon simulated data, and the second set compared sample path performance on real marketdata. In these experiments our intent was to focus on the benefit accrued from robustness;therefore, we wanted to avoid any user-defined variables, such as the minimum return in the robust minimum variance problem or the probability threshold in the robust VaRproblem. Thus, in our tests both the classical and robust portfolios were selected by solvingthe corresponding maximum Sharpe ratio problem. We expect the qualitative aspects of thecomputational results will carry over to the other portfolio selection problems. All the com-putations were performed using SeDuMi V1.03 (Strm 1999) within Matlab6.1R12 on a

Dell Precision 340 workstation running RedHat Linux 7.1. The details of the experimentalprocedure and the results are given below.

7.1. Computational results for simulated data. For our computational tests on sim-ulated data, we fixed the number of assets n = 500 and the number of factors m = 40. Asymmetric positive definite factor covariance matrix F was randomly generated, except thatwe ensured that the condition number of F, i.e., maxF/minF, was at most 20 by addinga suitable multiple of the identity. This factor covariance matrix was assumed to be knownand fixed. The nominal factor loading matrix V was also randomly generated. The covari-

ance matrix D of the residual returns was assumed to be certain (i.e., D = D = D) andset to D = 01diagVTFV; i.e., it was assumed that the linear model explains 90% of theasset variance.


26/39


The risk-free rate rf was set to 3, and the nominal asset returns i were chosen indepen-dently according to a uniform distribution on rf 2 rf +2. Next, we generated a sequenceof asset and factor return vectors according to the market model (1) in 2 for an investmentperiod of length p = 90 and used equations (62) and (63) in 5 to set the parameters V0,0, G, and . (Note that we did not estimate D from the data.) Next, the robust and theclassical portfolios r and m were computed by solving the robust maximum Sharpe ratioproblem (8) and its classical counterpart, respectively. Although the precise numbers usedin these simulation experiments were arbitrary, we expect that the qualitative aspects of theresults are not dependent on the precise values.

In the first set of simulation experiments we compared the performance of the robust andclassical portfolios as the confidence threshold (see 5 for details) was increased from001 to 095. The experimental results for three independent runs are shown in Figures 24.In each of the three figures, the top plot is the ratio of the mean Sharpe ratio of the robustportfolio to that of the classical portfolio. The mean Sharpe ratio of any portfolio isgiven by

0 rf1T

TVT0 FV0 The classical portfolio m maximizes the mean Sharpe ratio. The bottom plot in Figures24 is the ratio of the worst-case Sharpe ratio of the robust portfolio to that of the classicalportfolio. The worst-case Sharpe ratio of a portfolio is given by

minVSvSm DSd

rf1TTVFV+ D

where Sd, Sv and Sm are given by (2)(4). The robust portfolio r maximizes the worst-case Sharpe ratio. The performance on the three runs is almost identicalthe ratio of themean Sharpe ratios drops from approximately 1 to approximately 08 as increases from001 to 095 while the ratio of the worst-case Sharpe ratios increases from approximately 1to approximately 2. Thus, at a modest 20% reduction in the mean performance, the robust

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

1.8

Robust/Classical Mean Sharpe Ratios

Robust/Classical Worst Case Sharpe Ratios

Figure 2. Performance as a function of (Run 1).


27/39


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

2




framework delivers an impressive 200% increase in the worst-case performance. Notice thatthe drop in mean performance in run 3 (see Figure 4) is close to 25%, but the correspondingimprovement in the worst-case performance is close to 260%.

The second set of simulation experiments compared the performance of robust and clas-sical portfolios as a function of the upper bound on D. For this set of experiments, we set = 095 and D = 2 diagVT0 FV0, where 2 increased from 001 to 1. Since the perfor-mance of both the robust and classical portfolio was very sensitive to the sample path

particularly for large values of 2the results shown in Figures 57 were averaged over

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

2.6





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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

7



2


10 runs for every value of 2. As before, the top plot is the ratio of the mean performances

of the robust and classical portfolios and the bottom plot is the ratio of the worst-case per-

formances. (The three plots truncate at different values of 2 because we only compared

performances for values of 2 for which the worst-case Sharpe ratio of the classical port-

folio is nonnegative.) Again, the essential features of the performance were independent of

the particular runthe mean performance of the robust portfolio did not degrade signifi-

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

2.5



2



29/39


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

6


Robust/Classical of Worst Case Sharpe Ratios

2


cantly with the increase in noise variance, if it did so at all, and the worst-case performanceof the robust portfolio was significantly superior to that of the classical portfolio as the data

became noisy. This is not unexpected since the robust portfolios were designed to combatnoisy data.

Figure 8 compares the CPU time needed to solve the robust and classical maximumSharpe ratio problem as a function of the number of assets. For this comparison the number

of factors m

= 01n

, the risk-free rate rf was set to 3, and F were generated as described

above, the noise covariance D was set to 01diagVTFV, the confidence level was

0 200 400 600 800 1000 1200 1400 1600 1800 20000

100

200

300

400

500

600

Robust

Classical

CPU

Time(seconds)

Number of Assets n (m = 0.1n)

Figure 8. Complexity of robust and classical strategies.


30/39


set to 095, and the parameters 0 V0 G were computed from randomly generated

factor and asset return vectors for an investment period p = 2m. These experiments wereconducted using SeDuMi V1.03 (Strm 1999) within Matlab6.1R12 on a Dell Precision

340 machine. The running times were averaged over 100 randomly generated instances for

each problem size n. It is clear from Figure 8 that the computation time for the classicaland robust strategies is almost identical and grows quadratically with the problem size. We

should note, however, that algorithms based on the active set method can 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 approach on real market data. The

universe of assets that were chosen for investment were those currently ranked at the top

of each of 10 industry categories by Dow Jones in August 2000. In total there were n = 43assets 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 the covariance matrix of the asset returns; i.e., the total number of factors used was

m=

10. This choice 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 the covariance structure of the returns is a sophisticated industry, and our

choice of factors is by no means claimed to be the most appropriate.) The data sequence

Table 1. Assets

Aerospace Industry Telecommunication

AIR AAR corporation T AT&T

BA Boeing Corp. LU Lucent Technologies

LMT Lockheed Martin NOK Nokia

UTX United Technologies MOT Motorola

Semiconductor Computer Software

AMD Applied Materials ARBA Ariba

INTC Intel Corp. CMRC Commerce One Inc.

HIT Hitachi MSFT Microsoft

TXN Texas Instruments ORCL Oracle

Computer Hardware Internet and Online

DELL Dell Computer Corp. AKAM Akamai

PALM Palm Inc. AOL AOL Corp.

HWP Hewlett Packard CSCO Cisco Systems

IBM IBM Corp. NT Northern Telecom

SUNW Sun Microsystems PSIX PsiNet Inc.

Biotech and Pharmaceutical Utilities

BMV Bristol-Myers-Squibb ENE Enron Corporation

CRA Applera Corp.-Celera DUK Duke Energy CompanyCHIR Chiron Corp. EXC Exelon Corp.

LLV Eli Lilly and Co. PNW Pinnacle West

MRK Merck and Co.

Chemicals Industrial Goods

AVY Avery Denison Corp. FMC FMC Corp.

DD Du Pont GE General Electric

DOW Dow Chemical HON Honeywell

EMN Eastman Chemical Co. IR Ingersoll Rand


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Table 2. Base factors

DJA Dow Jones Composite Average

NDX Nasdaq 100

SPC Standard and Poors 500 Index (S&P500)

RUT Russell 2000TYX 30-year bond

consisted of daily asset returns from January 2, 1997 through December 29, 2000. Giventhat assets were selected in August 2000, the data sequence suffers from the survivorshipbias; i.e., we knew a priori that the companies we were considering in our universe were themajor stocks in their industry category in August 2000. It is expected that this bias wouldaffect both strategies in a similar manner; therefore, relative results are still meaningful.

A complete description of the experimental procedure is as follows. The entire datasequence was divided into investment periods of length p = 90 days. For each investmentperiod t, we first estimated the covariance matrix R of the asset returns based on themarket data of the previous p trading days and extracted the eigenvectors corresponding to

the 5 largest eigenvalues (if an asset did not exist during the entire period it was removedfrom consideration). These eigenvectors together with the base market indices defined thefactors for a particular period, and their returns were used to estimate the factor covariancematrix F. Next, the equations (62) and (63) in 5 were used to set V0, 0, G, and .The bound di on the variance of the residual return was set to di = s2i , where s2i is givenby (51), and the risk-free rate rf was set to zero. Once all the parameters were set, therobust portfolio tr (resp. classical portfolio

tm) was computed by solving the robust (resp.

classical) maximum Sharpe ratio problem. The portfolio tr and tm were held constant for

the period t and then rebalanced to the portfolio t+1r and t+1m in period t + 1.

Let wtr (resp. wtm) denote the wealth at the end of period t of an investor who has an

initial wealth w0 and employs the robust (resp. classical) strategy. Then

wt+1r= tp


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0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

Relative Performance of Robust vs Classical Portfolios

Period

RelativePercentagePerformance

Figure 9. Evolution of relative wealth for = 095.

the performance of the robust and classical strategies is quite close. For intermediate values,the dip at t = 7 is reduced, but unimpressive performance over the other periods dragsthe overall relative performance of the robust strategy down. The performance improves as

1 (for = 099 the robust strategy generates a final wealth that is 50% larger than thatgenerated by the classical strategy). Figure 12 plots the final wealth ratio as a function of (results were obtained for = 01, 02, , 09, 095, 099). It is clear that the performanceis not monotonic in .

An important aspect of any investment strategy is the cost of implementing it. Since

we are interested in comparing the costs of implementing the robust strategy with that of

0 1 2 3 4 5 6 7 8 9 1020

10

0

10

20

30

40


Period

RelativePe

rcentagePerformance

Figure 10. Evolution of relative wealth for = 07.


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0 1 2 3 4 5 6 7 8 9 1040

20

0

20

40

60

80


Period

RelativePercentagePerformance

= 0.2

= 0.4

= 0.6

= 0.8

= 0.95

Figure 11. Evolution of relative wealth as a function of .

implementing the classical strategy we will quantify the transaction costs byt t1

1.

In Figure 13 we plot the ratio of the costs, i.e.tr t1r 1tm t1m 1, for a

confidence threshold of = 095. The average cost is 09623; i.e., the transaction costsincurred by the robust strategy were approximately 4% less that that incurred by the classical

strategy. Figure 14 plots the same quantity for = 07 and now the average cost is 10057;i.e., as the robust strategy becomes less conservative it pays more in transaction costs.

Figure 15 shows the average cost as a function of . The average cost remains almost

constant at a value slightly greater than 1 until > 09, and then it decreases monotonically.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120

10

0

10

20

30

40

50

60


FinalRe

lativePercentageGain

Figure 12. Final relative wealth as a function of .


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2 3 4 5 6 7 8 9 100.4

0.6

0.8

1

1.2

1.4

Relative Cost of Robust vs Classical Portfolios

Period

Average Cost = 0.9623

RelativeCost

Figure 13. Relative cost per period for = 095.

7.3. Summary of computation results. The summary of our simulation experiments

and the experiments with real market data is as follows:

(a) The mean performance of the robust portfolios does not significantly degrade as the

confidence level is increased. Even at = 095, the relative loss of the robust portfoliosis only about 20%. 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

portfolios.

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

Relative Cost of Robust vs Classical Portfolios

Period

Average Cost = 1.0057

RelativeCost

Figure 14. Relative cost per period for = 07.


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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

Average Relative Cost of Robust vs Classical Portfolios

RelativeCost

Figure 15. Average cost vs .

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

These computational experiments, particularly those with the real data sequence, are byno means comprehensive. For one, the problem sizem = 40 n = 500 for the simulationexperiments and m

=10 n

=43 for market data experimentswas small. For the robust

strategy to be acceptable one would have to ensure that the complexity of the robust opti-mization problems is not significantly higher than that of the classical problems. A simplecomparison of run times suggests that this is indeed the case.

The experimental results on the real data suggests (see Figure 11) that if one wants therobust strategy to consistently outperform the classical strategy one would want to choose 1. However, such a choice of makes the robust strategy extremely conservative,which would hamper its performance if the noise in the model was low. Since the noiseis not known a priori, the correct choice of remains a vexing problem. Our preliminaryexperiments suggest that the factor model we used was probably noisy. More extensiveexperiments have to be conducted before one can assert that this is almost always the case(e.g., using better factors may significantly change the results), and therefore, setting 1 is a good choice. We would expect that in practice one would have to adjust dynamically by comparing the robust strategy with the classical benchmark. Also, we havenot tested robust portfolios based on our model in 6 which incorporates uncertainty in thefactor covariance matrix. These and other experimental studies are planned.

Appendix: Probabilistic interpretation. In this section, we interpret the choice of theuncertainty sets Sm, Sv and Sf in terms of the implied probabilistic guarantees on the risk-return performance.

First consider the case where the factor covariance F is fixed. Define Sm and Svfor a confidence level using (56) and (57). Let be the optimal solution of the robustmaximum Sharpe ratio problem (37) and let s be the corresponding Sharpe ratio.


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Since T 1 for all Sm, it follows that T 1 for V in the jointuncertainty set S defined in (55). Similarly, we have that

TVTFV 1

s

V

Sv

TVTFV 1s

V S

Therefore, we have that the Sharpe ratio of the portfolio :

TTVTFV

s V S

i.e., the Sharpe ratio is at least as large as s with confidence n. Similar probabilisticguarantees can be provided for the performance of the optimal solutions of robust minimumvariance, maximum return and VaR problems. Computing probabilistic guarantees on theperformance of the robust portfolio when the uncertainty sets are defined by (62)(63) isleft to the reader.

Thus, in contrast to the classical Markowitz portfolio selection, the robust formulationallows one to impose confidence thresholds on the value of the optimization problem. Settinga low value for results in a high value for the corresponding robust Sharpe ratio s,but the confidence that the corresponding optimal portfolio would indeed achieve theSharpe ratio is low. On the other hand, a high value for would imply a lower Sharpe ratios, but it would ensure that the corresponding optimal portfolio will achieve the Sharperatio with a higher confidence. Thus, the parameter can be viewed as a surrogate for riskaversion.

Next, consider the case where the factor covariance matrix F is uncertain. Suppose oneassumes the following Bayesian setup. The market parameters V and F are assumed tobe a priori independent and distributed according to a noninformative conjugate prior (seeGreene 1990, Anderson 1984, for appropriate choices for the conjugate prior). Suppose also

that the residual return is independent of, V, and F. Then the market model (1) impliesthat the conditional distribution f S R 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 V F = f B Ff S V B(87)

The a posteriori distribution f V F S B is, therefore, given by

f V F S B = f B Ff Ff S V Bf Vf S R

(88)

= c1f B Ff F c2f S V Bf Vwhere c1 c2 are suitable normalizing constants. From (88) it follows that F and V are

a posteriori independent. Therefore, the uncertainty sets for V F can be represented asa Cartesian product S Sf1 .Suppose the confidence threshold is set to p 2m, where p is the CDF of

a p +1/2 p 1/2 random variable (for a discussion of the upper bound on theachievable confidence, see 6). Set F0 = Fml and set by solving (76). Then F Sf1 with confidence m. As before, let S be the n confidence set for V. Let be theoptimal solution of the robust Sharpe ratio problem with uncertain covariance and let sbe the corresponding value. Then the confidence that V F S Sf1 is m+n,i.e., with a confidence level m+n, the realized Sharpe ratio of is at least as largeas s.


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

to probabilistic guarantees.

Acknowledgments. The first authors research was partially supported by DOE GrantGE-FG01-92ER-25126 and NSF Grants DMS-94-14438, CDA-97-26385, and DMS-01-04282. The second authors research was partially supported by NSF Grants CCR-00-09972and DMS-01-04282.

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