Download - Robust Portfolio Selection Problems
-
8/4/2019 Robust Portfolio Selection Problems
1/39
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.
1
0364-765X/03/2801/0001/$05.001526-5471 electronic ISSN, 2003, INFORMS
-
8/4/2019 Robust Portfolio Selection Problems
2/39
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
-
8/4/2019 Robust Portfolio Selection Problems
3/39
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 .
-
8/4/2019 Robust Portfolio Selection Problems
4/39
4 D. GOLDFARB AND G. IYENGAR
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
-
8/4/2019 Robust Portfolio Selection Problems
5/39
ROBUST PORTFOLIO SELECTION PROBLEMS 5
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
-
8/4/2019 Robust Portfolio Selection Problems
6/39
6 D. GOLDFARB AND G. IYENGAR
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)
-
8/4/2019 Robust Portfolio Selection Problems
7/39
ROBUST PORTFOLIO SELECTION PROBLEMS 7
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).
-
8/4/2019 Robust Portfolio Selection Problems
8/39
8 D. GOLDFARB AND G. IYENGAR
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:
-
8/4/2019 Robust Portfolio Selection Problems
9/39
ROBUST PORTFOLIO SELECTION PROBLEMS 9
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)
-
8/4/2019 Robust Portfolio Selection Problems
10/39
10 D. GOLDFARB AND G. IYENGAR
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.
-
8/4/2019 Robust Portfolio Selection Problems
11/39
ROBUST PORTFOLIO SELECTION PROBLEMS 11
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)
-
8/4/2019 Robust Portfolio Selection Problems
12/39
12 D. GOLDFARB AND G. IYENGAR
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.
-
8/4/2019 Robust Portfolio Selection Problems
13/39
ROBUST PORTFOLIO SELECTION PROBLEMS 13
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)
-
8/4/2019 Robust Portfolio Selection Problems
14/39
14 D. GOLDFARB AND G. IYENGAR
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)
-
8/4/2019 Robust Portfolio Selection Problems
15/39
ROBUST PORTFOLIO SELECTION PROBLEMS 15
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.
-
8/4/2019 Robust Portfolio Selection Problems
16/39
16 D. GOLDFARB AND G. IYENGAR
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
-
8/4/2019 Robust Portfolio Selection Problems
17/39
ROBUST PORTFOLIO SELECTION PROBLEMS 17
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
-
8/4/2019 Robust Portfolio Selection Problems
18/39
-
8/4/2019 Robust Portfolio Selection Problems
19/39
ROBUST PORTFOLIO SELECTION PROBLEMS 19
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).
-
8/4/2019 Robust Portfolio Selection Problems
20/39
20 D. GOLDFARB AND G. IYENGAR
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
-
8/4/2019 Robust Portfolio Selection Problems
21/39
ROBUST PORTFOLIO SELECTION PROBLEMS 21
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)
-
8/4/2019 Robust Portfolio Selection Problems
22/39
22 D. GOLDFARB AND G. IYENGAR
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.
-
8/4/2019 Robust Portfolio Selection Problems
23/39
ROBUST PORTFOLIO SELECTION PROBLEMS 23
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.
-
8/4/2019 Robust Portfolio Selection Problems
24/39
-
8/4/2019 Robust Portfolio Selection Problems
25/39
ROBUST PORTFOLIO SELECTION PROBLEMS 25
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.
-
8/4/2019 Robust Portfolio Selection Problems
26/39
26 D. GOLDFARB AND G. IYENGAR
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).
-
8/4/2019 Robust Portfolio Selection Problems
27/39
ROBUST PORTFOLIO SELECTION PROBLEMS 27
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
Robust/Classical Mean Sharpe Ratios
Robust/Classical Worst Case Sharpe Ratios
Figure 3. Performance as a function of (Run 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
Robust/Classical Mean Sharpe Ratios
Robust/Classical Worst Case Sharpe Ratios
Figure 4. Performance as a function of (Run 3).
-
8/4/2019 Robust Portfolio Selection Problems
28/39
28 D. GOLDFARB AND G. IYENGAR
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
Robust/Classical Mean Sharpe Ratios
Robust/Classical Worst Case Sharpe Ratios
2
Figure 5. Performance as a function of (Run 1).
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
Robust/Classical Mean Sharpe Ratios
Robust/Classical Worst Case Sharpe Ratios
2
Figure 6. Performance as a function of (Run 2).
-
8/4/2019 Robust Portfolio Selection Problems
29/39
ROBUST PORTFOLIO SELECTION PROBLEMS 29
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 Mean Sharpe Ratios
Robust/Classical of Worst Case Sharpe Ratios
2
Figure 7. Performance as a function of (Run 3).
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.
-
8/4/2019 Robust Portfolio Selection Problems
30/39
30 D. GOLDFARB AND G. IYENGAR
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
-
8/4/2019 Robust Portfolio Selection Problems
31/39
ROBUST PORTFOLIO SELECTION PROBLEMS 31
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
-
8/4/2019 Robust Portfolio Selection Problems
32/39
32 D. GOLDFARB AND G. IYENGAR
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
Relative Performance of Robust vs Classical Portfolios
Period
RelativePe
rcentagePerformance
Figure 10. Evolution of relative wealth for = 07.
-
8/4/2019 Robust Portfolio Selection Problems
33/39
ROBUST PORTFOLIO SELECTION PROBLEMS 33
0 1 2 3 4 5 6 7 8 9 1040
20
0
20
40
60
80
Relative Performance of Robust vs Classical Portfolios
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
Relative Performance of Robust vs Classical Portfolios
FinalRe
lativePercentageGain
Figure 12. Final relative wealth as a function of .
-
8/4/2019 Robust Portfolio Selection Problems
34/39
34 D. GOLDFARB AND G. IYENGAR
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.
-
8/4/2019 Robust Portfolio Selection Problems
35/39
ROBUST PORTFOLIO SELECTION PROBLEMS 35
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.
-
8/4/2019 Robust Portfolio Selection Problems
36/39
36 D. GOLDFARB AND G. IYENGAR
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.
-
8/4/2019 Robust Portfolio Selection Problems
37/39
ROBUST PORTFOLIO SELECTION PROBLEMS 37
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.
References
Anderson, T. W. 1984. An Introduction to Multivariate Statistical Analysis, 2nd ed. John Wiley & Sons, New York.
Andersson, F., H. Mausser, D. Rosen, S. Uryasev. 2001. Credit risk optimization with conditional value-at-risk
criterion. Math. Programming, Ser. B 89 273291.
Ben-Tal, A., T. Margalit, A. Nemirovski. 2000. Robust modeling of multi-stage portfolio problems. H. Frenk,
K. Roos, T. Terlsky, S. Zhang, eds. High Performance Optimization. Kluwer Academic Publisher,
Dordrecht, The Netherlands, 303328.
, A. Nemirovski. 1998. Robust convex optimization. Math. Oper. Res. 23(4) 769805.
, . 1999. Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1) 113., . 2001. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Appli-
cations, MPS/SIAM Series on Optimization. SIAM, Philadelphia, PA.
Bertsimas, D., J. Sethuraman. 2000. Moment problems and semidefinite optimization. H. Wolkowicz, R. Saigal,
L. Vandenberghe, eds. Handbook of Semidefinite Programming. Kluwer Academic Publishers, Dordrecht,
The Netherlands.
Black, F., R. Litterman. 1990. Asset allocation: Combining investor views with market equilibrium: Technical
report. Fixed Income Research, Goldman, Sachs & Co., New York.
Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan. 1994. Linear Matrix Inequalities in System and Control Theory .
SIAM, Philadelphia, PA.
Broadie, M. 1993. Computing efficient frontiers using estimated parameters. Ann. Oper. Res. 45 2158.
Chopra, V. K. 1993. Improving optimization. J. Investing 8(Fall) 5159.
, C. R. Hensel, A. L. Turner. 1993. Massaging mean-variance inputs: Returns from alternative investment
strategies in the 1980s. Management. Sci. 39(7) 845855.
, W. T. Ziemba. 1993. The effect of errors in means, variances and covariances on optimal portfolio choice.
J. Portfolio Management. (Winter) 611.Efron, B., R. Tibshirani. 1993. An Introduction to Bootstrap, No. 57, Monographs on Statistics and Applied
Probability. Chapman & Hall, New York.
El Ghaoui, L., H. Lebret. 1997. Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix
Anal. Appl. 18(4) 10351064.
, N. Niculescu eds. 1999. Recent Advances on LMI Methods in Control . SIAM, Philadelphia, PA.
, M. Oks, F. Oustry. 2002. Worst-case value-at-risk and robust portfolio optimization: A conic programming
approach. Oper. Res. Forthcoming.
, F. Oustry, H. Lebret. 1998. Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9(1)
3352.
Frost, P. A., J. E. Savarino. 1986. An empirical Bayes approach to efficient portfolio selection. J. Fin. Quan. Anal.
21 293305.
, . (1988). For better performance: Constrain portfolio weights. J. Portfolio Management. 15 2934.
W. H. Greene. 1990. Econometric Analysis. Macmillan Pub. Co., New York.
Halldrsson, B. V., R. H. Ttnc. 2000. An interior-point method for a class of saddle point problems: Technical
report. Carnegie Mellon University, Pittsburgh, PA.
Hansen, L. P., T. J. Sargent. 2001. Robust control and model uncertainty. Amer. Econom. Rev. 91 6066.
Huber, P. J. 1981. Robust Statistics. John Wiley & Sons, New York.
Kast, R., E. Luciano, L. Peccati. 1998. VaR and Optimization. Workshop on Preferences and Decisions.
Klein, R. W., V. S. Bawa. 1976. The effect of estimation risk on optimal portfolio choice. J. Financial. Econom.
3 215231.
Ledoit, O. 1996. A well-conditioned estimator for large-dimensional covariance matrices: Technical report. Ander-
son School, UCLA, Los Angeles, CA.
Lintner, J. 1965. Valuation of risky assets and the selection of risky investments in stock portfolios and capital
budgets. Rev. Econom. Statist. 47 1337.
Lobo, M. S., L. Vandenberghe, S. Boyd, H. Lebret. 1998. Applications of second-order cone programming. Linear
Algebra Appl. 284(13) 193228.
-
8/4/2019 Robust Portfolio Selection Problems
38/39
38 D. GOLDFARB AND G. IYENGAR
Markowitz, H. M. 1952. Portfolio selection. J. Finance 7 7791.
. 1959. Portfolio Selection. Wiley, New York.
Mausser, H., D. Rosen. 1999. Beyond VaR: From measuring risk to managing risk. ALGO Res. Quart. 1(2) 520.
Michaud, R. O. 1998. Efficient Asset Management: A Practical Guide to Stock Portfolio Management and Asset
Allocation, Financial Management Association, Survey and Synthesis Series. HBS Press, Boston, MA.
Mossin, J. 1966. Equilibrium in capital asset markets. Econometrica 34(4) 768783.Nesterov, Y., A. Nemirovski. 1993. Interior-Point Polynomial Algorithms in Convex Programming. SIAM,
Philadelphia, PA.
Schafer, J. L. 1997. Analysis of Incomplete Multivariate Data. Chapman and Hall, New York.
Sharpe, W. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. J. Finance 19(3)
425442.
Strm, J. 1999. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods
and Software 1112 625653.
Zhou, K., J. Doyle, K. Glover. 1996. Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ.
Ziemba, W. T., J. M. Mulvey, eds. 1998. Worldwide Asset and Liability Modeling. Cambridge University Press,
Cambridge, U.K.
D. Goldfarb: IEOR Department, Columbia University, New York, New York 10027; e-mail: gold@ieor.
columbia.edu
G. Iyengar: IEOR Department, Columbia University, New York, New York 10027; e-mail: [email protected].
edu
-
8/4/2019 Robust Portfolio Selection Problems
39/39