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Robust Bayesian AllocationAttilio Meucci1
this version: May 12, 2011latest revision of article and code at http://symmys.com/node/102
Abstract
Using the Bayesian posterior distribution of the market parameters we defineself-adjusting uncertainty regions for the robust mean-variance problem. Undera normal-inverse-Wishart conjugate assumption for the market, the ensuingrobust Bayesian mean-variance optimal portfolios are shrunk by the aversion toestimation risk toward the global minimum variance portfolio.After discussing the theory, we test robust Bayesian allocations in a simu-
lation study and in an application to the management of sectors of the S&P500.Fully commented code is available at http://symmys.com/node/102
JEL Classification: C1, G11
Keywords: estimation risk, Bayesian estimation, MCMC, robust optimiza-tion, location-dispersion ellipsoid, classical equivalent, shrinkage, global mini-mum variance portfolio, equally-weighted portfolio, quantitative portfolio man-agement, asset allocation
1The author is grateful to Ralf Werner and Jordi Serra
1
1 IntroductionThe classical approach to asset allocation is a two-step process: first the marketdistribution is estimated, then an optimization is performed, as if the estimateddistribution were the true market distribution. Since this is not the case, theclassical "optimal" allocation is not truly optimal. More importantly, sincethe optimization process is extremely sensitive to the input parameters, thesub-optimality due to estimation risk can be dramatic, see Jobson and Korkie(1980), Best and Grauer (1991), Chopra and Ziemba (1993).Bayesian theory provides a way to limit the sensitivity of the final allocation
to the input parameters by shrinking the estimate of the market parameterstoward the investor’s prior, see Bawa, Brown, and Klein (1979), Jorion (1986),Pastor and Stambaugh (2002).Similarly, the approach of Black and Litterman (1990) uses Bayes’ rule to
shrink the general equilibrium distribution of the market toward the investor’sviews.The theory of robust optimization provides a different approach to dealing
with estimation risk: the investor chooses the best allocation in the worst marketwithin a given uncertainty range, see Goldfarb and Iyengar (2003), Halldorssonand Tutuncu (2003), Ceria and Stubbs (2004).Robust allocations are guaranteed to perform adequately for all the markets
within the given uncertainty range. Nevertheless, the choice of this range isquite arbitrary. Furthermore, the investor’s prior knowledge, a key ingredientin any allocation decision, is not taken in consideration.Using the Bayesian approach to estimation we can naturally identify a suit-
able uncertainty range for the market parameters, namely the location-dispersionellipsoid of their posterior distribution. Robust Bayesian allocations are the so-lutions to a robust optimization problem that uses as uncertainty range theBayesian location-dispersion ellipsoid. Similarly to robust allocations, these al-locations account for estimation risk over a whole range of market parameters.Similarly to Bayesian decisions, these allocations include the investor’s priorknowledge in the optimization process within a sound and self-adjusting statis-tical framework.Robust Bayesian allocations are discussed in De Santis and Foresi (2002),
where the Bayesian setting is provided by the Black-Litterman posterior dis-tribution. Here we consider robust Bayesian decisions that also account forthe estimation error in the covariances and that, unlike in the Black-Littermanframework, explicitly process the information from the market, namely the ob-served time series of the past returns. As it turns out, the multi-parameter,non-conically constrained mean-variance optimization simplifies to a parsimo-nious Bayesian efficient frontier that resembles the classical frontier, except thatthe classical parameterization in terms of the exposure to market risk becomesin this context a parameterization in terms of the exposure to both market riskand estimation risk.In Section 2 we introduce the general robust Bayesian mean-variance ap-
proach to asset allocation. In Section 3 we compute explicitly the Bayesian
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location-dispersion ellipsoids for the market parameters under the standardnormal-inverse-Wishart conjugate assumption for the market. In Section 4 wesolve the ensuing mean-variance problem and we compute the robust Bayesianoptimal portfolios. In this section we also perform two empirical tests, a simulation-based test and a back-test on portfolios of sectors of the S&P 500. Section 6concludes.An appendix details all the technicalities. Fully commented code is available
at http://symmys.com/node/102
2 General robust Bayesianmean-variance frame-work
The classical mean-variance problem reads:
w(i) = argmaxw
w0µ (1)
subject to½w ∈ Cw0Σw ≤ v(i).
In this expression w is the N -dimensional vector of relative portfolio weights; Cis a set of investment constraints; the set
©v(1), . . . , v(I)
ªis a significative grid
of target variances of the return on the portfolio, where the return at time t fora horizon τ of an asset that at the generic time t trades at the price Pt is definedas Rt,τ ≡ Pt/Pt−τ − 1; and µ and Σ represent respectively the expected valuesand the covariances of the returns on the N securities in the market relative tothe investment horizon:
µ ≡ E {RT+τ,τ} , Σ ≡ Cov {RT+τ,τ} . (2)
The robust version of the mean-variance problem (1) reads:
w(i) = argmaxw
(minµ∈Θµ
{w0µ})
(3)
subject to
(w ∈ CmaxΣ∈ΘΣ
{w0Σw} ≤ v(i),
where bΘµ and bΘΣ are suitable uncertainty regions for µ and Σ respectively.
3
exp erien ce p o sterio r d en sity
in fo rm atio n
( )pof θ
( )prf θp rio r d en sity
Figure 1: Bayesian approach to parameter estimation
The Bayesian framework defines uncertainty sets in a natural way. Indeed,in the Bayesian framework the unknown generic market parameters θ (e.g. µor Σ) are random variables. The likelihood that the parameters assume givenvalues is described by the posterior probability density function fpo (θ), whichis determined by the information available at the time the allocation takes placeand by the investor’s experience and respective confidence, modeled in terms ofa prior probability density function fpr (θ), see Figure 1.The region where the posterior distribution displays a higher concentration
deserves more attention than the tails of the distribution: this region is a naturalchoice for the uncertainty set, which in the Bayesian literature is known ascredibility set.
4
ellip soid
m arg in al p o sterio r
ceµΘ µ
lo catio n -d isp ersio n
( )pof µ
2µ1µ
d en sity
Figure 2: Bayesian posterior distribution and uncertainty set
Consider first µ. A credibility set, i.e. a region where the posterior dis-tribution displays a higher concentration, can be represented by the location-dispersion ellipsoid of the marginal posterior distribution of µ, see Figure 2 forthe case N ≡ 2: bΘµ ≡ ©µ : (µ− bµce)0 S−1µ (µ− bµce) ≤ q2µ
ª. (4)
In this expression qµ is the radius factor for the ellipsoid; bµce is a classical-equivalent estimator such as the expected value or the mode of the marginalposterior distribution of µ; and Sµ is a scatter matrix such as the covariancematrix or the modal dispersion of the marginal posterior distribution of µ.
ceΣ
1 1Σ1 2Σ
2 2Σ
qΣΘ
qΣΘ
p o sitiv itysu rface
Figure 3: Bayesian location-dispersion ellipsoid for covariance
5
Similarly, consider Σ. A credibility set, i.e. a region where the posteriordistribution displays a higher concentration, can be represented by the location-dispersion ellipsoid of the marginal posterior distribution of Σ:
bΘΣ ≡ ½Σ : vech hΣ− bΣcei0 S−1Σ vechhΣ− bΣcei ≤ q2Σ
¾. (5)
In this expression vech is the operator that stacks the columns of a matrixskipping the redundant entries above the diagonal; qΣ is the radius factor forthe ellipsoid; bΣce is a classical-equivalent estimator such as the expected valueor the mode of the marginal posterior distribution of Σ; and Sµ is a scattermatrix such as the covariance matrix or the modal dispersion of the marginalposterior distribution of vech [Σ]. The matrices Σ in this ellipsoid are alwayssymmetric, because the vech operator only spans the non-redundant elementsof a matrix. When the radius factor qΣ is not too large the matrices Σ in thisellipsoid are also positive definite: indeed, positivity is a continuous propertyand bΣce is positive definite, see Figure 3 for the case N ≡ 2, which is completelydetermined by three entries.The robust Bayesian mean-variance approach to allocation consists in using
the Bayesian elliptical uncertainty sets (4) and (5) in the robust mean-varianceallocation problem (3). Notice that this problem is parametrized by the radiusfactor qµ, the radius factor qΣ and the target variance v(i).The term qµ represents aversion to estimation risk for the expected values
µ: considering a larger ellipsoid increases the chances of including the true, un-known value of µ within the ellipsoid. The value of qµ is typically set accordingto the quantile of the chi-square distribution with N degrees of freedom:
q2µ ≡ Qχ2N(pµ) . (6)
Indeed, under the normal hypothesis for the posterior of µ the square Maha-lanobis distance in (4) is chi-square distributed. A confidence level pµ ≡ 5%corresponds to an investor who is not too concerned with poorly estimating µ;a confidence level pµ ≡ 25% corresponds to a conservative investor who is veryconcerned with properly estimating µ.Similarly, the term qΣ represents aversion to estimation risk for the covari-
ances Σ: considering a larger ellipsoid increases the chances of including thetrue, unknown value of Σ within the ellipsoid. The value of qΣ can be set ac-cording to the quantile of the chi-square distribution with N (N + 1) /2 degreesof freedom:
q2Σ ≡ Qχ2N(N+1)/2
(pΣ) . (7)
This follows from a heuristic argument. Indeed, for a ν-dimensional randomvariable Z with fully arbitrary distribution the following identity for the Maha-lanobis distance holds
M ≡ (Z− E {Z})0Cov {Z}−1 (Z− E {Z})0 ≡νXi=1
Y 2i , (8)
6
whereY ≡ C−1 (Z− E {Z}) and C is the Cholesky decomposition of the covari-ance Cov {Z} ≡ CC0. It is easy to check that E {Yi} = 0, Var {Yi} = 1. Further-more, all the Yi are uncorrelated, although they are not necessarily independentof each other. Therefore we expect M to behave similarly to a chi-square dis-tribution with ν degrees of freedom, especially when ν = N (N + 1) /2 is ratherlarge. A confidence level pΣ ≡ 5% corresponds to an investor who is not tooconcerned with poorly estimating Σ; a confidence level pΣ ≡ 25% correspondsto a conservative investor who is very concerned with properly estimating Σ.Finally, the term v(i) represents exposure to market risk. As a result, in
principle, the robust Bayesian mean-variance efficient frontier should consti-tute a three-dimensional surface in the N -dimensional space of the allocations,parametrized by qµ, qΣ and v.
3 Specification of a market modelIn order to determine the Bayesian elliptical uncertainty sets (4) and (5) weneed to compute the posterior distributions of µ and Σ. To do so, we needto make parametric assumptions on the market dynamics and model our priorknowledge of the parameters that steer the market dynamics. In this sectionwe consider "conjugate" assumptions for the dynamics and the prior which giverise to analytical expressions for the posterior distribution of the market.We make the following assumptions: first, the market consists of equity-like
securities for which the returns are independently and identically distributedacross time; second, the estimation interval is the same as the investment hori-zon; third, the returns are normally distributed:
Rt,τ |µ,Σ ∼ N(µ,Σ) . (9)
Furthermore, we model the investor’s prior experience as a normal-inverse-Wishart distribution:
µ|Σ ∼ Nµµ0,Σ
T0
¶, Σ−1 ∼W
µν0,Σ−10ν0
¶. (10)
In this expression (µ0,Σ0) represent the investor’s experience on the parameters,whereas (T0, ν0) represent the respective confidence.Under the above hypotheses it is possible to compute the posterior distrib-
ution of µ and Σ analytically, see Aitchison and Dunsmore (1975) or Meucci(2005) for all the computations. First of all, the information from the mar-ket is summarized in the sample mean and the sample covariance of the pastrealizations of the returns:
bµ ≡ 1
T
TXt=1
rt,τ , bΣ ≡ 1
T
TXt=1
(rt,τ − bµ) (rt,τ − bµ) .The posterior distribution of µ and Σ, like the prior distribution (10), is also
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normal-inverse-Wishart, where the respective parameters read:
T1 ≡ T0 + T (11)
µ1 ≡ 1
T1[T0µ0 + T bµ] (12)
ν1 ≡ ν0 + T (13)
Σ1 ≡ 1
ν1
"ν0Σ0 + T bΣ+ (µ0 − bµ) (µ0 − bµ)01
T +1T0
#. (14)
The marginal posterior distribution of µ is multivariate Student t. From itsexpression it is possible to compute explicitly the classical-equivalent estimatorand the scatter matrix that appear in the location-dispersion ellipsoid (4), seeMeucci (2005):
bµce = µ1 (15)
Sµ =1
T1
ν1ν1 − 2
Σ1. (16)
It is also possible to compute explicitly the classical-equivalent estimator andthe scatter matrix of the inverse-Wishart marginal posterior distribution of Σthat appear in the location-dispersion ellipsoid (5), see Meucci (2005):
bΣce =ν1
ν1 +N + 1Σ1 (17)
SΣ =2ν21
(ν1 +N + 1)3
¡D0N
¡Σ−11 ⊗Σ−11
¢DN
¢−1. (18)
In this expression DN is the duplication matrix that reinstates the redundantentries above the diagonal of a symmetric matrix, see Magnus and Neudecker(1999), and ⊗ is the Kronecker product.We stress that the above specifications are flexible enough to describe "sim-
ple" priors on "simple" markets. However, as far as the views are concerned, thenormal-inverse-Wishart assumption (10) prevents the specification of differentconfidence levels on correlations and standard deviations. As far as the marketis concerned, the conditionally normal i.i.d. assumption (9) is sufficient to modelthe returns on asset classes such as stock sectors or mutual funds at mediuminvestment horizons. At short horizons, e.g. intra-day, or with different assetclasses, e.g. individual stocks or hedge funds, fat tails and market asymmetriesplay a major role. Furthermore, independence across time is no longer valid, asthe effect of volatility clustering becomes important.In these more complex situations where the conjugate normal-inverse Wishart
assumption is not suitable to model the market dynamics and the views, theposterior distribution of µ and Σ and the location-dispersion ellipsoids must becomputed numerically by means of Markov chain Monte Carlo techniques, seee.g. Geweke (2005).
8
4 Optimal portfoliosIn the technical appendix we prove that the robust Bayesian mean-varianceproblem ensuing from the specifications (9)-(10), i.e. the robust problem (3)where the Bayesian elliptical uncertainty sets (4) and (5) are specified by (15)-(18), simplifies as follows:
w(i)rB = argmax
w∈Cw0Σ1w≤γ(i)Σ
nw0µ1 − γµ
pw0Σ1w
o(19)
where:
γµ ≡
sq2µT1
ν1ν1 − 2
(20)
γ(i)Σ ≡ v(i)
ν1ν1+N+1
+q
2ν21q2Σ
(ν1+N+1)3
. (21)
2Under standard regularity assumptions for the investment constraints C themaximization (19) can be can be cast in the form of a second-order cone pro-gramming problem. Therefore the robust Bayesian frontier can be computednumerically, see Ben-Tal and Nemirovski (2001).Alternatively, the above result shows that the generic three-dimensional ro-
bust Bayesian efficient frontier degenerates to a one-parameter family, whichcan be parametrized by a single positive multiplier λ as follows:
w(i)rB ∈ w (λ) = argmax
w∈C
nw0µ1 − λ
pw0Σ1w
o. (22)
In other words, the a-priori three-dimensional robust Bayesian efficient frontiercollapses to a line. We stress that in markets other than (9) and with priorstructures more general than (10) the robust Bayesian efficient frontier, whichmust be computed by means of Markov chain Monte Carlo techniques, remainsthree-dimensional.Notice the self-adjusting nature of the Bayesian setting. From (11)-(14) the
expected values µ1 and the covariance matrix Σ1 that determine the efficientallocations (22) are mixtures of the investor’s prior knowledge (µ0,Σ0) and of
information from the market³bµ, bΣ´: the balance between these components is
steered by the relative weight of the confidence in the investor’s prior parameters,represented respectively by (T0, ν0), and the amount of information, representedby number of observations T in the time series.In particular, when the number of observations T is large with respect to
the confidence levels T0 and ν0 in the investor’s prior, the expected values µ12We do not need to worry about the positivity condition in Figure 3, because under our
assumptions the max of Σ in (3) always lies in the positivity region.
9
tend to the sample mean bµ and the covariance matrix Σ1 tends to the samplecovariance bΣ. Therefore we obtain a sample-based efficient frontier:
w (λ) = argmaxw∈C
nw0bµ− λ
pw0 bΣwo . (23)
Similarly, when the confidence levels T0 and ν0 in the investor’s prior are largewith respect to the number of observations T , the expected values µ1 tend tothe prior µ0 and the covariance matrix Σ1 tends to the prior Σ0. Therefore weobtain a prior efficient frontier that disregards any information from the market:
w (λ) = argmaxw∈C
nw0µ0 − λ
pw0Σ0w
o. (24)
The robust Bayesian frontier (22), as well as its limit cases (23) and (24),is apparently purely Bayesian, i.e. non-robust. However, estimation risk entersthe picture through the multiplier λ, which is determined by the scalars (20)and (21). It is easy to check that the value of λ is directly related to the aversionto estimation risk for µ, namely qµ, and to the aversion to estimation risk forΣ, namely qΣ, and inversely related to the exposure to market risk v(i).Accordingly, the term under the square root in (22) represents both estima-
tion risk and market risk and the coefficient λ represents aversion to both typesof risk. In the classical or purely Bayesian setting "risk" only refers to marketrisk, whereas in the robust Bayesian setting "risk" blends both market risk andestimation risk. As the aversion to estimation risk or market risk increases,investors ride down the frontier and shrink their allocation towards the globalminimum variance portfolio:
wMV = argminw∈C
{w0Σ1w} . (25)
The minimum-variance portfolio plays a special role in the shrinkage literature,see Jagannathan and Ma (2003). Shrinking to the global minimum varianceportfolio in order to diminish estimation risk is also explored in Kan and Zhou(2006), where this choice is imposed exogenously.
5 Empirical resultsIn this section we run two empirical tests. In both cases we assume that theinvestor is bound by the standard budget constraint w01 = 1 and no-short-saleconstraintw ≥ 0. Fully commented code is available at http://symmys.com/node/102The first test is a simulation study: we consider an artificial normal market
and we evaluate the robust Bayesian allocations in the true mean-variance planeof coordinates, i.e. the plane of coordinates that can be computed when the trueparameters underlying the market distribution are known. We assume N ≡ 40asset classes and T ≡ 52 observations in the time series. We factor the covariance
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matrix Σ into a homogeneous correlation matrix
C ≡
⎛⎜⎜⎜⎜⎝1 θ · · · θ
θ. . .
......
. . . θθ · · · θ 1
⎞⎟⎟⎟⎟⎠ ,where θ ≡ 0.7, and equally-spaced volatilities:
σ ≡ 0.1, σ +∆, . . . , σ −∆, σ ≡ 0.4.
We define the expected returns µ in such a way that a mean-variance optimiza-tion would yield an equally-weighed portfolio:
µ ≡ 2.5×Σ 1N.
We define the prior as follows: Σ0 is equal to the sample covariance on theprincipal diagonal and is null otherwise; as for the prior expected returns we set
µ0 ≡ 0.5×Σ01
N;
We set the confidence in the prior as twice the confidence in the sample: T0 ≡ν0 ≡ 2T . We set the aversion to estimation risk in (20)-(21) as in (6) and (7),where pµ ≡ pΣ ≡ 0.1. We set the target variance in (21) as the sample varianceof an aggressive portfolio whose sample expected return is 4/5 of the maximumsample expected return.
Figure 4: Shrinkage effect of robust Bayesian decisions
We run a Monte Carlo experiment and we report in Figure 4 the results. Thesample-based allocation is broadly scattered in inefficient regions. The purely
11
Bayesian portfolios are shrunk toward the prior: therefore they are less scatteredand more efficient, although the prior differs significantly from the true marketparameters. The robust-Bayesian allocations is a subset of the Bayesian frontierthat is further shrunk toward the global minimum variance portfolio and evenmore closely tight to the right of the efficient frontier.
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 0 . 2
- 0 . 1
0
0 . 1
0 . 2
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0- 0 . 2
- 0 . 1
0
0 . 1
0 . 2 R o b u s t B a y e s i a n
S a m p l e - b a s e d
Figure 5: Weekly returns of sample-based and robust Bayesian allocations
In the second experiment we compute optimal robust Bayesian portfolios ofsectors of the S&P 500. We consider an investment horizon of one week andthus we base our estimates on weekly returns. All the assumptions are the sameas in the simulation tests, except of course that the true market parameters areunknown. Similarly to the extensive analysis of DeMiguel, Garlappi, and Uppal(2005) we consider rolling estimates of the market parameters over a period ofone year. Then we compare the ensuing robust Bayesian allocation with thepurely sample-based allocation.
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 00
1
2
3
4
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 00
1
2
3
4 R o b u s t B a y e s i a n
S a m p l e - b a s e d
Figure 6: Cumulative P&L of sample-based and robust Bayesian allocations
In Figure 5 we plot the weekly returns of the two strategies and in Figure6 the respective cumulative P&L. As expected, the robust Bayesian allocationis more conservative. Although the final gain of the sample-based allocationis larger, its maximum drawdown largely exceed that of the robust Bayesian
12
strategy. We stress that this example is one of the least favorable to the robustBayesian approach. In other cases, such as using more than one year of data inthe rolling estimation of the market parameters, the robust Bayesian strategyclearly dominates the sample-based allocation, which incur substantial losses.
6 ConclusionsThe robust Bayesian approach to allocation displays the optimality featuresof robust optimization as well as the self-adjusting Bayesian mechanisms thataccount for the investor’s prior knowledge within a sound statistical framework.Under fairly standard assumptions for the market, the robust Bayesian
mean-variance optimal portfolios reduce to a purely Bayesian optimal port-folio which is shrunk toward the Bayesian minimum variance portfolio by theaversion to estimation risk.As customary in the Bayesian framework, the above Bayesian portfolios
are a mix of purely sample-based allocations, which completely disregard theinvestor’s prior knowledge, and purely prior allocations, which completely disre-gard the information from the market. The interplay between these two compo-nents is steered by the relative weight of the confidence in the prior with respectto the amount of information from the market.Please refer to the code available at http://symmys.com/node/102 for more
details.
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Bawa, V. S., S. J. Brown, and R. W. Klein, 1979, Estimation Risk and OptimalPorfolio Choice (North Holland).
Ben-Tal, A., and A. Nemirovski, 2001, Lectures on modern convex optimization:analysis, algorithms, and engineering applications (Society for Industrial andApplied Mathematics).
Best, M. J., and R. R. Grauer, 1991, On the sensitivity of mean-variance-efficientportfolios to changes in asset means: Some analytical and computationalresults, Review of Financial Studies 4, 315—342.
Black, F., and R. Litterman, 1990, Asset allocation: combining investor viewswith market equilibrium, Goldman Sachs Fixed Income Research.
Ceria, S., and R. A. Stubbs, 2004, Incorporating estimation errors into portfolioselection: Robust efficient frontiers, Axioma Inc. Technical Report.
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Chopra, V., and W. T. Ziemba, 1993, The effects of errors in means, variances,and covariances on optimal portfolio choice, Journal of Portfolio Managementpp. 6—11.
De Santis, G., and S. Foresi, 2002, Robust optimization, Goldman Sachs Tech-nical Report.
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Halldorsson, B. V., and R. H. Tutuncu, 2003, An interior-point method for aclass of saddle-point problems, Journal of Optimization Theory and Applica-tions 116, 559—590.
Jagannathan, R., and T. Ma, 2003, Risk reduction in large portfolios: Whyimposing the wrong constraints helps, Journal of Finance 58, 1651—1683.
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7 Technical appendixIn this appendix we show how to simplify the max-min expression for the efficientfrontier and how to reduce the problem in the SOCP form.
7.1 Worst-case scenario for µ
Consider the ellipsoid (4) under the specifications (15)-(16):
bΘµ ≡ ½µ : (µ− µ1)0Σ−11 (µ− µ1) ≤1
T1
ν1ν1 − 2
q2µ
¾. (26)
Consider the spectral decomposition of the dispersion parameter:
Σ1 ≡ FΓ1/2Γ1/2F0, (27)
where Γ is the diagonal matrix of the eigenvalues sorted in decreasing order andF is the juxtaposition of the respective eigenvectors.We can write (26) as follows:
bΘµ ≡ ½µ : (µ− µ1)0FΓ−1/2Γ−1/2F0 (µ− µ1) ≤ 1
T1
ν1ν1 − 2
q2µ
¾. (28)
Define the new variable:
u ≡µ1
T1
ν1ν1 − 2
q2µ
¶−1/2Γ−1/2F0 (µ− µ1) , (29)
which implies
µ = µ1 +
µ1
T1
ν1ν1 − 2
q2µ
¶1/2FΓ1/2u. (30)
We can write (28) as follows:
bΘµ ≡ (µ1 +µ 1T1 ν1ν1 − 2
q2µ
¶1/2FΓ1/2u, u0u ≤ 1
). (31)
Since
w0µ =
*w,µ1 +
µ1
T1
ν1ν1 − 2
q2µ
¶1/2FΓ1/2u
+(32)
= hw,µ1i+*µ
1
T1
ν1ν1 − 2
q2µ
¶1/2Γ1/2F0w,u
+,
we obtain:
minΣ∈Θµ
{w0µ} = hw,µ1i+ minu0u≤1
*µ1
T1
ν1ν1 − 2
q2µ
¶1/2Γ1/2F0w,u
+(33)
= w0µ1 −µ1
T1
ν1ν1 − 2
q2µ
¶1/2 °°°Γ1/2F0w°°° .15
Recalling (27) this becomes:
minΣ∈Θµ
{w0µ} = w0µ1 −µ1
T1
ν1ν1 − 2
q2µ
¶1/2pw0Σ1w. (34)
7.2 Worst-case scenario for Σ
Consider the ellipsoid (5):
bΘΣ ≡ ½Σ : vech hΣ− bΣcei0 S−1Σ vechhΣ− bΣcei ≤ q2Σ
¾, (35)
under the specifications (17)-(18). Consider the spectral decomposition of therescaled dispersion parameter (18):¡
D0N
¡Σ−11 ⊗Σ−11
¢DN
¢−1 ≡ EΛE0, (36)
where Λ is the diagonal matrix of the eigenvalues sorted in decreasing orderand E is the juxtaposition of the respective eigenvectors. We can write (35) asfollows:
bΘΣ ≡½vech
hΣ− bΣcei0EΛ−1/2 (37)
Λ−1/2E0 vechhΣ− bΣcei ≤ 2ν21q
2Σ
(ν1 +N + 1)3
).
Define the new variable:
u ≡Ã
2ν21q2Σ
(ν1 +N + 1)3
!−1/2Λ−1/2E0 vech
hΣ− bΣcei , (38)
which implies:
vech [Σ] ≡ vechhbΣcei+Ã 2ν21q
2Σ
(ν1 +N + 1)3
!1/2EΛ1/2u. (39)
We can write (37) as follows:
bΘΣ ≡⎧⎨⎩vech hbΣcei+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2EΛ1/2u, u0u ≤ 1
⎫⎬⎭ . (40)
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From (39) we obtain:
w0Σw = (w0 ⊗w0) vec [Σ] (41)
= (w0 ⊗w0)DN vech [Σ]
=DD0N (w
0 ⊗w0)0 , vechhbΣcei
+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2EΛ1/2u
+= w0 bΣcew
+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2 DΛ1/2E0D0
N (w0 ⊗w0)
0,uE.
Substituting (17) in (41) we obtain:
maxΣ∈ΘΣ
{w0Σw} =ν1
ν1 +N + 1w0Σ1w (42)
+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2maxu0u≤1
DΛ1/2E0D0
N (w0 ⊗w0)
0,uE
=ν1
ν1 +N + 1w0Σ1w
+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2 °°°Λ1/2E0D0N (w
0 ⊗w0)0°°° .
To simplify this expression, consider the pseudo inverse eD of the duplicationmatrix: eDNDN = IN(N+1)/2. (43)
It is possible to show that:
(D0NADN )
−1= eDNA
−1 eD0N (44)
and(w0 ⊗w0)DN
eDN = (w0 ⊗w0) , (45)
see Magnus and Neudecker (1999).
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Now consider the square of the norm in (42). Using (44) and (45) we obtain:
a ≡°°°Λ1/2E0D0
N (w0 ⊗w0)
0°°°2 (46)
= (w0 ⊗w0)DNEΛ1/2Λ1/2E0D0
N (w0 ⊗w0)0
= (w0 ⊗w0)DN
¡D0N
¡Σ−11 ⊗Σ−11
¢DN
¢−1D0N (w
0 ⊗w0)0
= (w0 ⊗w0)DNeDN (Σ1 ⊗Σ1) eD0
ND0N (w
0 ⊗w0)0
= (w0 ⊗w0)DNeDN (Σ1 ⊗Σ1)
h(w0 ⊗w0)
³DN
eDN
´i0= (w0 ⊗w0) (Σ1 ⊗Σ1) (w0 ⊗w0)
0
= (w0Σ1w)⊗ (w0Σ1w) = (w0Σ1w)2 .
Therefore (42) yields:
maxΣ∈ΘΣ
w0Σw =
⎡⎣ ν1ν1 +N + 1
+
Ã2ν21q
2Σ
(ν1 +N + 1)3
!1/2⎤⎦ (w0Σ1w) . (47)
7.3 Robust Bayesian mean-variance problem
Substituting (34) and (47) in (3) and using the definitions (20) and (21) weobtain the robust Bayesian mean-variance problem (19).Recalling (27), we can write (19) as follows:
w(i) = argmaxw
nw0µ1 − γµ
°°°Γ1/2F0w°°°o (48)
subject to
(w ∈ C°°°Γ1/2F0w°°° ≤qγ
(i)Σ .
This is equivalent to: ³w(i), z∗
´= argmax
w{w0µ1 − z} (49)
subject to
⎧⎪⎪⎨⎪⎪⎩w ∈ C°°°Γ1/2F0w°°° ≤ z/γµ°°°Γ1/2F0w°°° ≤qγ
(i)Σ .
If the investment constraints C are at most quadratic this is a second order coneprogramming problem.
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