A Black-Litterman asset allocation model under Elliptical distributions Yugu Xiao ∗†
Emiliano A. Valdez ‡
August 28, 2010
Abstract In optimal portfolio allocation, Black and Litterman (1992) provide for a pioneering framework of allowing to incorporate investors’ views based on a prior distribution to derive a posterior distribution of portfolio returns and optimal asset allocations. Meucci (2005) rephrases the model in terms of investors’ views on the market, rather than just the market parameters as in the original Black and Litterman (1992). This market-based version is believed to be much more parsimonious and allows for a more natural extension to directly input views in a non-Normal market. This paper extends Meucci’s market-based version of the BlackLitterman model to the case when returns in the market fall within the class of Elliptical distributions, while also importantly preserving the equilibrium-based assumption in the model. Here within this class for which the Normal distribution is a special case, we develop the explicit form of the posterior distribution after considering proper conditional conjugate-type prior distributions. This resulting posterior allows us to obtain solutions to optimization problems of asset allocation based on a variety of risk measures (e.g. Mean-Variance, Mean-VaR, Mean-Conditional VaR). Elliptical models of portfolio returns have recently crept into the financial literature because of its greater flexibility to accommodate larger tails. As a numerical demonstration, we examine how these principles work in a portfolio with international stock indices. Keywords: Optimal asset allocation, Black-Litterman model, risk measures, Elliptical distributions.
∗ Corresponding author: xiao yugu@yahoo.com.cn. Acknowledgement: Dr. Y. Xiao acknowledges the financial support of the National Philosophy and Social Science Foundation grant (No.07BTJ002) and the National Nature Science Foundation grant (No.10871201) of China. † School of Statistics, Renmin University of China, Beijing, P.R. China. ‡ Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA.
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1 Introduction Consider the case where we have an n × 1 vector of asset returns X whose distribution could be represented by its multivariate density function fX . Its expected value will be denoted by the vector E(X) = π and its covariance matrix by Cov(X) = Σ. Assume that, in addition, we have a set of k investors’ views of the market specified in a linear form as follows: V = PX, where P, for which we can call the “picks”, is a k × n matrix of (constant) asset weights within each investor’s view. Each investor’s view would depend on the observed value of the vector of asset returns x so that it is necessary to specify a model for V|x which could be represented by its conditional density function fV|x with expected value E(V|x) = Px. In some sense, as noted by Meucci (2005), the investors’ views V helps to smoothen the risk that may result from misestimating the distribution of the asset returns X. In the Bayesian context, one can then view the conditional distribution of V|x as the prior distribution. Thus, the investors’ views are incorporated into the return distribution by evaluating the posterior distribution of X|v using the relationship: f X| v ( x|v ) ∝ f V | x ( v |x) f X ( x) . The Black and Litterman (1992) model combines active investment views as well as equilibrium views through this Bayesian approach. In effect, it provides for a flexible and straightforward technique for portfolio managers to express judgments on linear combinations of market returns and immediately obtain the market distribution reflecting these views in a consistent manner. Several studies inspired by this framework further advanced our understanding and implementation of the Black-Litterman framework. For example, the work by Walters (2009) is a comprehensive survey of the Black-Litterman model together with extensions made in the literature. The k investors’ views specified according to the “picks” could either be a relative view or an absolute view. For a relative view, the weights will add up to 0 while for an absolute view, the sum of the weights will be equal to 1. To illustrate, consider the case where we have four assets together with two investors’ views. One is a relative view in which the investor believes that Asset 1 will outperform Asset 3 by V1 , that is X1 − X3 = V1 . The other is an absolute view in which the investor believes that Asset 4 will return V2 , that is X4 = V2 . These views are then specified as follows: P=
1 0 −1 0 0 0 0 1
!
Therefore, given a realization of x = ( x1 , x2 , x3 , x4 )′ , the investors’ views will give a realization of v|x = ( x1 − x3 , x4 )′ . To reflect the uncertainty associated with these views, we shall denote the covariance by Cov(V|x) = Q, 2
a k × k matrix which would give a measure of the investors’ confidence level on these views. According to our illustration, for example, if the two investors believe their above relative and absolute views with confidence at levels ω1 and ω2 , respectively, then Q=
ω1 0 0 ω2
!
.
The zeroes on the non-diagonals would reflect uncorrelated views. Generally, we would expect the variance of these views to be inversely related to the investors confidence in the views. The basic Black-Litterman model does not provide an intuitive way to quantify this relationship. Meucci (2005) proposes that it may be convenient to set this uncertainty matrix according to Q=
1 PΣP′ , c
where the scatter structure of the uncertainty is directly inherited from the dispersion Σ of the asset returns and c is a constant that represents an overall level of confidence in the view. In the Black-Litterman model, the computation of both the equilibrium portfolio and the prior distribution rely on the assumption of a Normal market. For the Normal market, both the asset returns X and the investors’ views V|x assume multivariate Normal distributions as X ∼ Nn (π, Σ) and V|x ∼ Nk (Px, Q), where N denotes multivariate Normal distribution, with the subscript denoting the corresponding dimension of the random vector. Using the conditional property of a multivariate Normal, it is a rather straightforward exercise to show that the posterior distribution of X|v is also a multivariate Normal with mean vector M µBL = π + ΣP′ (Q + PΣP′ )−1 (v − Pπ )
(1)
and covariance matrix M = Σ − ΣP′ (Q + PΣP′ )−1 PΣ. ΣBL
(2)
Notice that investors’ views affect both the mean and the covariance of asset returns. Note that the above construction differs slightly from the original Black and Litterman (1992) model. It starts with the market specification based on the hierarchical model X|µ ∼ Nn (µ, Σ)
µ ∼ Nn (π, τΣ)
where π represents the expectation for µ and τΣ, the uncertainty about this expectation. Investors’ views are incorporated based on the specification that V|µ ∼ Nk (Pµ, Q) 3
so that this results in an expression for the expected return in the market, conditional on the investors’ views, as follows: µ
µ|V ∼ Nn (µBL , ΣBL ),
(3)
where it was shown that µBL = π + τΣP′ (τPΣP′ + Q)−1 (v − Pπ ) and µ
ΣBL = τπ − τ 2 ΣP′ (τPΣP′ + Q)−1 PΣ. Finally, we have X|V ∼ Nn (µBL , ΣBL ) µ
where ΣBL = Σ + ΣBL . Once the posterior distribution of asset returns that accounts for the investors’ views has been determined, we can then use the result to derive optimal asset allocations. In an asset allocation decision problem, we are usually faced with the problem of optimizing a function of the return in the portfolio of assets together with a the choice of a risk measure. The decision is to find the allocation, say α, which is a vector of weights representing the number of securities to purchase in a portfolio. The classical mean-variance portfolio trade-off is an example of such an optimization problem. See, for example, Ingersoll (1987). There has been a surge of interest in models of asset returns departing from the Normality assumption. Several empirical studies have shown how inadequate the distribution is because, in reality, it appears that asset returns exhibit heavier tails and occasionally with high peaks. One alternative class of distributions that has increased in popularity, both in finance and insurance, is the class of multivariate Elliptical distributions, for which obviously the multivariate Normal is a special case. Member of this class shares many of the interesting and convenient properties of the Normal distribution without limiting the fatness of the tail or the peak of the distribution. See, for example, the book by Fang et al. (1990) which provides a comprehensive survey of this class of distributions and the work of Landsman and Valdez (2003) which examines its applicability for a particular risk measure. In the finance literature, we can find the work of Owen and Rabinovitch (1983) which proves CAPM for the Elliptical market and Bingham and Kiesel (2002) which provides empirical evidence of a satisfactory fit of members of this class for modeling asset returns. In other findings, special attention has been paid to the multivariate generalized version of the Student-t model, which is often regarded to be an appropriate model for stock returns; see, for example, Aparicio and Estrada (2001) and McDonald (1996). As yet another alternative distribution, Giacometti et al. (2007) improved the classical BlackLitterman model by considering the Stable distributions for asset returns. The various subsequent works of Meucci (2006a), Meucci (2006b), and Meucci (2008b), extended the framework to a more generic form of the market distribution. All these works considered as case studies also considered risk measures beyond the variance such as value-at-risk (VaR) and conditional value-at-risk 4
(CVaR) in the portfolio optimization allocation problem. When there is departure from Normality with alternative portfolio risk measures, there is the increased possibility of having to perform simulation, rather than having explicit form, to derive at decision solutions to the optimization problems. Such would increase the error for misestimating the allocations simply as a result of errors arising from the simulation. Our objective in this paper is to further the Meucci’s market-based version of the BlackLitterman model to the case where we have a market distribution that falls within the class of Elliptical distributions. This class of distributions inherit many of the interesting properties of the Normal. As a consequence, by considering a proper conditional conjugate-type for a prior distribution, we are able to express an explicit form of the posterior distribution of asset returns after accounting for investors’ views. Applying the theory of Elliptical distributions within the context of portfolio optimization is not new to the literature; see, for example Landsman (2008) and Landsman (2010). Applying the theory within the Black-Litterman is new and as demonstrated in this article, we are able to obtain the posterior asset allocation weights for optimization problems with alternative risk measures just as those conveniently considered in the Normal market. In particular, we considered the optimal allocation problems that involved mean-variance, mean-VaR and mean-conditional VaR. In addition, the Elliptical market allows us to preserve the equilibrium-based assumption for which several investors find very attractive, although the Black-Litterman model can be enhanced to non-Normal markets where there is not necessarily the Capital Asset Pricing Model (CAPM) equilibrium. See Meucci (2008a) and Meucci (2008b). Interestingly, as noted by Ingersoll (1987), a sufficient condition for a CAPM equilibrium is that the returns must be Elliptically distributed. This is not at all a surprising result because indeed, within the class of Elliptical distributions, the linear correlation remains to be the primary measure of dependence. We have organized the remainder of this article as follows. In Section 2, we provide for a preliminary discussion about Elliptical distributions, emphasizing properties that we find most useful for this article. In Section 3, we extend the Meucci’s market-based version of Black-Litterman model to the Elliptical market. By considering proper conditional conjugate-type prior distributions, we demonstrate a derivation of the resulting posterior distribution after accounting for investors’ views. Section 4 then utilizes the result in Section 3 to derive the posterior asset allocation weights for alternative risk measures. We numerically illustrate these results in Section 5. We conclude in Section 6.
2 A brief overview of Elliptical distributions The class of Elliptical distributions provides for a generalization of the multivariate Normal distribution and is a rich class that share many of its tractable and interesting statistical properties. Apart from the multivariate Normal, members of this class include, but are not limited to, the multivariate Student-t, multivariate Cauchy, multivariate Logistic, and multivariate Stable. In the one dimension, this class coincides with the class of symmetric distributions. Unlike the Normal distribution, however, other members provide flexibility on the tails and its peak. While this class of distributions was introduced by Kelker (1970), a statistical monograph by Fang et al. (1990) provides for a comprehensive survey of this class and its various characterizations. 5
There are a variety of equivalent ways to define members belonging to the class of elliptical distributions. First, consider the case of an n × 1 random vector X = ( X1 , . . . , Xn )′ . For our purposes, we shall define an Elliptical random vector in terms of its density, provided it exists. We use the notation X ∼ ECn (µ, D, gn ) to indicate that X is an n-dimensional random vector that belongs to the class of Elliptical distributions if its density can be expressed as fX (x; µ, D) = |D|−1/2 gn (x − µ)′ D−1 (x − µ) ,
(4)
for some function gn (·), called the density generator, defined for non-negative real numbers. µ ∈ Rn is a location vector and D is an n × n (positive-definite) dispersion matrix. Furthermore, if the density in (4) exists, then the density generator satisfies Z ∞ 0
un/2−1 g(n) (u)du =
Γ(n/2) . π n/2
To illustrate, consider the following four members of the class of Elliptical distributions: Example 2.1. Multivariate Normal distribution. The density generator for a multivariate Normal is given by gn (u) = cn e−u/2 , where cn = (2π )−n/2 is a normalizing constant. We shall write X ∼ Nn (µ, D) and its joint density can then be expressed as f X ( x) = c n |D |
−1/2
1 ′ −1 exp − (x − µ) D (x − µ) . 2
Example 2.2. Multivariate Student-t distribution. The density generator for a multivariate Studentt is given by u −(n+m)/2 gn ( u ) = c n 1 + , m where cn =
(πm)−n/2 Γ((m + n)/2) Γ(m/2)
and m usually refers to the degrees of freedom. We shall write X ∼ tn (µ, D, m) and its joint density can then be expressed as −(n+m)/2 (x − µ ) ′ D−1 (x − µ ) . fX (x) = cn |D|−1/2 1 + m Example 2.3. Multivariate Logistic distribution. The density generator for a multivariate Logistic
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is given by gn ( u ) = c n
e− u , (1 + e − u )2
where Γ(n/2) cn = π n/2
Z ∞ 0
un/2−1
e− u du. (1 + e − u )2
We shall write X ∼ MLn (µ, D) and its joint density can then be expressed as f X ( x) = c n |D |
−1/2
exp −(x − µ)′ D−1 (x − µ)
{1 + exp [−(x − µ)′ D−1 (x − µ)]}
2
.
Example 2.4. Multivariate Exponential Power distribution. The density generator for a multivariate Exponential Power is given by s
gn (u) = cn e−ru , where cn =
sΓ(n/2) π n/2 Γ(n/2s)
rn/2s .
Its joint density can then be expressed as n h is o fX (x) = cn |D|−1/2 exp r −(x − µ)′ D−1 (x − µ) . In the special case where µ = 0, a vector of zeroes, and D = In , the identity matrix, then X ∼ ECn (0, I, gn ) and is referred to as the spherical distribution with density gn (x′ x), where x ∈ Rn . In the univariate case where n = 1, we have X ∼ EC1 (0, 1, g1 ) has the density g1 ( x2 ).
Any linear combinations of Elliptical random vectors will also belong to the class of Elliptical distributions. To illustrate, for any vector of real numbers, a ∈ Rn , a linear combination a′ X is equal to kakX1 . See Theorem 2.4 in Fang et al. (1990), an important property that can be used to determine the distribution of an asset portfolio. Such linear combinations can even be extended to cases such as when A is some m × n matrix of rank m ≤ n and b some m-dimensional columnvector, then AX + b ∼ ECm ( Aµ + b, ADA′ , gm ). See, for example, Landsman and Valdez (2003). In Arellano-Valle et al. (2006), it has been shown that for each k ≤ n, the k-dimensional marginal distribution also belongs to the class of Elliptical distributions and its corresponding
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density generator satisfies π (n−k)/2 gk ( u ) = Γ((n − k)/2)
Z ∞ u
(r − u)(n−k)/2−1 gn (r)dr, for u ≥ 0.
(5)
This result can also be noted by partitioning the random vector X = (Xk ′ , X′n−k )′ , into two random vectors Xk and Xn−k . Without loss of generality, by defining the partitioned matrix
999
A = Ik 0k,n−k ,
the linear combination AX gives the vector Xk so that it immediately follows that Xk ∼ ECk (µk , Σk , gk ).
The conditional random vector Xk |Xn−k , for any k < n, also belongs to the class of Elliptical distributions. To see this, notice that its conditional density can be expressed as gn (x′k xk + x′n−k xn−k ) gn ( x′ x) f Xk | Xn − k ( xk |xn − k ) ∝ = gn−k (x′n−k xn−k ) gn−k (x′n−k xn−k )
In this case, just as shown in Arellano-Valle et al. (2006), the corresponding conditional density generator has the expression gk (u; x) =
gn ( u + x ) , for u, x ≥ 0. gn − k ( x )
(6)
In this paper, we assume that the vector of asset returns X ∼ ECn (π, D, gn ). For the same reason, our setting on the linear views is as in Meucci (2005) except using the dispersion matrix D instead of the covariance matrix Cov(X), for which in the Elliptical case, the two are not always equivalent. As pointed out by Bawa (1975), the scale parameter (and not the variance) is the natural measure of dispersion so that D is a measure of risk that is more appropriately suitable for the model.
3 Deriving the posterior for the Elliptical distribution In this section, we formulate the extension of the Meucci’s market-based version of the BlackLitterman model to the case where we assume an Elliptical market. In particular, we derive the explicit form of the posterior distribution by considering proper conditional conjugate-type prior distributions. More generalized conditional conjugate type distributions for Elliptical distribution are also discussed in Arellano-Valle et al. (2006). The method of constructing the proof in our Proposition 3.1 is quite similar to that in Theorem 2.2 of Branco et al. (2000). We find that the resulting mathematical expression of our problem is analogous to it, however, the linear calibration problem discussed in their paper is in an entirely different context from ours. Proposition 3.1 Let the random vector be distributed as X ∼ ECn (π, D, gn ) and the conditional
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random vector with prior distribution V|x ∼ ECk (Px, Q, gk (·; p(x)), where p ( x) = ( x − π ) ′ D − 1 ( x − π ) .
The posterior distribution of X|v is an Elliptical distribution with ECn (dBL , DBL , gn (·; q(v)), where dBL = π + DP′ (Q + PDP′ )−1 (v − Pπ ),
DBL = D − DP′ (Q + PDP′ )−1 PD, and
q(v) = (v − Pπ )′ (Q + PDP′ )−1 (v − Pπ )
Proof. Following formulas (5) and (6), we have gk (u; x) gn ( x) = gn+k (u + x) and fV,X (v, x) = fV|X (v|x) f X (x) ∝ | D |−1/2 |Q|−1/2 gk+n (q(v, x)),
(7)
where q(v, x) = (v − Px)′ Q−1 (v − Px) + p(x). Now, define the augmented vectors and matrices y=
v x
!
Pπ π
, d=
!
, and H =
Q + PDP′ PD DP′ D
!
.
After some algebraic manipulation on the matrices (details are attached in the appendix), we are able to obtain q(v, x) = (y − d)′ H−1 (y − d). From the relationship in (7), it implies that the vector (V, X) ∼ ECk+n (d, H, gk+n ). According to Corollary 5 in Cambanis et al. (1981), it immediately follows we have the form of the conditional location vector and the conditional dispersion matrix as expressed in the proposition. From Lemma 1 in Szablowski (1990), we can obtain an expression for the conditional covariance matrix in our proposition: Cov(X|v) = DBL Ck
1 (v − Pπ )′ (Q + PDP′ )−1 (v − Pπ ) 2
where function Ck : R+ → R+ satisfies the following condition 1 Ck ( x/2) gk ( x) = 2
Z ∞ x
gk (t)dt.
To illustrate, consider the following examples. Example 3.1. The Normal model. As illustrated in Example 2.1, in the case we have the Normal distribution, the density generator gn (u) = (2π )−n/2 e−u/2 so that Ck ( x/2) = 1, which does not depend on n nor k. Here, as discussed in Section 1, we have X ∼ Nn (π, Σ), V|x ∼ Nk (Px, Q), 9
M , Σ M ). The location and we can obtain the same result as in Meucci (2008a), i.e. X|v ∼ Nn (µBL BL dispersion parameters are given as in (1) and (2).
Example 3.2 The generalized Student-t model. An n-dimensional Elliptical vector X is said to have a generalized multivariate Student-t distribution if its density generator can be expressed as gn (u) = c(n, r)λr/2 (λ + u)−(n+r )/2, u ≥ 0, where c(n, r) =
π −n/2 Γ((n + r)/2) . Γ(r/2)
We can write X ∼ tn (π, D; λ, r). It can be shown than in this case, we have the covariance matrix Cov(X) = (λ/(r − 2))D. The generalized Student-t reduces to the ordinary n-dimensional Student-t distribution with r degrees of freedom when λ = r. See Arellano-Valle and Bolfarine (1995). Since gn+k (u + x) = c(n + k, r)λr/2 (λ + u + x)−(n+K+r )/2, u ≥ 0, it follows from (6) that we have gk (u; x) = c(k, r + n)(λ + x)(r +n)/2 (λ + x + u)−(k+r +n)/2, u ≥ 0. Now, assuming X ∼ tn (π, D; λ, r) and considering the conjugate-type conditional distribution given by V|x ∼ tk (Px, Q; λ + q(x), r + n), from Proposition 3.1, we obtain X|v ∼ tn (dBL , DBL ; λ + q(v), r + k), and indeed, the conditional covariance matrix in this case is explicitly expressed as Cov(X|v) =
1 (λ + (v − Pπ )′ (Q + PDP′ )−1 (v − Pπ ))DBL . k+r−2
4 Portfolio asset allocation for various risk measures Finally in this section, we discuss the posterior asset allocation weights resulting from portfolio optimization under various alternative risk measures. We consider portfolio selection based on mean-variance, mean-VaR and mean-conditional VaR. We show that computing the posterior asset allocation weights in the Elliptical market based on these various risk measures are as convenient as the Normal market. In portfolio asset allocation, we are interested in determining the proportion wi that will be invested in asset i given we are choosing from a set of n assets with asset return vector X. We shall denote the proportion of all asset allocations as an n × 1 vector w, with each entry representing the corresponding proportion.
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4.1 Unconstrained mean-variance trade-off Let us start with the simple Markowitz (1952) type asset allocation problem. Assume that all investors maximize wealth according to a quadratic utility function and that the optimization is unconstrained as expressed below: δ max w′ (E(X) − R f 1n ) − w′ Cov(X)w , w 2
(8)
where δ is the risk aversion coefficient, R f denotes the risk-free rate and 1n = (1, 1, . . . , 1)′ , an n × 1 vector of 1’s. It is straightforward to show that for the first order condition, it yields that w∗ = (δCov(X))−1 E(X − R f 1n ),
(9)
where w∗ denotes the optimal portfolio weight, the solution to (8). Most of the literature on the Black-Litterman model discuss the unconstrained mean-variance trade-off, and explains how to use a good proxy for the market portfolio to provide for a neutral reference point for asset allocation. Once we have a reference weight w∗ , implied equilibrium excess returns, E(X) − R f 1n , can be derived by a reverse optimization from the benchmark weights according to the allocation in (9). See, for example, Walters (2009). Here, we focus on the difference between the asset allocation weights of two different markets. Example 4.1. Assuming the mean return and covariance for the prior market return is µ and Σ, let us use two models to describe the two markets: XN ∼ N(µ, Σ) and XEC ∼ EC(µ, D, gn ). Here, XN and XEC have the same covariance matrix, and the dispersion matrix satisfies Σ = Cov(X N ) = Cov(XEC ) = a0 D, where a0 is a constant. Then the asset allocation weights for the two markets are both equal to wN = wEC = (δΣ)−1 (µ − R f 1n ). Assume the parameters P, v and c of the investors’ views are the same for the two markets, and the measures of the uncertainty of these views are QN = (1/c)PΣP′ and QEC = (1/c)PDP′ , respectively. It can be shown that the posterior asset allocation weights are ∗N M −1 M wBL = (δΣBL ) (µBL − R f 1n )
and M ∗EC − R f 1n ) , = (δDBL a1 )−1 (µBL wBL ′
where a1 = Ck ( 21 (v − Pµ) (Q + PDP′ )−1 (v − Pµ)). Finally, we can obtain the relationship ∗EC wBL 1 . = ∗ N a1 a0 wBL
According to this, investor’s holdings in the risky assets are proportional for these two cases.
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4.2 Mean-Conditional VaR trade-off For a random return variable R, the Value-at-Risk VaR p ( R) is implicitly defined from the solution of P( R ≤ −VaR p ( R)) = p, for some probability level between 0 and 1. The conditional VaR, denoted by CVaR p ( R), is defined to be CVaR p ( R) = −E( R| R ≤ −VaR p ( R)). The −VaR p ( R) is p-quantile of the distribution of the return variable R, and p represents the maximum probability of loss that the investor is willing to accept. On the other hand, CVaR p ( R) is interpreted as the expected worse loss. The conditional VaR coincides with the expected short¨ fall (ES) and the tail conditional expectation (TCE) for continuous distributions; see Hurlimann (2003). According to the mean-conditional VaR trade-off, the manager minimizes the expected shortfall subject to a set of linear constraints min CVaR p (w′ X) ,
(10)
w∈A
where A is the set of linear constraints. Lemma 4.1 Let X ∼ EC(µ, D, gn ). For any w ∈ Rn , we have
CVaR p (w′ X) =
√
w′ Dw CVaR p (Y ) − w′ µ,
(11)
where Y is a spherical random variable, i.e. Y ∼ EC1 (0, 1, g1 ), and that 1 ¯ 1 2 CVaR p (Y ) = G y , p 2 q where G¯ ( x) = G (∞) − G ( x) if G (∞) < ∞, G ( x) = FY (yq ) =
Z yq
−∞
g1 (u2 )du = q, and q = 1 − p.
Z x 0
(12)
g1 (2u)du, yq is the q-quantile satisfying
Proof. The proof follows immediately from equations (8) and (9) of Section 4.2 in Stoyanov et al. (2006), together with Theorem 1 in Landsman and Valdez (2003). After blending the investors’ views, from Lemma 4.1, the posterior asset allocation weights are the solutions to the following minimization problem: min
w∈ABL
p
M w′ DBL w CVaR p (YBL ) − w′ µBL ,
(13)
where YBL ∼ EC1 (0, 1, g1 (·; q(v)), and ABL is the same linear constraints referred previously as A, except for the location vector and dispersion matrix. Note that since the objective function in (13) is convex, see Landsman (2010), and the set of constraints ABL is a linear system, the optimization is a classic convex programming problem; see Boyd and Vandenberghe (2004). The global solution therefore exists and is unique, and can be calculated using several efficient numerical algorithms. 12
For a special case, if we have a system of affine constraints, we yield the explicit optimal solution by Landsman (2008). His paper presents an explicit closed form solution of the problem of minimizing the root of a quadratic functional subject to a system of affine constraints. Example 4.2. For an application, we consider a very interesting case. Assume the manager minimizes the expected shortfall subject to the full-investment constraint together with the constraint of a target expected return r. In this case, we can express the constraints as A = {w|w′ 1n = 1, w′ E(X) = r}, allowing for short selling. For this special case, the solution to (11) coincides with the solution mean-dispersion optimal problem as discussed in Landsman (2008). Furthermore, the optimal posterior asset allocation weight as expressed in (13) can be solved from min w′ DBL w,
(14)
w ∈ABL
M = r }. Interestingly, this means that, in order to be able to where ABL = {w|w′ 1n = 1, w′ µBL obtain the target return, the optimal posterior asset allocation weight must be independent of the tail risk measure CVaR(YBL ) even after blending the linear investors’ views.
Example 4.3. Continuing from Example 3.2, if we assume X ∼ tn (π, D; λ, r) and the conjugatetype conditional distribution V|x ∼ tk (Px, Q; λ + q(x), r + n), then we have the posterior X|v ∼ tn (dBL , DBL ; λ + q(v), r + k). Consequently, the asset allocation problem in (10) can be similarly expressed as min c(1, r) w ∈A
√
yq (1−r )/2 √ ′ λ 1+ ( w Dw) − w′ µ, ( r − 1) p λ
(15)
where yq is the q-quantile of the probability density function of Y, Y ∼ t1 (0, 1; λ, r). In fact, if Y ∼ t1 (0, 1; λ, r), then g1 (u) = c(1, r)λr/2 (λ + u)−(r +1)/2. Since G ( x) =
Z x 0
√
" # 2x (1−r )/2 λ g1 (2u)du = c(1, r) 1− 1+ , r−1 λ
(16)
and by Lemma (4.1), we get the desired result. Therefore, for YBL ∼ t1 (0, 1; λ + q(v), r + k), the allocation solution to (13) also has the corresponding expression to formula (15).
4.3 Mean-VaR trade-off Assume that, in order to account for market downside risk, the manager monitors the meanVaR trade-off of his positions under a set of linear constraints. Here we have what is commonly referred to as the mean-VaR trade-off allocation problem expressed as max E(w′ X) − δVaR p (w′ X)
(17)
w ∈A
w ′ X − w ′ E( X ) √ ∼ EC1 (0, 1, g1 ) and w′ Dw ′ −VaR p (w′ X) − w′ E(X) w X − w′ E(X) ′ ′ √ √ P(w X < −VaR p (w X)) = P < = p, w′ Dw w′ Dw
where A is the set of linear constraints. Because
13
the VaR of portfolio returns equals
√ −VaR p (w′ X) = −( w′ Dw)VaR p (Y ) + w′ E(X), where Y is a spherical random variable. Blending the investors’ views, the posterior asset allocation weight is then solved from M max (1 + δ)w′ µBL − δ(
w ∈ABL
p
w′ DBL w)VaR p (YBL ),
(18)
where YBL ∼ EC1 (0, 1, g1 (·; q(v)). We observe that the above optimization is analogous to (13) except for the coefficients in the objective function.
5 A numerical illustration In this section, we apply our model to a simplified, yet non-trivial, portfolio allocation problem. We demonstrate the impact of having two types of investors’ views on the posterior asset allocation. We determine the allocation weights using the mean-conditional VaR optimization problem as discussed in section (4.2) with the maximum probability of loss of p = 0.05 that the investor would accept. For the constraints set, we assume the standard long-only, full-investment constraints together with the constraint of a minimum target expected return r. In effect, we have for the constraint ABL = {w|w ≥ 0, w′ 1n = 1, w′ µBL ≥ r}. We adapt the data from Meucci (2006a). He considers the situation where we have an internationalequity fund manager, whose investment horizon is one week and whose market is represented by the returns on n = 4 international stock indices: the US S&P 500 (X1 ), the UK FTSE 100 (X2 ), the French Cac 40 (X3 ) and the German Dax (X4 ). The data can be modeled by a Student-t distribuψ −2 tion, i.e. the asset returns X ∼ tn (µ, ψ Σ; ψ, ψ), where ψ is the degrees of freedom. The estimates for the parameter ψ and the (rescaled) covariance matrix Σ are given below:
ψ = 5, Σ = 10−3 ×
0.376 0.253 0.333 0.397
0.253 0.360 0.360 0.396
0.333 0.360 0.600 0.578
0.397 0.396 0.578 0.775
.
Because of the short horizon, we appropriately assume a zero weekly risk free return. From (9), the expected value of the returns can be determined from an equilibrium argument, where an equally-weighted portfolio allocation of 1/4 is assumed and δ = 2.5, as in Meucci (2006a). As a result, we have µ = δΣweq = 10−3 × (0.849, 0.856, 1.169, 1.341)′ , where weq = (1/4, 1/4, 1/4, 1/4)′ . In the first scenario, consider an absolute view in which the investor believes that German Dax (X4 ) will return 0.0011 smaller than the equilibrium return of 0.001341. Here we have P = (0, 0, 0, 1) and v = 0.0011. We arbitrarily set c = 0.5. 14
For a second scenario, consider a relative view in which the investor believes that French Cac 40 (X3 ) will outperform German Dax (X4 ) by 0.0005. Here we have P = (0, 0, 1, −1) and v = 0.0005. Again, we set c = 0.5. Tables 1 and 2 displays the comparison of the posterior return vectors and the posterior dispersion matrices for the absolute and relative investors’s views. Table 3 provides for the resulting portfolio allocations, again comparing the two views but at the same time, comparing various target returns. Table 1: Return vectors (×10−3 )
S&P FTSE Cac Dax
Implied equilibrium return vector µ 0.849 0.856 1.169 1.341
Posterior return vector ( 1) µBL of the absolute view 0.808 0.815 1.109 1.261
Posterior return vector ( 2) µBL of the relative view 0.784 0.819 1.192 1.140
Table 2: Dispersion matrix (×10−3 )
S&P FTSE Cac Dax
Prior dispersion matrix D 0.226 0.152 0.200 0.238 0.216 0.216 0.238 0.360 0.347 0.465
Posterior dispersion matrix ( 1) DBL of the absolute view 0.185 0.111 0.141 0.159 0.176 0.157 0.158 0.274 0.231 0.310
Posterior dispersion matrix ( 2) DBL of the relative view 0.222 0.150 0.201 0.227 0.215 0.217 0.231 0.360 0.351 0.430
Table 3: Portfolio Weights (%)
Target return (×10−3 ) S&P FTSE Cac Dax
Prior allocation weight
Posterior allocation weight ( 1) w BL of the absolute view
Posterior allocation weight ( 2) w BL of the relative view
0.90 42.7 42.4 15.0 0.0
0.90 35.1 34.7 17.4 12.7
0.90 36.0 38.9 25.1 0.0
0.95 36.9 36.7 18.3 8.1
1.00 31.1 31.1 21.5 16.2
1.05 25.4 25.5 24.8 24.3
0.95 28.4 28.3 22.4 21.0
1.00 21.7 21.9 26.8 29.5
1.05 15.2 15.6 30.7 38.5
0.95 28.6 33.6 37.8 0.0
1.00 21.5 27.9 50.6 0.0
1.05 15.0 21.6 63.4 0.0
In Table 3, we observe that the allocation weight for the German Dax increases although its posterior expected return decreased. The reason is each individual return is linked to the other returns via the covariance matrix of returns. A single view causes the return of every asset in the portfolio to decrease as shown in Table 1. And in order to obtain a higher target return, not surprisingly, the manager has to take more risk so that the weights for the German Dax and the French Cac 40 increase as the target return increases. See 3 which displays the optimal posterior 15
asset allocation after blending the relative view. Figure 1 shows the optimal allocation weights prior to incorporating the investor’s views. Figure 2 shows the result of blending in the absolute view. Blending the relative view provides for a very interesting result. The manager decides to move the allocation of German Dax to French Cac 40. This is consistent with our intuition because the posterior return for French Cac 40 is higher than that of the German Dax as shown in Table 1 and the posterior dispersion is lower as shown in Table 2.
6 Conclusion Allowing for investors’ views in the modeling of return distributions can sometimes help alleviate the model misestimation risk. The early work of Black and Litterman (1992) is believed to be the first to develop a framework flexible to accommodate investors’ views into the model specification. The framework is indeed simple and falls within the realm of Bayesian statistics. In this article, we make our contribution to extend the Black-Litterman model by departing from the usual Normal market and by extending it to the larger class of Elliptical distributions. This class has the advantages of allowing for more heavy tails and higher peaks than the ordinary Normal, while at the same time, preserving several tractable statistical properties of the Normal distribution. Indeed, the Normal belongs to this class of distributions. We worked with Meucci’s market-based version of Black-Litterman model and made the extension to the Elliptical market. In the paper, we present the explicit posterior distribution, after blending investors’ views, by considering proper conditional conjugate-type for prior distributions. Once the posterior is determined, we can then solve optimal asset allocation problems for various alternative risk measures that were equally convenient as in the Normal market.
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Appendix In this appendix, we provide additional details of the proof for the following result used in Proposition 3.1:
(v − Px)′ Q−1 (v − Px) + (x − π )′ D−1 (x − π ) = (y − d)′ H−1 (y − d) where y=
v x
!
Pπ π
, d=
!
, and H =
Q + PDP′ PD DP′ D
!
.
Begin by letting L=H
−1
=
L11 L12 L21 L22
!
to be the inverse of the partitioned matrix H=
H11 H12 H21 H22
!
Since we have D − DP′ (Q + PDP′ )−1 PD = (D−1 + P′ Q−1 P)−1 , it follows that −1 −1 L11 = H11.2 = H11 − H12 H22 H21 = [(Q + PDP′ ) − PDD−1DP′ ]−1 = Q−1 , −1 −1 L12 = −H11.2 H12 H22 = −Q−1PDD−1 = −Q−1 P, −1 −1 L21 = −H22 H21 H11.2 = −D−1DP′ Q−1 = −P′ Q−1, and −1 −1 L22 = H22.1 = (H22 − H21 H11 H12 )−1 = D−1 + P′ Q−1 P.
18
Thus, we have
(y − d) ′ H−1 (y − d)
= (v′ − π ′ P′ )Q−1 (v − Pπ ) + 2(x′ − π ′ )(−P′ Q−1 )(v − Pπ ) + (x′ − π ′ )(P′ Q−1P)(x − π ) +(x′ − π ′ )D−1 (x − π )
= v′ Q−1 v − 2x′ P′ Q−1v + x′ P′ Q−1Px + (x′ − π ′ )D−1 (x − π ) = (v − Px)′ Q−1 (v − Px) + (x − π )′ D−1 (x − π ).
for which the result follows.
19
Figure 1: The optimal prior asset allocation weight
Figure 2: The optimal posterior asset allocation weight after blending the absolute view
Figure 3: The optimal posterior asset allocation weight after blending the relative view
20