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Stable distributions in the

Black-Litterman approach to the

asset allocation

December 13, 2005

Rosella Giacometti1, Marida Bertocchi1, Svetlozar T. Rachev2 and FrankJ. Fabozzi3

Acknowledgments.The authors acknowledge the support given by researchprojects MIUR 60% 2003 Simulation models for complex portfolio alloca-tion and MIUR 60% 2004 Models for energy pricing, by research grantsfrom Division of Mathematical, Life and Physical Sciences, College of Let-

ters and Science, University of California, Santa Barbara and the DeutschenForschungsgemeinschaft.

1Department of Mathematics, Statistics,Computer Science and Applications, BergamoUniversity, Via dei Caniana, 2, Bergamo 24127, Italy

2School of Economics and Business Engineering, University of Karlsruhe, Postfach6980, 76128 Karlasruhe, Germany and Department of Statistics and Applied Probability,University of California, Santa Barbara, CA 93106-3110, USA

3Yale School of Management, 135 Prospect Street, Box 208200, New Haven, Connecti-cut 06520-8200, USA

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Stable distributions in the Black-Litterman approach tothe asset allocation

Abstract. The integration of quantitative asset allocation models and the

judgment of portfolio managers and analysts (i.e., qualitative view) dates

back to papers by Black and Litterman [4], [5], [6]. In this paper we improve

the classical Black-Litterman model by applying more realistic models for

asset returns (the normal, the t-student, and the stable distributions) and

by using alternative risk measures (dispersion-based risk measures, value at

risk, conditional value at risk). Results are reported for monthly data and

goodness of the models are tested through a rolling window of fixed size along

a fixed horizon. Finally, we find that incorporation of the views of investors

into the model provides information as to how the different distributional

hypotheses can impact the optimal composition of the portfolio.

Key Words. Black-Litterman model, risk measures, return distributions.

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Stable distributions in the Black-Litterman approach to

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1 Introduction

The mean-variance model for portfolio management as formulated by Markowitz

[18] is probably one of the most known and cited financial model. Despite

its introduction in 1952, there are several reasons cited by academics and

practitioners as why its use is not widespread. Some of the major reasons

are the scarcity of diversification [12] or highly concentrated portfolios and

the sensitivity of the solution to inputs (especially to estimation errors of the

mean, see Kallberg and Ziemba [16], [17], Best and Grauer [3], Michaud [21])

and the approximation errors in the solution of the maximization problem.

The integration of quantitative asset allocation models and the judgment

of portfolio managers and analysts (i.e., qualitative view) has been motivated

by various discussions on increasing the usefulness of quantitative models for

global portfolio management. The framework dates back to papers by Black

and Litterman [4], [5], [6] that led to development of extensions of the frame-

work proposed by members of both the academic and practitioner commu-

nities. Subsequent research has explained the advantages of this framework,

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what is now popularly referred to as Black-Litterman model (BL model here-

after), as well as the models main characteristics 1.

In most of these papers, there are explicit or implicit assumptions that

logaritmic returns on N asset classes are multivariate Gaussian distributed,

an assumption consistent with other mainstream theories in finance such as

the standard Black-Scholes model [7]. However, there are numerous empirical

studies 2 that show that in many cases logaritmic returns are quite far from

being normally distributed, especially for high frequency data. Many recent

papers (see Ortobelli et al. [24], [25], Bertocchi et al. [2]) show that sta-

ble Paretian distributions are suitable for the autoregressive portfolio return

process in the framework of asset allocation problem over a fixed horizon.

In this paper we investigate whether the BL model can be enhanced

by using the stable Paretian distributions as a statistical tool for asset re-

turns. We use as a portfolio of assets a subset, duly constructed, of the

S&P500 benchmark. We generalize the procedure of the BL model allowing

the introduction of dispersion matrices obtained from multivariate Gaussian

distribution, symmetric t-Student, and -stable distributions for computing1See the papers by Fusai and Meucci [10], Satchell and Scowcroft [30], He and Litterman

[13] and the books by Litterman [15] and Meucci [23].2See Embrechts et al. [8], Rachev and Mittnik [28] and the references therein, Mittnick

and Paolella [22], Panorska et al. [27], Tokat et al. [33], Tokat and Schwartz [32].

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the equilibrium returns. Moreover, three different measures of risk (variance,

value at risk and conditional value at risk) are considered. Results are re-

ported for monthly data and goodness of the models are tested through a

rolling window of fixed size along a fixed horizon. Finally, our analysis shows

that the incorporation of the views of investors into the model provides in-

formation as to how the different distributional hypotheses can impact the

optimal composition of the portfolio.

2 The -stable distribution

The -stable distributions describe a general class of distribution functions.

A random variable X is - stable distributed if it has a domain of attraction:

that is, there exists a sequence of i.i.d. random variables {Yi}i, a sequenceof positive real values {di}i and a sequence of real values {ai}i such that

as n +

1

dn

ni=1

Yi + andX (1)

where

d

points out the convergence in the distribution. Thus, the -stable random variables describe a general class of distributions including the

leptokurtic and asymmetric ones.

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The -stable distribution is identified by four parameters: the index of

stability (0, 2] which is the parameter of the kurtosis, the skewness pa-

rameter [1, 1]; and + which are, respectively, the location

and the dispersion parameter. If X is a random variable whose distribu-

tion is -stable, we use the following notation to underline the parameter

dependence

Xd=S( , , ) (2)

The stable distribution is normal, when = 2 and it is leptokurtotic

when < 2. A positive skewness ( > 0) identifies distributions with right

fat tails, while a negative skewness ( < 0) typically characterizes distrib-

utions with left fat tails. Therefore, the stable density functions synthesize

the distributional forms empirically observed in the real data. The Maximum

Likelihood Estimation (MLE) procedure used to approximate stable parame-

ters is described by Rachev and Mittnik [28]). Unfortunately the density of

stable distributions cannot be express in closed form. Thus, in order to value

the density function, it is necessary to invert the characteristic function.

In the case where the vector r = [r1, r2, . . . rn] of returns is sub-Gaussian

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-stable distributed with 1 < < 2, then the characteristic function of r

assumes the following form:

r(t) = E(exp(it

r)) = exp((tVt) 2 + itE(r)) (3)

For the dispersion matrix V = [v2ij ] we use the following estimation (see

Ortobelli and al. [26] and Lamantia et al. [14])

v2ij = ( vjj)2qA(q)

1

T

Tk=1

rik|rik|q1sgn( rjk) (4)

where rjk = rjk E(rj) is the k-th centered observation of the j-th asset,

A(q) =(1 q

2)

2q(1 q)( q+1

2)

and 1 < q <

vjj = (A(p)1

n

nk=1

|rjk |p)2p , 1 < p < 2 (5)

where rjk = rjk E(rj) is the k-th centered observation of the j-th asset.

2.1 t-Student distribution

Analogously to the normal distribution, the t-Student distribution depends

on an n - dimensional location parameter u which corresponds to the peak of

the distribution and on a nn symmetric dispersion matrix V that influences

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the shape of the distribution around the peak. Moreover, the degree of free-

dom of the distribution, , provides information on the relation between the

peak and the tails. However, as the number of degrees of freedom increases,

the t-Student distribution approaches closely the normal distribution.

The multivariate t-Student probability density function is given by:

v,u,V(x) = ()n

2( +n

2)

(2