Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution

Abstract

The returns on most financial assets exhibit kurtosis and many also have probability distributions that possess skewness as well. In this paper a general multivariate model for the probability distribution of assets returns, which incorporates both kurtosis and skewness, is described. It is based on the multivariate extended skew-Student-t distribution. Salient features of the distribution are described and these are applied to the task of asset pricing. The paper shows that the market model is non-linear in general and that the sensitivity of asset returns to return on the market portfolio is not the same as the conventional beta, although this measure does arise in special cases. It is shown that the variance of asset returns is time varying and depends on the squared deviation of market portfolio return from its location parameter. The first order conditions for portfolio selection are described. Expected utility maximisers will select portfolios from an efficient surface, which is an analogue of the familiar meanvariance frontier, and which may be implemented using quadratic programming.

How skewness influences optimal allocation in a risky asset?

“…; see Adcock and Shutes (2001). Since R is the sum of two independent variables, the skewness is just the sum of the skewness of addenda, so …”

“…We now model the risky asset with the extended version of a skew-normal distribution proposed by Adcock and Shutes (2001). We denote the skew-normal …”

“…The motivation for this paper is the growing body of empirical literature that documents the inconsistency of the normality assumption, especially with respect to the significant skewness observed in asset returns (see, e.g., Peiró, 1999;Su and Hung, 2011;Xu et al, 2011). We model the risky return with a skew-normal distribution (see Azzalini, 1985;Adcock and Shutes, 2001) that has many attractive features for modeling real asset returns (see Adcock, 2007;Harvey et al, 2010). A skew-normal variable is defined as a Gaussian perturbed via the addition of a skewness shock given by a truncated Gaussian.…”

CAPM with various utility functions: Theoretical developments and application to international data

“…This issue has led many authors to propose alternative forms of the probability density function than the Gaussian. For example, Simaan (1993) proposed a non-spherical distribution, Adcock and Shutes (1999) assumed that the financial asset returns distribution follows a multivariate Skew-Normal, Rachev and Mitnik (2000) defined a Levy-Pareto stable distribution for returns. It seems thus necessary to define an approach that accounts for the first four moments.…”

A closed-form solution of the multi-period portfolio choice problem for a quadratic utility function

“…A number of papers are devoted to questions like, e.g., how an optimal portfolio can be constructed, monitored, and/or estimated by using historical data (see, e.g., Alexander and Baptista (2004) , Golosnoy and Schmid (2007), Bodnar (2009)), what is the influence of parameter uncertainty on the portfolio performance (cf., Okhrin and Schmid (2006) , Bodnar and Schmid (2008)), how do the asset returns influence the portfolio choice (see, e.g., Jondeau and Rockinger (2006), Mencía and Sentana (2009), Adcock (2009), Harvey et al (2010), Amenguala and Sentana (2010)), how is it possible to estimate the characteristics of the distribution of the asset returns (see, e.g., Jorion (1986), Wang (2005), Frahm and Memmel (2010)), how can the structure of optimal portfolio be statistically justified (Gibbons et al (1989), Britten-Jones (1999), Bodnar and Schmid (2009)). …”