An alternative stochastic model for linear portfolios

Abstract

Stock market returns often tend to follow a non-normal probability distribution due to extreme losses in the tails. These cause fatter tails than normal and consequently heavy-tailed probability distributions are mostly used for modeling returns. In this work, we consider the generalized T (GT) distribution which can be heavy-tailed through its parameters and propose to use it in modeling the random stock returns. The GT distribution also contains the normal, Student t and generalized Laplace distributions as special or limiting cases of the shape parameters. The closed form expressions for the important risk measures are obtained. In case of a portfolio modeling, a multivariate extension of the distribution within the class of elliptical distributions is used. The tail mean-variance portfolio model based on the multivariate GT distribution is developed and optimal portfolio problem is solved. Risk measures for the random return of an asset whose density function is a mixture of the GT densities are obtained. The computability of all expressions derived is shown, and a real data application consisting of seven real stocks from the same sector is given.

Keywords:

• Conditional value-at-risk
• Distribution
• Generalized T (GT)
• Portfolio
• Risk measure
• Tail variance

Acknowledgments

The authors express their sincere thanks to an anonymous reviewer for the valuable comments and suggestions on an earlier version of this manuscript.