A stable CAPM in the presence of heavy-tailed distributions

Abstract

One of the main assumptions of CAPM is the normality of returns. A powerful statistical for the Gaussian assumption is the Central Limit Theorem (CLT), which states that the sum of a large number of independent and identically distributed (iid) random variables (r.v.'s) from a finite-variance distribution will tend to be normally distributed. Due to the influential works of Mandelbrot (1963), however, the stable non-GAussian, or rather, α-stable distribution has often been considered to be a more realistic one for asset returns than the normal ditribution. This is because asser returns are typically fat—tailed adn excessively peaked around zero—phenomena that can be captured by α-stable distributions with α < 2. The α-stable distributional assumption is a generalization rather than an alternative to the Gaussian distribution, since the latter is a special case of the former. According to Generalized CLT, the limiting distribution of the sum of a large number of iid r.v.'s must be a stable distribution, see Zolotarev (1986).