Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale

Abstract

The capital asset pricing model (CAPM) is often based on the Gaussianity or normality assumption. However, such an assumption is frequently violated in practical situations. In this paper, we introduce the symmetric CAPM considering distributions with lighter or heavier tails than the normal distribution. These distributions are symmetric and belong to the family of elliptical distributions. We pay special attention to the family members related to the normal, power-exponential, and Student-t cases, with the power-exponential distribution being particularly considered, as it has not been explored widely. Based on these cases, the expectation-maximization algorithm can be used to facilitate the estimation of model parameters utilizing the maximum likelihood method. In addition, we derive the leverage and local influence methods to carry out diagnostics in the symmetric CAPM. We conduct a detailed case study to apply the obtained results estimating the systematic risk of the financial assets of a Chilean company with real data. We employ the Akaike information criterion to conclude that the studied models provide better results than the CAPM under Gaussianity.