Fat Tails and the Capital Asset Pricing Model

Abstract

Much of the theory of modern finance is based on the assumption that returns follow a multivariate normal distribution. It continues to be widely used mainly for its convenience and tractability despite the growing body of research that suggests that returns on financial assets have distributions with fat tails. Since 1982, there have been many papers which model fat tails using versions and/or developments of the ARCH family of models. More recently, several empirical studies have provided support for the idea that returns on a financial asset follow a Student’s t distribution. However, to date the majority, although not all, of this work has been univariate in nature.

In this paper, we propose the multivariate Student t distribution as a model for asset returns. In addition to its motivation as an empirical model, the paper uses the fact that the multivariate Student distribution arises as a consequence of a Bayesian approach applied to the standard multivariate normal model. The paper describes the general model and its key properties, including some of the implications for portfolio selection. Also described is a model which is the Student form of the market model. This model is similar in structure to the familiar market model, but possesses the property that stock specific volatility is time varying. The Student market model is fitted to monthly data on returns on S&P500 stocks. The paper shows that, for the time period considered, the estimated degrees of freedom in the multivariate Student model is approximately 25 and that there is a substantial effect on estimates of beta and stock specific volatility.