The Limits of Diversification and the Pricing of Tail Risk

Abstract

This paper adopts a copula approach at assessing the dependence structure of the U.S. equity market. Seven types of copulas are considered: Gaussian, Student t, Clayton, rotated Clayton, Gumbel, rotated Gumbel and BB4. By adopting a twenty-two year sample of daily returns on the seventeen Fama and French (1993) industry portfolios, it is found that the Stundet t copula provides the best representation of the dependence structure of portfolio returns. This result suggests that equity returns have thicker tails than those implied by the Gaussian distribution. In addition, it is found that the Gaussian, Gumbel and rotated Clayton copulas overstate the benefits of diversification in the economy, while the rotated Gumbel and Clayton copulas tend to understate them. Since risk aversion and efficient markets suggest that investors should demand a premium for bearing additional ‘tail risks’, this paper provides an initial approach at assessing whether different levels of tail dependence imply different market premiums. The results show that negative tail dependence is associated with positive premiums and that positive tail dependence is associated with negative premiums. However, when the covariance with the market is taken into account, the significance of these premiums dissipates, especially for symmetric distributions.

Keywords: Asset Pricing, Copulas, Diversification, Maximum Likelihood

JEL Classification: G11, G12, C12, C13, C51