Black-Litterman Model for Continuous Distributions

Abstract

The Black-Litterman methodology of portfolio optimization, developed at the turn of the 1990s, combines statistical information on asset returns with investor's views within the Markowitz mean-variance framework. The main assumption underlying the Black-Litterman model is that asset returns and investor's views are multivariate normally distributed. However, the empirical research demonstrates that the distribution of asset returns has fat tails and is asymmetric, which contradicts normality. Moreover, recent advances in risk measurement advocate replacing the variance by risk measures that take account of tail behavior of the portfolio return distribution. This paper extends Black-Litterman theory into general continuous distributions with the risk measured by deviation conditional Value-at-Risk. Using ideas from the Black-Litterman methodology, we design analytical and numerical methods (with variance reduction techniques) for the inverse portfolio optimization that extracts in a stable way statistical information from historical data. We introduce a quantitative model for stating investor's views and blending them consistently with the market information via Bayes formula. Analytical (for elliptical distributions) and numerical methods are presented. The results are illustrated by numerical examples which demonstrate significant impact of the choice of distributions on optimal portfolio weights to the extent that the classical Black-Litterman procedure cannot be viewed as an adequate approximation. This is a new and significantly improved version of the paper.