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, empirical research demonstrates that the distribution of asset returns has fat tails and is asymmetric, which contradicts normality. 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 the Black–Litterman model into general continuous distributions and deviation measures of risk. Using ideas from the Black–Litterman methodology, we design numerical methods (with variance reduction techniques) for the inverse portfolio optimization that extracts statistical information from historical data in a stable way. We introduce a quantitative model for stating investor’s views and blending them consistently with the market information. The theory is complemented by efficient numerical methods with the implementation distributed in the form of publicly available R packages. We conduct practical tests, 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.