A Black-Litterman Asset Allocation Model Under Elliptical Distributions

Abstract

In optimal portfolio allocation, Black and Litterman (1992) provide for a pioneering framework of allowing to incorporate investors’ views based on a prior distribution to derive a posterior distribution of portfolio returns and optimal asset allocations. Meucci (2005) rephrases the model in terms of investors’ views on the market, rather than just the market parameters as in the original Black and Litterman (1992). This market-based version is believed to be much more parsimonious and allows for a natural extension to directly input views in a non-Normal market. This paper extends Meucci’s market-based version of the Black-Litterman model to the case when returns in the market fall within the class of Elliptical distributions, while also importantly preserving the equilibrium-based assumption in the model. Here within this class for which the Normal distribution is a special case, we develop the explicit form of the posterior distribution after considering proper conditional conjugate-type prior distributions. This resulting posterior allows us to obtain solutions to optimization problems of asset allocation based on a variety of risk measures (e.g. Mean-Variance, Mean-VaR, Mean-Conditional VaR). Elliptical models of portfolio returns have recently crept into the financial literature because of its greater flexibility to accommodate larger tails. As a numerical demonstration, we examine how these principles work in a portfolio with international stock indices.

Keywords: optimal asset allocation, Black-Litterman model, risk measures, Elliptical distributions

JEL Classification: G11, G20