A Black-Litterman asset allocation model under Elliptical distributions

In optimal portfolio allocation, Black and Litterman [ Financ. Anal. J. , 1992, 48 , 28-43] 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 [ Risk and Asset Allocation , 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 [ Financ. Anal. J. , 1992, 48 , 28-43]. This market-based version is believed to be much more parsimonious and allows for a more 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 their greater flexibility to accommodate larger tails. As numerical demonstrations, we examine how these principles work in a portfolio with international stock indices and show why models with more flexible tails can affect the asset allocation.