Uncertainty in the Black-Litterman model: A practical note

Deriving an optimal asset allocation for institutional investors hinges crucially on the quality of inputs used in the optimization. If the mean vector and the covariance matrix are known with certainty, the classical mean-variance optimization of Markowitz (1952) produces optimal portfolios. If, however, both and are estimated with uncertainty, mean-variance optimization tends to maximize estimation error, as shown in Michaud (1989). The Black-Litterman model (Black and Litterman (1991, 1992)), a derivation of the Bayesian methods developed in academia, has particular practical appeal for institutional investors. It allows the specification of views and an uncertainty about these views, which are combined with equilibrium returns and incorporated consistently to arrive at and .These new parameters can then be used in the portfolio optimization process. In the Black-Litterman model, however, uncertainty about the equilibrium returns is specified with an overall scalar uncertainty parameter, which is difficult to set and introduces rigidity.We propose a slight modification of the Black-Litterman model to allow the specification of uncertainty in a flexible way not only in individual views, but also in the equilibrium returns of every asset entering the model. Our modification is an "add-on" to the traditional framework, which allows to adjust the uncertainty individually and is still permitting the Black-Litterman approach as a special case.