A Generalized Black–Litterman Model

The Black–Litterman model provides a framework for combining the forecasts of a backward-looking equilibrium model with the views of (several) forward-looking experts in a portfolio allocation decision. The classical version uses the capital asset pricing model to specify expected returns, and assumes that expert views are unbiased noisy observations of future returns. It combines the two using Bayes’ rule and the portfolio allocation decision is made on the basis of the updated forecast. The classical Black–Litterman model assumes that the equilibrium and expert models are accurately specified. This is generally not the case, however, and there may be substantial efficiency loss if misspecification is ignored. In this paper, we formulate a generalized Black–Litterman model that accounts for both misspecification and bias in the equilibrium and expert models. We show how to calibrate this model using historical view and return data, and study the value of our generalized model for portfolio construction. More generally, this paper shows how the views of multiple experts can be modeled as a Bayesian graphical model and estimated using historical data, which may be of interest in applications that involve the aggregation of expert opinions for the purpose of decision making.