Encyclopedia of Financial Models I

Bayesian Techniques and the Black-Litterman Model

PETTER N. KOLM, PhD

Director of the Mathematics in Finance Masters Program and Clinical Associate Professor, Courant Institute of Mathematical Sciences, New York University

FRANK J. FABOZZI, PhD, CFA, CPA

Professor of Finance, EDHEC Business School

SERGIO M. FOCARDI, PhD

Partner, The Intertek Group

Abstract: Investment policies constructed using inferior estimates, such as sample means and sample covariance matrices, typically perform very poorly in practice. Besides introducing spurious changes in portfolio weights each time the portfolio is rebalanced, this undesirable property also results in unnecessary turnover and increased transaction costs. These phenomena are not necessarily a sign that portfolio optimization does not work, but rather that the modern portfolio theory framework is very sensitive to the accuracy of inputs. There are different ways to address this issue. On the estimation side, one can try to produce more robust estimates of the input parameters for the optimization problems. This is most often achieved by using estimators that are less sensitive to outliers, and possibly, other sampling errors, such as Bayesian and shrinkage estimators. On the modeling side, one can constrain portfolio weights, use portfolio resampling, or apply robust or stochastic optimization techniques to specify scenarios or ranges of values for parameters estimated from data, thus incorporating uncertainty into the optimization process ...