The Black-Litterman Model: Active Risk Targeting and the Parameter Tau

Abstract

While the Black-Litterman (BL) model of expected returns is well-known and widely used throughout the investment management industry, there is apparent persistent confusion over certain aspects of the model, with a number of publications offering various explanations and clarifications as to how the model works in practice. The parameter tau has proved to be a particularly confounding feature of the model: a wide range of opinions and suggestions on how to interpret and quantify tau has accumulated in the literature, and includes some harsh criticism of the BL model specifically related to tau. In this article we consider tau in the context of active risk. First, we show how tau is directly related to the level of active risk implicit in BL expected returns, and that tau can be calibrated to target a desired level of active risk, but only up to a certain maximum. We next introduce an alternative derivation of the BL model that provides a more direct approach to active risk targeting, allows for an unlimited range of active risk, and for which tau is irrelevant, and we show that an investor who targets a specific level of active risk using the BL model does not need to consider tau at all.