The Black–Litterman model: active risk targeting and the parameter tau

There is apparent persistent confusion over certain aspects of Black–Litterman expected returns, with a number of publications offering various explanations, clarifications, and criticisms as to how the model works in practice. The parameter tau (τ) 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 condemnation of Black–Litterman specifically related to tau. This article presents a simple interpretation of tau, shows that it is directly related to the level of active risk implicit in the Black–Litterman model, and is easily calibrated so that Black–Litterman expected returns produce portfolios with targeted levels of active risk. The main contribution is an alternative derivation of Black–Litterman that affords a direct way to target active risk without requiring a specific value for tau. This derivation reveals that portfolio construction using Black–Litterman is equivalent to creating a mean-variance optimal portfolio of active strategies that is then overlaid onto a benchmark portfolio, and by targeting active risk directly, users of the Black–Litterman model do not need to consider tau at all.