A demystification of the Black–Litterman model: Managing quantitative and traditional portfolio construction

Abstract

have a well-defined n-dimensional covariance matrix ͚; in particular, their covariance matrix is non-singular. If the returns for period t are denoted by r t , we shall write E(r) to mean expected forecasted returns. This is shorthand for E(r t ϩ 1 ͉ ᑣ t), where ᑣ t refers to all information up to and including period t. A second related concept is the (n ϫ 1) vector ␲ representing equilibrium excess returns, either in terms of a theory such as the capital asset pricing model (CAPM) or in the sense of the prevailing supply of value-weighted assets. The latter interpretation corresponds to a global market portfolio demonetised in domestic currency. Algebraically, assuming the validity of the CAPM, ␲ ϭ ␤(␮ m Ϫ r f) where ␮ m is the return on the global market in domestic currency, r f is the riskless (cash) domestic rate of return, ␤ is an (n ϫ 1) vector of asset betas, where ␤ ϭ Cov(r, rЈ w m)/␴ 2 m where rЈ w m is the return on the global market, w m are the weights on the global market, determined by market values, and ␴ 2 m is the variance of the rate of return on the world market. If we let ͚ ϭ Cov(r, rЈ) be the covariance matrix of the n asset classes, then ␲ ϭ ␦͚w m where ␦ ϭ (␮ m Ϫ r f)/␴ 2 m is a positive constant. If returns were arithmetic with no reinvestment, ␦ would be invariant to time, since both numerator and denominator would be linear in time. However, if compounding is present, there may be some time effect. In this paper, we shall only consider approaches is the Black–Litterman (BL) model (Black and Litterman, 1991, 1992). This is based on a Bayesian methodology which effectively updates currently held opinions with data to form new opinions. This framework allows the judgmental managers to give their views/forecasts, these views are added to the quantitative model and the final forecasts reflect a blend of both viewpoints. A lucid discussion of the model appears in Lee (1999). Given the importance of this model, however, there appears to be no readable description of the mathematics underlying it. The purpose of this paper is to present such a description. In the second and third sections we describe the workings of the model and present some examples. In the fourth section we present an alternative formulation which takes into account prior beliefs on volatility. In the second and third sections, particular attention is paid to the interesting issue of how to connect the subjective views of our managers into information usable in the model.