in the direction of the view portfolio-that is, a proportional increase in the market portfolio offset by a short position in German equities. The model provides the appropriate weights on the view portfolio, given the stated expected return on the portfolio and the degree of confidence in that view. The model balances the contributions to expected return of the view portfolio and the market portfolio against their contributions to overall portfolio risk. The result is transparent and intuitive. How does this approach differ from the badly behaved approach of the standard optimizer? In both cases the unconstrained optimal portfolio, zf:1", is given by the same matrix equation: w* = kZ'V* (7.2) where K = Risk aversion parameter S = Covariance matrix of excess returns u,* = Vector of expected excess returns The difference between the Black-Litterman approach and the previous approach is that rather than specifying the expected excess returns directly, we define view portfolios, specify expected returns and degrees of confidence in the view portfolios, and apply the following Black-Litterman formula:3 \i* = [(xS)"1 + P'U"1 PY^xiym + P'O^Q] (7.3) This formula creates an expected excess return vector, u,;i", from the information in k views: Pu. = Q + e (7.4) and in a prior reflecting equilibrium: ji = n + ee (7.5) In these formulas P is a k X n matrix specifying k view portfolios in terms of their weights on the n assets. Q is a ^-vector expressing the expected excess returns on the k view portfolios. Q is the covariance matrix of the random variables representing the uncertainty in the views, n is the w-vector of equilibrium risk premiums. Finally, % scales the covariance matrix of returns in order to specify the covariance matrix of the zero-mean distribution for ee. Let us look at the Black-Litterman expected excess returns. These expected excess returns and their deviations from equilibrium are given in Table 7.6. In 3This formula was derived in the paper "Global Portfolio Optimization," by Fischer Black and Robert Litterman, Financial Analysts Journal, September-October 1992, pages 28^43. In a subsequent paper, "A Demystification of the Black-Litterman Model: Managing Quantitative and Traditional Portfolio Construction," published in the Journal of Asset Management, 2000, vol. 1, no. 2, pages 138-150, Stephen Satchell and Alan Scowcroft extend the analysis.