Goldman Sachs Fixed Income Research, 1991 Fischer Black Robert Litterman Executive Summary A year ago, Goldman Sachs introduced a quantitative model that offered an innovative approach to the management of fixed income portfolios.* It provided a mechanism for investors to make global asset allocation decisions by combining their views on expected returns with Fischer Blackâs âuniversal hedgingâ equilibrium. Given an investorâs views about interest rates and exchange rates, this initial version of the Black-Litterman Global Asset Allocation Model has been used to generate portfolios with optimal weights in bonds in different countries and the optimal degree of currency exposure. In this paper, we describe an updated version of the Black- Litterman Model that incorporates equities as well as bonds and currencies. The new version of the model will be especially useful to portfolio managers who make global asset allocation decisions across equity and fixed income markets, but it will also have advantages for pure fixed income managers. The addition of the equity asset class to the model allows us to use an equilibrium based on both bonds and equities. This equilibrium is more desirable from a theoretical standpoint because it incorporates a larger fraction of the universe of investment assets. In our model (as in any Capital Asset Pricing Model equilibrium), the equilibrium expected excess return on an asset is proportional to the covariance of the assetâs return with the return of the market portfolio. Even for pure fixed income managers, it is useful to use as broad a measure of the âmarket portfolioâ as is practical. As we described in our earlier paper, the equilibrium is important in the model because it provides a neutral reference point for expected returns. This allows the investor to express views only for the assets that he desires; views for the other assets are derived from the equilibrium. By providing a center of gravity for expected returns, the equilibrium makes the modelâs portfolios more balanced than those from standard quantitative asset allocation models. Standard models tend to choose unbalanced portfolios unless artificial constraints are imposed on portfolio composition. I. Introduction Investors with global portfolios of equities and bonds are Igenerally aware that their asset allocation decisions â the proportions of funds that they invest in the asset classes of different countries and the degrees of currency hedging âare the most important investment decisions they make. In attempting to decide on the appropriate allocation, they are usually comfortable with the simplifying assumption that their objective is to maximize expected return for any given level of risk, subject in most cases to various types of constraints. Given the straightforward mathematics of this optimization problem, the many correlations among global asset classes required in measuring risk, and the large amounts of money involved, one might expect that in todayâs computerized world, quantitative models would play a dominant role in this global allocation process. Unfortunately, when investors have tried to use quantitative models to help optimize this critical allocation decision, the unreasonable nature of the results has often thwarted their efforts.2 When investors impose no constraints, asset weights in the optimized portfolios almost always ordain large short positions in many assets. When constraints rule out short positions, the models often prescribe âcornerâ solutions with zero weights in many assets, as well as unreasonably large weights in the assets of markets with small capitalizations. These unreasonable results have stemmed from two wellrecognized problems. First, expected returns are very difficult to estimate. Investors typically have views about absolute or relative returns in only a few markets. In order to use a standard optimization model, however, they must state a set of expected returns for all assets and currencies. Thus, they must augment their views with a set of auxiliary assumptions, and the historical returns that portfolio managers often use for this purpose provide poor guides to future returns. Second, the optimal portfolio asset weights and currency positions in standard asset allocation models are extremely sensitive to the expected return assumptions. The two problems compound each other because the standard model has no way to distinguish strongly held views from auxiliary assumptions, and given its sensitivity to the expected returns, the optimal portfolio it generates often appears to have little or no relation to the views that the investor wishes to express. Confronting these problems, investors are often disappointed when they attempt to use a standard asset allocation model. Our experience has been that in practice, despite the obvious conceptual attractions of a quantitative approach, few global investment managers regularly allow quantitative models to play a major role in their asset allocation decisions. In this paper we describe an approach that provides an intuitive solution to these two problems that have plagued the use of quantitative asset allocation models. The key to our approach is the combining of two established tenets of modern portfolio theory: the mean-variance optimization framework of Markowitz (1952) and the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965). We allow the investor to combine his views about the outlook for global equities, bonds, and currencies with the risk premiums generated by Blackâs (1989) global version of the CAPM equilibrium. These equilibrium risk premiums are the excess returns that equate the supply and demand for global assets and currencies. http://www.hss.caltech.edu/media/from-carthage/filer/262.pdf

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