as is regression analysis, where the coefficients represent the return to a portfolio with unit exposure to the signal. These portfolios can be equal-weighted, cap-weighted, or even risk-weighted depending on the model's ultimate purpose. Finally, the return forecasting model should be tested using a realistic simulation that controls the target level of risk, takes account of transaction costs, and imposes appropriate constraints (e.g., the non-negativity constraint for long-only portfolios). In our experience, many promising return-forecasting signals fail to add value in realistic back tests-either because they involve excessive trading; work only for small, illiquid stocks; or contain information that is already captured by other components of the model. The third step in building a return-forecasting model is determining each signal's weight. When computing expected returns, more weight should be put on signals that, over time, have been more stable, generated higher and more consistent returns, and provided superior diversification benefits. Maintaining exposures to signals that change slowly requires less trading, and hence lower costs, than is the case for signals that change rapidly. Other things being equal, a stable signal (such as the ratio of book-to-market equity) should get more weight than a less stable signal (such as one-month price reversal). High, consistent returns are essential to a profitable, low-risk investment strategy; hence, signals that generate high returns with little risk should get more weight than signals that produce lower returns with higher risk. Finally, signals with more diversified payoffs should get more weight because they can hedge overall performance when other signals in the model perform poorly. The last step in forecasting returns is to make sure the forecasts are reasonable and internally consistent by comparing them with equilibrium views. Return forecasts that ignore equilibrium expectations can create problems in the portfolio construction step. Seemingly reasonable return forecasts can cause an optimizer to maximize errors rather than expected returns, producing extreme, unbalanced portfolios. The problem is caused by return forecasts that are inconsistent with the assumed correlations across stocks. If two stocks (or subportfolios) are highly correlated, then the equilibrium expectation is that their returns should be similar; otherwise, the optimizer will treat the pair of stocks as a (near) arbitrage opportunity by going extremely long the high-return stock and extremely short the low-return stock. However, with hundreds of stocks, it is not always obvious whether certain stocks, or combinations of stocks, are highly correlated and therefore ought to have similar return forecasts. The Black-Litterman model was specifically designed to alleviate this problem. It blends a model's raw return forecasts with equilibrium expected returns-which are the returns that would make the benchmark optimal for a given risk model-to produce internally consistent return forecasts that reflect the manager's (or model's) views yet are consistent with the risk model. (For a discussion of how to use the Black-Litterman model to incorporate equilibrium views into a return-fore casting model, please see Chapter 7.) Forecasting Risks In a portfolio context, the risk of a single stock is a function of the variance of its returns, as well as the covariances between its returns and the returns of other