risk model, is used to measure the risk of a portfolio. For EPM, investors rarely estimate the full variance-covariance matrix directly because the number of individual elements is too large, and for a well-behaved (i.e., nonsingular) matrix, the number of observations used to estimate the matrix must significantly exceed the number of stocks in the matrix/ Instead, most equity portfolio managers use a factor risk model in which individual variances and covariances are expressed as a function of a small set of stock characteristics-such as industry membership, size, and leverage. This greatly reduces the number of unknown risk parameters that the manager needs to estimate. When developing an equity factor risk model, it is important to include all of the variables used to forecast returns among the (potentially larger) set of variables used to forecast risks. This way, the risk model "sees" all of the potential risks in an investment strategy, both those managers are willing to accept and those they would like to avoid. Further, a mismatch between the variables in the return and risk models can produce less efficient portfolios in the optimizer For instance, suppose a return model comprises two factors, each with 50 percent weight: the book-to-price (B/P) ratio and return on equity (ROE). Suppose the risk model, on the other hand, has only one factor: B/P. When forming a portfolio, the optimizer will manage risk only for the factors in the risk model-B/P but not ROE. This inconsistency between the return and risk models can lead to portfolios with extreme positions and higher-than-expected risk. The portfolio will not reflect the original 50-50 weights on the two return factors because the optimizer will dampen the exposure to B/P, but not to ROE. In addition, the risk model's estimate of tracking error will be too low because it will not capture any risk from the portfolio's exposure to ROE. The most direct way to avoid these two problems is to make sure all of the factors in the return model are also included in the risk model (although the converse does not need to be true-there can be risk factors without expected returns). A final issue to consider when developing or selecting a risk model is the frequency of data used in the estimation process. Many popular risk models use monthly returns, whereas some portfolio managers (including us) have developed proprietary risk models that use daily returns. Clearly, when estimating variances and covariances, the more observations, the better. High-frequency data produce more observations and hence more precise and reliable estimates. Further, by giving more weight to recent observations, estimates can be more responsive to changing economic conditions. As a result, risk models that use high-frequency returns should provide more accurate risk estimates. (For a detailed discussion of factor risk models, please see Chapter 20.) Forecasting Transaction Costs Although often overlooked, accurate trade-cost estimates are critical to the EPM process. After all, what really matters is not the gross return a portfolio might £With N stocks, the variance-covariance matrix has N(N + l)/2 elements, consisting of N variances and N(N - l)/2 covariances. For an S&P 500 portfolio, for instance, there are 500 x (500 + l)/2 = 125,250 unknown parameters to estimate-500 variances and 124,750 covariances.