among assets is captured exclusively by the correlation among factors and the asset exposures. In ex ante risk analysis we are interested in forecasts of covariance matrices of factor returns and specific returns. Let I^(t \ t - 1) and A*(t \ t - 1) denote conditional estimates (forecast) of covariance matrices of factor and specific returns, respectively. Forecasts of Yf'{t \ t - 1) and k?'(t \ t - 1) may be obtained by different methods-an important point to remember. Therefore, the forecast of the asset return covariance matrix, which is used to estimate total risk and tracking error, may be actually a combination of two different forecast covariance matrices. Next, we explain how forecasts of the factor return covariance matrices are generated. Factor Return Covariance Matrix Forecasts There are a variety of different methodologies that can be employed to estimate factor return covariance matrices. In this section, we explain one particular methodology that has gained widespread use. When forecasting covariances among factor returns we place relatively more weight on recent returns by weighting the data exponentially. This methodology is consistent with the empirical research that shows that the volatilities of financial returns tend to cluster over time. Exponentially weighted covariance matrices are constructed as follows: Step 1 Start with time series of daily returns on, say, 10 factors over the prior two years (504 days). Let i^(504) with element f£(t) (rth row, kxh column of), denote a 504 X 10 matrix of factor returns (each row represents one day of factor returns and each column represents a time series of a different factor return). Moreover, the first row of i^(504) denotes the most recent day's factor returns whereas the last row represents the factor returns occurring 504 days ago. Each column of Ff{504) is mean-centered (it has subtracted from it the equally weighted sample mean [taken over time]). Step 2 Weight the factor returns in F*(504) so that the weight applied to returns at some past date is half the value it is currently. For example, suppose we set the half-life-the time it takes the weight to reach one-half its current value-to 25 days. In this case, we would apply the weight X = 1 to the most recent day's factor returns in row 1 of i^(504), X1 to the previous day's return, A.2 to returns from two days ago, and so on, until 25 days prior, X25 = 0.50. Solving for the weight X, we get X = 0.501/25 = 0.97. Now, when we form the covariance matrix estimate, we normalize the weights so that they sum to one. We construct new weights such that at days ago their value, a^, is given by X< ;=0 where T = 504