... V"/504/ic(504)_ Step 3 An exponentially weighted covariance matrix forecast is given by £(t\t-l)=F(t)TF(t) The principal advantage of using exponentially weighted forecasts is that they allow the covariance matrix to react quickly to recent market movements. However, some portfolio managers may find that the exponentially weighted covariance matrix forecasts are unreasonably volatile. In this case, we can decrease the decay rate so as to more evenly distribute the weight across historical observations. As discussed at the beginning of this section, within the context of a linear factor model, the estimation of the total return covariance matrix requires that we estimate (1) the covariance matrix of factor returns and (2) the covariance matrix of specific returns. Next, we discuss the estimation of the covariance matrix of specific returns. Specific Return Covariance Matrix Forecasts Specific risk estimates are a function of the estimate of the specific return covariance matrix. The specific return covariance matrix is simply a matrix of zeros with specific return variances along the diagonal. That is, in the calculation of specific risk, it is assumed that specific returns are uncorrelated with each other. We write the forecast for the specific return covariance matrix of N assets at time t as A(t\t-1) = 5f(Hf-l) 0 5j(t\t-l) 0 0 0 0 5&(flf-l) (20.41; where the variance of the nth specific return at time t is given by b2n(t I t -1). Note that, unlike the factor return covariance matrix,15 the specific return covariance matrix has the same dimension as the number of assets (returns). Practitioners apply 15Recall that the dimension of the factor return covariance matrix is based on the number of factors.