1, 2,3,4) Fk{t) = Monthly factor returns from time t- 1 to t {k = 1, 2) Bnk(t - 1) = Factor loadings that are known at time t - 1 (i.e., at the beginning of the tt\\ month). These loadings measure the sensitivity between the factor returns and the original set of four returns (n = 1, 2, 3, 4; k = 1, 2) u (t) = nxh stock's specific return from time t- 1 to t Using matrix notation, we can write (20.2) in a more condensed format: R(t) = Bit - l)F(t) + u{t) (20.3) where R(t) =4x1 vector of excess stock returns from t - 1 to t F(t) =2x1 vector of factor returns from t - 1 to t B(t - 1) = 4 X 2 matrix of factor loadings that are known at time t - 1 u(t) =4x1 vector of stock-specific returns from t- 1 to t (it is assumed that these returns are uncorrelated with one another) We use the factor model presented in (20.3) to write the covariance matrix of excess returns, V(t), in terms of variances and covariances of the factor returns and the security-specific returns. Taking the variance of (20.3), we get V(t) = B(t - l)l(t)B(t- 1)T + A{t) (20.4) where £(?) =2x2 covariance matrix of factor returns A(t) =4x4 covariance matrix of specific returns (we assume that specific returns are uncorrelated; therefore, A(t) is a diagonal matrix with specific return variances as elements) Equation (20.4) shows how the covariance matrix of stock returns can be written in terms of the covariance matrix of factor returns and the covariance matrix of stock-specific returns. Next, we describe how we can estimate Z(t), A(t), and the co-variance matrix of stock returns. Assume for the moment that the factor loadings matrix B(t - 1) is known at time t - 1 and that we have 60 months of history on factor returns F{t) (t =1,2, ... , 60). We can form an estimate of the stock return covariance matrix as follows. 1.    Use the historical time series of factor returns over the past 60 months to estimate the factor return covariance matrix, S(f). 2.    Use (20.3) to construct a time series of stock-specific returns that are defined as u{t) = R{t) - B(t -1 )F{t). This involves generating a 4 X 1 vector of specific returns, u(t), each month (one month at a time) over the 60-month estimation period. 3.    Use the time series of stock-specific returns to estimate the stock-specific co-variance matrix A{t). By assuming zero correlation among specific returns, this simply requires the estimation of stock-specific variances. 4.    An estimate of the stock return covariance matrix is given by V(t) = B(t - 1) t(t)B(t - 1)T + A(t). Note that we are not restricted to estimate 2(f) and A(t) in any particular way.