that asset returns have exposures to the following classes of factors: investment styles, industries, countries, and currencies. Let's rewrite (20.56) so that the exchange rate returns appear as factors. To do so we add columns of ones and zeros to the exposures matrix B(t - 1) and rows (of returns) to the vector F*{t). R(t) =B(t-l )F(t) + u<{t) (20.57) where F{t) = [F{t)\E (t) + xc(t)] and B(t- 1) incorporates the exposures to currency factors. In practice, we may choose to ignore the cross term xc(t). The forecast covariance matrix of asset returns, as of time t - 1, is based on forecasting the variance of the N-vector R(t - 1) as specified by equation (20.57). The forecast covariance matrix of R{t- 1) is V(t\t-l)=B(t-h)l(t\t-l)B(t-b)T+A(t\t-l) (20.58) where h > 1 and £(? I t - 1) is the covariance matrix of factor returns which include investment styles, industries, countries, and currencies. The notation "t I t - 1" reads as "the time t forecast given information up to and including time t - 1." A(t\t - 1) is the specific return variance matrix. We can think of the factor return co-variance matrix as a four-by-four block expressed thus: Ind & IS Cty & IS Ccy & IS Industry (Ind) Cty & Ind Ccy & Ind Ind & Cty Countries (Cty) Ccy & Cty Ind & Ccy Cty & Ccy Currencies (Ccy) l{t\t-l) = Investment styles (IS) IS &Ind IS & Cty IS & Ccy Equation (20.59) shows that each class of factors represents a block along the diagonal of S(t I t-1). The off-diagonal elements involve the interaction among the factor returns. When we measure the risk of a portfolio, the part coming from factors is, in effect, a sum of components of the matrix in (20.59) that are weighted by the factor exposures, that is, B(t - h). Equipped with expressions for the covariance matrix of stock returns, we can formulate the expression for the variance of the managed and active portfolios. POPtfoliO Risk Measures The variance of the managed portfolio return is given by the expression <51p{t) = WP{t)TV(t\t-l)wP{t) = bP{t - \]L(t 11 - l)bP{t -If + wP{t - XfUt 11 - l)ivP{t-l) yzu-bU> where bp(t - 1) = ivp{t- l)TB(t -h). Equation (20.60) provides a measure of a managed portfolio's total risk (squared). In practice, this number is usually reported in standard deviation terms, that is, O (t). The portfolio's factor and specific risk components are given by