market portfolio. ejt) = Mean-zero, random component known as residual return. This return is uncorrected with the market return, a (t) = Alpha-captures difference between market component and asset return. Using the definition, r (t) = wm(t - 1 )TR(t), we can derive an expression for predicted beta. For the Kth asset, its predicted beta is: P(f|f-l)=y(flf-1)T"r(f-1) (20.94) <5m(tf where v{t I t - 1) is an N X 1 vector of covariances, representing the Kth row of the covariance matrix V(t \ t - 1). To better understand the definition of predicted beta, consider the case where the market portfolio consists of two assets and we are interested in finding the predicted beta for these two assets. In this case, r (t) = w(t - 1 )R1(t) + w(t - 1 )R1(t) and Pi(flf-1) = M\t-\) = O^ (t \t -l)w?(t-l) + G?2(t\t-l)w2(t-l) om(tf o111(t\t-X}w?(t-X) + G11(t\t-l)w1{t-l) om(tf (20.95; In terms of a linear factor model, the expression for predicted beta is given by P(r I r-l) = om(tf V(t\t-l)wm{t-l) (20.96 _[B(t-lfWt-lMt-DTy'(t-l) | A(t\t-l)wm(t-l) om(t)2 om(tf Here, p(£ I t - 1) is an N X 1 vector of predicted betas. A portfolio's beta is defined as the weighted average of individual betas p (?) = wp{t - 1)TP(? I t - 1). Similar results hold for the benchmark and active portfolios. Note from (20.96) that predicted beta consists of two components-factor and specific. This is a useful result because we can determine the source of a portfolio's beta. If a large amount of a portfolio's beta is due to factor exposure, then we can reduce its overall exposure to the market by reducing its factor exposure. On the other hand, if the specific risk component dominates a portfolio beta, then a manager should not focus on factors to reduce market exposure. Predicted beta is also known as fundamental beta since its value depends on asset exposures to fundamental factors through the term B(t - h). Another mea-