(19.8) is a decomposition of the return on the managed portfolio. Decompositions of active, benchmark, market, or other types of portfolio returns are derived in an analogous fashion. The only difference is that different portfolio weights are used. Consider an example with an active portfolio consisting of three assets and a linear factor model with two common factors. In this case, K = 2, N = 3, and the decomposition of the portfolio's active return can be written as: rj (t) = w{ (t - T)R( (t) + w$(t- l)Ri (t) + wa2(t- l)Rl (t) = S((t) + Si(t) +w?(t-l)i4(t) + w%(t-T)u£2(t)+wl(t-T)ul{t) i19-9) 1-------- >--------- ' i------------------------------------------ >------------------------------------- ' Factor contribution Specific contribution In the above discussion we provide a simple decomposition of return. That is, assuming a linear factor model, the total return on an arbitrary portfolio can be attributed to exposures to factors such as investment styles, industries, and countries, and to returns specific to individual assets. Within the factor model-based approach, a more sophisticated decomposition of total return first separates out the expected market-related exposure. This approach works as follows.3 Start with an estimate of the portfolio's total return in excess of the local risk-free rate. A portfolio's local excess return can be written as fp (?) - ff (t). It is the sum of the benchmark portfolio's excess return, rb (t) - n (?), and the active portfolio return, ri{t)-rb{t). Alternatively expressed, 4(t)-rf{t) = \rl(t)-4{t)\ + \4(t)-rf{t)\ (19.10) The total active return can be written as the sum of (1) the expected active return and (2) the exceptional active return. The expected active return is defined as the product of the active beta and the expected long-run return on the relevant market. Mathematically, the expected active return is written as Pactive(t) X rjag~!ua(t) where Pactive(?) is defined as the difference between the managed portfolio's beta and the benchmark portfolio's beta. When the benchmark is the same as the market portfolio, the benchmark portfolio's beta is 1. The long-run expected return on the relevant market may be based on history or fixed at some annualized amount such as 10 percent. Expected active return is the part of active return that is consistent with the market. For example, suppose that the portfolio manager's active beta (difference between managed beta and benchmark beta) is zero. In this case, the portfolio manager would not expect to out- or underperform the market in the long run. 3Reference: R. C. Grinold and R. N. Kahn, 1999, Active Portfolio Management: A Quantitative Approach for Producing Superior Returns and Selecting Superior Returns and Controlling Risk, 2nd Edition, New York: McGraw-Hill.