it allows portfolio managers to view their portfolios' risk in a variety of ways. Important Note: Contribution to Risk by Sector Sectors contain one or more industries. In practice, it is common to report exposures by sector as well as contribution to tracking error by sector. For a particular sector, its exposure is simply the sum of all industry exposures that belong to that sector. Similarly, for a particular sector, the sum of its respective industry contributions to risk is equal to that sector's contribution to risk. For example, suppose that a sector consists of two industries-industry A and industry B-and a portfolio has an active exposure of 25 percent to industry A and -25 percent (underweight) to industry B. Since these industries are (the only industries) in the same sector, the portfolio's sector exposure is zero. Moreover, suppose that both industries have a contribution to risk of 30 percent. (Note that the total contributions to risk over both industries is not 100 percent because we are assuming that the portfolio has exposure to other industries.) When we add the two industry contributions to risk, we find that the contribution to risk from the sector is 60 percent. Mathematically, we have established the following result. Let there be St (s = 1, . . . , St) industries in the zth sector. In this example, the sum of the industry exposures for a particular sector is zero: that is, 5>f(*-u=o (20-91> Now, the contribution to tracking error from a sector computed as the sum of contributions to tracking error by the individual industries is given by Y^-^xAFCTE, <20-92> In our example, (20.92) is equal to 0.60. Predicted Beta A portfolio manager seeking a forward-looking view of how the managed portfolio varies with a market or benchmark portfolio can look at the portfolio's predicted beta. In order to derive a portfolio's predicted beta, we refer back to the market model introduced in the section on macroeconomic factors. The market model for the nxh asset return is RJt) = ajt) + $n(t)rjt) + ejt) (20.93) where RJt) = Excess return on the ?zth asset over the risk-free rate at time t. rm{t) = Return on a market portfolio over the risk-free rate.