asset. As before, we search for the capital allocation line with the highest reward-to-variability ratio (that is, the steepest slope) as shown in Figure 8.11. The CAL that is supported by the optimal portfolio, P, is tangent to the efficient frontier. This CAL dominates all alternative feasible lines (the broken lines that are drawn through the frontier). Portfolio P, therefore, is the optimal risky portfolio. Finally, in the last part of the problem the individual investor chooses the appropriate mix between the optimal risky portfolio P and T-bills, exactly as in Figure 8.8. Now let us consider each part of the portfolio construction problem in more detail. In the first part of the problem, risk-return analysis, the portfolio manager needs as inputs a set of II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001           CHAPTER 8 Optimal Risky Portfolios 227     estimates for the expected returns of each security and a set of estimates for the covariance matrix. (In Part V on security analysis we will examine the security valuation techniques and methods of financial analysis that analysts use. For now, we will assume that analysts already have spent the time and resources to prepare the inputs.) Suppose that the horizon of the portfolio plan is one year. Therefore, all estimates pertain to a one-year holding period return. Our security analysts cover n securities. As of now, time zero, we observed these security prices: P0, . . . , P0. The analysts derive estimates for each 1 n securitys expected rate of return by forecasting end-of-year (time 1) prices: E(P1), . . . , E(P1), and the expected dividends for the period: E(D1), . . . , E(Dn). The set of expected rates of return is then computed from E(P1) E(D ) P0 E(ri) 0 i The covariances among the rates of return on the analyzed securities (the covariance ma- trix) usually are estimated from historical data. Another method is to use a scenario analysis of possible returns from all securities instead of, or as a supplement to, historical