the portfolio proportions, wi, the expected return, E(rp), and the standard deviation, p. Let us restate what our portfolio manager has done so far. The estimates generated by the analysts were transformed into a set of expected rates of return and a covariance matrix. This group of estimates we shall call the input list. This input list is then fed into the opti- mization program. Before we proceed to the second step of choosing the optimal risky portfolio from the frontier set, let us consider a practical point. Some clients may be subject to additional con- straints. For example, many institutions are prohibited from taking short positions in any asset. For these clients the portfolio manager will add to the program constraints that rule out negative (short) positions in the search for efficient portfolios. In this special case it is possible that single assets may be, in and of themselves, efficient risky portfolios. For ex- ample, the asset with the highest expected return will be a frontier portfolio because, with- out the opportunity of short sales, the only way to obtain that rate of return is to hold the asset as ones entire risky portfolio. Short-sale restrictions are by no means the only such constraints. For example, some clients may want to ensure a minimal level of expected dividend yield from the optimal port- II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001           CHAPTER 8 Optimal Risky Portfolios 229     folio. In this case the input list will be expanded to include a set of expected dividend yields d1, . . . , dn and the optimization program will include an additional constraint that ensures that the expected dividend yield of the portfolio will equal or exceed the desired level, d. Portfolio managers can tailor the efficient set to conform to any desire of the client. Of course, any constraint carries a price tag in the sense that an efficient frontier constructed subject to extra constraints will offer a reward-to-variability ratio inferior to that of a less constrained one. The client should be made aware of this cost and should carefully consider constraints that are not mandated by law. Another type of constraint is aimed at ruling out investments in industries or countries considered ethically or politically undesirable. This is referred to as socially responsible investing, which entails a cost in the form of a lower reward-to-variability on the resultant constrained, optimal portfolio. This cost can be justifiably viewed as a contribution to the underlying cause.       8.5 A SPREADSHEET MODEL