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Setting the Cut-off Rate |C*)

As discussed earlier, C* is the cut-off rate. All securities whose excess-return-to-risk ratio is above the cut-off rate are selected and all whose ratios are below are rejected. The value of C* is computed from the characteristics of all of the securities that belong in the optimum portfolio. To determine C* it is necessary to calculate its value as if there were different numbers of securities in the optimum portfolio. Designate C. as a candidate for C*. I he value of Ci is calculated when i securities are assumed to belong to the optimal portfolio.

Since securities are ranked from highest excess return to Beta to lowest, we know that if a particular security belongs in the optimal portfolio, all higher ranked securities also belong in the optimal portfolio. We proceed to calculate values of a variable Ci (the procedure is outlined below) as if the first ranked security was in the optimal portfolio (i = I), then the first and second ranked securities were in the optimal portfolio (i = 2), then the first, second, and third ranked securities were in the optimal portfolio (i = 3), and so forth. These C are candidates for C*. We know we have found the optimum C;—that is

C*—when all securities used in the calculation of Ci have excess returns to Beta above C. and all securities not used to calculate C. have excess returns to Beta below Cf. For example, column 7 of Table 9.2 shows the C. for alternative values of i. Examining the table shows that C5 is the only value of C; for which all securities used in the calculation of i (1 through 5 in the table) have a ratio of excess return to Beta above C; and all securities not used in the calculation of C. (6 through 10 in the table) have an excess return to Beta ratio below Cr C5 serves the role of a cut-off rate in the way a cut-off rate was defined earlier. In particular, C5 is the only Ci that when used as a cut-off rate selects only the stocks used to construct it. There will always be one and only one C. with this property and it is C*.

Calculating the Cut-off Rate C*

Recall that stocks are ranked by excess return to risk from highest to lowest. For a portfolio of i stocks Cl is given by

(j2m = the variance in the market index u2j = the variance of a stock's movement that is not associated with the movement of the market index. This is usually referred to as a stock's unsystematic risk.

This looks horrible. But a moment's reflection combined with a peek at the example below will show that it is not as hard to compute as it appears. While Equation (9.1) is the form that should actually be used to compute C;, this expression can be stated in a mathematically equivalent way that clarifies the meaning of C2

P;p = the expected change in the rate of return on stock i associated with a 1% change in the return on the optimal portfolio Rp = the expected return on the optimal portfolio All other terms as before.

Pip and Rp are, of course, not known until the optimal portfolio is determined. Hence, Equation (9.2) could not be used to actually determine the optimum portfolio; rather, Equation (9.1) must be used. However, this expression for C; is useful in interpreting the economic significance of our procedure. Recall that securities are added to the portfolio as long as

where

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