A Security Analyst Works for a Large Institution That Uses $\bar { R } _ { \mathrm { A } } = 14 \%$

A security analyst works for a large institution that uses the single-index model as part of its portfolio-management scheme.The security analyst believes the following values are relevant for the four stocks she follows $\bar { R } _ { \mathrm { A } } = 14 \%$ ; $\bar { R } _ { B } = 12 \%$ ; $\bar { R } _ { C } = 8 \%$ ; $\bar { R } _ { D } = 11 \%$ ; $\beta _ { \mathrm { A } } = 2.0$ ; $\beta _ { g } = 1.5$ ; $\beta _ { C } = 1.0$ ; $\beta _ { D } = 1.0$ ; $\sigma _ { \text {eA } } ^ { 2 } = 15$ ; $\sigma _ { e B } ^ { 2 } = 7.5$ ; $\sigma _ { e c } ^ { 2 } = 9$ ; $\sigma _ { e D } ^ { 2 } = 10$ .
The institution assumes that the risk-free rate is 6%,and short selling is not allowed.The institution accepts the Sharpe single-index model and uses the procedure described by Elton,Gruber and Padberg (EGP)to determine the optimum risky-asset portfolio for the institution to hold.The procedure is to compute $$Z _ { i } = \frac { \beta _ { i } } { \sigma _ { e i } ^ { 2 } } \times \left[ ( \text { ranking criterion for asset } i ) - C ^ { * } \right]$$ where the ranking criterion is as described by EGP and where C^* depends on all risky assets the institution holds.The institution's management has determined that C^* = 3.
a. Which stocks that the analyst follows will be held in the institution's optimum portfolio?
b. If the sum of the Z_i's for all the institution's stocks in the optimum portfolio is equal to 4, what fraction of the institution's optimum portfolio will each of the stocks that the analyst follows represent?
c. Why should
$\sigma _ { ei} ^ { 2 }$ (diversifiable risk) enter into the optimal solution?