Investments, tenth edition

 Figure 7.7 

The opportunity set of the debt and equity funds with the optimal 

CAL and the optimal risky portfolio  

 Using our data, the solution for the optimal risky portfolio is   

w

D

5

(8

2 5)400 2 (13 2 5)72

(8

2 5)400 1 (13 2 5)144 2 (8 2 5 1 13 2 5)72

5 .40

w

E

 5 1 2 .40 5 .60  

 Example  7.2 

Optimal Risky Portfolio 

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218 

P A R T   I I

  Portfolio Theory and Practice

 In Chapter 6 we found the optimal  complete  portfolio given an optimal  risky   portfolio 

and the CAL generated by a combination of this portfolio and T-bills. Now that we have 

constructed the optimal risky portfolio,  P,  we can use the individual investor’s degree of 

risk aversion,  A,  to calculate the optimal proportion of the complete portfolio to invest in 

the risky component. 

 

 The expected return and standard deviation of this optimal risky portfolio are   

E (r

P 

) 5 (.4 3 8) 1 (.6 3 13) 5 11%

s

P

 5 [(.4

2

 3 144) 1 (.6

2

 3 400) 1 (2 3 .4 3 .6 3 72)]

1/2

 5 14.2%  

 This asset allocation produces an optimal risky portfolio whose CAL has a slope of   

S

P

5

11

2 5

14.2

5 .42 

which is the Sharpe ratio of portfolio  P.  Notice that this slope exceeds the slope of any 

of the other feasible portfolios that we have considered, as it must if it is to be the slope 

of the best feasible CAL. 

  

7

 Notice that we express returns as decimals in Equation 7.14. This is necessary when using the risk aversion 

parameter,  A,  to solve for capital allocation. 

 Now that asset allocation is decided, we can find each investor’s optimal capital allo-

cation. An investor with a coefficient of risk aversion  A    5   4 would take a position in 

portfolio  P  of  

7

  

    

 

y

5

E(r

P

)

2 r

f

As

P

2

5

.11

2 .05

4

3 .142

2

5 .7439 

 (7.14)  

Thus the investor will invest 74.39% of his or her wealth in portfolio  P  and 25.61% 

in T-bills. Portfolio  P  consists of 40% in bonds, so the fraction of wealth in bonds will 

be  yw  

 D 

   5  .4  3  .7439  5  .2976, or 29.76%. Similarly, the investment in stocks will be 

 yw  

 E 

   5  .6  3  .7439  5  .4463, or 44.63%. The graphical solution of this asset allocation 

problem is presented in  Figures 7.8  and  7.9.