Investments, tenth edition

  (1)  

 y  

  (2)   

E  (  r  

 C 

  )  

  (3)  

   s    

 C 

  

  (4)  

 U   5   E(r)    2  ½  A    s    

 

2

 

  

 0  

.070  

  0 

 .0700 

 0.1  

.078  

.022  

.0770 

 0.2  

.086  

.044  

.0821 

 0.3  

.094  

.066  

.0853 

 0.4  

.102  

.088  

.0865 

 0.5  

.110  

.110  

.0858 

 0.6  

.118  

.132  

.0832 

 0.7  

.126  

.154  

.0786 

 0.8  

.134  

.176  

.0720 

 0.9  

.142  

.198  

.0636 

 1.0  

.150  

.220  

.0532 

 Table 6.4 

 Utility levels for 

various positions in 

risky assets ( y ) for 

an investor with 

risk aversion  A  5 4 

5

 The derivative with respect to  y   equals     E(r

P

) 2 r

f

2 yAs

P

2

.  Setting this expression equal to zero and solving for 

y  yields Equation 6.7. 

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  C H A P T E R  

6

  Capital Allocation to Risky Assets 

183

 This solution shows that the optimal position in the risky asset is  inversely   proportional 

to the level of risk aversion and the level of risk (as measured by the variance) and directly 

proportional to the risk premium offered by the risky asset. 

 

 Figure 6.6 

Utility as a function of allocation to the risky asset,  y   

Utility

0

0

0.2

0.4

0.6

Allocation to Risky Asset, y

0.8

1

1.2

.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

 A graphical way of presenting this decision problem is to use indifference curve 

 analysis. To illustrate how to build an indifference curve, consider an investor with 

risk aversion  A 5 4 who currently holds all her wealth in a risk-free portfolio yielding 

 Using our numerical example [ r  

 f 

  5 7%,  E ( r  

 P 

 ) 5 15%, and  s  

 P 

  5 22%], and expressing all 

returns as decimals, the optimal solution for an investor with a coefficient of risk aver-

sion  A  5 4 is   

y* 5

.15 2 .07

4 3 .22

2

5

.41  

 In other words, this particular investor will invest 41% of the investment budget in the 

risky asset and 59% in the risk-free asset. As we saw in  Figure 6.6 , this is the value of  y

at which utility is maximized. 

 With 41% invested in the risky portfolio, the expected return and standard deviation 

of the complete portfolio are   

 

E(r

C

) 5 7 1

3.41 3 (15 2 7)4 5 10.28%

 

 s

C

5

.41 3 22 5 9.02%  

 The risk premium of the complete portfolio is  E ( r  

 C 

 )  2   r  

 f 

  5 3.28%, which is obtained by 

taking on a portfolio with a standard deviation of 9.02%. Notice that 3.28/9.02 = .36, 

which is the reward-to-volatility (Sharpe) ratio of any complete portfolio given the 

parameters of this example. 

 Example  6.4 

Capital Allocation 

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184

P A R T   I I

  Portfolio Theory and Practice

r  

 f 

  5 5%. Because the variance of such a portfolio is zero, Equation 6.1 tells us that its 

utility value is  U   5  .05. Now we find the expected return the investor would require 

to maintain the  same  level of utility when holding a risky portfolio, say, with  s  5 1%. 

We use Equation 6.1 to find how much  E ( r ) must increase to compensate for the higher 

value of  s :   

 U 5 E(r) 2 ½ 3 A 3 s

2

 .05 5 E(r) 2 ½ 3 4 3 .01

2

  

 This implies that the necessary expected return increases to   

  

Required 

E(r) 5 .05 1 ½ 3 A 3 s

2

 

 5 .05 1 ½ 3 4 3 .01

2

5 .0502

 

 (6.8)   

 We can repeat this calculation for other levels of  s , each time finding the value of  E ( r ) 

necessary to maintain  U  5 .05. This process will yield all combinations of expected return 

and volatility with utility level of .05; plotting these combinations gives us the indifference 

curve. 

 We can readily generate an investor’s indifference curves using a spreadsheet.  Table 6.5  

contains risk–return combinations with utility values of .05 and .09 for two investors, one 

with  A   5  2 and the other with  A   5  4. The plot of these indifference curves appears in 

 Figure 6.7 . Notice that the intercepts of the indifference curves are at .05 and .09, exactly 

the level of utility corresponding to the two curves.  

 Any investor would prefer a portfolio on the higher indifference curve with a higher cer-

tainty equivalent (utility). Portfolios on higher indifference curves offer a higher expected 

return for any given level of risk. For example, both indifference curves for  A  5 2 have 

the same shape, but for any level of volatility, a portfolio on the curve with utility of .09 

offers an expected return 4% greater than the corresponding portfolio on the lower curve, 

for which  U  5 .05. 

 

 

Figure  6.7 

 demonstrates that more risk-averse investors have steeper indifference 

curves than less risk-averse investors. Steeper curves mean that investors require a greater 

increase in expected return to compensate for an increase in  portfolio risk.