Investments, tenth edition

  

  A   5 2  

  A   5 4  

    s    

  U  5  .05  

  U   5 .09  

  U   5 .05  

  U   5 .09  

  0 

 .0500  

.0900  

.050  

.090 

 .05  

.0525  

.0925  

.055  

.095 

 .10  

.0600  

.1000  

.070  

.110 

 .15  

.0725  

.1125  

.095  

.135 

 .20  

.0900  

.1300  

.130  

.170 

 .25  

.1125  

.1525  

.175  

.215 

 .30  

.1400  

.1800  

.230  

.270 

 .35  

.1725  

.2125  

.295  

.335 

 .40  

.2100  

.2500  

.370  

.410 

 .45  

.2525  

.2925  

.455  

.495 

 .50  

.3000  

.3400  

.550  

.590 

 Table 6.5 

 Spreadsheet 

 calculations of 

indifference curves 

(Entries in columns 

2–4 are expected 

returns necessary 

to provide specified 

utility value.) 

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  C H A P T E R  

6

  Capital Allocation to Risky Assets 

185

 

Higher indifference curves cor-

respond to higher levels of utility. 

The investor thus attempts to find 

the complete portfolio on the high-

est possible indifference curve. 

When we superimpose plots of 

indifference curves on the invest-

ment opportunity set represented 

by the capital allocation line as 

in  

Figure  6.8 

, we can identify the 

 highest possible  indifference curve 

that still touches the CAL. That 

indifference curve is tangent to the 

CAL, and the tangency point corre-

sponds to the standard deviation and 

expected return of the optimal com-

plete portfolio. 

  To  illustrate,   Table  6.6   provides 

calculations for four indifference 

curves (with  

utility levels of .07, 

.078, .08653, and .094) for an inves-

tor with  

A   5  4. Columns (2)–(5) 

use Equation 6.8 to calculate the 

expected return that must be paired 

 Figure 6.7 

Indifference curves for  U  5 .05 and  U  5 .09 with  A  5 2 

and  A  5 4  

E(r)

0

U 

= .09

A 

= 4

A 

= 4

A 

= 2

A 

= 2

U 

= .05

.10

.20

.30

.40

.50

σ

.60

.40

.20

σ

c

 

= .0902

σ

P

 

= .22

σ

E(r)

E(r

P

) 

= .15

E(r

c

) 

= .1028

r

f

  

= .07

C

P

CAL

0

U 

= .094

U 

= .08653

U 

= .078

U 

= .07

 Figure 6.8 

Finding the optimal complete portfolio by using  indifference curves  

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186

P A R T   I I

  Portfolio Theory and Practice

with the standard deviation in column (1) to provide the utility value corresponding to each 

curve. Column (6) uses Equation 6.5 to calculate  E ( r  

 C 

 ) on the CAL for the standard devia-

tion  s  

 C 

  in column (1):   

E(r

C

) 5 r

f

1

3E(r

P

) 2 r

f

4

 

s

C

s

P

5 7 1

315 2 74

 

s

C

22

   

  Figure 6.8  graphs the four indifference curves and the CAL. The graph reveals that the 

indifference curve with  U  5 .08653 is tangent to the CAL; the tangency point corresponds 

to the complete portfolio that maximizes utility. The tangency point occurs at  s  

 C 

  5 9.02% 

and  E ( r  

 C 

 )  5  10.28%, the risk–return parameters of the optimal complete portfolio with 

y * 5 0.41. These values match our algebraic solution using Equation 6.7. 

 We conclude that the choice for  y *, the fraction of overall investment funds to place in 

the risky portfolio, is determined by risk aversion (the slope of indifference curves) and the 

Sharpe ratio (the slope of the opportunity set). 

 In sum, capital allocation determines the complete portfolio, which constitutes the 

investor’s entire wealth. Portfolio  P  represents all-wealth-at-risk. Hence, when returns are 

normally distributed, standard deviation is the appropriate measure of risk. In future chap-

ters we will consider augmenting  P  with “good” additions, meaning assets that improve 

the feasible risk-return trade-off. The risk of these potential additions will have to be mea-

sured by their  incremental  effect on the standard deviation of  P.  

  Nonnormal Returns 

 In the foregoing analysis we assumed normality of returns by taking the standard devia-

tion as the appropriate measure of risk. But as we discussed in Chapter 5, departures from 

normality could result in extreme losses with far greater likelihood than would be plausible 

under a normal distribution. These exposures, which are typically measured by value at 

risk (VaR) or expected shortfall (ES), also would be important to investors. 

 Therefore, an appropriate extension of our analysis would be to present investors with 

forecasts of VaR and ES. Taking the capital allocation from the normal-based analysis 

as a benchmark, investors facing fat-tailed distributions might consider reducing their 

allocation to the risky portfolio in favor of an increase in the allocation to the risk-free 

vehicle.