Investments, tenth edition

 Figure 6.4 

The investment opportunity set with a risky asset and 

a risk-free asset in the expected return–standard  deviation plane  

E(r)

 

σ

P

 

= 22%

E(r

P

) 

= 15%

σ

r

ƒ

 

= 7%

F

P

E(r

P

) 

− r

ƒ

 

= 8%

CAL 

= Capital

           Allocation

           Line

 S 

= 8/22

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bod61671_ch06_168-204.indd   179

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180 

P A R T   I I

  Portfolio Theory and Practice

standard deviation at the rate of 22%, according to Equation 6.4. The extra return per extra risk 

is thus 8/22 5 .36. 

 

To derive the exact equation for the straight line between  

F 

 and  

P, 

 we rearrange 

Equation 6.4 to find that  y  5  s  

 C 

 / s  

 P 

 , and we substitute for  y  in Equation 6.3 to describe the 

expected return–standard deviation trade-off:   

 

 E(r

C

) 5 r

f

1 y

3E(r

P

) 2 r

f

4

 

  5 r

f

1

s

C

s

P

 

3E(r

P

) 2 r

f

4 5 7 1

8

22

 

s

C

 

 (6.5)

   

 Thus the expected return of the complete portfolio as a function of its standard devia-

tion is a straight line, with intercept  r  

 f  

   and  slope   

 

S 5

E(r

P

) 2 r

f

s

P

5

8

22

 

 (6.6)   

  Figure 6.4   graphs  the   investment opportunity set,  which is the set of feasible expected 

return and standard deviation pairs of all portfolios resulting from different values of  y.   The 

graph is a straight line originating at  r  

 f 

  and going through the point labeled  P.  

  This straight line is called the    capital  allocation  line    (CAL). It depicts all the risk–return 

combinations available to investors. The slope of the CAL, denoted  S,  equals the increase 

in the expected return of the complete portfolio per unit of additional standard deviation—

in other words, incremental return per incremental risk. For this reason, the slope is called 

the    reward-to-volatility  ratio    .  It also is called the Sharpe ratio (see Chapter 5). 

 A portfolio equally divided between the risky asset and the risk-free asset, that is, where 

 y  5 .5, will have an expected rate of return of  E ( r  

 C 

 ) 5 7 1 .5  3  8 5 11%, implying a risk 

premium of 4%, and a standard deviation of  s  

 C 

  5 .5  3  22 5 11%. It will plot on the line 

 FP  midway between  F  and  P.  The reward-to-volatility ratio is  S  5 4/11 5 .36, precisely 

the same as that of portfolio  P.  

 What about points on the CAL to the right of portfolio  P ? If investors can borrow at the 

(risk-free) rate of  r  

 f 

  5 7%, they can construct portfolios that may be plotted on the CAL 

to the right of  P.  

 

 

 Can the reward-to-volatility (Sharpe) ratio,  S  5 [ E ( r  

 C 

 )  2   r  

 f 

 ]/ s  

 C  

 , of any combination of the risky asset and 

the risk-free asset be different from the ratio for the risky asset taken alone, [ E ( r  

 P 

 )  2   r  

 f 

 ]/ s  

 P 

 , which in this 

case is .36? 

 CONCEPT CHECK 

6.5 

 Suppose the investment budget is $300,000 and our investor borrows an additional 

$120,000, investing the total available funds in the risky asset. This is a  levered  position 

in the risky asset, financed in part by borrowing. In that case   

y 5

420,000

300,000

5

1.4  

 and 1  2   y  5 1  2  1.4 5  2 .4, reflecting a short (borrowing) position in the risk-free asset. 

Rather than lending at a 7% interest rate, the investor borrows at 7%. The distribution 

of the portfolio rate of return still exhibits the same reward-to-volatility ratio:   

 

E(r

C

) 5 7% 1 (1.4 3 8%) 5 18.2%