Investments, tenth edition

 Figure 6.3 

Spread between 3-month CD and T-bill rates  

1970

1972

1974

1976

1978

1980

1982

1984

1986

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

2008

2010

2012

2014

OPEC I

OPEC ll

Penn Square

Market Crash

LTCM

Credit Crisis

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Percentage Points

  

2

 See    http://www.icifactbook.org/  , Section 4 of Data Tables. 

    6.4 

Portfolios of One Risky Asset 

and a Risk-Free Asset 

  In this section we examine the feasible risk–return combinations available to investors 

when the choice of the risky portfolio has already been made. This is the “technical” part 

of capital allocation. In the next section we address the “personal” part of the problem—

the individual’s choice of the best risk–return combination from the feasible set. 

 Suppose the investor has already decided on the composition of the risky portfolio,  P.

Now the concern is with capital allocation, that is, the proportion of the investment budget, 

y,  to be allocated to  P.  The remaining proportion, 1  2   y,  is to be invested in the risk-free 

asset,  F.  

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  C H A P T E R  

6

  Capital Allocation to Risky Assets 

179

 Denote the risky rate of return of  P  by  r  

 P 

 , its expected rate of return by  E ( r  

 P 

 ), and its 

standard deviation by  s  

 P 

 . The rate of return on the risk-free asset is denoted as  r  

 f 

 . In the 

numerical example we assume that  E ( r  

 P 

 )  5  15%,   s  

 P 

   5  22%, and the risk-free rate is 

 r  

 f 

  5 7%. Thus the risk premium on the risky asset is  E ( r  

 P 

 )  2   r  

 f 

  5 8%. 

 With a proportion,  y,  in the risky portfolio, and 1  2   y  in the risk-free asset, the rate of 

return on the  complete  portfolio, denoted  C,  is  r  

 C 

   where   

 

r

C

5 yr

P

1 (1 2 y)r

f

 

 (6.2)   

 Taking the expectation of this portfolio’s rate of return,   

 

 E(r

C

) 5 yE(r

P

) 1 (1 2 y)r

f

 

  5 r

f

1 y

3E(r

P

) 2 r

f

4 5 7 1 y(15 2 7) 

 

(6.3)

   

 This result is easily interpreted. The base rate of return for any portfolio is the risk-free 

rate. In addition, the portfolio is  expected  to earn a proportion,  y,  of the risk premium of the 

risky portfolio,  E ( r  

 P 

 )  2   r  

 f  

 . Investors are assumed risk averse and unwilling to take a risky 

position without a positive risk premium. 

 With a proportion  y  in a risky asset, the standard deviation of the complete portfolio is 

the standard deviation of the risky asset multiplied by the weight,  y,  of the risky asset in 

that portfolio.  

3

   Because the standard deviation of the risky portfolio is  s  

 P 

  5 22%,   

 

s

C

5 ys

P

5 22y 

 (6.4)    

 which makes sense because the standard deviation of the portfolio is proportional to both 

the standard deviation of the risky asset and the proportion invested in it. In sum, the 

expected return of the complete portfolio is  E ( r  

 C 

 ) 5  r  

 f 

  1  y [ E ( r  

 P 

 )  2   r  

 f 

 ] 5 7 1 8 y  and the 

standard deviation is  s  

 C 

  5 22 y.  

 The next step is to plot the  portfolio characteristics (with various choices for  y ) in the 

expected return–standard deviation plane in  Figure  6.4 . The risk-free asset,  F,   appears 

on the vertical axis because its standard 

deviation is zero. The risky asset,  

P,   is 

plotted with a standard deviation  of  22%, 

and expected return of 15%. If an investor 

chooses to invest solely in the risky asset, 

then  y   5  1.0, and the complete portfolio 

is  P.  If the chosen position is  y  5 0, then 

1  2   y  5 1.0, and the complete portfolio is 

the risk-free portfolio  F.  

 What about the more interesting mid-

range portfolios where  y  lies between 0 

and 1? These portfolios will graph on the 

straight line connecting points  

F  and  P.  

The slope of that line is [ E ( r  

 P 

 )   2     r  

 f 

 ]/ s  

 P 

  

(rise/run), in this case, 8/22. 

 

The conclusion is straightforward. 

Increasing the fraction of the overall port-

folio invested in the risky asset increases 

expected return at a rate of 8%, according 

to Equation 6.3. It also increases portfolio 

3

 This is an application of a basic rule from statistics: If you multiply a random variable by a constant, the standard 

deviation is multiplied by the same constant. In our application, the random variable is the rate of return on the 

risky asset, and the constant is the fraction of that asset in the complete portfolio. We will elaborate on the rules 

for portfolio return and risk in the following chapter.