Investments, tenth edition

A

n

B

n

Unique Risk

Market Risk

σ

σ

Figure 7.1 

Portfolio risk as a function of the number of stocks in the portfolio  

Panel A:  All risk is firm specific.  Panel B:  Some risk is systematic, or marketwide.

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P A R T   I I

  Portfolio Theory and Practice

 In the last section we considered naive diversification using equally weighted portfolios 

of several securities. It is time now to study  efficient  diversification, whereby we construct 

risky portfolios to provide the lowest possible risk for any given level of expected return. 

 Portfolios of two risky assets are relatively easy to analyze, and they illustrate the prin-

ciples and considerations that apply to portfolios of many assets. It makes sense to think 

about a two-asset portfolio as an asset allocation decision, and so we consider two mutual 

funds, a bond portfolio specializing in long-term debt securities, denoted  D,  and a stock 

fund that specializes in equity securities,  E.   Table 7.1  lists the parameters describing the 

rate-of-return distribution of these funds.  

 A proportion denoted by  w  

 D 

  is invested in the bond fund, and the remainder, 1  2   w  

 D 

 , 

denoted  w  

 E 

 , is invested in the stock fund. The rate of return on this portfolio,  r  

 p 

 , will be  

2

      

 

r

p

5 w

D

r

D

1 w

E

r

E

 

 (7.1)  

where  r  

 D 

  is the rate of return on the debt fund and  r  

 E 

  is the rate of return on the equity fund. 

 The expected return on the portfolio is a weighted average of expected returns on the 

component securities with portfolio proportions as weights:   

 

E (r

p

)

5 w

D

E (r

D

)

1 w

E

E (r

E

) 

 (7.2)   

 The variance of the two-asset portfolio is   

 

s

p

2

5 w

D

2

s

D

2

1 w

E

2

s

E

2

1 2w

D

w

E

 Cov(r

D

, r

E

) 

 (7.3)   

 Our first observation is that the variance of the portfolio, unlike the expected return, is 

 not  a weighted average of the individual asset variances. To understand the formula for the 

portfolio variance more clearly, recall that the covariance of a variable with itself is the 

variance of that variable; that is   

 Cov(r

D

, r

D

)

5 a

scenarios

Pr(scenario)

3r

D

2 E(r

D

)

4 3r

D

2 E(r

D

)

4

 

5 a

scenarios

Pr (scenario)

3r

D

2 E(r

D

)

4

2

 (7.4)

 

 

5 s

D

2

 

   

Therefore, another way to write the variance of the portfolio is   

 

s

p

2

5 w

D

w

D

 Cov(r

D

, r

D

)

1 w

E

w

E

 Cov(r

E

, r

E

)

1 2w

D

w

E

 Cov(r

D

, r

E

) 

 (7.5)  

    7.2 

Portfolios of Two Risky Assets