Investments, tenth edition

 Figure 10.1 

Excess returns as a function of the systematic factor.  Panel A,

 Well-diversified portfolio  A. Panel B,  Single stock ( S ).  

0

F

A

10

Excess Return (%)

0

F

S

10

Excess Return (%)

A

B

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  Arbitrage Pricing Theory and Multifactor Models of Risk and Return

331

  Residual SD of Each    Stock   5   50%  

  Residual SD of Each    Stock   5   100%  

  N  

 SD( e

P

 ) 

  N  

 SD( e

P

 ) 

  Equal weights:    w  

 i 

     5   1/  N  

 4  

25.00  

4  

50.00 

 60  

6.45  

60  

12.91 

 200  

3.54  

200  

7.07 

 1,000  

1.58  

1,000  

3.16 

 10,000  

0.50  

10,000  

1.00 

  Sets of four relative weights:   w  

1

     5   0.65,    w  

2

     5   0.2,   w  

3

     5   0.1,   w  

4

     5   0.05  

 4  

36.23  

4  

72.46 

 60  

9.35  

60  

18.71 

 200  

5.12  

200  

10.25 

 1,000  

2.29  

1,000  

4.58 

 10,000  

0.72  

10,000  

1.45 

 Table 10.1 

 Residual variance with even and uneven portfolio weights 

Perfect correlation means that in a plot of expected return versus standard deviation (such 

as Figure 7.5), any two well-diversified portfolios lie on a straight line. We will see later 

that this common line is the CML.  

  Diversification and Residual Risk in Practice 

 What is the effect of diversification on portfolio residual SD  in practice,  where portfo-

lio size is not unlimited? In reality, we may find (annualized) residual SDs as high as 

50% for large stocks and even 100% for small stocks. To illustrate the impact of diversi-

fication, we examine portfolios of two configurations. One portfolio is equally weighted; 

this achieves the highest benefits of diversification with equal-SD stocks. For compari-

son, we form the other portfolio using far-from-equal weights. We select stocks in groups 

of four, with relative weights in each group of 70%, 15%, 10%, and 5%. The highest 

weight is 14 times greater than the lowest, which will severely reduce potential benefits of 

diversification. However, extended diversification in which we add to the portfolio more 

and more groups of four stocks with the same relative weights will overcome this prob-

lem because the highest portfolio weight still falls with additional diversification. In an 

equally weighted 1,000-stock portfolio, each weight is 0.1%; in the unequally weighted 

portfolio, with 1,000/4  5  250 groups of four stocks, the highest and lowest weights are 

70%/250  5  0.28% and 5%/250  5  0.02%,  respectively. 

 What is a large portfolio? Many widely held ETFs each include hundreds of stocks, and 

some funds such as the Wilshire 5000 hold thousands. These portfolios are accessible to 

the public since the annual expense ratios of investment companies that offer such funds 

are of the order of only 10 basis points. Thus a portfolio of 1,000 stocks is not unheard of, 

but a portfolio of 10,000 stocks is. 

  Table 10.1  shows portfolio residual SD as a function of the number of stocks. Equally 

weighted, 1,000-stock portfolios achieve small but not negligible standard deviations of 

1.58% when residual risk is 50% and 3.16% when residual risk is 100%. The SDs for the 

unbalanced portfolios are about double these values. For 10,000-stock portfolios, the SDs 

are negligible, verifying that diversification can eliminate risk even in very unbalanced 

portfolios, at least in principle, if the investment universe is large enough.   

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332 

P A R T   I I I

  Equilibrium in Capital Markets