S S Covr rj
j= 1 i= 1 j*i we can express portfolio variance as of =  a2
F 17
Now examine the effect of diversification. When the average covariance among security returns is zero, as it is when all risk is firmspecific, portfolio variance can be driven to zero. We see this from equation 8A.3: The second term on the righthand side will be zero in this
PART II Portfolio Theory scenario, while the first term approaches zero as n becomes larger. Hence when security returns are uncorrelated, the power of diversification to limit portfolio risk is unlimited.
However, the more important case is the one in which economywide risk factors impart positive correlation among stock returns. In this case, as the portfolio becomes more highly diversified (n increases) portfolio variance remains positive. Although firmspecific risk, represented by the first term in equation 8A.3, is still diversified away, the second term simply approaches Cov as n becomes greater. [Note that (n  1)/n = 1  1/n, which approaches 1 for large n.] Thus the irreducible risk of a diversified portfolio depends on the covariance of the returns of the component securities, which in turn is a function of the importance of systematic factors in the economy.
To see further the fundamental relationship between systematic risk and security correlations, suppose for simplicity that all securities have a common standard deviation, ct, and all security pairs have a common correlation coefficient, p. Then the covariance between all pairs of securities is pa2, and equation 8A.3 becomes
lJ n
The effect of correlation is now explicit. When p = 0, we again obtain the insurance principle, where portfolio variance approaches zero as n becomes greater. For p > 0, however, portfolio variance remains positive. In fact, for p = 1, portfolio variance equals ct2 regardless of n, demonstrating that diversification is of no benefit: In the case of perfect correlation, all risk is systematic. More generally, as n becomes greater, equation 8A.4 shows that systematic risk becomes pa2.
Table 8A.1 presents portfolio standard deviation as we include evergreater numbers of securities in the portfolio for two cases, p = 0 and p = .40. The table takes ct to be 50%. As one would expect, portfolio risk is greater when p = .40. More surprising, perhaps, is that portfolio risk diminishes far less rapidly as n increases in the positive correlation case. The correlation among security returns limits the power of diversification.
Note that for a 100security portfolio, the standard deviation is 5% in the uncorrelated case—still significant compared to the potential of zero standard deviation. For p = .40, the standard deviation is high, 31.86%, yet it is very ^ close to undiversifiable systematic risk in t2 = V.4 X 502 = 31.62%. At this point, further the infinitesized security universe, Vp( diversification is of little value.
We also gain an important insight from this exercise. When we hold diversified portfolios, the contribution to portfolio risk of a particular security will depend on the covariance of that security's return with those of other securities, and not on the security's variance. As we shall see in Chapter 9, this implies that fair risk premiums also should depend on covariances rather than total variability of returns.
CONCEPT CHECK ^ QUESTION A.1
Suppose that the universe of available risky securities consists of a large number of stocks, identically distributed with E(r) = 15%, ct = 60%, and a common correlation coefficient of p = .5.
a. What is the expected return and standard deviation of an equally weighted risky portfolio of 25 stocks?
b. What is the smallest number of stocks necessary to generate an efficient portfolio with a standard deviation equal to or smaller than 43%?
c. What is the systematic risk in this security universe?
d. If Tbills are available and yield 10%, what is the slope of the CAL?
CHAPTER 8 Optimal Risky Portfolios
Table 8A.1 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes
CHAPTER 8 Optimal Risky Portfolios
Table 8A.1 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes
P = 
0 
P = 
.4  
Universe 
ODtimal Portfolio 
Standard 
Reduction 
Standard 
Reduction 
Size n 
Proportion 1/n (%) 
Deviation (%) 
in a 
Deviation (%) 
in a 
1 
100 
50.00 
14.64 
50.00 
8.17 
2 
50 
35.36 
41.83  
5 
20 
22.36 
1.95 
36.06 
0.70 
6 
16.67 
20.41 
35.36  
10 
10 
15.81 
0.73 
33.91 
0.20 
11 
9.09 
15.08 
33.71  
20 
5 
11.18 
0.27 
32.79 
0.06 
21 
4.76 
10.91 
32.73  
100 
1 
5.00 
0.02 
31.86 
0.00 
101 
0.99 
4.98 
31.86 
A.1. The parameters are E(r) = 15, a = 60, and the correlation between any pair of stocks is p = .5.
a. The portfolio expected return is invariant to the size of the portfolio because all stocks have identical expected returns. The standard deviation of a portfolio with n = 25 stocks is aP = [a2/n + p X a2(n  1)/n]1/2
b. Because the stocks are identical, efficient portfolios are equally weighted. To obtain a standard deviation of 43%, we need to solve for n:
3,600 1,800 49
1,800n 36.73
1,800
Thus we need 37 stocks and will come in with volatility slightly under the target.
c. As n gets very large, the variance of an efficient (equally weighted) portfolio diminishes, leaving only the variance that comes from the covariances among stocks, that is aP = VpXa2 = V.5 X 602 = 42.43
Note that with 25 stocks we came within .84% of the systematic risk, that is, the nonsystematic risk of a portfolio of 25 stocks is .84%. With 37 stocks the standard deviation is 43%, of which nonsystematic risk is .57%.
d. If the riskfree is 10%, then the risk premium on any size portfolio is 15 — 10 = 5%. The standard deviation of a welldiversified portfolio is (practically) 42.43%; hence the slope of the CAL is
SOLUTIONS TO CONCEPT CHECKS
PART II Portfolio Theory
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