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 firm-specific 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 ap- proaches 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 im- portance of systematic factors in the economy. To see further the fundamental relationship between systematic risk and security corre- lations, suppose for simplicity that all securities have a common standard deviation, , and all security pairs have a common correlation coefficient, . Then the covariance between all pairs of securities is 2, and equation 8A.3 becomes   2 1 2 n 1 2   (8A.4) p n n   The effect of correlation is now explicit. When 0, we again obtain the insurance principle, where portfolio variance approaches zero as n becomes greater. For > 0, how- ever, portfolio variance remains positive. In fact, for 1, portfolio variance equals 2 re- gardless 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 2. Table 8A.1 presents portfolio standard deviation as we include ever-greater numbers of securities in the portfolio for two cases, 0 and .40. The table takes to be 50%. As one would expect, portfolio risk is greater when .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 100-security portfolio, the standard deviation is 5% in the uncorrelated case-still significant compared to the potential of zero standard deviation. For .40, the standard deviation is high, 31.86%, yet it is very close to undiversifiable systematic risk in the infinite-sized security universe, 2 .4 502 31.62%. At this point, further diversification is of little value. We also gain an important insight from this exercise. When we hold diversified portfo- lios, the contribution to portfolio risk of a particular security will depend on the covariance of that securitys return with those of other securities, and not on the securitys variance. As we shall see in Chapter 9, this implies that fair risk premiums also should depend on co- variances rather than total variability of returns.