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assumption that at once eases our com- putational burden and offers significant new insights into the nature of system- atic risk versus firm-specific risk. This abstraction is the notion of an “index model,” specifying the process by which security returns are generated. Our discussion of the index model also introduces the concept of multifactor models of security returns, a concept at the heart of contemporary investment theory and its applications. 292 III. Equilibrium In Capital Markets 10. Single−Index and Multifactor Models The McGraw−Hill Companies, 2001 CHAPTER 10 Single-Index and Multifactor Models 293 10.1 A SINGLE-INDEX SECURITY MARKET Systematic Risk versus Firm-Specific Risk The success of a portfolio selection rule depends on the quality of the input list, that is, the estimates of expected security returns and the covariance matrix. In the long run, efficient portfolios will beat portfolios with less reliable input lists and consequently inferior re- ward-to-risk trade-offs. Suppose your security analysts can thoroughly analyze 50 stocks. This means that your input list will include the following: n 50 estimates of expected returns n 50 estimates of variances (n2 n)/2 1,225 estimates of covariances 1,325 estimates This is a formidable task, particularly in light of the fact that a 50-security portfolio is relatively small. Doubling n to 100 will nearly quadruple the number of estimates to 5,150. If n 3,000, roughly the number of NYSE stocks, we need more than 4.5 million
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