Chapter 8 - CHAPTER 8 INDEX MODELS 8.1 A SINGLE-INDEX...

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CHAPTER 8 INDEX MODELS 8.1 A SINGLE-INDEX SECURITY MARKET 1. The Input List of the Markowitz Model 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 reward-to-risk trade-offs. Introducing a model that simplifies the way we describe the sources of security risk allows us to use a smaller, consistent set of estimates of risk parameters and risk premiums. The simplification emerges because positive covariances among security returns arise from the same economic forces that affect the fortunes of most firms. Some samples of common economic factors are business cycles, interest rates, technological changes, and cost of natural resources . All these (interrelated) factors affect almost all firms. Thus unexpected changes in these variables cause, simultaneously, unexpected changes in the rates of return on the entire stock market. By decomposing uncertainty into system- wide versus firm-specific sources, we vastly simplify the problem of estimating covariance and correlation. Suppose that we summarize all relevant economic factors by one macroeconomic factor, m , and assume that it affects all firms. We further assume that, beyond this common effect, all remaining uncertainty in stock returns is firm specific; that is, there is no other source of correlation between securities. Firm-specific events would include new inventions, deaths of key employees, and other factors that affect the fortune of the individual firm without affecting the broad economy in a measurable way. 2. Normality of Returns and Systematic Risk We can summarize the distinction between macroeconomic and firm-specific factors by writing the holding period return (HPR) on security i as: i e m ) E(r r i i , (1) where m is the impact of the unanticipated macro events (surprises) on the security’s return during the holding period, and e i is the impact of the unanticipated firm-specific events (surprises). Both m and e i have a mean of zero because each represents the impact of unanticipated events, which by definition must average out to zero, and standard deviation of m and e . Most important is the fact that m and e i are uncorrelated; thus the variance of r i arises from two uncorrelated sources, systematic and firm-specific. Therefore: ) (e σ σ σ i 2 2 M 2 i (2) The common factor, m , generates correlation across securities, because all securities will respond to the same macroeconomic news, while the firm-specific surprises, captured by and e i , are assumed to be uncorrelated across firms. Because m is also uncorrelated with any of the firm-specific surprises, the covariance between any two securities i and j is: 1
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2 m j i j i ) e , e Cov( ) r , Cov(r m m (3) We can gain further insight by recognizing that some securities (firms) will be more sensitive than others to macroeconomic shocks. We can capture this refinement by assigning each firm a sensitivity
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