Chapter 10_Hand-out 9

Chapter 10_Hand-out 9 - CHAPTER 10 ARBITRAGE PRICING THEORY...

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CHAPTER 10 ARBITRAGE PRICING THEORY AND MULTIFACTOR MODLES OF RISK AND RETURN 10.1 MULTIFACTOR MODELS: AN OVERVIEW 1. Multifactor Models of Security Returns The index model introduced earlier in this handout give us a way of decomposing stock variability into market or systematic versus firm-specific effects that can be diversified in large portfolios. In the index model, the return on the market portfolio summarizes the aggregate impact of macro factors. In reality, however, systematic risk is not due to one source, but instead derives from uncertainty in many economy-wide factors such as business-cycle risk, interest or inflation rate risk, energy price risk, etc. How can we improve on the single-index model but still maintain the useful dichotomy between systematic and diversifiable risk? It is easy to see that models that allow for several systematic factors – multifactor models – can provide better description of security returns. Let’s illustrate with a two-factor model. Suppose the two most important macroeconomic sources of risk are uncertainties surrounding the state of the business cycle, news of which we again assume is reflected in the rate of return on a broad index such as the S&P 500, and unanticipated changes in interest rates, which may be captured by the return on a T-bond portfolio. The return on any stock will respond to both sources of macro risk as well as to its own firm-specific influences. Therefore, we can generalize the single-index model into a two-factor model describing the excess rate of return on a stock i in some time period t as follows: it TBt iTB Mt iM i it e R β R β α R + + + = , (1) where β TB is the sensitivity of the stock’ excess return to that of the T-bond portfolio, and R TB is the excess return of the T-bill portfolio in month t . The two indexes on the right-hand side of the equation capture the effect of the two systematic factors in the economy; thus they play the role of the market index in the single-index model. As before, e t reflects firm-specific influences in period t . How will the security market line of the CAPM generalize once we recognize the presence of multiple sources of systematic risk? Perhaps, not surprisingly, a multifactor index model gives rise to a multifactor security market line (SML) in which the risk premium is determined by the exposure to each systematic risk 1
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factor and by a risk premium associated with each of those factors. Such a multifactor CAPM was first presented by Merton (1973). For example, in a two- factor economy in which the risk exposure can be measure by the equation (1), the expected rate of return on a security i would be express as follows: ] r ) [E(r β ] r ) [E(r β r ) E(r f TB iTB f M iM f i - + - + = (2) It is clear that the equation above is a generalization of the simple security market line. In the single-factor SML, the benchmark risk premium is given by the risk premium of the market portfolio, but once we generalize to multiple risk
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This note was uploaded on 09/25/2011 for the course FINA 4320 taught by Professor John during the Spring '11 term at Houston Baptist.

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Chapter 10_Hand-out 9 - CHAPTER 10 ARBITRAGE PRICING THEORY...

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