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Chapter Ten Arbitrage Pricing Theory and Multifactor Models of Risk and Return INVESTMENTS | BODIE, KANE, MARCUS Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Single Factor Model • Returns on a security come from two sources: – Common macro-economic factor – Firm specific events • Possible common macro-economic factors – Gross Domestic Product Growth – Interest Rates 10-2 INVESTMENTS | BODIE, KANE, MARCUS Single Factor Model Equation 𝑅𝑖 = 𝐸 𝑅𝑖 + β𝑖 𝐹 + 𝑒𝑖 Ri = Excess return on security βi= Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (F could be positive or negative but has expected value of zero) ei = Firm specific events (zero expected value) 10-3 INVESTMENTS | BODIE, KANE, MARCUS Multifactor Models • Use more than one factor in addition to market return – Examples include gross domestic product, expected inflation, interest rates, etc. – Estimate a beta or factor loading for each factor using multiple regression. 10-4 INVESTMENTS | BODIE, KANE, MARCUS Multifactor Model Equation Ri E Ri iGDPGDP iIR IR ei Ri = Excess return for security i βGDP = Factor sensitivity for GDP βIR = Factor sensitivity for Interest Rate ei = Firm specific events 10-5 INVESTMENTS | BODIE, KANE, MARCUS Interpretation The expected return on a security is the sum of: 1.The risk-free rate 2.The sensitivity to GDP times the risk premium for bearing GDP risk 3.The sensitivity to interest rate risk times the risk premium for bearing interest rate risk 10-6 INVESTMENTS | BODIE, KANE, MARCUS Arbitrage Pricing Theory • Arbitrage occurs if there is a zero investment portfolio with a sure profit. Since no investment is required, investors can create large positions to obtain large profits. 10-7 INVESTMENTS | BODIE, KANE, MARCUS Arbitrage Pricing Theory • Regardless of wealth or risk aversion, investors will want an infinite position in the risk-free arbitrage portfolio. • In efficient markets, profitable arbitrage opportunities will quickly disappear. 10-8 INVESTMENTS | BODIE, KANE, MARCUS APT & Well-Diversified Portfolios RP = E (RP) + bPF + eP F = some factor • For a well-diversified portfolio, eP – approaches zero as the number of securities in the portfolio increases – and their associated weights decrease 10-9 INVESTMENTS | BODIE, KANE, MARCUS Figure 10.1 Returns as a Function of the Systematic Factor 10-10 INVESTMENTS | BODIE, KANE, MARCUS Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity 10-11 INVESTMENTS | BODIE, KANE, MARCUS Figure 10.3 An Arbitrage Opportunity 10-12 INVESTMENTS | BODIE, KANE, MARCUS No-Arbitrage Equation of APT 10-13 INVESTMENTS | BODIE, KANE, MARCUS the APT, the CAPM and the Index Model APT • Assumes a welldiversified portfolio, but residual risk is still a factor. • Does not assume investors are meanvariance optimizers. • Uses an observable, market index • Reveals arbitrage opportunities 10-14 CAPM • Model is based on an inherently unobservable “market” portfolio. • Rests on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium. INVESTMENTS | BODIE, KANE, MARCUS Multifactor APT • Use of more than a single systematic factor • Requires formation of factor portfolios • What factors? – Factors that are important to performance of the general economy – What about firm characteristics? 10-15 INVESTMENTS | BODIE, KANE, MARCUS Two-Factor Model • The multifactor APT is similar to the onefactor case. 𝑅𝑖 = 𝐸 𝑅𝑖 + β𝑖1 𝐹1 + β𝑖2 𝐹2 + 𝑒𝑖 10-16 INVESTMENTS | BODIE, KANE, MARCUS Two-Factor Model • Track with diversified factor portfolios: – beta=1 for one of the factors and 0 for all other factors. • The factor portfolios track a particular source of macroeconomic risk, but are uncorrelated with other sources of risk. 10-17 INVESTMENTS | BODIE, KANE, MARCUS Fama-French Three-Factor Model • SMB = Small Minus Big (firm size) • HML = High Minus Low (book-to-market ratio) • Are these firm characteristics correlated with actual (but currently unknown) systematic risk factors? Rit i iM RMt iSMBSMBt iHMLHMLt eit 10-18 INVESTMENTS | BODIE, KANE, MARCUS The Multifactor CAPM and the APT • A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge • The APT is largely silent on where to look for priced sources of risk 10-19 INVESTMENTS | BODIE, KANE, MARCUS