1、CHAPTER 10Arbitrage Pricing Theory and Multifactor Models of Risk and ReturnSingle 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 Single Factor Model Equati
2、onri=Return on security i=Factor sensitivity or factor loading or factor betaF=Surprise in macro-economic factor(F could be positive or negative but has expected value of zero)ei=Firm specific events(zero expected value)()iiiirE rFeMultifactor Models Use more than one factor in addition to market re
3、turn Examples include gross domestic product,expected inflation,interest rates,etc.Estimate a beta or factor loading for each factor using multiple regression.Multifactor Model Equationri =Return for security iGDP=Factor sensitivity for GDP IR =Factor sensitivity for Interest Rate ei=Firm specific e
4、vents iiIRiGDPiieIRGDPrErMultifactor SML Models GDP=Factor sensitivity for GDP RPGDP=Risk premium for GDP IR =Factor sensitivity for Interest RateRPIR =Risk premium for Interest Rateii IRiIRGDPiGDPfiRPRPrrEInterpretationThe expected return on a security is the sum of:1.The risk-free rate2.The sensit
5、ivity to GDP times the risk premium for bearing GDP risk3.The sensitivity to interest rate risk times the risk premium for bearing interest rate riskArbitrage Pricing Theory Arbitrage occurs if there is a zero investment portfolio with a sure profit.Since no investment is required,investors can crea
6、te large positions to obtain large profits.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.APT&Well-Diversified PortfoliosrP=E(r
7、P)+PF+ePF=some factor For a well-diversified portfolio,eP approaches zero as the number of securities in the portfolio increases and their associated weights decreaseFigure 10.1 Returns as a Function of the Systematic FactorFigure 10.2 Returns as a Function of the Systematic Factor:An Arbitrage Oppo
8、rtunityFigure 10.3 An Arbitrage OpportunityFigure 10.4 The Security Market LineAPT Model APT applies to well diversified portfolios and not necessarily to individual stocks.With APT it is possible for some individual stocks to be mispriced-not lie on the SML.APT can be extended to multifactor models
9、.APT and CAPMAPT Equilibrium means no arbitrage opportunities.APT equilibrium is quickly restored even if only a few investors recognize an arbitrage opportunity.The expected returnbeta relationship can be derived without using the true market portfolio.CAPM Model is based on an inherently unobserva
10、ble“market”portfolio.Rests on mean-variance efficiency.The actions of many small investors restore CAPM equilibrium.CAPM describes equilibrium for all assets.Multifactor APT Use of more than a single systematic factor Requires formation of factor portfolios What factors?Factors that are important to
11、 performance of the general economy What about firm characteristics?Two-Factor Model The multifactor APT is similar to the one-factor case.1122()iiiiirE rFFeTwo-Factor Model Track with diversified factor portfolios:beta=1 for one of the factors and 0 for all other factors.The factor portfolios track
12、 a particular source of macroeconomic risk,but are uncorrelated with other sources of risk.Where Should We Look for Factors?Need important systematic risk factors Chen,Roll,and Ross used industrial production,expected inflation,unanticipated inflation,excess return on corporate bonds,and excess retu
13、rn on government bonds.Fama and French used firm characteristics that proxy for systematic risk factors.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?ittiHMLtiSMBMtiMiiteHMLSMBRrThe 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
