®1999 South-Western College Publishing 1 Chapter 10 Index Models And The Arbitrage Pricing Theory

®1999 South-Western College Publishing 2 Single Index Model (SIM) Stock’s Rate of ReturnStock’s Rate of Return –Percentage change in the index (I) Common factorCommon factor –Changes related to firm-specific events (e i ) On average = 0On average = 0 Any given period, it can be + or -Any given period, it can be + or -

®1999 South-Western College Publishing 3 SIM Calculations R i = Constant + Common-Factor + Firm-SpecificR i = Constant + Common-Factor + Firm-Specific News News News News R i =  i +  i I + e iR i =  i +  i I + e i  I = R i -  i I  I = R i -  i I

®1999 South-Western College Publishing 4 ii RiRi I The Return Components Firm-Specific News Realized Return Average Return With Common-Factor News ii

®1999 South-Western College Publishing 5 Why Does SIM Reduce Computations? It Decreases the Number of Calculations of CovariancesIt Decreases the Number of Calculations of Covariances FromFrom Cov(R i, R j ) = Cov(  i +  i I + e i,  j +  j I + e j ) ToTo Cov (R i, R j ) = (  i  j  2 I ) BecauseBecause Cov(e i, e j ) = 0 and Cov(I, e i ) = 0

®1999 South-Western College Publishing 6 Portfolio Risk Systematic Risk Common-Factor Risk Undiversifiable Unsystematic Risk Firm-Specific Risk Diversifiable Investors are not rewarded for unsystematic risk

®1999 South-Western College Publishing 7 Systematic RiskSystematic Risk –Inflation rate –Unemployment rate –Interest rate Unsystematic RiskUnsystematic Risk –Resignation of the president –Change in dividends –New discovery

®1999 South-Western College Publishing 8 Return On A Portfolio Portfolio + Portfolio Return + Portfolio Return Intercept Due to Due to Market Factor Firm-Specific Market Factor Firm-Specific Factors Factors

®1999 South-Western College Publishing 9 Risk And E(R) With SIM Start With E(R i ) =  I +  i  E(I) + E(e i )

®1999 South-Western College Publishing n 22 With Diversification Unsystematic Risk Systematic Risk

®1999 South-Western College Publishing 11 Risk And E(R) With SIM E(e i ) = 0 R i = E(R i ) +  I [I - E(I)] + e i Just Like APT

®1999 South-Western College Publishing 12 APT Linear Risk-Return RelationshipLinear Risk-Return Relationship No Arbitrage OpportunitiesNo Arbitrage Opportunities EquilibriumEquilibrium

®1999 South-Western College Publishing 13 Examples Of Arbitrage Borrow at 5% and Save at 6%Borrow at 5% and Save at 6% Selling Stock ShortSelling Stock Short What Causes the Arbitrage?What Causes the Arbitrage? –Zero out-of-pocket investment –Return is always nonnegative

®1999 South-Western College Publishing 14 APT Assumptions Rates of Return Depend on the Return- Generation ProcessRates of Return Depend on the Return- Generation Process –Common factors and some noise –Derives the risk-return relationship Large Number of Assets in the EconomyLarge Number of Assets in the Economy Short Sale Allowed With ProceedsShort Sale Allowed With Proceeds Investors Prefer More Wealth to LessInvestors Prefer More Wealth to Less

®1999 South-Western College Publishing 15 APT’s Common Factors Used I - E(I) Instead of Just IUsed I - E(I) Instead of Just I Called the Surprise FactorCalled the Surprise Factor Measures the Difference BetweenMeasures the Difference Between –Expectations –Actual outcomes

®1999 South-Western College Publishing 16 Firm’s Beta The Larger the Beta The Larger the Effect of the Surprise On the Firm’s Return

®1999 South-Western College Publishing 17 Linear APT Relationship No Arbitrage OpportunitiesNo Arbitrage Opportunities –Equilibrium Using the Zero BetaUsing the Zero Beta Using Zero Investment PortfolioUsing Zero Investment Portfolio Linear RelationshipLinear Relationship E(R i ) = a 0 + a 1  iE(R i ) = a 0 + a 1  i

®1999 South-Western College Publishing 18 a0a Beta Rate of Return Illustration

®1999 South-Western College Publishing 19 APT And CAPM Yield Same ResultsYield Same Results Different AssumptionsDifferent Assumptions –APT is less restrictive Disadvantage APTDisadvantage APT –Fails to identify common factors

®1999 South-Western College Publishing 20 Multifactor APT Suggested FactorsSuggested Factors –Default premium –Term structure –Inflation –Corporate profits –Market risk E(R i ) = a 0 + a 1  1 + a 2  2 + … + a k  kE(R i ) = a 0 + a 1  1 + a 2  2 + … + a k  k