2. ®1999 South-Western College Publishing 1 Chapter 10 Index Models And The Arbitrage Pricing Theory

3. ®1999 South-Western College Publishing 2 Index Models and APT Provide Potential Solutions To: –Estimation problems in implementing MPT –Shortcomings of CAPM

4. ®1999 South-Western College Publishing 3 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 -

5. ®1999 South-Western College Publishing 4 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 Note: CAPM is a specific form of SIMNote: CAPM is a specific form of SIM

6. ®1999 South-Western College Publishing 5 ii RiRi I The Return Components Firm-Specific News Realized Return Average Return With Common-Factor News ii

7. ®1999 South-Western College Publishing 6 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 Because (by assumption)Because (by assumption) Cov(e i, e j ) = 0 and Cov(I, e i ) = 0

8. ®1999 South-Western College Publishing 7 Estimation Issues Results of portfolio allocation depend on accurate statistical inputsResults of portfolio allocation depend on accurate statistical inputs Estimates ofEstimates of –Expected returns –Standard deviation –Correlation coefficient Among entire set of assetsAmong entire set of assets With 100 assets, 4,950 correlation estimatesWith 100 assets, 4,950 correlation estimates Estimation risk refers to potential errorsEstimation risk refers to potential errors

9. ®1999 South-Western College Publishing 8 Estimation Issues With the assumption that stock returns can be described by a single market model, the number of estimated inputs required for the covariances reduces to the number of assets (one beta for each) plus one (the variance of the Index’s return)With the assumption that stock returns can be described by a single market model, the number of estimated inputs required for the covariances reduces to the number of assets (one beta for each) plus one (the variance of the Index’s return) No. of estimated inputs required for covariances:No. of estimated inputs required for covariances: No. of stocks W/o simplification Under SIM 5106 104511 1004,950101 500124,750501

10. ®1999 South-Western College Publishing 9 Estimation Issues Benefits of using SIM for estimating Eff. FrontierBenefits of using SIM for estimating Eff. Frontier –Fewer estimated inputs implies less chance for estimation error (true Eff. Frontier is more likely to be near where it is estimated to be) Costs of using SIMCosts of using SIM –Potentially unrealistic, oversimplified assumptions –Ignores the potential for “industry effects” –different stocks in the same industry will tend to move together in ways that are separate from what the market as a whole is doing –But, market effects are still the strongest

11. ®1999 South-Western College Publishing 10 Portfolio Risk Systematic Risk Common-Factor Risk Undiversifiable Unsystematic Risk Firm-Specific Risk Diversifiable Investors are not rewarded for unsystematic risk

12. ®1999 South-Western College Publishing 11 Systematic RiskSystematic Risk –Inflation rate –Unemployment rate –Interest rate Unsystematic RiskUnsystematic Risk –Resignation of the president –Change in dividends –New discovery

13. ®1999 South-Western College Publishing 12 Return On A Portfolio Portfolio + Portfolio Return + Portfolio Return Intercept Due to Due to Market Factor Firm-Specific Market Factor Firm-Specific Factors Factors

14. ®1999 South-Western College Publishing 13 Risk And E(R) With SIM Start With E(R i ) =  I +  i  E(I) + E(e i )

15. ®1999 South-Western College Publishing 14 30 n 22 With Diversification Unsystematic Risk Systematic Risk

16. ®1999 South-Western College Publishing 15 Risk And E(R) With SIM E(e i ) = 0 R i = E(R i ) +  I [I - E(I)] + e i Just Like APT

17. ®1999 South-Western College Publishing 16 APT Linear Risk-Return RelationshipLinear Risk-Return Relationship No Arbitrage OpportunitiesNo Arbitrage Opportunities Equilibrium ModelEquilibrium Model

18. ®1999 South-Western College Publishing 17 What Is Arbitrage? Borrow at 5% and Save at 6%Borrow at 5% and Save at 6% Simultaneously buying stock cheaply on one market and selling it short on another higher-quoted marketSimultaneously buying stock cheaply on one market and selling it short on another higher-quoted market What Causes the Arbitrage?What Causes the Arbitrage? –Zero out-of-pocket investment –Return is always nonnegative

19. ®1999 South-Western College Publishing 18 Arbitrage Pricing Theory (APT) Arbitrage is a process of buying a lower priced asset and selling a higher priced asset, both of similar risk, and capturing the difference in arbitrage profitsArbitrage is a process of buying a lower priced asset and selling a higher priced asset, both of similar risk, and capturing the difference in arbitrage profits The general arbitrage principle states that two identical securities will sell at identical pricesThe general arbitrage principle states that two identical securities will sell at identical prices Price differences will immediately disappear as arbitrage takes placePrice differences will immediately disappear as arbitrage takes place

20. ®1999 South-Western College Publishing 19 Arbitrage Pricing Theory (APT) CAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmarkCAPM is criticized because of the difficulties in selecting a proxy for the market portfolio as a benchmark An alternative pricing theory with fewer assumptions was developed:An alternative pricing theory with fewer assumptions was developed: Arbitrage Pricing TheoryArbitrage Pricing Theory

21. ®1999 South-Western College Publishing 20 Assumptions of Arbitrage Pricing Theory (APT) 1. Capital markets are perfectly competitive 2. Investors always prefer more wealth to less wealth with certainty 3. The stochastic process generating asset returns can be presented as K factor model –Common factors plus some noise –Describes the risk-return relationship Other Assumptions:Other Assumptions: –Large number of assets in the economy –Short sales allowed with proceeds

22. ®1999 South-Western College Publishing 21 Assumptions of CAPM That Were Not Required by APT APT does not assume A market portfolio that contains all risky assets, and is mean-variance efficientA market portfolio that contains all risky assets, and is mean-variance efficient Normally distributed security returnsNormally distributed security returns Quadratic utility functionQuadratic utility function

23. ®1999 South-Western College Publishing 22 Arbitrage Pricing Theory (APT) For i = 1 to N where: Ri = return on asset i during a specified time period Ei = expected return for asset i bik = reaction in asset i’s returns to movements in the kth common factor δk = a common factor with a zero mean that influences the returns on all assets εi = a unique effect on asset i’s return that, by assumption, is completely diversifiable in large portfolios and has a mean of zero N = number of assets

24. ®1999 South-Western College Publishing 23 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets:

25. ®1999 South-Western College Publishing 24 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation

26. ®1999 South-Western College Publishing 25 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP

27. ®1999 South-Western College Publishing 26 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP –Major political upheavals

28. ®1999 South-Western College Publishing 27 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP –Major political upheavals –Changes in interest rates

29. ®1999 South-Western College Publishing 28 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP –Major political upheavals –Changes in interest rates –And many more….

30. ®1999 South-Western College Publishing 29 Arbitrage Pricing Theory (APT) Multiple factors expected to have an impact on all assets: Multiple factors expected to have an impact on all assets: –Inflation –Growth in GNP –Major political upheavals –Changes in interest rates –And many more…. Contrast with CAPM insistence that only beta is relevant

31. ®1999 South-Western College Publishing 30 Arbitrage Pricing Theory (APT) b ik (beta(i,k)) determines how each asset reacts to this common factor Each asset may be affected by growth in GNP, but the effects will differ In application of the theory, the factors are not identified Similar to the CAPM, the unique effects ( ε i ) are independent and will be diversified away in a large portfolio

32. ®1999 South-Western College Publishing 31 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

33. ®1999 South-Western College Publishing 32 Firm’s Beta The Larger the Beta The Larger the Effect of the Surprise On the Firm’s Return

34. ®1999 South-Western College Publishing 33 Arbitrage Pricing Theory (APT) APT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified awayAPT assumes that, in equilibrium, the return on a zero-investment, zero-systematic-risk portfolio is zero when the unique effects are diversified away The expected return on any asset i (E i ) can be expressed as:The expected return on any asset i (E i ) can be expressed as:

35. ®1999 South-Western College Publishing 34 Arbitrage Pricing Theory (APT) where: = the expected return on an asset with zero systematic risk: = the risk premium related to each of the common factors - for example the risk premium related to interest rate risk b ik = the pricing relationship between the risk premium and asset i - that is how responsive asset i is to this common factor k

36. ®1999 South-Western College Publishing 35 Example of Two Stocks and a Two-Factor Model λ 0 = the rate of return on a zero-systematic-risk asset (zero beta: b ik =0) is 3 percent ( λ 0 = 0.03 ) λ 1 = changes in the rate of inflation. The risk premium related to this factor is 1 percent for every 1 percent change in the rate ( λ 1 = 0.01 ) λ 2 = percent growth in real GNP. The average risk premium related to this factor is 2 percent for every 1 percent change in the rate ( λ 2 = 0.02 )

37. ®1999 South-Western College Publishing 36 Example of Two Stocks and a Two-Factor Model = the response of asset X to changes in the rate of inflation is 0.50 = the response of asset Y to changes in the rate of inflation is 2.00 = the response of asset X to changes in the growth rate of real GNP is 1.50 = the response of asset Y to changes in the growth rate of real GNP is 1.75

38. ®1999 South-Western College Publishing 37 Example of Two Stocks and a Two-Factor Model =.03 + (.01)b i1 + (.02)b i2 =.03 + (.01)b i1 + (.02)b i2 E x =.03 + (.01)(0.50) + (.02)(1.50) E x =.03 + (.01)(0.50) + (.02)(1.50) =.065 = 6.5% =.065 = 6.5% E y =.03 + (.01)(2.00) + (.02)(1.75) E y =.03 + (.01)(2.00) + (.02)(1.75) =.085 = 8.5% =.085 = 8.5%

39. ®1999 South-Western College Publishing 38 APT vs. CAPM Similar ResultsSimilar Results –Both yield a linear risk-return relationship Advantage of APTAdvantage of APT –More realistic, less restrictive assumptions –Allows for multiple risk factors (e.g., industry effects) Disadvantage of APTDisadvantage of APT –Fails to identify common factors

40. ®1999 South-Western College Publishing 39 Multifactor APT Suggested FactorsSuggested Factors –Default premium –Term structure –Inflation –Corporate profits –Market risk E(R i ) = λ 0 + λ 1  i1 + λ 2  i2 + … + λ k  ikE(R i ) = λ 0 + λ 1  i1 + λ 2  i2 + … + λ k  ik

41. ®1999 South-Western College Publishing 40 Empirical Tests of the APT Studies by Roll and Ross and by Chen support APT by explaining different rates of return with some better results than CAPMStudies by Roll and Ross and by Chen support APT by explaining different rates of return with some better results than CAPM Reinganum’s study did not explain small- firm resultsReinganum’s study did not explain small- firm results Dhrymes and Shanken question the usefulness of APT because it was not possible to identify the factorsDhrymes and Shanken question the usefulness of APT because it was not possible to identify the factors

42. ®1999 South-Western College Publishing 41 Roll-Ross Study 1. Estimate the expected returns and the factor coefficients from time-series data on individual asset returns 2. Use these estimates to test the basic cross- sectional pricing conclusion implied by the APT

43. ®1999 South-Western College Publishing 42 Extensions of the Roll-Ross Study Cho, Elton, and Gruber examined the number of factors in the return-generating process that were pricedCho, Elton, and Gruber examined the number of factors in the return-generating process that were priced Dhrymes, Friend, and Gultekin (DFG) reexamined techniques and their limitations and found the number of factors varies with the size of the portfolioDhrymes, Friend, and Gultekin (DFG) reexamined techniques and their limitations and found the number of factors varies with the size of the portfolio

44. ®1999 South-Western College Publishing 43 The APT and Anomalies Small-firm effectSmall-firm effect Reinganum - results inconsistent with the APT Chen - supported the APT model over CAPM January anomalyJanuary anomaly Gultekin - APT not better than CAPM Burmeister and McElroy - effect not captured by model, but still rejected CAPM in favor of APT APT and inflationAPT and inflation Elton, Gruber, and Rentzler - analyzed real returns

45. ®1999 South-Western College Publishing 44 The Shanken Challenge to Testability of the APT If returns are not explained by a model, it is not considered rejection of a model; however if the factors do explain returns, it is considered supportIf returns are not explained by a model, it is not considered rejection of a model; however if the factors do explain returns, it is considered support APT has no advantage because the factors need not be observable, so equivalent sets may conform to different factor structuresAPT has no advantage because the factors need not be observable, so equivalent sets may conform to different factor structures Empirical formulation of the APT may yield different implications regarding the expected returns for a given set of securitiesEmpirical formulation of the APT may yield different implications regarding the expected returns for a given set of securities Thus, the theory cannot explain differential returns between securities because it cannot identify the relevant factor structure that explains the differential returnsThus, the theory cannot explain differential returns between securities because it cannot identify the relevant factor structure that explains the differential returns

46. ®1999 South-Western College Publishing 45 Alternative Testing Techniques Jobson proposes APT testing with a multivariate linear regression modelJobson proposes APT testing with a multivariate linear regression model Brown and Weinstein propose using a bilinear paradigmBrown and Weinstein propose using a bilinear paradigm Others propose new methodologiesOthers propose new methodologies