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An Empirical Investigation Of The Arbitrage Pricing Theory - Ross and Roll Paper

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- 1. An Empirical Investigation of the Arbitrage Pricing Theory - Roll and Ross<br />
- 2. Agenda<br />APT – An introduction<br />Purpose of Study<br />Methodology<br />APT and its Testability<br />APT Testing<br />Empirical Results<br />APT against a specific alternative<br />Test for equivalence of Factor structure across group<br />Conclusion<br />
- 3. Arbitrage<br />Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit.<br />Since no investment is required, an investor can create large positions to secure large levels of profit.<br />In efficient markets, profitable arbitrage opportunities will quickly disappear<br />
- 4. Arbitrage Pricing Theory<br />APT is a testable alternative to CAPM<br />APT aims to explain correlations between returns by developing a model of where those correlations come from<br />Difference between APT and original Sharpe CAPM<br />APT Allows more that one generating factor<br />APT demonstrate no arbitrage profits – Linear relationship between each assets expected return and its return’s response amplitudes<br />
- 5. Contd.<br />Improvement over CAPM<br />Based on linear Return Generating Principle and requires no utility assumptions beyond monotonicity and concavity <br />Hold in both single period and multiple period<br />Do away with the assumption of market portfolio being mean variance efficient<br />
- 6. Purpose of Study <br />First study that tests empirically the APT model developed by Ross in 1976.<br />Researches casts doubt on CAPM ability to explain asset returns<br />The study was done to investigate the existence of different factors that are “priced” or they are associated with risk premium<br />
- 7. Methodology<br />Data – Daily returns from 1962 to 72, alphabetically and grouped into 42 groups of 30 securities each <br />Factor analysis (factor loadings) as statistical technique to estimate b coefficients.<br />Tests: (all repeated for each of the 42 groups)<br />From the time series of returns, within each of the 42 groups, they compute a sample product-moment covariance matrix;<br />A Maximum Likelihood Estimation factor analysis estimates the number of factors and the matrix of loadings (b’s);<br />These factor loadings are then used as independent variables to explain the cross-sectional variation of individual stock returns;<br />Estimates from the cross-sectional model are used to measure the size and significance of risk premium associated with the estimated factors.<br />
- 8. Contd.<br />Section I – Discussion of the unique testable features of APT<br />Section II – Basic Tests<br />Section III – APT is compared against specific alternative hypothesis that “own variance influences expected returns”<br />Section IV – Test of consistency of APT across group of Assets<br />
- 9. SECTION I<br />
- 10. The APT and Its Testability<br />Assumptions<br />Perfectly competitive and frictionless assets market<br />Random return generation<br />Model:<br /> ‾ri= Ei+bi1δ1 + …. + bik‾δk + ‾εi,<br />Where, Ei = expected return on ith asset<br />δ = factors explaining systematic risk<br /> b = Beta coefficient <br />‾εi = Noise term – error related to unsystematic risk<br />Possible Systematic Factors – Economic aggregates, GNP, Interest rate etc. <br />
- 11. Contd.<br />If the error terms were omitted then the previous equations states that each asset i has returns ri that are a linear combination of the returns on a risk-less asset and the returns on k other factors<br />The linearity makes it possible to create perfectly substitutable portfolios and hence the APT states that there are only a few systematic components of risk existing. <br />
- 12. APT from Return generating process<br />Consider an individual who wishes to alter his current portfolio.<br />The new portfolio will differ from old one by investment proportion xi<br />Decision will depend on arbitrage portfolio investigation<br />x‾r = ∑ixi‾ri<br />= xE +(xb1)‾δ1 + … + (xbk)‾δk + x‾ε<br />Now if x is chosen in a way that there is no systematic risk then xbj= ∑ixibij= 0<br />And also the error term will tend to zero if the law of large numbers is applied, thus x‾r= xE<br />It shows that we can choose portfolios with no systematic and unsystematic risk. This is not possible.<br />
- 13. Estimation of expected returns<br />Multifactor<br />Ei= λ0+ λ1bi1+……………..+ λkbikfor all i<br />For riskless asset, b0j = 0 <br /> E0 = λ0<br />Ei – E0 =λ1bi1+……………..+ λkbik<br />If there is a single factor<br />Ei – E0 =λbi <br />Ei= systematic risk<br />
- 14. Continued……..<br />Market portfolio as a systematic risk<br />CAPM includes all of the universe of available assets in market portfolio whereas APT considers only the subsets of the sets of all returns.<br />Stochastic models is convenient for APT.<br />Critical assumption : returns be generated over the shortest trading period<br />
- 15. APT Testing<br />Two step procedure<br />Expected returns and factor betas are estimated from time series data on individual security returns.<br />Using the above estimates the basic cross-sectional pricing relationship:<br /> Ho: There exist non-zero constants (E0, λ1, λ2, … λk)<br />
- 16. Estimation of factor coefficients<br />V = BΛB’ + D<br />where, <br />B: [bij] factor loading matrix<br />Λ : factor covariance matrix<br />D : diagonal matrix of own asset variances <br />If G is an orthogonal transformation martix GG’ =I<br />V = BΛB’ + D<br /> = BGG’Λ GG’B’ + D<br /> = (BG) (G’ΛG) (BG’) + D<br />Ei – E0 = B λ<br />Ei – E0 = (BG) (G’λ)<br /> The APT concludes that excess expected returns lie in the space spanned by the factor loadings.<br />
- 17. Testing of hypothesis<br />Ei =E0 + λ1bi1+……………..+ λkbik<br />sample errors<br />ˆEi = Ei + ei<br />ˆby = by + βy<br />Under null hypothesis, cross sectional regression for any period will be of the form<br />ˆEi = Ei + ei<br /> = E0 + λ1bi1+……………..+ λkbik + ei<br />= E0 + λ1bi1+……………..+ λkbik + ξi<br />Where the regression error<br />ξi= ei– (λ1 β i1+……………..+ λk βik) (refer pg.1085)<br />
- 18. SECTION II<br />
- 19. Data<br />Source: Center for research in Security prices (University of Chicago)<br />Selection Criteria: Alphabetical order (30 companies each group listed on NYSE or AMSE)<br />Time Horizon: July 1962-Dec 1972<br />Basic Data Unit: Return adjusted for capital changes and dividends<br />Maximum Sample Size: 2619 daily returns, variations in no. of observations<br />No. of groups: 42<br />
- 20. Estimating the factor model<br />The following procedure were performed for all the groups and Tabulated <br />Covariance matrices of returns of individual assets for each group<br />Maximum likelihood factor analysis: to estimate the no. of factors and matrix<br />Above factors are used to run a multi factor regression (Cross sectional model)<br />Expected returns and Variations<br />Estimation of size and significance of Risk premium<br />
- 21. Stage1.Covariance matrices of returns of individual assets for each group<br />Every element in the covariance matrix was divided by one-half of the largest of the 30 individual variances: <br />To prevent rounding error<br />No effect on the results since factor analysis is scale free<br />
- 22. Stage2: Maximum likelihood factor analysis<br />Optimization Technique used (Joreskog and Sorbom)<br />Method is usually preferable since more is known about its statistical properties<br />M.L.E. provides the capability of estimating the number of factors<br />Specify the no. of factors (k<30)<br />Solve for the M.L. condition on a covariance matrix generated by k factors<br />
- 23. Contd.<br />Calculate an alternative value of likelihood functions (Without any restrictions)<br />Calculate Likelihood ratio (1st/ 2nd)<br />2 log (likelihood ratio) ~ Chi Sq distribution<br />If (Cal) > (Tab), more than k factors re required <br />Add factors till the Chi Sq. statistic shows 50% probability level<br />However for ease in further analysis we will use some more factors<br />
- 24. Findings<br />No. of factors used: 5<br />Chi Sq.: 246.1, df: 295, prob. Level: (.98)<br />Table: 2<br />Results: 5 factors is a conservative figure<br />Statistical dependence because of Covariance<br />Model: Equation (6) <br />
- 25. Test of APT<br />Equation: 8<br />Run a simple OLS cross sectional regression: Equation 9<br />Modification in the equation because of biasness (Fama & MacBeth): Equation (10), GLS estimation of Risk premia<br />Further Analysis<br />
- 26. Results<br />Table III<br />Part 1:<br /> L0 = 6 %<br />Conclusion: At least three factors are important for pricing<br />Part 2:<br /> L0 : Estimated istead of Assumed<br />Conclusion: At least two factors are significant<br />So, Overestimation in above part 1 above<br />
- 27. SECTION III<br />
- 28. Testing APT against a specific alternative<br />Hypothesis:<br />Other variables (apart from the ones found to be “priced”) are associated with non-zero risk-premia, even though they are not related to undiversifiable risk. <br />The total variance of individual returns, or the “own” variance.<br /><ul><li>If APT is valid, total variance should not affect expected returns (diversifiable and such would be eliminated by portfolio formation), and its non-diversifiable part would depend only on the factor loadings and factor variances</li></li></ul><li>Procedure<br />Regress cross-sectional estimates of expected returns on the 5 factor loadings from the previous section and on the SD of individual returns.<br />Result – Table IV pg 1094<br />Results show that the “own” SD has significant explanatory power of the cross-section of returns.<br />Note : Positive dependence across groups<br />
- 29. Questioning results<br />Miller and Scholes - Skewness can create dependence between the sample mean and sample standard deviation. <br />Testing for skewness – Individual daily returns are highly skewed (96.3%). (table V)<br />Skewness is cross-sectionally correlated positively with the mean return and standard deviation.<br />Result – Dependence can not be removed by exploiting the measured skewness. <br />
- 30. Correcting for skewness<br />By estimating each parameters from a different set of observations.<br />No sampling covariation.<br />Cross asset population relationship would remain.<br />Results – Table VI pg 1097<br />9 out of 42 groups exhibit a significant t-stat for s.<br />The effect can further be reduced by inserting more days between observations.<br />
- 31. SECTION IV<br />
- 32. Test Of equivalence of Factor Structure across group<br />In previous sections assets were splited into groups - results were suspected to have spurious sampling dependence among groups<br />No method of determining whether the same factors are there across group or different<br />But intercept must be same across group<br />
- 33. Hotelling’s T2 Test<br />Reason – time series data which could have correlation among security returns across groups<br />Data – 19 time series for Z(g/2) having 400 observations each<br /> H0 : E( ) = 0, g = 2, 4, …. , 38<br />Simultaneous observations were done<br />Results: no evidence that intercept terms are different.<br />
- 34. Issues<br />Test is quite weak<br />Very low explanatory power in daily cross sectional regression<br />Sampling variation is quite large<br />
- 35. Conclusion<br />The empirical data support the APT against both unspecified alternative – a very weak test and specified alternative that own variance has an independent explanatory effect on excess returns.<br />Empirical anomalies could be reexamined – <br />Ex - APT shows the explanatory power of price-earnings ratio for excess returns, acts as a surrogate for factor loadings.<br />An effort must be directed at identifying a more meaningful set of sufficient statistics for the underlying factors.<br />At the end it’s the systematic variability along that effects the expected returns. <br />

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