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