Testing CAPM

NES EFM 2005/6 2 Plan Up to now: analysis of return predictability Up to now: analysis of return predictability –Main conclusion: need a better risk model explaining cross-sectional differences in returns Today: is CAPM beta a sufficient description of risks? Today: is CAPM beta a sufficient description of risks? –Time-series tests –Cross-sectional tests –Anomalies and their interpretation

NES EFM 2005/6 3 CAPM Sharpe-Lintner CAPM: Sharpe-Lintner CAPM: E t-1 [R i,t ] = R F + β i (E t-1 [R M,t ] – R F ) Black (zero-beta) CAPM: Black (zero-beta) CAPM: E t-1 [R i,t ] = E t-1 [R Z,t ] + β i (E t-1 [R M,t ] – E t-1 [R Z,t ]) Single-period model for expected returns, implying that Single-period model for expected returns, implying that –The intercept is zero –Beta fully captures cross-sectional variation in expected returns Testing CAPM = checking that market portfolio is on the mean-variance frontier Testing CAPM = checking that market portfolio is on the mean-variance frontier –Mean-variance efficiency tests

NES EFM 2005/6 4 Testing CAPM Standard assumptions for testing CAPM Standard assumptions for testing CAPM –Rational expectations for R i,t, R M,t, R Z,t : Ex ante ex post Ex ante ex post E.g., R i,t = E t-1 R i,t + e i,t, where e is white noise E.g., R i,t = E t-1 R i,t + e i,t, where e is white noise –Constant beta Testable equations: Testable equations: R i,t -R F = β i (R M,t -R F ) + ε i,t, R i,t = (1-β i )R Z,t + β i R M,t + ε i,t, –where E t-1 (ε i,t )=0, E t-1 (R M,t ε i,t )=0, E t-1 (R Z,t ε i,t )=0, E t-1 (ε i,t, ε i,t+j )=0 (j0)

NES EFM 2005/6 5 Time-series tests Sharpe-Lintner CAPM: Sharpe-Lintner CAPM: R i,t -R F = α i + β i (R M,t -R F ) + ε i,t (+ δ i X i,t-1 ) –H 0 : α i =0 for any i=1,…,N (δ i =0) Strong assumptions: R i,t ~ IID Normal Strong assumptions: R i,t ~ IID Normal –Estimate by ML, same as OLS Finite-sample F-test, which can be rewritten in terms of Sharpe ratios Finite-sample F-test, which can be rewritten in terms of Sharpe ratios –Alternatively: Wald test or LR test Weaker assumptions: allow non-normality, heteroscedasticity, auto-correlation of returns Weaker assumptions: allow non-normality, heteroscedasticity, auto-correlation of returns –Test by GMM

NES EFM 2005/6 6 Time-series tests (cont.) Black (zero-beta) CAPM: Black (zero-beta) CAPM: R i,t = α i + β i R M,t + ε i,t, –H 0 : there exists γ s.t. α i =(1-β i )γ for any i=1,…,N Strong assumptions: R i,t ~ IID Normal Strong assumptions: R i,t ~ IID Normal –LR test with finite-sample adjustment Performance of tests: Performance of tests: –The size is correct after the finite-sample adjustment –The power is fine for small N relative to T

NES EFM 2005/6 7 Results Early tests: did not reject CAPM Early tests: did not reject CAPM Gibbons, Ross, and Shanken (1989) Gibbons, Ross, and Shanken (1989) –Data: US, , monthly returns of 11 industry portfolios, VW-CRSP market index –For each individual portfolio, standard CAPM is not rejected –Joint test rejects CAPM CLM, Table 5.3 CLM, Table 5.3 –Data: US, , monthly returns of 10 size portfolios, VW-CRSP market index –Joint test rejects CAPM, esp. in the earlier part of the sample period

NES EFM 2005/6 8 Cross-sectional tests Main idea: Main idea: R i,t = γ 0 + γ 1 β i + ε i,t (+γ 2 X i,t ) H 0 : asset returns lie on the security market line H 0 : asset returns lie on the security market line –γ 0 = R F, –γ 1 = mean(R M -R F ) > 0, –γ 2 = 0 Two-stage procedure (Fama-MacBeth, 1973): Two-stage procedure (Fama-MacBeth, 1973): –Time-series regressions to estimate beta –Cross-sectional regressions period-by-period

NES EFM 2005/6 9 Time-series regressions R i,t = α i + β i R M,t + ε i,t First 5y period: First 5y period: –Estimate betas for individual stocks, form 20 beta- sorted portfolios with equal number of stocks Second 5y period: Second 5y period: –Recalculate betas of the stocks, assign average stock betas to the portfolios Third 5y period: Third 5y period: –Each month, run cross-sectional regressions

NES EFM 2005/6 10 Cross-sectional regressions R i,t -R F = γ 0 + γ 1 β i + γ 2 β 2 i + γ 3 σ i + ε i,t Running this regression for each month t, one gets the time series of coefficients γ 0,t, γ 1,t, … Running this regression for each month t, one gets the time series of coefficients γ 0,t, γ 1,t, … Compute mean and std of γs from these time series: Compute mean and std of γs from these time series: –No need for s.e. of coefficients in the cross-sectional regressions! –Shankens correction for the error-in-variables problem Assuming normal IID returns, t-test Assuming normal IID returns, t-test

NES EFM 2005/6 11 Why is Fama-MacBeth approach popular in finance? Period-by-period cross-sectional regressions instead of one panel regression Period-by-period cross-sectional regressions instead of one panel regression –The time series of coefficients => can estimate the mean value of the coefficient and its s.e. over the full period or subperiods –If coefficients are constant over time, this is equivalent to FE panel regression Simple: Simple: –Avoids estimation of s.e. in the cross-sectional regressions –Esp. valuable in presence of cross-correlation Flexible: Flexible: –Easy to accommodate additional regressors –Easy to generalize to Black CAPM

NES EFM 2005/6 12 Results Until late 1970s: CAPM is not rejected Until late 1970s: CAPM is not rejected –But: betas are unstable over time Since late 70s: multiple anomalies, fishing license on CAPM Since late 70s: multiple anomalies, fishing license on CAPM –Standard Fama-MacBeth procedure for a given stock characteristic X: Estimate betas of portfolios of stocks sorted by X Estimate betas of portfolios of stocks sorted by X Cross-sectional regressions of the ptf excess returns on estimated betas and X Cross-sectional regressions of the ptf excess returns on estimated betas and X Reinganum (1981): Reinganum (1981): –No relation between betas and average returns for beta- sorted portfolios in in the US

NES EFM 2005/6 13 Asset pricing anomalies VariablePremium's sign Reinganum (1983)January dummy+ French (1980)Monday dummy- Basu (1977, 1983)E/P+ Stattman (1980)Book-to-market: BE/ME+ Banz (1981)Size: ME- Bhandari (1988)Leverage: D/E+ Jegadeesh & Titman (1993)Momentum: 6m-1y return+ De Bondt & Thaler (1985)Contrarian: 3y-5y return- Brennan et al. (1996)Liquidity: trading volume-

NES EFM 2005/6 14 Interpretation of anomalies Technical explanations Technical explanations –There are no real anomalies Multiple risk factors Multiple risk factors –Anomalous variables proxy additional risk factors Irrational investor behavior Irrational investor behavior

NES EFM 2005/6 15 Technical explanations: Rolls critique For any ex post MVE portfolio, pricing equations suffice automatically For any ex post MVE portfolio, pricing equations suffice automatically It is impossible to test CAPM, since any market index is not complete It is impossible to test CAPM, since any market index is not complete Response to Rolls critique Response to Rolls critique –Stambaugh (1982): similar results if add to stock index bonds and real estate: unable to reject zero-beta CAPM –Shanken (1987): if correlation between stock index and true global index exceeds , CAPM is rejected Counter-argument: Counter-argument: –Roll and Ross (1994): even when stock market index is not far from the frontier, CAPM can be rejected

NES EFM 2005/6 16 Technical explanations: Data snooping bias Only the successful results (out of many investigated variables) are published Only the successful results (out of many investigated variables) are published –Subsequent studies using variables correlated with those that were found significant before are also likely to reject CAPM Out-of-sample evidence: Out-of-sample evidence: –Post-publication performance in US: premiums get smaller (size, turn of the year effects) or disappear (the week-end, dividend yield effects) –Pre-1963 performance in US (Davis, Fama, and French, 2001): similar value premium, which subsumes the size effect –Other countries (Fama&French, 1998): value premium in 13 developed countries

NES EFM 2005/6 17 Technical explanations (cont.) Error-in-variables problem: Error-in-variables problem: –Betas are measured imprecisely –Anomalous variables are correlated with true betas Sample selection problem Sample selection problem –Survivor bias: the smallest stocks with low returns are excluded Sensitivity to the data frequency: Sensitivity to the data frequency: –CAPM not rejected with annual data Mechanical relation between prices and returns (Berk, 1995) Mechanical relation between prices and returns (Berk, 1995) –Purely random cross-variation in the current prices (P t ) automatically implies higher returns (R t =P t+1 /P t ) for low-price stocks and vice versa

NES EFM 2005/6 18 Multiple risk factors Some anomalies are correlated with each other: Some anomalies are correlated with each other: –E.g., size and January effects Ball (1978): Ball (1978): –The value effect indicates a fault in CAPM rather than market inefficiency, since the value characteristics are stable and easy to observe => low info costs and turnover

NES EFM 2005/6 19 Multiple risk factors (cont.) Chan and Chen (1991): Chan and Chen (1991): –Small firms bear a higher risk of distress, since they are more sensitive to macroeconomic changes and are less likely to survive adverse economic conditions Lewellen (2002): Lewellen (2002): –The momentum effect exists for large diversified portfolios of stocks sorted by size and BE/ME => cant be explained by behavioral biases in info processing

NES EFM 2005/6 20 Irrational investor behavior Investors overreact to bad earnings => temporary undervaluation of value firms Investors overreact to bad earnings => temporary undervaluation of value firms La Porta et al. (1987): La Porta et al. (1987): –The size premium is the highest after bad earnings announcements

Testing CAPM: is beta dead ?

NES EFM 2005/6 22 Quiz Why is it important to have evidence based on non-US data? Why is it important to have evidence based on non-US data? Is Fama-MacBeth approach useful in other areas than testing CAPM? Is Fama-MacBeth approach useful in other areas than testing CAPM? What is Roll and Ross (1994) argument ? What is Roll and Ross (1994) argument ?

NES EFM 2005/6 23 Plan Joint test of anomalies Joint test of anomalies Conditional tests of CAPM Conditional tests of CAPM

NES EFM 2005/6 24 Fama and French (1992) " The cross-section of expected stock returns ", a.k.a. "Beta is dead article Evaluate joint roles of market beta, size, E/P, leverage, and BE/ME in explaining cross-sectional variation in US stock returns Evaluate joint roles of market beta, size, E/P, leverage, and BE/ME in explaining cross-sectional variation in US stock returns

NES EFM 2005/6 25 Data All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in Monthly return data (CRSP) Monthly return data (CRSP) Annual financial statement data (COMPUSTAT) Annual financial statement data (COMPUSTAT) –Used with a 6m gap Market index: the CRSP value-wtd portfolio of stocks in the three exchanges Market index: the CRSP value-wtd portfolio of stocks in the three exchanges –Alternatively: EW and VW portfolio of NYSE stocks, similar results (unreported)

NES EFM 2005/6 26 Data (cont.) Anomaly variables: Anomaly variables: –Size: ln(ME) –Book-to-market: ln(BE/ME) –Leverage: ln(A/ME) or ln(A/BE) –Earnings-to-price: E/P dummy (1 if E<0) or E(+)/P E/P is a proxy for future earnings only when E>0 E/P is a proxy for future earnings only when E>0

NES EFM 2005/6 27 Methodology Each year t, in June: Each year t, in June: –Determine the NYSE decile breakpoints for size (ME), divide all stocks to 10 size portfolios –Divide each size portfolio into 10 portfolios based on pre-ranking betas estimated over 60 past months –Measure post-ranking monthly returns of 100 size-beta EW portfolios for the next 12 months Measure full-period betas of 100 size-beta portfolios Measure full-period betas of 100 size-beta portfolios Run Fama-MacBeth (month-by-month) CS regressions of the individual stock excess returns on betas, size, etc. Run Fama-MacBeth (month-by-month) CS regressions of the individual stock excess returns on betas, size, etc. –Assign to each stock a post-ranking beta of its portfolio

NES EFM 2005/6 28 Results Table 1: characteristics of 100 size-beta portfolios Table 1: characteristics of 100 size-beta portfolios –Panel A: enough variation in returns, small (but not high-beta) stocks earn higher returns –Panel B: enough variation in post-ranking betas, strong negative correlation (on average, ) between size and beta; in each size decile, post-ranking betas capture the ordering of pre-ranking betas –Panel C: in any size decile, the average size is similar across beta-sorted portfolios

NES EFM 2005/6 29 Results (cont.) Table 2: characteristics of portfolios sorted by size or by pre-ranking beta Table 2: characteristics of portfolios sorted by size or by pre-ranking beta –When sorted by size alone: strong negative relation between size and returns, strong positive relation between betas and returns –When sorted by betas alone: no clear relation between betas and returns!

NES EFM 2005/6 30 Results (cont.) Table 3: Fama-MacBeth regressions Table 3: Fama-MacBeth regressions –Even when alone, beta fails to explain returns! –Size has reliable negative relation with returns –Book-to-market has even stronger (positive) relation –Market and book leverage have significant, but opposite effect on returns (+/-) Since coefficients are close in absolute value, this is just another manifestation of book-to-market effect! Since coefficients are close in absolute value, this is just another manifestation of book-to-market effect! –Earnings-to-price: U-shape, but the significance is killed by size and BE/ME

NES EFM 2005/6 31 Authors conclusions Beta is dead: no relation between beta and average returns in Beta is dead: no relation between beta and average returns in –Other variables correlated with true betas? But: beta fails even when alone But: beta fails even when alone Though: shouldnt beta be significant because of high negative correlation with size? Though: shouldnt beta be significant because of high negative correlation with size? –Noisy beta estimates? But: post-ranking betas have low s.e. (most below 0.05) But: post-ranking betas have low s.e. (most below 0.05) But: close correspondence between pre- and post-ranking betas for the beta-sorted portfolios But: close correspondence between pre- and post-ranking betas for the beta-sorted portfolios But: same results if use 5y pre-ranking or 5y post-ranking betas But: same results if use 5y pre-ranking or 5y post-ranking betas

NES EFM 2005/6 32 Authors conclusions Robustness: Robustness: –Similar results in subsamples –Similar results for NYSE stocks in Suggest a new model for average returns, with size and book-to-market equity Suggest a new model for average returns, with size and book-to-market equity –This combination explains well CS variation in returns and absorbs other anomalies

NES EFM 2005/6 33 Discussion Hard to separate size effects from CAPM Hard to separate size effects from CAPM –Size and beta are highly correlated –Since size is measured precisely, and beta is estimated with large measurement error, size may well subsume the role of beta! Once more, Roll and Ross (1994): Once more, Roll and Ross (1994): –Even portfolios deviating only slightly (within the sampling error) from mean-variance efficiency may produce a flat relation between expected returns and beta

NES EFM 2005/6 34 Further research Conditional CAPM Conditional CAPM –The anomaly variables may proxy for time-varying market risk exposures Consumption-based CAPM Consumption-based CAPM –The anomaly variables may proxy for consumption betas Multifactor models Multifactor models –The anomaly variables may proxy for time-varying risk exposures to multiple factors

NES EFM 2005/6 35 Ferson and Harvey (1998) "Fundamental determinants of national equity market returns: A perspective on conditional asset pricing" Conduct conditional tests of CAPM on the country level Conduct conditional tests of CAPM on the country level –Relating the instruments to alpha and beta –Global vs local instruments

NES EFM 2005/6 36 Data Monthly returns on MSCI stock indices of 21 developed countries, Monthly returns on MSCI stock indices of 21 developed countries, Risk-free rate: US 30-day T-bill Risk-free rate: US 30-day T-bill Common instruments (lagged): Common instruments (lagged): –World market return and dividend yield –The G10 vs USD FX return –30d Eurodollar deposit rate, 90-30d Eurodollar term spread Local instruments: Local instruments: –Valuation ratios: E/P, P/CF, P/BV, D/P –Financial: 60m volatility and 6m momentum –Macro: GDP per capita and inflation (relative to OECD), long-term interest rate, term spread, credit risks

NES EFM 2005/6 37 Methodology The return-generating process (rational expectations) The return-generating process (rational expectations) r i,t+1 = E t [r i,t+1 ] + β i,t (r M,t+1 -E t [r M,t+1 ]) + ε i,t+1, –where E t (ε i,t+1 ) = 0, E t (R M,t+1 ε i,t+1 ) = 0, –r i,t+1 is country is excess return in US dollars Model for conditional expected returns and betas: Model for conditional expected returns and betas: E t [r i,t+1 ] = α i,t + β i,t E t [r M,t+1 ], –where β i,t = β 0i + β 1i Z t + β 2i A i,t, – α i,t = α 0i + α 1i Z t + α 2i A i,t –Z t are global (world) instruments –A i,t are local (country-specific) instruments

NES EFM 2005/6 38 Methodology (cont.) Estimation by GMM (or OLS): Estimation by GMM (or OLS): r i,t+1 =(α 0i +α 1i Z t +α 2i A i,t ) + (β 0i +β 1i Z t +β 2i A i,t ) r M,t+1 +ε i,t+1 –H 0 : α i = 0 Two-factor model: Two-factor model: –Adding foreign exchange risk

NES EFM 2005/6 39 Results Table 2: conditional betas, joint tests for the groups of attributes Table 2: conditional betas, joint tests for the groups of attributes –For most countries, betas are time-varying –The impact of global instruments on betas is subsumed by local variables –Most important country-specific instruments: for market betas: E/P, inflation, and long-term interest rate for market betas: E/P, inflation, and long-term interest rate for FX betas: inflation and credit risks for FX betas: inflation and credit risks

NES EFM 2005/6 40 Results (cont.) Table 3: conditional alphas in a two-factor model, leaving only important instruments for betas Table 3: conditional alphas in a two-factor model, leaving only important instruments for betas –For most countries, alphas are time-varying –Panel B, jointly significant variables across the countries: E/P, P/CF, P/BV, volatility, inflation, long- term interest rate, and term spread –Panel C, economic significance: typical abnormal return (in response to 1σ change in X) around 1-2% per month

NES EFM 2005/6 41 Results (cont.) Table 5: cross-sectional explanatory power of lagged attributes Table 5: cross-sectional explanatory power of lagged attributes –The raw attributes alone produce low R2 –The explanatory power of attributes as instruments for risk is much greater than for mispricing –Some attributes enter mainly as instruments for beta (e.g., E/P) or alpha (e.g., momentum)

NES EFM 2005/6 42 Methodology II Fama-MacBeth approach with conditional alpha and betas in a two-factor model: for each month, Fama-MacBeth approach with conditional alpha and betas in a two-factor model: for each month, Estimate time-series regression with 60 prior months using one attribute at a time Estimate time-series regression with 60 prior months using one attribute at a time r i,t+1 = (α 0i + α 1i A i,t ) + (β 0i + β 1i A i,t ) r W,t+1 + ε i,t+1, –where r W,t+1 is a vector of the world market return and FX rate Estimate WLS cross-sectional regression using the fitted values of alpha and/or betas as well as raw attributes: Estimate WLS cross-sectional regression using the fitted values of alpha and/or betas as well as raw attributes: r i,t+1 = γ 0,t+1 + γ 1,t+1 a i,t+1 + γ 2,t+1 b i,t+1 + γ 3,t+1 A i,t + e i,t+1

NES EFM 2005/6 43 Conclusions Strong support for the conditional asset pricing model Strong support for the conditional asset pricing model Local attributes drive out global information variables in models of conditional betas Local attributes drive out global information variables in models of conditional betas The explanatory power of attributes as instruments for risk is much greater than for mispricing The explanatory power of attributes as instruments for risk is much greater than for mispricing The relation of the attributes to expected returns and risks is different across countries The relation of the attributes to expected returns and risks is different across countries