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Testing CAPM. Plan. 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? Time-series tests Cross-sectional tests Anomalies and their interpretation. - PowerPoint PPT Presentation
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Testing CAPMTesting CAPM
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Plan Plan
Up to now: analysis of return Up to now: analysis of return predictabilitypredictability– Main conclusion: need a better risk model Main conclusion: need a better risk model
explaining cross-sectional differences in explaining cross-sectional differences in returns returns
Today: is CAPM beta a sufficient Today: is CAPM beta a sufficient description of risks?description of risks?– Time-series testsTime-series tests– Cross-sectional testsCross-sectional tests– Anomalies and their interpretationAnomalies and their interpretation
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CAPMCAPM
Sharpe-Lintner CAPM: Sharpe-Lintner CAPM:
EEt-1t-1[R[Ri,ti,t] = R] = RFF + β + βii (E (Et-1t-1[R[RM,tM,t] – R] – RFF)) Black (zero-beta) CAPM: Black (zero-beta) CAPM:
EEt-1t-1[R[Ri,ti,t] = E] = Et-1t-1[R[RZ,tZ,t] + β] + βii (E (Et-1t-1[R[RM,tM,t] – E] – Et-1t-1[R[RZ,tZ,t])]) Single-period model for expected returns, Single-period model for expected returns,
implying thatimplying that– The intercept is zeroThe intercept is zero– Beta fully captures cross-sectional variation in Beta fully captures cross-sectional variation in
expected returnsexpected returns Testing CAPM = checking that market portfolio Testing CAPM = checking that market portfolio
is on the mean-variance frontieris on the mean-variance frontier– ‘‘Mean-variance efficiencyMean-variance efficiency’ tests’ tests
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Testing Testing CAPMCAPM
Standard assumptions for testing CAPMStandard assumptions for testing CAPM– Rational expectations for Rational expectations for RRi,ti,t, R, RM,tM,t, R, RZ,tZ,t: :
Ex ante Ex ante → → ex postex post E.g., RE.g., Ri,ti,t = E = Et-1t-1RRi,ti,t + e + ei,ti,t, where e is white noise, where e is white noise
– Constant betaConstant beta Testable equations:Testable equations:
RRi,ti,t-R-RFF = β = βii(R(RM,tM,t-R-RFF) + ε) + εi,ti,t,,
RRi,ti,t = (1-β = (1-βii)R)RZ,tZ,t + β + βiiRRM,tM,t + ε + εi,ti,t,,– where Ewhere Et-1t-1(ε(εi,ti,t)=0, E)=0, Et-1t-1(R(RM,tM,tεεi,ti,t)=0, E)=0, Et-1t-1(R(RZ,tZ,tεεi,ti,t)=0, )=0,
EEt-1t-1(ε(εi,ti,t, ε, εi,t+ji,t+j)=0 (j≠0))=0 (j≠0)
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Time-series testsTime-series tests
Sharpe-Lintner CAPM:Sharpe-Lintner CAPM:RRi,ti,t-R-RFF = α = αii + β + βii(R(RM,tM,t-R-RFF) + ε) + εi,ti,t (+ δ (+ δiiXXi,t-1i,t-1))– HH00: α: αii=0 for any i=1,…,N (δ=0 for any i=1,…,N (δii=0)=0)
Strong assumptions: RStrong assumptions: Ri,ti,t ~ IID Normal ~ IID Normal– Estimate by ML, same as OLSEstimate by ML, same as OLS
Finite-sample F-test, which can be rewritten in terms Finite-sample F-test, which can be rewritten in terms of Sharpe ratiosof Sharpe ratios
– Alternatively: Wald test or LR testAlternatively: Wald test or LR test Weaker assumptions: allow non-normality, Weaker assumptions: allow non-normality,
heteroscedasticity, auto-correlation of heteroscedasticity, auto-correlation of returnsreturns– Test by GMMTest by GMM
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Time-series testsTime-series tests (cont.)(cont.) Black (zero-beta) CAPM:Black (zero-beta) CAPM:
RRi,ti,t = α = αii + β + βiiRRM,tM,t + ε + εi,ti,t,,– HH00: there exists γ s.t. α: there exists γ s.t. αii=(1-β=(1-βii)γ for any )γ for any
i=1,…,Ni=1,…,N Strong assumptions: RStrong assumptions: Ri,ti,t ~ IID Normal ~ IID Normal
– LR test with finite-sample adjustmentLR test with finite-sample adjustment Performance of tests:Performance of tests:
– The size is correct after the finite-sample The size is correct after the finite-sample adjustmentadjustment
– The power is fine for small N relative to TThe power is fine for small N relative to T
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ResultsResults
Early tests: did not reject CAPMEarly tests: did not reject CAPM Gibbons, Ross, and Shanken (1989)Gibbons, Ross, and Shanken (1989)
– Data: US, 1926-1982, monthly returns of 11 Data: US, 1926-1982, monthly returns of 11 industry portfolios, VW-CRSP market indexindustry portfolios, VW-CRSP market index
– For each individual portfolio, standard For each individual portfolio, standard CAPM is not rejectedCAPM is not rejected
– Joint test rejects CAPMJoint test rejects CAPM CLM, Table 5.3CLM, Table 5.3
– Data: US, 1965-1994, monthly returns of 10 Data: US, 1965-1994, monthly returns of 10 size portfolios, VW-CRSP market indexsize portfolios, VW-CRSP market index
– Joint test rejects CAPM, esp. in the earlier Joint test rejects CAPM, esp. in the earlier part of the sample periodpart of the sample period
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Cross-sectional testsCross-sectional tests
Main idea: Main idea: RRi,ti,t = γ = γ00 + γ + γ11ββii + ε + εi,ti,t (+γ (+γ22XXi,ti,t))
HH00: asset returns lie on the security : asset returns lie on the security market linemarket line– γγ00 = R = RFF, , – γγ11 = mean(R = mean(RMM-R-RFF) > 0, ) > 0, – γγ22 = 0 = 0
Two-stage procedure (Fama-MacBeth, Two-stage procedure (Fama-MacBeth, 1973):1973):– Time-series Time-series regressionsregressions to estimate betato estimate beta– Cross-sectional Cross-sectional regressions period-by-periodregressions period-by-period
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Time-series Time-series regressionsregressions
RRi,ti,t = α = αii + β + βiiRRM,tM,t + ε + εi,ti,t First 5y period: First 5y period:
– Estimate betas for individual stocks, form Estimate betas for individual stocks, form 20 beta-sorted portfolios with equal number 20 beta-sorted portfolios with equal number of stocksof stocks
Second 5y period: Second 5y period: – Recalculate betas of the stocks, assign Recalculate betas of the stocks, assign
average stock betas to the portfoliosaverage stock betas to the portfolios Third 5y period: Third 5y period:
– Each month, run cross-sectional regressionsEach month, run cross-sectional regressions
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Cross-sectional Cross-sectional regressionsregressions
RRi,ti,t-R-RFF = γ = γ00 + γ + γ11ββii + γ + γ22ββ22ii + γ + γ33σσii + ε + εi,ti,t
Running this regression for each month Running this regression for each month tt, , one gets the time series of coefficients one gets the time series of coefficients γγ0,t0,t, γ, γ1,t1,t, …, …
Compute mean and std of γ’s from these Compute mean and std of γ’s from these time series:time series:– No need for s.e. of coefficients in the cross-No need for s.e. of coefficients in the cross-
sectional regressions!sectional regressions!– Shanken’s correction for the ‘Shanken’s correction for the ‘error-in-error-in-
variablesvariables’ problem’ problem
Assuming normal IID returns, t-testAssuming normal IID returns, t-test
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Why is Fama-MacBeth Why is Fama-MacBeth approach popular in approach popular in finance?finance? Period-by-period cross-sectional regressions Period-by-period cross-sectional regressions
instead of one panel regressioninstead of one panel regression– The time series of coefficients => can estimate the The time series of coefficients => can estimate the
mean value of the coefficient and its s.e. over the mean value of the coefficient and its s.e. over the full period or subperiodsfull period or subperiods
– If coefficients are constant over time, this is If coefficients are constant over time, this is equivalent to FE panel regressionequivalent to FE panel regression
Simple: Simple: – Avoids estimation of s.e. in the cross-sectional Avoids estimation of s.e. in the cross-sectional
regressionsregressions– Esp. valuable in presence of cross-correlationEsp. valuable in presence of cross-correlation
Flexible:Flexible:– Easy to accommodate additional regressorsEasy to accommodate additional regressors– Easy to generalize to Black CAPMEasy to generalize to Black CAPM
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ResultsResults
Until late 1970s: CAPM is not rejectedUntil late 1970s: CAPM is not rejected– But: betas are unstable over timeBut: betas are unstable over time
Since late 70s: multiple anomalies, Since late 70s: multiple anomalies, ““fishing licensefishing license” on CAPM” on CAPM– Standard Fama-MacBeth procedure for a Standard Fama-MacBeth procedure for a
given stock characteristic X:given stock characteristic X: Estimate betas of portfolios of stocks sorted by XEstimate betas of portfolios of stocks sorted by X Cross-sectional regressions of the ptf excess Cross-sectional regressions of the ptf excess
returns on estimated betas and Xreturns on estimated betas and X Reinganum (1981): Reinganum (1981):
– No relation between betas and average No relation between betas and average returns for beta-sorted portfolios in 1964-returns for beta-sorted portfolios in 1964-1979 in the US1979 in the US
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Asset pricing Asset pricing anomaliesanomalies
Variable Premium'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 -
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Interpretation of Interpretation of anomaliesanomalies Technical explanationsTechnical explanations
– There are no real anomaliesThere are no real anomalies Multiple risk factorsMultiple risk factors
– Anomalous variables proxy Anomalous variables proxy additional risk factorsadditional risk factors
Irrational investor behaviorIrrational investor behavior
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Technical Technical explanations: Roll’s explanations: Roll’s critique critique For any ex post MVE portfolio, pricing equations For any ex post MVE portfolio, pricing equations
suffice automaticallysuffice automatically It is impossible to test CAPM, since any market It is impossible to test CAPM, since any market
index index is not completeis not complete Response to Roll’s critiqueResponse to Roll’s critique
– Stambaugh (1982): similar results if add to stock index Stambaugh (1982): similar results if add to stock index bonds and real estate: unable to reject zero-beta CAPMbonds and real estate: unable to reject zero-beta CAPM
– Shanken (1987): if correlation between stock index and Shanken (1987): if correlation between stock index and true global index exceeds 0.7-0.8, CAPM is rejectedtrue global index exceeds 0.7-0.8, CAPM is rejected
Counter-argument:Counter-argument:– Roll and Ross (1994): even when stock market index is Roll and Ross (1994): even when stock market index is
not far from the frontier, CAPM can be rejectednot far from the frontier, CAPM can be rejected
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Technical Technical explanations: Data explanations: Data snooping bias snooping bias Only the successful results (out of many Only the successful results (out of many
investigated variables) are publishedinvestigated variables) are published– Subsequent studies using variables correlated with Subsequent studies using variables correlated with
those that were found significant before are also those that were found significant before are also likely to reject CAPMlikely to reject CAPM
Out-of-sample evidence:Out-of-sample evidence:– Post-publication performance in US: premiums get Post-publication performance in US: premiums get
smaller (size, turn of the year effects) or disappear smaller (size, turn of the year effects) or disappear (the week-end, dividend yield effects)(the week-end, dividend yield effects)
– Pre-1963 performance in US (Davis, Fama, and Pre-1963 performance in US (Davis, Fama, and French, 2001): similar value premium, which French, 2001): similar value premium, which subsumes the size effectsubsumes the size effect
– Other countries (Fama&French, 1998): value Other countries (Fama&French, 1998): value premium in 13 developed countriespremium in 13 developed countries
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Technical explanations Technical explanations (cont.)(cont.) Error-in-variables problem: Error-in-variables problem:
– Betas are measured impreciselyBetas are measured imprecisely– Anomalous variables are correlated with true betasAnomalous variables are correlated with true betas
Sample selection problemSample selection problem– Survivor bias: the smallest stocks with low returns Survivor bias: the smallest stocks with low returns
are excludedare excluded Sensitivity to the data frequency: Sensitivity to the data frequency:
– CAPM not rejected with annual dataCAPM not rejected with annual data Mechanical relation between prices and Mechanical relation between prices and
returns (Berk, 1995)returns (Berk, 1995)– Purely random cross-variation in the current prices Purely random cross-variation in the current prices
(P(Ptt) automatically implies higher returns ) automatically implies higher returns (R(Rtt=P=Pt+1t+1/P/Ptt) for low-price stocks and vice versa) for low-price stocks and vice versa
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Multiple risk factorsMultiple risk factors
Some anomalies are correlated with Some anomalies are correlated with each other: each other: – E.g., size and January effectsE.g., size and January effects
Ball (1978): Ball (1978): – The value effect indicates a fault in The value effect indicates a fault in
CAPM rather than market inefficiency, CAPM rather than market inefficiency, since the value characteristics are since the value characteristics are stable and easy to observe => low info stable and easy to observe => low info costs and turnovercosts and turnover
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Multiple risk factors Multiple risk factors (cont.)(cont.) Chan and Chen (1991): Chan and Chen (1991):
– Small firms bear a higher risk of Small firms bear a higher risk of distressdistress, , since they are more sensitive to since they are more sensitive to macroeconomic changes and are less likely macroeconomic changes and are less likely to survive adverse economic conditionsto survive adverse economic conditions
Lewellen (2002): Lewellen (2002): – The momentum effect exists for large The momentum effect exists for large
diversified portfolios of stocks sorted by diversified portfolios of stocks sorted by size and BE/ME => can’t be explained by size and BE/ME => can’t be explained by behavioral biases in info processingbehavioral biases in info processing
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Irrational investor Irrational investor behaviorbehavior Investors overreact to bad Investors overreact to bad
earnings => temporary earnings => temporary undervaluation of undervaluation of value value firmsfirms
La Porta et al. (1987): La Porta et al. (1987): – The size premium is the highest The size premium is the highest
after bad earnings announcementsafter bad earnings announcements
Testing CAPM:Testing CAPM:is beta dead ?is beta dead ?
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Fama and FrenchFama and French (1992)(1992)
""The cross-section of expected The cross-section of expected stock returnsstock returns", ", a.k.a. a.k.a. "Beta is "Beta is dead“ articledead“ article
Evaluate joint roles of market Evaluate joint roles of market beta, size, E/P, leverage, and beta, size, E/P, leverage, and BE/ME in explaining cross-BE/ME in explaining cross-sectional variation in US stock sectional variation in US stock returnsreturns
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DataData
All non-financial firms in NYSE, AMEX, All non-financial firms in NYSE, AMEX, and (after 1972) NASDAQ in 1963-1990and (after 1972) NASDAQ in 1963-1990
Monthly return data (CRSP)Monthly return data (CRSP) Annual financial statement data Annual financial statement data
(COMPUSTAT)(COMPUSTAT)– Used with a 6m gapUsed with a 6m gap
Market index: the CRSP value-wtd Market index: the CRSP value-wtd portfolio of stocks in the three exchangesportfolio of stocks in the three exchanges– Alternatively: EW and VW portfolio of NYSE Alternatively: EW and VW portfolio of NYSE
stocks, similar results (unreported)stocks, similar results (unreported)
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Data (cont.)Data (cont.)
‘‘Anomaly’ variables:Anomaly’ variables:– Size: ln(ME)Size: ln(ME)– Book-to-market: ln(BE/ME)Book-to-market: ln(BE/ME)– Leverage: ln(A/ME) or ln(A/BE)Leverage: ln(A/ME) or ln(A/BE)– Earnings-to-price: E/P dummy (1 if Earnings-to-price: E/P dummy (1 if
E<0) or E(+)/PE<0) or E(+)/P E/P is a proxy for future earnings only E/P is a proxy for future earnings only
when E>0when E>0
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MethodologyMethodology
Each year Each year tt, in June:, in June:– Determine the NYSE decile breakpoints for size (ME), Determine the NYSE decile breakpoints for size (ME),
divide all stocks to 10 size portfoliosdivide all stocks to 10 size portfolios– Divide each size portfolio into 10 portfolios based on Divide each size portfolio into 10 portfolios based on
pre-ranking pre-ranking betas estimated over 60 past monthsbetas estimated over 60 past months– Measure Measure post-ranking post-ranking monthly returns of 100 size-beta monthly returns of 100 size-beta
EW portfolios for the next 12 monthsEW portfolios for the next 12 months Measure full-period betas of 100 size-beta Measure full-period betas of 100 size-beta
portfoliosportfolios Run Fama-MacBeth (month-by-month) CS Run Fama-MacBeth (month-by-month) CS
regressions of the regressions of the individual stock individual stock excess excess returns on betas, size, etc.returns on betas, size, etc.– Assign to each stock a post-ranking beta of its portfolioAssign to each stock a post-ranking beta of its portfolio
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ResultsResults
Table 1: characteristics of 100 size-beta Table 1: characteristics of 100 size-beta portfoliosportfolios– Panel A: enough variation in returns, small Panel A: enough variation in returns, small
(but not high-beta) stocks earn higher (but not high-beta) stocks earn higher returnsreturns
– Panel B: enough variation in post-ranking Panel B: enough variation in post-ranking betas, strong negative correlation (on betas, strong negative correlation (on average, -0.988) between size and beta; in average, -0.988) between size and beta; in each size decile, post-ranking betas capture each size decile, post-ranking betas capture the ordering of pre-ranking betasthe ordering of pre-ranking betas
– Panel C: in any size decile, the average size Panel C: in any size decile, the average size is similar across beta-sorted portfoliosis similar across beta-sorted portfolios
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Results (cont.)Results (cont.)
Table 2: characteristics of Table 2: characteristics of portfolios sorted by size or by pre-portfolios sorted by size or by pre-ranking betaranking beta– When sorted by size alone: strong When sorted by size alone: strong
negative relation between size and negative relation between size and returns, strong positive relation returns, strong positive relation between betas and returnsbetween betas and returns
– When sorted by betas alone: no When sorted by betas alone: no clear relation between betas and clear relation between betas and returns!returns!
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Results (cont.)Results (cont.)
Table 3: Fama-MacBeth regressions Table 3: Fama-MacBeth regressions – Even when alone, beta fails to explain returns!Even when alone, beta fails to explain returns!– Size has reliable negative relation with returnsSize has reliable negative relation with returns– Book-to-market has even stronger (positive) Book-to-market has even stronger (positive)
relationrelation– Market and book leverage have significant, Market and book leverage have significant,
but opposite effect on returns (+/-)but opposite effect on returns (+/-) Since coefficients are close in absolute value, this is Since coefficients are close in absolute value, this is
just another manifestation of book-to-market effect!just another manifestation of book-to-market effect!
– Earnings-to-price: U-shape, but the Earnings-to-price: U-shape, but the significance is killed by size and BE/MEsignificance is killed by size and BE/ME
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Authors’ conclusionsAuthors’ conclusions
““Beta is dead”: no relation between beta Beta is dead”: no relation between beta and average returns in 1963-1990and average returns in 1963-1990– Other variables correlated with true betas?Other variables correlated with true betas?
But: beta fails even when aloneBut: beta fails even when alone Though: shouldn’t beta be significant because of high Though: shouldn’t beta be significant because of high
negative correlation with size?negative correlation with size?
– Noisy beta estimates?Noisy beta estimates? But: post-ranking betas have low s.e. (most below But: post-ranking betas have low s.e. (most below
0.05)0.05) But: close correspondence between pre- and post-But: close correspondence between pre- and post-
ranking betas for the beta-sorted portfoliosranking betas for the beta-sorted portfolios But: same results if use 5y pre-ranking or 5y post-But: same results if use 5y pre-ranking or 5y post-
ranking betasranking betas
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Authors’ conclusionsAuthors’ conclusions
Robustness:Robustness:– Similar results in subsamplesSimilar results in subsamples– Similar results for NYSE stocks in 1941-Similar results for NYSE stocks in 1941-
19901990 Suggest a new model for average Suggest a new model for average
returns, with size and book-to-market returns, with size and book-to-market equityequity– This combination explains well CS variation This combination explains well CS variation
in returns and absorbs other anomaliesin returns and absorbs other anomalies
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DiscussionDiscussion
Hard to separate size effects from CAPMHard to separate size effects from CAPM– Size and beta are highly correlatedSize and beta are highly correlated– Since size is measured precisely, and beta is Since size is measured precisely, and beta is
estimated with large measurement error, size estimated with large measurement error, size may well subsume the role of beta!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 Even portfolios deviating only slightly (within
the sampling error) from mean-variance the sampling error) from mean-variance efficiency may produce a flat relation between efficiency may produce a flat relation between expected returns and betaexpected returns and beta
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Further researchFurther research
Conditional CAPMConditional CAPM– The ‘anomaly’ variables may proxy for time-The ‘anomaly’ variables may proxy for time-
varying market risk exposuresvarying market risk exposures Consumption-based CAPMConsumption-based CAPM
– The ‘anomaly’ variables may proxy for The ‘anomaly’ variables may proxy for consumption betasconsumption betas
Multifactor modelsMultifactor models– The ‘anomaly’ variables may proxy for time-The ‘anomaly’ variables may proxy for time-
varying risk exposures to multiple factorsvarying risk exposures to multiple factors
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Ferson and HarveyFerson and Harvey (1998)(1998)
"Fundamental determinants of "Fundamental determinants of national equity market returns: national equity market returns: A perspective on conditional A perspective on conditional asset pricingasset pricing""
Conditional tests of CAPM on the Conditional tests of CAPM on the country levelcountry level– Monthly returns on MSCI stock Monthly returns on MSCI stock
indices of 21 developed countries, indices of 21 developed countries, 1970-19931970-1993
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Time series approachTime series approach
rri,t+1i,t+1=(α=(α0i0i+α’+α’1i1iZZtt+α’+α’2i2iAAi,ti,t) + (β) + (β0i0i+β’+β’1i1iZZtt+β’+β’2i2iAAi,ti,t) ) rrM,t+1M,t+1++εεi,t+1i,t+1
ZZtt are global instruments are global instruments– World market return, dividend yield, FX, interest World market return, dividend yield, FX, interest
ratesrates
AAi,ti,t are local (country-specific) instruments are local (country-specific) instruments– E/P, D/P, 60m volatility, 6m momentum, GDP per E/P, D/P, 60m volatility, 6m momentum, GDP per
capita, inflation, interest ratescapita, inflation, interest rates
HH00: α: αii = 0 = 0
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Results: for mostResults: for most countriescountries
Betas are time-varying, mostly due Betas are time-varying, mostly due to local variablesto local variables– E/P, inflation, long-term interest rateE/P, inflation, long-term interest rate
Alphas are time-varying, due toAlphas are time-varying, due to– E/P, P/CF, P/BV, volatility, inflation, E/P, P/CF, P/BV, volatility, inflation,
long-term interest rate, and term long-term interest rate, and term spreadspread
– Economic significance: typical Economic significance: typical abnormal return (in response to 1σ abnormal return (in response to 1σ change in X) around 1-2% per monthchange in X) around 1-2% per month
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Fama-MacBeth Fama-MacBeth approachapproach Each month:Each month: Estimate time-series regression with 60 prior Estimate time-series regression with 60 prior
months using one local instrumentmonths using one local instrument
rri,t+1i,t+1 = ( = (αα0i0i + + αα1i1iAAi,ti,t) + () + (ββ0i0i + + ββ1i1iAAi,ti,t) r) rM,t+1M,t+1 + + εεi,t+1i,t+1
Estimate WLS cross-sectional regression using Estimate WLS cross-sectional regression using the fitted values of alpha and beta and the the fitted values of alpha and beta and the attribute:attribute:
rri,t+1i,t+1 = = γγ0,t+10,t+1++γγ1,t+11,t+1aai,t+1i,t+1++γγ’’2,t+12,t+1bbi,t+1i,t+1++γγ3,t+13,t+1AAi,ti,t+e+ei,t+1i,t+1
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ResultsResults
The explanatory power of The explanatory power of attributes as instruments for risk attributes as instruments for risk is much greater than for is much greater than for mispricingmispricing
Some attributes enter mainly as Some attributes enter mainly as instruments for beta (e.g., E/P) or instruments for beta (e.g., E/P) or alpha (e.g., momentum)alpha (e.g., momentum)
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ConclusionsConclusions
Strong support for the conditional asset Strong support for the conditional asset pricing modelpricing model
Local attributes drive out global Local attributes drive out global information variables in models of information variables in models of conditional betasconditional betas
The explanatory power of attributes as The explanatory power of attributes as instruments for risk is much greater than instruments for risk is much greater than for mispricingfor mispricing
The relation of the attributes to expected The relation of the attributes to expected returns and risks is different across returns and risks is different across countriescountries