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LECTURE 9 : EMPRICIAL EVIDENCE : CAPM AND APT. (Asset Pricing and Portfolio Theory). Contents. How can the CAPM be tested by using time series approach by using cross sectional approach Fama and MacBeth (1973) approach. CAPM : Time Series Tests.

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Lecture 9 empricial evidence capm and apt l.jpg


(Asset Pricing and Portfolio Theory)

Contents l.jpg

  • How can the CAPM be tested

    • by using time series approach

    • by using cross sectional approach

  • Fama and MacBeth (1973) approach

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CAPM : Time Series Tests

  • (ERi – rf)t = ai + bi(ERm – rf)t + eit

    Null hypothesis : ai = 0

    Assume individual returns are iid, but allow contemporaneous correlation across assets : E(eit, ejt) ≠ 0

    Estimation : maximum likelihood using panel data (N assets, T time periods)

  • Early studies in the 1970s find ai = 0, but later studies find the opposite.

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CAPM : Cross-Section Tests

  • Useing the Security Market Line (SML)

  • Hypothesis : average returns (in a cross section of stocks) depend linearly (and solely) on asset betas

  • Problems :

    • Individual stock returns are so volatile that one cannot reject the null hypothesis that average returns across different stocks are the same

      (s ≈ 30 – 80% p.a., hence cannot reject the null that average returns across different stocks are the same.)

    • Betas are measured with errors

      (to overcome these problems form portfolios)

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CAPM : Cross-Section Tests (Cont.)

  • Two Stage procedure

    1st stage : Time series regression, estimation of betas

    (ERi – rf)t = ai + bi(ERm – rf)t + eit

    (Time series regression has to be repeated for each fund, using ‘say’ 60 observations.)

    2nd stage :

    ERi = l0 + l1bhati + ui

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Cochane (2001)

  • Testing the CAPM

    Sort all stocks on the NYSE into 10 portfolios on basis of size, plus 2 bond portfolios

    1st stage : Estimating the betas for each of the 12 portfolios using time series data

    2nd stage : sample average returns regressed against the 12 estimated betas

    CAPM seems to do a reasonable ‘job’.

    (but if sorted by book to market value, decile returns are not explained by market beta).

    Results : see graph on next slide

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Size-Sorted Value-Weighted Decile Portfolio (NYSE – from 1947)

Smallest firm



OLS cross-section regression



Mean excess return (%)


CAPM prediction = line fitted through

NYSE value weighted return at A


Corporate Bonds


Gov. bonds










Market Betas, i,m

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Fama and MacBeth Rolling Regression 1947)

  • Much used regression methodology that involves ‘rolling’ cross-sectional regressions

  • Suppose

    • N stock returns for any single month

    • T months

      1st stage : For each months (t) estimate : (simplified equation)

      R(t)i = a(t) + g(t)bi + d(t)Zi + e(t)i

      (For any month H0 : a(t) = d(t) = 0 and g(t) > 0)

      Will get T a’s, g’s, d’s

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Fama and MacBeth Rolling Regression (Cont.) 1947)

  • 2nd stage : Testing the time series properties of the parameters to see if a E(a(t)) = 0 and g E(g(t)) > 0

  • If returns are niid, the following test can be conducted

which is t-distributed

where s(.) is the standard deviation of the monthly

estimates and is are the number of months.

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Data and Model 1947)

The data

  • Sample period : 1934 – June 1968, monthly data

  • NYSE common stock

    The model

    R(t)i = g0(t) + g1(t)bi + g2(t)bi2 + g3(t)si + e(t)i

    where s = measure of risk of security i not related to bi

    (e.g. Rit = ai + biRmt + eit with si = SD(eit))

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Hypotheses Tested 1947)

Hyp. 1 : Linearity

E(g2(t)) = 0

Hyp. 2 : No systematic effects of non-beta risk

E(g3(t)) = 0

Hyp. 3 : Positive return-risk trade off

E(g1(t)) = [ERm(t) – ER0(t)] > 0

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Main Results 1947)

  • Coefficient and residuals from Risk and return regression are consistent with “efficient capital markets”.

  • On average, positive trade off between risk and return

    (g1 (Hypothesis 3) is positive and statistically significant for the whole sample period.)

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Roll’s Critique 1947)

  • Roll (1977) shows that for any portfolio that is efficient ex-post, in sample of data there is an exact relationship between the mean returns and betas.

    • If market portfolio is mean-variance efficient, then the CAPM/SML must hold in the sample

    • Violation of SML implies market portfolio chosen by researcher is not true ‘market portfolio’.

    • (true market portfolio should include maybe : land, commodities, human capital as well as stocks and bonds)

  • Why proceed ? How far can a particular empirical model explain equilibrium returns ?

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Fama and French (1993) 1947)

  • US : cross sectional data

  • Sample : July 1963 to Dec. 1991 (monthly data)

  • 25 ‘size and value’ sorted portfolios, monthly time series returns on US stocks are explained by a 3-factor model

    Rit = b1iRmt + b2,iSMBt + b3iHMLt

    Rbari = lmb1i + lSMBb2i + lHMLb3i

    where Ri = excess return on asset i

    Rm = excess return on the market

    Rbar = mean return

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Fama and French (1993) (Cont.) 1947)

  • Variables mimicking ‘portfolios’ for

    • Book-to-market ratio (HML)

      • (Low B/M stocks have persistently high earnings)

      • Variable defined as return on high book to market stocks minus low book to market stocks

      • mimicks the risk factor in returns related to ‘distressed stocks’

    • Size (SMB)

      • Variable is defined as difference between return on small stock portfolio and large stock portfolio

      • mimicks the risk factor in returns related to size

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Fama and French (1993) Findings 1947)

  • For the 25 portfolios : sorted (i.) by size and (ii.) value (BMV) and (iii.) by value (BMV) and size.

    • Market betas are clustered in range 0.8 and 1.5

    • Average monthly returns are between 0.25 and 1

      (If CAPM is correct – expect positive correlation)

  • Sorting portfolios by book to market rejects the CAPM (see graph)

  • Success of Fama-French 3 factor model can be seen by comparing the predicted (average) returns with the actual returns (see graph)

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Average Excess Returns and Market Beta 1947)

Returns sorted by BMV,

within given size quintile

Returns sorted by size,

within given BMV quintile

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Actual and Predicted Average Returns FF - 3 Factor Model 1947)

Growth stocks (ie. Low book to market

for 5 different size categories)

Value stocks

(ie. High book to

market for 5 different

size categories)

The two lines connect portfolios of different size categories, within a given book-to-market category. We only connect the points within the highest and lowest BMV categories. If we had joined up point for the other BMV quintiles, the lines would show a positive relationship, like that for the value stocks – showing that the predicted returns from the Fama-French 3 factor model broadly predict average returns on portfolios sorted by size and BMV.

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Fama and French (1993) Findings (Cont.) 1947)

  • If the R-squared of the 25 portfolios is 100%, then the 3 factors would perfectly mimic the 25 portfolio average returns

     APT model

  • R2 lies between 0.83 and 0.97

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Summary 1947)

  • 2 stage procedure to test CAPM/APT

  • Empirical test on CAPM/APT use portfolios to minimise the errors in measuring betas

  • CAPM beta explains the difference in average (cross-section) returns between stocks and bonds, but not within portfolios of stocks

  • Fama and French book to market and size variables should be included as additional risk factors to explain cross-section of average stock returns

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References 1947)

  • Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial Economics’, Chapters 8

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References 1947)

  • Cochrane, J.H. (2001) ‘Asset Pricing’, Princeton University Press

  • Fama, E.F. and MacBeth, J.D. (1973) ‘Risk, Return and Equilibrium : Empirical Tests’, Journal of Political Economy, pp. 607- 636

  • Roll, R. (1977) ‘A Critique of Asset Pricing Theory’s Tests’, Journal of Financial Economics, Vol. 4, pp. 1073-1103.

  • Fama, E.F. and French, K. (1993) ‘Common Risk Factors in the Returns on Stocks and Bonds’, Journal of Financial Economics, Vol. 33, pp. 3-56.

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References 1947)

  • More information about the supplementary variables HML and SMB can be found on Kenneth French’s website