# Empirical Evidence : CAPM and APT

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

(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.
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)
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 + l1 bhati + ui
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
Mean excess return (%)
Size-Sorted Value-Weighted Decile
Portfolio (NYSE – from 1947)
14
OLS cross-section regression
Smallest firm
decile
10
A
6
CAPM prediction = line fitted through
NYSE value weighted return at A
Corporate Bonds
Gov. bonds
2
0
-2
0
0.2
0.4
0.6
0.8 1.0
1.2
Market Betas, bi,m
1.4
Fama and MacBeth (1973)
Fama and MacBeth
Rolling Regression


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) &gt; 0)
Will get T a’s, g’s, d’s
Fama and MacBeth Rolling
Regression (Cont.)


2nd stage : Testing the time series
properties of the parameters to see if a 
E(a(t)) = 0 and g  E(g(t)) &gt; 0
If returns are niid, the following test can be
conducted
which is t-distributed
gˆ
t gˆ j 
sgˆ j  / n
 
where s(.) is the standard deviation of the monthly
estimates and is are the number of months.
Data and Model
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))
Hypotheses Tested
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)] &gt; 0
Main Results


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.)
Roll’s Critique

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 ?
CAPM, Multifactor Models
and APT
Fama and French (1993)
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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
Fama and French (1993)
(Cont.)

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
Fama and French (1993)
Findings

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)
Average Excess Returns
and Market Beta
1.2
Returns sorted by BMV,
within given size quintile
Excess return
1
0.8
0.6
Returns sorted by size,
within given BMV quintile
0.4
0.2
0
0
0.2
0.4
0.6
0.8
Beta on Market
1
1.2
1.4
1.6
Actual and Predicted Average
Returns FF - 3 Factor Model
Actual 'mean' excess return
1.2
1
Growth stocks (ie. Low book to market
for 5 different size categories)
0.8
0.6
Value stocks
(ie. High book to
market for 5 different
size categories)
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
Predicted 'mean' excess return
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 FamaFrench 3 factor model broadly predict average returns on portfolios sorted by size and BMV.
Fama and French (1993)
Findings (Cont.)
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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
Summary
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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
References

Cuthbertson, K. and Nitzsche, D. (2004)
‘Quantitative Financial Economics’,
Chapters 8
References
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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.
References
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