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are not the same as the bij in (3.56).) Substituting for hj from (3.58) and (3.5
and rearranging
ERi = r @,*(ER"- r )

where pt = bilpl bi2p2. Thus the two-factor APT model with the factors
mined by the betas of the CAPM yields an equation for the expected return
which is of the form given by the standard CAPM itself. Hence, in testing t
with each other. (The argument is easily generalised to the k factor case.)
The APT model involves some rather subtle arguments and it is not easily
at an intuitive level. The main elements of the APT model are:
It provides a structure for determining equilibrium returns based on con
portfolio that has zero risk (i.e. zero-beta portfolio) and requires no cash
Arbitrage arguments imply that such a riskless portfolio has an actual an
return of zero.
The above conditions, plus the assumptions of linear factor weightings
enough number of securities to give an infinitely small (zero) specific
orthogonality restrictions to be placed on the parameters of the expec
equation. These restrictions give rise to an expected returns equation th
on the factor loadings (bij) and the indices of risk ( h j s ) .
The APT does not rely on any assumptions about utility functions or
consider only the mean and variance of prospective portfolios. The APT doe
require homogeneous expectations.
The CAPM is not necessarily inconsistent with even a multi-index APT bu
it may not be consistent with the APT.
The APT contains arbitrary elements when its empirical implementation is
(e.g. what are the appropriate factors F j ? Are the bij constant over time
be difficult to interpret (e.g. there are no a priori restrictions on the sign
and h j ) .
The CAPM is more restrictive than the APT but it has a more immedia
appeal and is somewhat easier to test in that the 'factor' is more easily 'pin
(e.g. in the standard CAPM it is the excess return on the market portfolio

This section discusses some early tests of the CAPM and APT models and a
simple (yet theoretically naive) model known as the single index model. We
to the question of the appropriate methods to use in testing models of equilibr
in future chapters.

3.5.1 The Single Index Model SIM
This is not really a 'model' in the sense that it embodies any behavioural hyp
it is merely a statistical assumption that the return on any security Rir may be
represented as a linear function of a single (economic) variable I , (e.g. GNP)
+ 6iIt +
= 6i
Rit &it

where Eir is white noise and (3.61) holds for i = 1, 2 . . . .m securities and f
periods. If the unexplained element of the return & j r for any security i is inde
cov(l,, E j t ) = 0 for all i and t
Under the above assumptions, unbiased estimates of (e;,6;) for each securit
portfolio of securities) can be obtained by an OLS regression using (3.61) on
data for Rj, and I,.
The popularity of the SIM arises from the fact that it considerably reduces
of parameters (or inputs) in order to calculate the mean and variance of a p
n securities and hence to calculate the efficient portfolio. Given our assump
easy to show that

= 6; a, i-aEi

For a portfolio of n securities we have

n n

;=I j=1
i= 1

To calculate the optimal proportions x:, in general requires
n - expected returns ER,
n - variances ai
n(n - 1)/2 - covariances
as ˜inputs™. However, if the SIM is a good statistical description of asset returns
as ˜inputs™ n values of (ei, 6i, a,) and values for EI, and U; in order to cal
and a; using (3.64). However, to calculate all the covariance terms aij no
information is required (see equation 3.64) and compared with the general cas
˜save™ on n(n - 1)/2 calculations: if n is large this is a considerable advan
SIM. When the SIM is used in this way the ˜single index™ I, is often taken
actual (ex-post) return on the market portfolio RY with variance a (replacing
The expected value of the market return ER? and 0 might then be based o
sample averages over a recent data period. However, as we see below, th
poor representation of expected returns and in particular the independence a
E(EjEj) = 0, rarely holds in practice. The reason for this is that if R; for two
securities depends on more than one index then Ei and E j will not be uncorre
is a case of omitted variables bias with a common omitted variable in each eq
another way, it is unlikely that shocks or news which influence shares in por
not also influence the returns on portfolio B.
ERi, - rt = pi(ERY - r t ) (standard CAPM)
E(R;, - $) = piE(RY - q) (zero-beta CAPM)
where we assume in both models that is constant over time. If we now assu
expectations (see Chapter 5 ) so that the difference between the out-turn and t
is random then:
Ri, = ERi, E;,

RY = ERY + w,
where E;, and w, are white noise (random) errors. Equations (3.66) and (3.67)
+ (standard CAPM)
(R;, - r , ) = p;(RY - r,) E;,

(Ri, - q)= p;(RY - q) &rt
+ (zero-beta CAPM)
= E i t - B j W f . Rearranging equations (3.68) and (3.69):
where E:,

+ BiRY +
Ri, = rt(l - pi) (standard CAPM)

Ri, = R“(1- p;) + BiRY + (zero-beta CAPM)

Comparing either the standard CAPM or the zero-beta CAPM with th
equation (3.61) with I, = ER? then it is easy to see why the SIM is defic
standard CAPM is true then
Sj = r,(l - pi)
and for the zero-beta CAPM:
= Rf(1 - p i )

Thus, even if B; is constant over time we would not expect 6j in the SIM to b
since r, or R: vary over time. Also since E:, and ET, depend on w,, the error in
the return on the market portfolio, then E:, and E;, in the SIM will be correl
CAPM is the true model. This violates a key assumption of the SIM. Finally
the CAPM is true and r, is correlated with RM then the SIM has an ˜omitted va
by standard econometric theory the OLS estimate of Si is biased and if the
p(r, R*) < 0 then 6i is biased downward and cannot be taken as a ˜good™
the true pi given by the CAPM. (In these circumstances the intercept 8i in
biased upwards.) Hence results from the SIM certainly throw no light on the
the CAPM or, put another way, if the CAPM is ˜true™ and given that r, (or R:
time then the SIM model is invalid.

Direct Tests of the CAPM
These often take the form of a two-stage procedure. Under the assumption
constant over time, a first-pass time series regression of (R;, - r,) on (RY -
constant term gives:
+ +
(Rir - rt) = a pi(RY - rr)
+ Jrlbi + Vi
Ri = 90
Comparing (3.75) with the standard CAPM relation

where a bar indicates the sample mean values. An even stronger test of
in the second-pass regression is to note that if there is an unbiased estimat
(i = 1 to k) then onZy @i should influence Ri. Under the null that the CAPM i
in the following second-pass cross-section regression:
+ + +
R; = QO J r l P i Q2P: Q;

we expect
H o : Q2 = +3 = 0
where o:i is an unbiased estimate of the (own) variance of security i from th
regression (3.76). If Q2 # 0 then there are said to be non-linearities in the secu
line. If Q3 # 0 then diversifiable risk affects the expected return on a securit
violations of the CAPM.
We can repeat the above tests for the zero-beta CAPM by replacing r,
particular, if the zero-beta CAPM is the true model

The first-pass time series regression (3.74) rearranged is:

Comparing (3.78) and (3.79)
= (R˜ - r ) ( l - p i )

Hence if the zero-beta CAPM is true rather than the standard CAPM then in th
regression (3.79) we expect ai > 0 when @i < 1 and vice versa, since we kno
theoretical discussion in section 3.1 that RZ - r > 0.
There are yet further econometric problems with the two-step approach
CAPM doesn™t rule out the possibility that the error terms may be heterosce
the variances of the error terms are not constant). In this case the OLS standar
incorrect and other estimation techniques need to be used (e.g. Hansen™s GMM
or some other form of GLS estimator). The CAPM does rule out the erro
being serially correlated over time unless the data frequency is finer than the
horizon for Ri, (e.g. weekly data and monthly expected returns). In the latter ca
estimator is required (see Chapter 19).
correlated with a securities™ error variance o:˜, then the latter serves as a proxy
#?i and hence if S i is measured with error, then .E˜ may be significant in the s
regression. Finally, note that if the error distribution of Eit is non-normal (e.g
skewed) then any estimation technique based on normality will produce biased
In particular positive skewness in the residuals of the cross-section regression
up as an association between residual risk and return even though in the true m
is no association.
The reader can see that there are acute econometric problems in trying s
to estimate the true betas from the first-pass regression and to obtain correct
ingful results in the second-pass regressions. These problems have been listed
explained or proved for those who are familiar with the econometric methodo
As an illustration of these early studies consider that of Black et a1 (1
monthly rates of return 1926-1966 in the first-pass time series regress
minimised the heteroscedasticity problem and the error in estimating the betas b
all stocks into a set of ten portfolios based on the size of the betas for individua
(i.e. the time series estimates of #?i for individual securities over a rollin
estimation period are used to assemble the portfolios). For the ten ˜size por
monthly return R i is regressed on R r over a period of 35 years

and the results are given in Table 3.1. In the second-pass regressions they ob

with the R squared, for the regression = 0.98. The non-zero intercept is not
with the standard CAPM but is consistent with the zero-beta CAPM. (At le
portfolio of stocks used here rather than for individual stocks.) Note that w

™IBble 3.1 Estimates of j3 for Ten Portfolios
˜˜ ˜ ˜ ˜ ˜

Portfolio Excess ReturdU) ai
˜˜ ˜ ˜˜

1 1.56 2.13 -0.08
2 1.38 1.77 -0.19
3 1.25 1.71 -0.06
4 1.16 1.63 -0.02
5 1.05 -0.05
6 -0.05
0.92 1.37
7 0.85 1.26 0.05
8 0.75 1.15 0.08
9 0.63 1.10 0.19
10 0.49 0.91 0.20
Market 1.42
(a) Percent per month.
Source: Black et a1 (1972)
where the data consist of 20 portfolios. The second-pass regression is repeate
month over 1935-68 to see how the q s vary over time. They find that the
the @2 and \I/3 estimates are not significantly different from zero and hence
standard CAPM. They also find that qj is not serially correlated over the 35
period for the monthly residuals, again weak support for the CAPM.
Our final illustrative second-pass regression uses a sampZe estimate of th
of the return on individual securities o:[= C,(Rj, R,)/(n- l)] and not t
variance from the first-pass regression. A representative result from Levy (19
+ +
Rj = 0.117 0.0086j 0.1976:
(14.2) (0.9)
R2 = 0.38, ( - ) = t statistic

and hence 6j provides no additional explanation of the expected return over
that provided by the own variance 0 This directly contradicts the CAPM
note that recent econometric research (see Pagan and Ullah (1988)) has shown
an estimate of the sample variance as a measure of the true (conditional) var
to biased estimates, so Levy™s early results are open to question. A more so
method of estimating variances and covariances is examined in Chapter 17.

The Post-Tax Standard CAPM
If individuals face different tax rates on dividend income and capital gains
with homogeneous expectations about pre-tax rates of return, each investor
different after-tax efficient frontier. However, in this extension of the CAPM
equilibrium ˜returns equation™ for all assets and all portfolios exists and is giv

where 6, = dividend yield of the market portfolio, Sj = dividend yield for
t = tax rate parameter depending on a weighted average of the tax rates for a
and their wealth.
From (3.84) any cross-section second-pass regression pi and the dividen
will affect expected returns. (The security market line (SML) is now a plan
dimensional (Ri, pi, S i ) space rather than a straight line.) When dividends a
a higher efective rate than capital gains (which is the case in the US and
capital gains tax is largely avoidable) then 5 > 0 and hence expected retur
with dividend yield. In this extended CAPM model, all investors hold a widely
portfolio, but those investors with high income tax rates will hold more lo
paying stocks in their portfolio (relative to the market portfolio holdings of an
who faces average tax rates).
However, the study by Litzenberger and Ramaswamy (1979) uses ˜superior™
likelihood methods and carefully considers the exact monthly timing of dividend
They find that over the period 1936-77 on monthly time series data:
Rj, - T = 0.0063 0.04218i1 + 0.23(Si, - rt)
[2.6] [1.9]
[ - ] = t statistic
The model (3.85) suggests that in (3.86) we expect a = r. When t, = 0 (i.e. ze
capital gains tax) then the tax parameter r is equal to the average income
their study Litzenberger and Ramaswamy run equation (3.86) over several dif
periods and they find 0.24 < a < 0.38. The true income tax rate is within thi
their results support the post-tax CAPM. In addition Litzenberger and Ramas
that over six subperiods during the 1936-77 time span the coefficient on Si
positive and is statistically significant in five of these subperiods. Unfortu
coefficient on P i t is statistically significant in only two of the subperiods and is
temporally unstable as it changes sign over different periods. It appears that th
yield is a better predictor of expected returns than is beta. Hence overall the
not provide support for the CAPM. We shall encounter the importance of th
yield in determining expected returns in more advanced studies in later chapt
This schematic overview of the conflicting results yielded by early tests of
can be summarised by noting
difficulties in using actual data to formulate the model correctly, part

measuring variances correctly
the assumption of constant betas (over time) may not be correct

econometric techniques have advanced so that these early studies may give

on balance the results (after consulting the original sources) suggest to this

the CAPM (particularly the zero-beta version) certainly has some empiric

Roll™s Critique
Again, it is worth mentioning Roll™s (1977) powerful critique of tests of th
CAPM based on the security market line (SML). He demonstrated that for an
that is efficient ex post (call it q) then in a sample of data there is an e
relationship between the mean return and beta. It follows that there is reall
testable implication of the zero-beta CAPM,namely that the market portfoli
variance efficient. If the market portfolio is mean-variance efficient then the
hold in the sample. Hence violations of the SML in empirical work may be
that the portfolio chosen by the researcher is not the true ˜market portfoli
the researcher is confident he has the true market portfolio (which may inc
commodities, human capital, as well as stocks and bonds) then tests based o
can explain equilibrium returns. Also once one recognises that the betas m
varying (see Chapter 17) this considerably weakens the force of Roll™s argum
is directed at the second-pass regressions on the SML.

Tests of the APT
Roll and Ross (1984) applied factor analysis to 42 groups of 30 stocks using
between 1962 and 1972. In their first-pass regressions they find that for most gr
five ˜factors™ provide a sufficiently good statistical explanation of Ri,. In the s
regression they find that three factors are sufficient. However, Dhrymes e
show that one problem in interpreting results from factor analysis is that the
statistically significant factors appears to increase as more securities are inclu
The second-pass regressions of Fama and MacBeth (1973) and Litzen
Ramaswamy (1979) reported above may be interpreted as second-pass regre
of the APT since they are of the form
+ + h2bi2
Ri = ho A1bil

where bj2 equals #, o, or 6j. These results are conflicting on whether the CA
a sufficient statistic for R , or whether additional ˜factors™ are required as imp
Sharpe (1982), Chen (1983), Roll and Ross (1984), and Chen et a1 (19
a wide variety of factors F , that might influence expected return in the firs
series regression. Such factors include the dividend yield, a long-short yield
government bonds and changes in industrial production. The estimated bjj fro
pass regressions are then used as cross-section variables in the second-pass
(3.87) for each month. Several of the hj are statistically significant thus su
multi-factor APT model. In Shape™s results the securities™ CAPM beta was
whereas for Chen et a1 (1986) the CAPM beta when added to the factors alread
in the APT returns equation contributed no additional explanatory power to t
pass regression. However, the second-pass regressions have a maximum R2 of
on monthly data so there is obviously a great deal of ˜noise™ in asset retur
makes it difficult to make firm inferences.
Overall this early empirical work on the APT is suggestive that more than
is important in determining assets returns. This is confirmed by more recent
US data (e.g. Shanken (1992) and Shanken and Weinstein (1990)) and UK
and Thomas (1994) and Poon and Taylor (1991), which use improved vari
two-stage regression tests.
Yet another approach to testing the APT uses the two basic equations:
+ +
Rif = E(Rj,) bijFj, &it

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