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to the portfolio held by the representative investor. Then the Treynor, Sharpe
indices may well provide a useful summary statistic of the relative perform
mutual fund against the S&P index (which one must admit may not be mea
efficient).

3.3.4 Performance of Mutual Funds
There have been a large number of studies of mutual fund performance using
three indices. Most studies have found that fund managers are unable system
beat the market and hence they do not outperform an unmanaged (yet diversified
such as the S&P index. Shawky (1982) examines the performance of 255 mu
different from zero in only 25 out of the 255 mutual funds studied and, o
had negative values of J i and only nine had positive values. Thus, out of mo
mutual funds only nine outperformed an unmanaged diversified portfolio such
index. Using Sharpe™s index, between 15 and 20 percent of mutual funds ou
an unmanaged portfolio, while for Treynor™s index the figure was slightly
around 33 percent outperforming the unmanaged portfolio. Hence, althoug
some mutual funds which outperform an unmanaged portfolio these are not
large in number.
Given the above evidence, there seems to be something of a paradox in
funds are highly popular, and their growth in Western developed nations thro
1970-1990 period has been substantial. While it is true that mutual funds
do not outperform the unmanaged S&P index, nevertheless, they do outperf
any unmanaged portfolio consisting of only a small number of shares. Rela
transaction costs for small investors often imply that investing in the S&P ind
viable alternative and hence they purchase mutual funds. The key results to e
this section are:
Sharpe™s reward-to-variability index is an appropriate performance measur
0

investor holds mutual fund shares plus a riskless asset.
Treynor™s and Jensen™s performance indices are appropriate when the
0

assumed to diversify his portfolio and holds both mutual fund shares, to
many other risky assets and the riskless asset.


3.4 THE ARBITRAGE PRICING THEORY (APT
An alternative to the CAPM in determining the expected rate of return on indivi
and on portfolios of stocks is the arbitrage pricing theory (APT). Broadly sp
APT implies that the return on a security can be broken down into an expe
and an unexpected or surprise component. For any individual stock this surpr
component can be further broken down into ˜general news™ that affects all
specific ˜news™ which affects only this particular stock. For example, news w
all stocks might be an unexpected announcement of an increase in interest r
government. News that affects the stocks of a specific industrial sector, for exam
be the invention of a new radar system which might be thought to influence th
industry but not other industries like chemicals and service industries. The AP
that ˜general news™ will affect the rate of return on all stocks but by differen
For example, a 1 percent unexpected rise in interest rates might affect the retur
of a company that was highly geared, more than that for a company that was
The APT, in one sense, is more general than the CAPM in that it allows a la
of factors to affect the rate of return on a particular security. In the CAPM the
only one factor that influences expected return, namely the covariance between
on the security and the return on the market portfolio.
stock and uit = the unexpected, surprise or news element.
We can further subdivide the surprise or news element Ujr into systematic
risk mt,that is risk that affects a large number of stocks each to a greater or le
and unsystematic (idiosyncratic or specific) risk & i f , which specifically affec
firm or a small group of firms:
+
Uit = mt Ejr

As in the case of the CAPM, the systematic risk cannot be diversified aw
this element of news or new information affects all companies. However, as
unsystematic or specific risk may be diversified away.
In order to make the APT operational we need some idea of what causes
risk. News about economy-wide variables are, for example, a government ann
that GDP is higher than expected or a sudden increase in interest rates by the Ce
These economy-wide factors F (indexed by j ) , may have different effects o
securities and this is reflected in the different values for the coefficients bij
given below:

+ bi2(F2r - EF2,) +
m, = C b i j ( F j - E F j ) , = b i l ( F 1 , - EF1,) ***

j

where the expectations operator E applies to information at time t - 1 or e
example, if for a particular fr the beta attached to the surprise in interest ra
im
to 0.5 then for every 1 percent that the interest rate rises above its expected
would increase the return on security i by half a percent (above its expected v
that the ˜betas™ here are not the same as the CAPM betas and hence are denot
than /?.(Later it will be shown that the betas of the APT may be reconciled wit
of the CAPM under certain restrictive assumptions.)
A crucial assumption of the APT is that the idiosyncratic or specific risk E
related across different securities:
COV(Ei, E j ) = 0
In fact, as we shall see below, specific risk can be diversified away by hold
number of securities.

Return on the Portfolio
For simplicity, suppose there is only one systematic risk factor, F,, and thre
in the portfolio. The return on a portfolio RP of three securities held in propo
I
by definition
3 3


i=l i=l
3 /3 \ 3
to be close to zero. In fact, as the number of securities increases the last te
right-hand side of (3.37) will approach zero and the specific risk will have be
fied away. Hence the return on the portfolio is made up of the expected retu
individual securities and the systematic risk as represented by the single eco
news term F , .

A More Formal Approach
The beauty of the APT is that it does not require any assumptions about uti
or that the mean and variance of a portfolio are the only two elements in the
objective function. The model is really a mechanism (an algorithm almost) that
to derive an expression for the expected return on a security (or a portfolio of
based on the idea that riskless arbitrage opportunities will be instantaneously
Not surprisingly, the APT does, however, require some (arbitrary) assumpt
assume that agents have homogeneous expectations and that the return Rit on
is linearly related to a set of k factors F j t :




where the bij are known as factor weights. Taking expectations of (3.38) and
EEit = 0, and subtracting it from itself
k
+ +
- EF j t )
= ERit bij ( Fj t
Rit &it
i
Equation (3.39) shows that although each security is affected by all the factors,
of any particular Fk depends on the value of bik and this is different for eac
This is the source of the covariance between returns Rit on different securities
Assume that we can continue adding factors to (3.39) until the unexplained
return E; is such that
= 0 for all i # j and all time periods
E(EiEj)

- E F j ) ] = 0 for all stocks and factors (and all t )
E[Ei(Fj

Respectively, (3.40) and (3.41) state that the unsysfematic (or specific) risk is un
across securities and is independent of the factors F, that is of systematic risk
the factors F are common across all securities. Now we perform an ˜experim
investors form a zero-beta portfolio with zero net investment. The zero-beta port
satisfy
n
C x i b i j = 0 for all j = 1 , 2 , .. . k.
i= 1
and the assumption of zero investment implies
n
CX;=O
i=l
can be earned by arbitrage. This arbitrage condition places a restriction on th
return of the portfolio, so using (3.34) we have:
k
n




Using (3.42) and the assumption that for a large well-diversified portfolio th
on the RHS approaches zero gives:



where the second equality holds by definition. Since this artificially constructe
has an actual rate of return RP equal to the expected return ER:, there is a zero
in its return and it is therefore riskless. Arbitrage arguments then suggest that t
return must be zero: n
ER; = C x i E R i , = 0
i=l
We now have to invoke a proof based on linear algebra. Given the conditi
(3.42), (3.43) and (3.46) which are known as orthogonality conditions then
shown that the expected return on any security i may be written as a linear c
of the factor weightings bij. For example, for a two-factor model:
+ hlbjl + h2bj2
ER, = ho
It was noted above that bjl and bi2 in (3.39) are specific to security i . The expe
on security i weights these security-specific betas by a weight hj that is the s
securities. Hence hj may be interpreted as the extra expected return required
a securities sensitivity to the jth
attribute (e.g. GNP or interest rates) of the s

Interpretation of the A j
Assume for the moment that the values of bil and bi2 are known. We can inte
as follows. Consider a zero-beta portfolio (i.e. bil and bi2 = 0 ) which has a
return ERZ.If riskless borrowing and lending exist then ERZ = r , the risk-free r
bjl = bi2 = 0 in (3.47) gives
A0 = ERZ (or r )

Next consider a portfolio having bil = 1 and bi2 = 0 with an expected return E
tuting in (3.47) we obtain
hi = ER1 - A0 = E(R1 - R Z )
+ bilE(R1 - R Z )+ bi2E(R2 - RI)
ERj = E R

Thus one interpretation of the APT is that the expected return on a security i
the sensitivity of the actual return to the factor loadings (i.e. the bij). In add
factor loading (e.g. bil) is ˜weighted™ by the expected excess return E(R1 -
the (excess) return on a portfolio whose beta with respect to the first factor
with respect to all other factors is zero. This portfolio with a ˜beta of 1™ theref
the unexpected movements in the factor F1.

3.4.1 Implementation of the APT
The APT may be summed up in two equations:




k
+
ERi, = ho bijhj
j= 1

where ho = r, or ER“. The APT may be implemented in the following (sty
A ˜first-pass™ time series regression of Ri, on a set of factors F j f (e.g. infla
growth, interest rates) will yield estimates of ai and the bil, bi2, etc. This can b
for i = 1 , 2 , . . .rn securities so that we have rn values for each of the betas, on
of the different securities. In the ˜second-pass™ regression the bi vary over the r
and are therefore the RHS variables in (3.53). Hence in equation (3.53) the
variables which are different across the rn securities. The hj are the same for al
and hence these can be estimated from the cross-section regression (3.53) of
bij (for i = 1,2, . . . rn). The risk-free rate is constant across securities and h
constant term in the cross-section regression.
The above estimation is a two-step procedure. There exists a superior pro
principle at least) whereby both equations (3.52) and (3.53) are estimated simu
This is known as factor analysis. Factor analysis chooses a subset of all the fac
that the covariance between each equation™s residuals is (close to) zero (i.e. E (
which is consistent with the theoretical assumption that the portfolio is fully
One stops adding factors Fj when the next factor adds ˜little™ additional e
Thus we simultaneously estimate the appropriate number of F j S and their cor
bijs. The hj are then estimated from the cross-section regression (3.53).
There are, however, problems in interpreting the results from factor anal
the signs on the bij and h j s are arbitrary and could be reversed (e.g. a positi
negative hj is statistically undistinguishable from a negative bi j and positive hj
there is a scaling problem in that the results still hold if the p i j are doubled
halved. Finally, if the regressions are repeated on different samples of data t
guarantee that the same factors will appear in the same order of importance.
arguments plus a few other minimal restrictive assumptions) nevertheless it is
implement and make operational. As well as the problems of interpretation of
h j which we cannot 'sign' apriori (i.e. either could be positive or negative)
also have problems in that the bij or hj may not be constant over time. In gen
applied work has concentrated on regressions of equation (3.52) in an effort
few factors that explain actual returns.

3.4.2 The CAPM and the APT
It must by now be clear to the reader that these two models of equilibrium
returns are based on rather different (behavioural) assumptions. The APT is oft
to as a multi-factor (or multi-index) model. The standard CAPM in this termin
be shown to be a very special case of the APT, namely a single-factor ver
APT, where the single factor is the expected return on the market portfolio E
return generating equation for security i is hypothesised to depend on only one
this factor is taken to be the return on the market portfolio, then the APT giv

+ bjR7 +
= aj
Rjt &it


This single index APT equation (3.54) can be shown to imply that the expecte
given by:
+
ERi, = rt b j ( E q - r t )
which conforms with the equilibrium return equation for the CAPM.
The standard CAPM may also be shown to be consistent with a multi-inde
see this consider the two-factor APT:


and
+ bjlh1 + bj2h2
ERj = r
Now hj (as we have seen) is the excess return on a portfolio with a bij of un
factor and zero on all other factors. Since h j is an excess return, then ifthe CA
= pl(ER"' - r )
)cl

= p2(ERm - r )
A2

where the pi are the CAPM betas. (Hence pi = cov(Ri, Rm)/var(Rm)- note
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


3.5 TESTING THE SINGLE INDEX MODEL,
THE CAPM AND THE APT
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


2
22
= 6; a, i-aEi
0
:




For a portfolio of n securities we have
n


i=l
n n
n


;=I j=1
i= 1

To calculate the optimal proportions x:, in general requires
n - expected returns ER,
2
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)
E:,

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

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 )
Si

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)
i
-
+ 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;
Jr30:

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 )
(Yi


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
˜˜ ˜ ˜ ˜ ˜




P
i
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
1.45
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
-
1.o
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:
(5.2)
(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)
WI
[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
2
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
2
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
0

measuring variances correctly
the assumption of constant betas (over time) may not be correct
0

econometric techniques have advanced so that these early studies may give
0

results
on balance the results (after consulting the original sources) suggest to this
0

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
353
..
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
analysis.
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
APT.
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:
k
+ +
Rif = E(Rj,) bijFj, &it
j=1

+ + hkbik
E(Ri,) = ho hlbil ***
Comparing (3.88) with regressions with unrestricted constant terms a, (a
ho = r):
+ +
(Ri, - r l ) = ai CbijFj, &it
Comparing (3.88) and (3.89) we see that there are non-linear restrictions,




in (3.89), if the APT is the correct model. In the combined time series cr
regressions we can apply non-linear least squares to (3.88) and test these restri
hs and bijs are jointly estimated in (3.88) and results on US portfolios (Mc
(1985)) suggest that the restrictions do not hold.
The key issues in testing and finding an acceptable empirical APT model a
how to measure the ˜news™ factors, F j f . Should one use first differences o
0

ables, or residuals from single equation regressions or VAR models, and
allow the coefficients in such models to vary, in order to mimic learning
(e.g. recursive least squares, time varying parameter models)?
are the set of factors Fjt and the resulting values of hj constant over differ
0

periods and across different portfolios (e.g. by size or by industry)? If
different in different sample periods then the price of risk for factor j is tim
(contrary to the theory) and the returns equation for R; is non-unique

Although there has been considerable progress in estimating and testing th
empirical evidence on the above issues is far from definitive.


36 SUMMARY
.
This chapter has extended our analysis of the CAPM and analysed an alterna
of equilibrium returns, the APT. It explained how the concepts behind the C
be used to construct alternative performance indicators and briefly looked at
empirical tests of the CAPM and tests of the APT. In addition, it showed how
variance approach can be used to derive an individual™s asset demand function
compared this approach to the CAPM.
4
I
Valuation Models
In this chapter we look at models that seek to determine how investors in
decide what is the correct, fundamental or fair value V, for a particular sto
decided on what is the fair value for the stock, market participants (in an efficie
should set the actual price P , equal to the fundamental value. If P , # V, then u
profit opportunities exist in the market. For example, if P , < V, then risk neutra
would buy the share since they anticipate they will make a capital gain as P , ris
its ˜correct value™ in the future. As investors purchase the share with Pi < V
would tend to lead to a rise in the current price as demand increases, so that
moves tawards its fundamental value. The above, of course, assumes that inve
margin have homogeneous expectations or more precisely that their subjective
probability distribution of the fundamental value reflects the ˜true™ underlying d
A key proposition running through this chapter is that stock returns and s
are closely linked. Indeed, alternative models of expected returns give rise t
expressions for the determination of fundamental yalue and hence stock prices
themes covered in the chapter are as follows:

We begin with a simple model where equilibrium expected stock returns a
to be constant and this implies that stock prices equal the discounted pre
(DPV) of future dividends, with a constant discount rate. This express
further simplified if one also assumes either a constant level of dividends or
dividend growth rate.
We explore the implications for the determination of stock prices when e
expected returns are assumed to vary over time and then show how the ris
of the CAPM helps to determine a (possibly time varying) discount rate i
formula for stock prices.
We discuss a somewhat more general CAPM than that discussed in Chap
3 which is known as the consumption CAPM (or C-CAPM). This model i
poral, in that investors are assumed to maximise expected utility of current
consumption. Financial assets allow the consumer to smooth his consumpt
over time, selling assets to finance consumption in ˜bad™ times and saving
times. Assets whose returns have a high negative covariance with consumpt
willingly held even though they have low expected returns. This is becaus
be ˜cashed in™ at a time when they are most needed, namely when consump
4.1 THE RATIONAL VALUATION FORMULA (RV
Expected Returns are Constant
One of the simplest assumptions one can make is that expected returns are co
expected return is defined as



where V, is the value of the stock at the end of time t and Dr+l are divi
+
between t and t 1. E, is the expectations operator based on information a
earlier, S2,. The superscript ˜e™ is equivalent to E,; it helps to simplify the not
used when no ambiguity is likely to arise, that is:


Assume investors are willing to hold the stock as long as it is expected to earn
return (= k). We can think of this ˜required return™ k as that rate of return
sufficient to compensate investors for the inherent riskiness of the stock:


The form of (4.2) is known as the fair game property of excess returns (see
The stochastic behaviour of R,+l - k is such that no abnormal returns are
average: the expected (conditional) excess return on the stock is zero:


Using (4.1) and (4.2) we obtain a differential (Euler) equation which dete
movement in ˜value™ over time:


+ k) with 0 < 6 < 1. Leading (4.4) one per
where 6 = discount factor = 1/(1


Now take expectations of (4.5) assuming information is only available up to


In deriving (4.6) we have used the law of iterated expectations:


The expectation formed today (t) of what one™s expectation will be tomorr
+
of the value at t 2 is the LHS of (4.7). This simply equals one™s expectatio
Vf+2 (i.e. the RHS of (4.7)) since one cannot know how one will alter one™s e
The next part of the solution requires substitution of (4.8) in (4.6):


By successive substitution


Now let N + 00 and hence aN -+ 0. If the expected growth in D is not explo
D;+N is finite and if V;+N is also finite then:



Equation (4.10) is known as a terminal condition or transversality condition
out rational speculative bubbles (see Chapter 7). Equation (4.9) then becomes

Vl = 6'D;+,
i=l
+ k ) we have derived (4.11) under the assumptions:
Where 6 = 1/(1
expected returns are constant
0

the law of iterated expectations (i.e. RE) holds for all investors
0

dividend growth is not explosive and the terminal condition holds
0

all investors have the same view (model) of the determinants of return
0

homogeneous expectations

Hence the correct or fundamental value V , of a share is the DPV of expe
dividends. If we add the assumption that
investors instantaneously set the actual market price P, equal to fundament
0

then we obtain the rational valuation formula (RVF) for stock prices with




If investors ensure (4.12) holds at all times then the Euler equation (4.6) an
formula can be expressed in terms of P I , the actual price (rather than V t ) ,
simplifies the notation, we use this whenever possible.
In the above analysis we did not distinguish between real and nominal va
indeed the mathematics goes through for either case: hence (4.12) is true w
variables are nominal or are all deflated by an aggregate goods price index. N
intuitive reasoning and causal empiricism suggest that expected real return
likely to be constant than expected nominal returns (not least because of g
If investors have a finite horizon, that is they are concerned with the price they
in the near future, does this alter our view of the determination of fundame
Consider the simple case of an investor with a one-period horizon:
Pf= 6D:+, + S2P;+,
+
The price today depends on the expected price at t 1. But how is this
+
determine the value P:+l at which he can sell at t l? If he is consistent (ra
should determine this in exactly the same way that he does for P,. That is


But by repeated forward induction each investor with a one-period horizon w
that P,+j is determined by the above Euler equation and hence today™s price
the DPV of dividends in allfutureperiods.
Thus even if some agents have a finite investment horizon they will still c
determine the fundamental value in such a way that it is equal to that of an in
has an infinite horizon. This is the usual counterargument to the view that i
have short horizons then price cannot reflect fundamental value (i.e. short-term
counterargument assumes, of course, that investors are homogeneous, that they a
the underlying equilibrium model of returns given by (4.2) is the true mode
˜push™ price immediately to its equilibrium value. Later in this section the assum
equilibrium returns are constant is relaxed and we find the RVF still holds. I
Chapter 8 allows non-rational agents or noise traders to influence returns and
model price may not equal fundamental value.

Expected Dividends are Constant
Let us simplify the RVF further by assuming a time series model for (real)
which has the property that the best forecast of all future (real) dividends is e
current level of dividends, that is the random walk model:

+ Wf+l
Dr+l = D,
where w,+1 is white noise. Under RE we have E,(w,+jlS2,)= 0 ( j 2 1) and E,
and hence the growth in dividends is expected to be zero. The RVF (4.12) then r
+ S + J2 + * . * ) D = S/(1 - S)D, = ( l / k ) D ,
P, = S(l t
Equation (4.13) predicts that the dividend price (DP) or dividend yield is
ratio
equal to the required (real) return, k. (Note that the ratio D/P is the same w
variables are measured in real or nominal terms.) If the required (real) return
(8 percent per annum) then the zero dividend growth model predicts a constan
price ratio which also equals 8 percent. The model (4.13) also predicts that the
change in stock prices P, equals the percentage change in current dividends. Pric
only occur when new information about dividends becomes available: the mod
dends). The conditional variance of prices therefore depends on the varian
about these fundamentals:




Much later in the book, Chapter 16 examines the more general case where the
prices depends not only on the volatility in dividends but also the volatility in t
rate. In fact, an attempt is made to ascertain whether the volatility in stock price
due to the volatility in dividends, or the discount factor.

Expected Dividend Growth is Constant
A time series model in which (real) dividends grow at a constant rate g is
model:
+
Df+1 = (1 + g)Dr W f + l
where w, is white noise and E(w,+llQ,) = 0. Expected dividend growth fro
easily seen to be equal to g.




Note that if the logarithm of dividends follows a random walk with drift para
then this also gives a constant expected growth rate for dividends (i.e. In D
+
In D, w,). The optimal forecasts of future dividends may be found by lea
one period
+ +
Df+2 = (1 g)Q+1 W f + 2



Hence by repeated substitution:


Substituting the forecast of future dividends from (4.18) in the rationa
formula gives:
00

+ g)™D,
P, = 6™(1
i=l

which after some simple algebra yields

P, = - ˜IDf with ( k - g ) > 0
(™ +




(k -g)
series properties of dividend payouts is such that it is reasonable (rational) fo
to expect that dividends grow at a constant rate. Equation (4.20) ˜collapses
when g = 0.

Time Varying Expected Returns
Suppose investors require a different expected return in each future period in
they will willingly hold a particular stock. (Why this may be the case is i
later.) Our model is therefore:
E,R,+l = kf+l
where we have a time subscript on k to indicate it is time varying, Repeating th
steps, involving forward substitution, gives:

+ &+1&+2D;+2 + b+1&+2&+3D;+3+ + & + N E t ( D r + N + P z
Pr = &+1DF+, ***




which can be written in more compact form (assuming the transversality condit




+
where 8,+; = 1/(1 k,+i). The current stock price therefore depends on all fu
tations of the discount rate &+, and all future expected dividends. Note that 0
for all periods and hence expected dividends $-for-$ have less influence on
stock price the further they accrue in the future. However, it is possible (b
unlikely) that an event announced today (e.g. a merger with another company
expected to have a substantial $ impact on dividends starting in say five y
In this case the announcement could have a large effect on the current stock
though it is relatively heavily discounted. Note that in a well-informed (˜efficie
one expects the stock price to respond immediately and completely to the ann
even though no dividends will actually be paid for five years. In contrast, if th
inefficient (e.g. noise traders are present) then the price might rise not only in
period but also in subsequent periods. Tests of the stock price response to anno
are known as event studies.
At the moment (4.23) is ˜non-operational™ since it involves unobservable ex
terms. We cannot calculate fundamental value (i.e. the RHS of (4.23)) and he
see if it corresponds to the observable current price P,.We need some ancillar
hypotheses about investors™ forecasts of dividends and the discount rate. It i
straightforward to develop forecasting equations for dividends (and hence prov
ical values for E,D,+j ( j = 1,2, . . .). For example on annual data an AR(1)
model for dividends fits the data quite well. The difficulty arises with the equil
+
of return k,(S, = 1/(1 k , ) ) which is required for investors willingly to hold
There are numerous competing models of equilibrium returns and next it will
how the CAPM can be combined with the rational valuation formula so tha
price is determined by a time varying discount rate.
folio). Merton (1973) developed this idea in an intertemporal framework and s
the (nominal) excess return over the risk-free rate, on the market portfolio, is p
to the expected variance of returns on the market portfolio


The expected return can be defined as comprising a risk-free return p
premium r p t :
+
ER): = r, r p ,
where r p , = h E r a ˜ , + ,Comparing (4.21) and (4.25) we see that according to
.
the required rate of return on the market portfolio is given by:


The equilibrium required return depends positively on the risk-free interest rat
the (non-diversifiable) risk of the market portfolio, as measured by its condition
If either:

agents do not perceive the market as risky (i.e. Eta:,+, = 0), or
0

agents are risk neutral (i.e. h = 0)
0


then the appropriate discount factor used by investors is the risk-free rate, r,. N
determine price using the RVF, investors in general must determine k, and hen
future values of the risk-free rate and the risk premium.

Individual Securities
Consider now the price of an individual security or a portfolio of assets which
of the market portfolio (e.g. shares of either all industrial companies or all bank
The CAPM implies that to be willingly held as part of a diversified portfolio th
(nominal) return on portfolio i is given by:
+ Bit (ErR:+l
EtRit+l = rt - rt )




Substituting from (4.24) for Erair+, have('):
we
+ AEr(oim)t+l
ErRir+l = rt
where a i m is the covariance between returns on asset i and the market portf
comparing (4.21) and (4.29) the equilibrium required rate of return on asset i


The covariance term may be time varying and hence so might the future disco
6 , + j in the RVF for an individual security (or portfolio of securities).
an individual stock as part of a wider portfolio is equal to the risk-free rate plu
for risk or riskpremium r p ,



Summary: The RVF
From the definition of the expected return on a stock it is possible via
equation and rational expectations to derive the rational valuation formula
stock prices.
Stock prices in an efficient and well-informed market are determined by t
expected future dividends and expected future discount rates.
If equilibrium expected returns are constant then the discount rate in t
constant.
If equilibrium (nominal) expected returns are given by the CAPM then th
factor in the RVF may be time varying and depends on the (nominal) ris
and a variance/covariance term.

4.1.2 The Consumption CAPM
In the one-period CAPM the individual investor™s objective function is assu
fully determined by the standard deviation and return on the portfolio. The in
to earn the highest expected return for any given level of portfolio risk.
An alternative view of the determination of equilibrium returns in a well
portfolio is provided by the intertemporal consumption CAPM (denoted
Here, the investor maximises expected utility which depends only on current
consumption (see Lucas (1978) and Mankiw and Shapiro (1986)). Financial as
role in this model in that they help to smooth consumption over time. Securiti
to transfer purchasing power from one period to another. If an agent had no
was not allowed to accumulate assets then his consumption would be determi
current income. If he holds assets, he can sell some of these to finance consum
his current income is low. A n individual asset is therefore more ˜desirable™ if i
expected to be high when consumption is expected to be low. Thus the system
the asset is determined by the covariance of the assets return with respect to co
(rather than its covariance with respect to the return on the market portfolio
˜basic™ CAPM). The C-CAPM is an intertemporal model unlike the basic
CAPM. In the C-CAPM, dividends and prices are all denominated in consum
and hence all analysis is in terms of real variables. In the C-CAPM the individu
maximises
where D,= dividends received (e.g. see Sheffrin, 1983), X , = holdings of s
+
time t which will yield dividend payments at t 1. The first term on the RHS
dividend income and the second term represents receipts from the sale of the
(shares). The individual must choose the amount of consumption today and the
taneously choose the quantity of risky assets to be carried forward into the ne

First-Order Condition
The first-order condition for maximising expected utility has the agent equatin
loss from a reduction in current consumption, with the additional gain in (
consumption next period. Lower consumption expenditure at time t allows inv
an asset which has an expected positive return and therefore yields extra re
future consumption. More formally, a $1 reduction in (real) consumption tod
+
utility by U™(C,) but results in an expected payout of $(1 EtRil+l) next pe
spent on next period™s consumption, the extra utility per (real) $ next period,
at a (real) rate 8, is 8,U™(Cr+1).Hence the total extra utility expected nex
+
E,[(1 R;r+1)6,U™(C,+I)]. equilibrium we have
In




is the marginal rate of substitution of current for future (discounted) consump
from the discount rate er, the MRS depends on the ratio of the marginal u
+
consumption at t 1 to that at time t. The MRS therefore depends on agents™
(˜tastes™) between consumption today versus consumption tomorrow and hen
be constant or vary only slowly over time (as tastes change).
We can now use equation (4.34) to
calculate the implied rational valuation formula (or DPV) for stock prices
0

obtain a relationship for the expected return in terms of the covarianc
0

consumption and asset returns, that is the C-CAPM

Consumption and the Rational Valuation Formula
The RVF for stock prices can be calculated by using (4.35) in the Euler equa
is then solved by forward substitution as in our earlier examples. The expect
defined as
Forward substitution yields (given a suitable terminal condition and restricti
sequence Sr+l,S,+2.. .):

+ Sf+lSf+2Df+2+ Sf+lSf+2Sf+3Q+3 + -
Vf = E , [Sf+lDf+l *


rj i
30




and it is easily seen that the marginal rate of substitution Sr+j plays the role
varying discount factor. Further simplification assuming 8 is constant yields:




Thus the discount factor for dividends in the C-CAPM depends on 0 and the m
+
of substitution of consumption at t j for consumption today. However, as 0 <
weight attached to the marginal utility of future consumption (relative to today
utility) declines, the further into the future the consumption is expected to a
worth noting that (4.38) reduces to the ˜constant discount rate™ case if agen
neutral. Under risk neutrality the utility function is linear and U ™ ( C )= consta
0
Consumption and the CAPM
To obtain the ˜usual™ covariance term, note that for any two random variables



Hence substituting for E(Ri,+l . Sf+l)in (4.34) and rearranging we have


The C-CAPM therefore predicts that the expected return on any asset i depen

negatively on the covariance of the return on asset i with the MRS of consum
0
varies inversely with the expected marginal rate of substitution of consum
0


The intuitive interpretation behind equations (4.35) and (4.39) is as follow
believed that tomorrow will bring ˜good times™ and a high level of consumption
diminishing marginal utility of consumption the additional utility value of this
tion will be low, next period. Hence Sr+l will be relatively low. According to
expected return on security i will then have to be higher than average, to persua
vidual to defer consumption today, hold the asset and carry the consumption o
forward to tomorrow, when its contribution to additional utility will be low. Fo
the equilibrium expected return on securities during the Great Depression when
tion was low must have been relatively high. Unless investors in the 1930s
negative covariance with S are very ˜risky™ and will be willingly held only if t
high expected return. This is because they would be sold to finance ˜high utili
consumption unless the investor is compensated by the high expected return.
Thus, the C-CAPM explains why different assets earn different equilibrium
also allows equilibrium returns to vary over time as agents™ marginal rate of s
varies over the business cycle. The C-CAPM therefore links asset returns w
economy (i.e. economic fundamentals).
In order to make the C-CAPM operational, we need to calibrate the ter
RHS of (4.39). To investigate this it is assumed that the intertemporal utili
to be maximised by the investor is additively separable over time (and with
leisure) and that the utility from future consumption U(C,+j) is discounted eac
a constant rate 8 (0 < 8 < 1). The specific form for the utility function in ea
exhibits diminishing marginal utility (and has a constant Arrow -Pratt measure
risk aversion, a):
c;
U ( C , )= -
-
f
f




O<a<l
1-Cr
It may be shown that with utility function (4.40) the unobservable covarian
(4.39) may be approximated as
)]
cov(Rit+l, St+l) = -a@[cov(Rir+lv
where g;+l = Cr+l/C,, the growth in consumption. The ˜unobservable™ covaria
proportional to the observable covariance between the return on asset i and the
consumption. Finally, substituting (4.41) in (4.39) we obtain an expression tha
in form to that for the basic CAPM:


where
Yor = (1 - ESr ) / E &
=
Ylr

The terms Yor and Ylr depend on the MRS of present for future consumption
is assumed to be constant then (4.42) provides a straightforward interpreta
C-CAPM. If the return on asset i has a ˜low™ covariance with consumption g
it will be willingly held, even if its expected return is relatively low. This is b
asset on average has a high return when consumption is low and hence can
finance current consumption which has a high marginal utility.
Equation (4.35) suggests a way in which we can test the C-CAPM model of e
returns. If we assume a constant relative risk aversion utility function, then (4.
assets i and j becomes (see Scott, (1991)):
estimate these equations jointly, using time series data, these cross-equation
provide a strong test of the C-CAPM. We do this in Chapter 6.

C-CAPM and the Discount Rate
The C-CAPM implies that in the equation underlying the derivation of the rat
ation formula, namely:
E&+1 = k + l

the required equilibrium (real) rate of return is given by


Hence the required (real) rate of return in the RVF would only be constant if the
term is expected to be constant in the future, which in general one would no
be the case.
Of course if one assumes risk neutrality (i.e. a linear utility function), th
aversion parameter a = 0, the MRS is constant and then (4.39) reduces to
˜constant™. Hence the RVF becomes the simple expression P , = E SiD;+i
,:
(real) discount factor 6 depends on the constant MRS.

Appra isa 1 of C-CAPM
For economic theorists the C-CAPM model has the advantage that it is firmly
in an intertemporal maximisation problem and is based on views about agents™
are reasonably uncontroversial (at least among economists), such as diminishin
utility from consumption. The theory (Grossman and Shiller, 1981) holds for a
portfolio of assets. It holds for any individual consumer who has the option o
in stocks even though he may not actually invest in stocks. It incorporates a
of uncertainty as long as these are reflected in consumption decisions. It also
any time horizon of returns (e.g. month, year). It is therefore a very general
equilibrium asset returns. However, sceptics would question some of the assu
the C-CAPM, for example:

(i) a constant discount rate (6) for future consumption and that utility onl
on the level of consumption in each future period. Critics might sugge
rate at which investors discount future utility may vary over time due to
optimism or pessimism about the future. More importantly perhaps, uti
likely to be influenced by the uncertainty attached to future consumptio
need to introduce second moments, namely the variance of consumptio
mathematical model).
(ii) Critics might argue that insurance companies and pension funds are not
in maximising intertemporal consumption of policyholders but in maximi
term profits or the size of the f r .As these agents are ˜big players™ in
im
market the C-CAPM may be an incomplete model of valuation.
rational expectations, homogeneous expectations by agents, etc.

When we discuss tests of the C-CAPM in Chapter 6, the above potential lim
the model should aid our interpretation of the empirical results.

Wealth in the Utility Function
A more general version of the CAPM than the C-CAPM may be obtained by
investors maximise the DPV of expected future real wealth W (see Scott (1
marginal utility of consumption is then replaced with the marginal utility of
U™(W,). If the intertemporal utility function is separable over time then U ™ ( W
and the model collapses to the C-CAPM. However, where this is not the ca
MRS of current for future real wealth which determines the discount factor i
In general, models based on intertemporal maximisation of consumption or w
in equations for fundamental value V, of the form:




with a constant real discount rate 6 (in the intertemporal utility function) an
consumption or real wealth. The first term in the square brackets is the MR
t + j and t ) . Since these models are based on utility functions involving real ˜
all the variables are also in real terms. However, with a small modification
a similar relationship for nominal fundamental value which then depends o
+
dividends and the MRS between $1 at time t j and $1 at time t .


SUMMARY
4.2
In this chapter we have concentrated on two main themes. The first is to demo
relationship between any model of equilibrium expected returns and the RV
for stock prices. The second is the development of a CAPM based on utility
main conclusions to emerge are:
Assuming E,Rt+l = k,+l where kt+l represents some equilibrium model o
returns we can then invoke the (rational expectations) chain rule of for
obtain the RVF for stock prices. In general, stock prices depend on th
expected future dividends and expected future discount rates. Any chang
fundamental variables will cause a change in stock prices.
Models such as the CAPM and C-CAPM, which give rise to equilibrium
are (potentially at least) time varying, also imply time varying discount
RVF of stock prices.
The C-CAPM is based on utility theory and is a very general intertempora
equilibrium asset returns. It implies that assets whose returns have a hig
ENDNOTE
1. There is a sleight of hand here, since for two random variables x and y
Ex/Ey.


FURTHER READING
An entertaining and informative history of the origin of modern finance is p
Bernstein (1992). There are many very good basic text books in finance dealin
issues in Part 1. A clear and simple exposition is Kolb (1995), with Levy
(1984) and Elton and Gruber (1993) covering similar ground at a more adva
Blake (1990) provides an intermediate approach bridging the gap between
the techniques used by practitioners. Applications of the basic concepts in
practical issues can also be found in issues of the Bank of England Quarter
publications of the US Federal Reserve Banks (many of which are provid
request) and professional journals such as the Journal of Portfolio Manageme
of Fhed Income, Financial Analysts Journal and .Risk.
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PART 2
I
I Efficiency, Predictability
l I and Volatility
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5
I
L The Efficient Markets Hypothes
The efficient markets hypothesis (EMH) may be expressed in a number of
ways (not all of which are equivaler;) and the differences between these
representations can easily become rather esoteric, technical and subtle (e.g.
(1989)). In this introductory section these technical issues will be avoided as far
and general terms will be used about the ideas which lie behind the concept of
market - the more technical aspects will be ˜filled in™ later, in this and other
When economists speak of capital markets as being eficient they usually mea
view asset prices and returns as being determined as the outcome of supply and
a competitive market, peopled by rational traders. These rational traders rapidly
any information that is relevant to the determination of asset prices or returns
dividend prospects) and adjust prices accordingly. Hence, individuals do not hav
comparative advantages in the acquisition of information. It follows that in su
there should be no opportunities for making a return on a stock that is in exce
payment for the riskiness of that stock. In short, abnormal profits from trad
be zero. Thus, agents process information efficiently and immediately incor
information into stock prices. If current and past information is immediately in
into current prices then only new information or ˜news™ should cause change
Since news is by definition unforecastable, then price changes (or returns)
unforecastable: no information at time t or earlier should help to improve t
of returns (or equivalently to reduce the forecast error made by the individ
independence of forecast errors from previous information is known as the ort
property and it is a widely used concept in testing the efficient markets hypot
The above are the basic ideas behind the EMH, but financial economists, b
academics, need to put these ideas into a testable form and this requires some ma
notation and terminology. In order to guide the reader through the maze that i
the procedure will be as follows:

In order to introduce readers to the concepts involved an overview is prov
0

basic ideas using fairly simple mathematics.
To motivate our subsequent technical analysis of the EMH an overview
0

the implications of the EMH (and violations of it). Consideration will be g
role of investment analysts, public policy issues concerning mergers and
capital adequacy, the cost of capital, the excess volatility of stock prices an
trading halts and margin requirements on stock transactions. In general, vi
It will be shown how general common sense ideas of ˜efficiency™ can b
0

mathematical form. For example, if investors use all available relevant info
forecast stock returns thus eliminating abnormal profits this can be shown
concepts such as a fair game, a martingale and a random walk.
Empirical tests of the EMH can be based either on survey data on expecta
0

a specific model of equilibrium expected returns (such as those discussed
chapters) and a fairly simple overview is given of such tests which are ex
detail in later chapters.
The basic concepts of the EMH using the stock market as an example are dis
the same general ideas are applicable to other financial instruments (e.g. bond
opt ions).


5.1 OVERVIEW
Under the EMH the stock price P, already incorporates all relevant informati
+
only reason for prices to change between time t and time t 1 is the arrival
or unanticipated events. Forecast errors, that is, ‚¬,+I = P,+l - E,P,+1 should th
zero on average and should be uncorrelated with any information 52, that was a
the time the forecast was made. The latter is often referred to as the rational e
(RE) element of the EMH and may be represented:


The forecast error is expected to be zero on average because prices only cha
arrival of ˜news™ which itself is a random variable, sometimes ˜good™ someti
The expected value of the forecast errur is zero


= 0 is that the forecast of P,+l is unbiased (i.e. o
A implication of
n ErEf+l
actual price equals expected price). Note that &,+I could also be (loosely) de
+
the unexpected profit (or loss) on holding the stock between t and t 1. Under
unexpected profits must be zero on average and this is represented by (5.la).
The statement that ˜the forecast error must be independent of any infor
available at time t (or earlier)™ is known as the orthogonalityproperty. It may
that if E, is serially correlated then the orthogonality property is violated. An
a serially correlated error term is the first-order autoregressive process, AR( 1)



where v, is a (white noise) random element (and by assumption is independen
mation at time t, Q,). The forecast error Er = P, - Er-IP, is known at time t
forms part of 52,. Equation (5.2) says that this period™s forecast error Er has ap
helps to forecast P,+1 as follows. Lag equation (5.1) one period and multip
giving
+
PPt = P(&-lPr) PE,

subtracting (5.1) from (5.3), rearranging and using v, = &,+I - PE, from (5.2)


We can see from (5.4) that when E is serially correlated, tomorrow™s price de
today™s price and is therefore (partly) forecastable from the information avail
(Note that the term in brackets being a change in expectations is not forecastab
fore, the assumption of ˜no serial correlation™ in E is really subsumed unde
assumption that information available today should be of no use in forecasting t
stock price (i.e. the orthogonality property).
Note that the EMH/RE assumption places no restrictions on the form of the
higher moments of the distribution of E,. For example, the variance of ˜ ˜ (de +
without violating RE. (This is an ARC
may be related to its past value, of,
see Chapter 17.) RE places restrictions only on the behaviour of the first m
expected value) of E ˜ .
The efficient markets hypothesis is often applied to the return on stocks, R,, a
that one cannot earn supernormal profits by buying and selling stocks. Thus a
similar to (5.1) applies to stock returns. Actual returns R,+l will sometimes be
sometimes below expected returns E,R,+1 but on average, unexpected retur
E r + l are zero:




The variable Er+l could also be described as the ˜forecast error™ of returns.
EMH we need a model of how investors form a view about expected returns. T
should be based on rational behaviour (somehow defined). For the moment ass
simple model where

(i) stocks pay no dividends, so that the expected return is the expected capit
to price changes
(ii) investors are willing to hold stocks as long as expected or required
constant, hence:
m r + 1 =k

Substituting in (5.5) in (5.3):
+ E,+l
Rf+l = k

where &,+I is white noise and independent of 52,. We may think of the re
of return k on the risky asset as consisting of a risk-free rate r and a risk pr
+
(i.e. k = r rp) and (5.7) assumes both of these are constant over time. S
Equation (5.9) is a random walk in the logarithm of P with drift term k. No
logarithm of) stock prices will only follow a random walk under the EMH if th
rate r and the risk premium v p are constant and dividends are zero. Often in
+
work the ˜price™ at t 1 is adjusted to include dividends paid between t and
when it is stated that ˜stock prices follow a random walk™, this usually applie
inclusive of dividends™. In some empirical work researchers may take the vie
stock return is dominated by capital gains (and losses) and hence will use qu
excluding dividends.
For daiZy changes in stock prices over a period of relative tranquillity (e.g
the crash of October 1987) it may appear a reasonable assumption that the ris
is a constant. However, when daily changes in stock prices are examined it
found that the error term is serially correlated and that the return varies on dif
of the week. In particular, price changes between Friday and Monday are smal
other days of the week. This is known as the weekend efect. It has also been
some™stocks that daily price changes in the month of January are different fro
other months. ˜Weekends™ and ˜January™ are clearly predictable events! Theref

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