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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
on stocks depend in a predictable way upon information readily available (i.e
of the week it is). This is a violation of the EMH under the assumption of a co
premium since returns are in part predictable. However, in the ˜real world™ it m
the case that this predictability implies that investors can earn supernormal p
transactions costs need to be taken into account.
It should be clear from the above discussion that in order to test the EMH
some view of how prices or rates of return are determined in the market.
require an economic model of the determination of equilibrium returns and a
Our test of whether agents use information efficiently is conditional on our hav
the correct model to explain either stock prices or the rate of return on such stoc

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