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or rejection of the efficient markets hypothesis may be either because we have
equilibrium ˜pricing model™ or because agents genuinely do not use information
As noted above, another way of describing the EMH is to say that in an effic
it is impossible for investors to make supernormal profits. Under the EMH inve
a return on each security which covers the riskiness of that security (and they
sufficient profits just to cover their costs in order to stay in the industry, th
business of buying and selling shares). However, there must not be opportunitie
to make abnormal profits by dealing in stocks. The latter is often referred to
game™ property .
One particular equilibrium valuation model is the CAPM. The CAPM pred
equilibrium the expected excess return on asset i should depend only on its b
expected excess return on the market.

Since the excess return on the market portfolio is constant at a particular po
then two assets will have different expected excess returns only if their respe
its risk class and this would be a violation of the EMH. However, note tha
be a violation of the EMH under the assumption that the CAPM is the true
equilibrium asset pricing. This again emphasises the fact that tests of the EM
involve the joint hypotheses (i) that agents use information rationally, (ii) th
use the same equilibrium model for asset pricing which happens to be the ˜tru

The view that the return on shares is determined by the actions of™rational a
competitive market and that equilibrium returns reflect all available public in
is probably quite a widely held view among financial economists. The slight
assertion, namely that stock prices also reflect their fundamental value (i.e. t
future dividends) is also widely held. What then are the implications of the EM
to the stock market?
As far as a risk averse investor is concerned the EMH means that he shou
˜buy and hold™ policy. He should spread his risks and hold the market portfo
20 or so shares that mimicethe market portfolio). Andrew Carnegie™s advice
your eggs in one basket and watch the basket™ should be avoided. The role for
analysts, if the EMH is correct, is very limited and would, for example, inclu

(i) advising on the choice of the 20 or so shares that mimic the market por
(ii) altering the proportion of wealth held in each asset to reflect the ma
portfolio weights (the xi* of the CAPM) which will alter over time. The
both as expected returns change and as the riskiness of each security rela
market changes (i.e. covariances of returns),
(iii) altering the portfolio as taxes change (e.g. if dividends are more highly
capital gains then for high rate income tax payers it is optimal, at the
move to shares which have low dividends and high expected capital gai
(iv) ˜shopping around™ in order to minimise transactions costs of buying and
Under EMH the current share price incorporates all relevant publicly availabl
tion, hence the investment analyst cannot pick winners by reanalysing publicl
information or by using trading rules (e.g. buy ˜low™, wait for a price rise and s
Thus the EMH implies that a major part of the current activities of investmen
is wasteful. We can go even further. The individual investor can buy a pro
an indexfund (e.g. mutual fund or unit trusts). The latter contains enough se
closely mimic the market portfolio and transactions costs for individual inv
extremely low (say less than 2 percent of the value of the portfolio). Practiti
as investment managers do not take kindly to the assertion that their skills
redundant. However, somewhat paradoxically they often support the view that
is ˜efficient™. But their use of ˜efficient™ is usually the assertion that the stock
low transactions costs and should be free of government intervention (e.g. z
duty, minimal regulations on trading positions and capital adequacy).
sell funds in particular sectors (e.g. chemicals or services) or specific geograp
(e.g. Japanese stocks). There is a marketing reason for this. If finance house
number of such funds, then they effectively hold the market portfolio, while the
can speculate on individual ˜packages™ of mutual funds. Also with this strategy
house will usually have at least one fund that it can boast has ˜beaten the ma

Takeovers, Conglomerates and Financial Institutions
Let us turn now to some public policy issues. The stock market is supposed to
˜correct™ signals for the allocation of real resources (i.e. fixed investment). O
proportion of corporate investment is financed from new issues (e.g. about 4 p
gross basis in the UK); nevertheless, the rate of return of a quoted company o
market provides a measure of the opportunity cost of funds corrected for risk
can be used in discounting future expected profits from a physical investm
(i.e. in investment appraisal). Other things equal, if profits from a firm™s new
project are expected to be high the existing share price will be ˜high™ and th
obtain its funds by issuing fewer shares. However, if the share price does
firndamentals but is influenced by whim or fads of ˜irrational™ investors the
is broken. An abnormally low share price which reflects ill-informed extrane
(e.g. irrational market prejudice) will then inhibit a firm from embarking on
rational calculation) is a viable investment project.
The above analysis also applies to takeovers. If the stock market is myo
only considers profits and dividends that accrue in the near future, then manag
of a takeover may distribute more in current dividends rather than using th
profits to undertake profitable real investment say on R&D expenditure. Th
will boost the share price if the market is myopic. This is generally known
termism™. A possible response by government to such short-termism might b
hostile takeovers (e.g. as in Japan). The impact of short-termism on share pr
also be exacerbated by an incentive system whereby part of a manager™s rem
is in the form of share options. See Chapter 8 for a discussion on the theo
empirical issues that are relevant to the debate on short-termism.
The opposite view to the above, namely that hostile takeovers are welfare
(i.e. in terms of the output and profits of the firm) requires the assumption th
are efficient and that takeovers enable ˜bad™ incumbent managers to be replac
scenario, the hostile bidder recognises that the incumbent ˜bad™ manageme
shareholders to mark down the firm™s share price. The hostile bidder pays
excess of the existing share price. After replacing the ˜bad™ managers and re
the firm, the ensuing higher future profits are just sufficient to compensate for
price he paid for the shares.
In the 1960s and 1970s there was a wave of conglomerate formation follo
1980s by leveraged buyouts and conglomerate breakups (i.e. ˜asset stripping™).
erate mergers were sometimes justified on the grounds that the acquisition o
firms by ˜firm A™ reduced risk to the shareholder who held A™s shares since the
erate™ constituted a diversified portfolio of firms. The latter is an analogous a
not diversify their share holdings.)
Note that if share prices do reflect fundamentals but ˜news™ occurs freque
expected to make a substantial impact on a firm™s future performance, then one
expect to observe highly volatile share prices, even if the market is efficient. H
on occasions such volatility had adverse implications for parts of the real ec
an ˜externality™) - for example, that a stock market crash led to insolvencies
institutions, a ˜credit crunch™ and less physical investment - this would at le
a prima facie argument for governments to try and limit share price moveme
closing markets for a ˜cooling-off period™). Also, where systemic risk is invo
˜run™ on one bank causes a run on other banks) one might be prepared to proh
institutions from holding ˜highly volatile™ assets such as shares (e.g. banks in
By definition, ˜news™ is random around zero. Hence ˜news™ will not influen
of share prices over a long horizon. Therefore except in exceptional circumsta
would be unlikely to cause panics leading to a ˜run™ on banks or financial ins
their depositors. However, if the market is inefficient and prices are subject to
˜irrational swings™ then stock price volatility may be greater than that predicte
efficient markets hypothesis. Here, a prima facie case for financial institutio
enough resources (reserves) to weather such storms seems stronger. This is on
for general capital adequacy rules applied to financial institutions. If there are
risks (i.e. a form of externality) then, in principle, government action is require
that the level of capital reflects the marginal social costs of the systematic
than the marginal private costs (for any individual financial institution).
What are the implications of market efficiency in stock and bond markets f
corporate finance? If the market is efficient then there is no point in delaying
investment project in the hope that ˜financing conditions will improve™ (i.e. th
price will be higher): under the EMH the current price is the correct price a
expected future earnings from the project. Also under the EMH the firm™s cos
cannot be lowered by a given mix of securities (e.g. by altering the proportio
and equity). The Modigliani-Miller theorem (in the absence of taxes and b
suggests that the cost of capital is independent of the capital mix (i.e. debt-e
in an efficient market. The issue of capital mix can also be applied to the matu
structure of debt. Since rates on long and short corporate bonds fully reflec
information the proportion of long debt to short-dated debt will also not alter
capital to the firm. For example, under the expectations hypothesis, low lon
high current short rates simply reflect an expectation of lower future short rates
no advantage ex ante to financing an investment project by issuing long bonds
˜rolling over™ a series of short bonds. (This is discussed in Chapter 9 on the term
It follows from the above arguments that the role of the corporate treas
˜active manager™ either as regards the choice over the appropriate ˜mix™ of
finance or in analysing the optimum time to float new stock or bond issues is f
the EMH. Of course, if the market is not efficient the corporate treasurer m
to ˜beat the market™ and he can also alter the stock market valuation of the
chosen dividend policy or by share repurchase schemes, etc.
case for government intervention. However, given uncertainty about the imp
government policies on the behaviour of economic agents, the government s
intervene if, on balance, it feels the expected return from its policies outweig
attached to such policies. Any model of market inefficiency needs to ascerta
from efficiency the market is on average and what implications this has for pu
decisions and economic welfare in general. This is a rather difficult task giv
knowledge, as subsequent chapters will show.

As previously mentioned, the EMH can be formally stated in a number of diff
We do not wish to get unduly embroiled in the finer points of these alterna
our main concern is to see how the hypothesis may be tested and used in un
the behaviour of asset prices and rates of return. However, some formal de
the EMH are required. To this end, let us begin with some properties of
mathematical expectations; we can then state the basic axioms of rational e
such as unbiasedness, orthogonality and the chain rule of forecasting. Next w
the concepts of a martingale and a fair game. We then have the basic tools
alternative representations and tests of the ˜efficient markets hypothesis™.

Mathematical Expectations
If X is a random variable (e.g. heights of males in the UK) which can ta
values X I , Xz, X3 . . . with probabilities Ttj then the expected value of X, deno
defined as


If X is a continuous random variable (-00 < X < 00) with a continuous
distribution f ( X ) (e.g. normal distribution) then

Conditional probability distributions or conditional density functions are used
in the RE literature. For example, a fair die has a probability of (1/6)th
on any number from 1 to 6. However, suppose a friend lets you know t
to be used is biased and lands on the number 6 for half the time and on
numbers equally for the remaining throws. Conditional on the information
friend you would then alter your probabilities to (1/2) for a 6 and (1/10) for the
five numbers. Your conditional expected value would therefore be differen
expected value from an unbiased die, since the associated probabilities (or
density function) are different. The conditional expectation based on the info
where f(X, 152,) is the conditional density function. A conditional expectati
viewed as an optimal forecast of the random variable X t , based on all relevant i
52,. The conditional forecast error is defined as where
= Xr+l - E(Xr+l 152,)

This (mathematical) conditional forecast error can be shown (always) to b
E(Er+llQ) = E(Xr+lIQ) - E(Xr+llQr) = 0

We can rearrange (5.14) as:

and hence reinterpret (5.16) as stating that the conditional expectations are a
forecast of the outturn value.
The second property of conditional mathematical expectations is that the fo
is uncorrelated with all information at time t or earlier which, stated mathema

This is known as the orthogonality property of conditional expectations. Th
reason why (5.17) holds is that if RI could be used to reduce the forecast erro
it could be used to improve the forecast: hence all relevant information coul
been used in forecasting X r + l . It also follows that an optimal conditional fore
where subsequent forecast errors are unpredictable.
Note that an optimal forecast need not necessarily predict X,+l accurately.
can be large and the conditional expectation E t X l + j may only explain a sm
the variation in actual X , + j . What is important is that the optimal forecast
improved upon (in the sense of using 5 2 r to reduce the forecast errors E t + j ) .
worth noting that it is only the behaviour of the mean of the forecast error th
restricted in (5.17). The variance of the conditional forecast error denoted E ( q
not be constant and indeed may in part be predictable. The latter is of import
discussing the implications for market volatility within the framework of the
Consider for a moment making a forecast in January (at time t ) as to what t
you will make in February (t 1) will be, about the outcome of the variable X
(i.e. X r + 2 ) . Mathematically this may be represented as
Er [Er+1 ( X t + 2 I fir+ 1

If information 52, at time t is used efficiently then you cannot predict today ho
change your forecast in the future, hence

where E , (Xl+l) is equivalent to E(X,+1(Rr).This is the rule of iterated e
which may be succinctly represented as:
- - - = Er
of outcomes. Economic agents are therefore assumed to behave as if they fo
subjective expectations as the mathematical expectations of the true model of th
This is generally referred to as ˜Muth-RE™ (Muth, 1961).
To get a feel for what this entails consider a simple supply and demand
say, wheat. The supply and demand curves are subject to random shocks (e
in the weather on the supply side and changes in ˜tastes™ on the demand side
based products such as cookies). The actual equilibrium price depends in p
actual value of such ˜shocks™ which will only be revealed after the market h
Conceptually the individual RE farmer has to determine his supply of wheat, at
and the expected supplies of wheat of all other farmers (based on known fact
technology, prices of inputs, etc.). He makes a similar calculation of the kno
influencing demand such as income, xf. He then solves for the expected e
price by setting the demand and supply shocks to their expected values of zero
farmers behave as if they use a competitive stochastic model of supply and de
difference between the equilibrium or expected price and the outturn price is
unforecastable ˜error™ due to the random shocks to the supply and demand fun
additional information available to the farmer can reduce such errors any furth
RE orthogonality property holds). The stochastic reduced form is

Pf+1 = p;+1 + Ef+1 = fcx;™ x ; ) + Ef+1
where P f ; , = f ( x f , x : ) is the equilibrium price based on the known factor
influence supply and demand. The forecast error is the random variable &
under Muth-RE the uncertainty or randomness in the economy (e.g. the weat
product innovations) gives rise to agents™ forecast errors for the actual equilib
To test whether agents™ actual subjective expectations obey the axioms of
ical conditional expectations we either need an accurate measure of individ
subjective expectations or we need to know the form of the true model of th
used by all agents. Survey data on expectations can provide a ˜noisy™ proxy v
each agent™s subjective expectations. If we are to test that actual forecast erro
properties of conditional mathematical expectations via the second method (i.e
true model of the economy) the researcher has to choose a particular model fr
the many available on the ˜economist™s shelf™ (e.g. Keynesian, monetarist, re
cycle, etc.). Clearly, a failure of the forecast errors from such a model to ob
axioms could be due to the researcher choosing the wrong model from the ˜s
is, agents in the real world actually use another model.) The latter can provid
nient alibi for a supporter of RE, since he can always claim that failure to con
axioms is not due to agents being non-rational but because the ˜wrong™ econo
was used.

Martingale and Fair Game Properties
Suppose we have a stochastic variable X , which has the property:
Thus a fair game has the property that the expected ˜return™ is zero given Q r .
trivially that if X, is a martingale y,+l = X,+l - X, is a fair game. A fair game
sometimes referred to as a martingale difference. A fair game is such that th
return is zero. For example, tossing an (unbiased) coin with a payout of $1
and minus $1 for a tail is a fair game. The fair game property implies that t
to the random variable yt is zero on average even though the agent uses a
information Ql in making his forecast.
One definition of the EMH is that it embodies the fair game property for
stock returns yr+l = R,+l - Rf,,, where R:+l is the equilibrium return give
economic model of the supply and demand for risky assets (e.g. CAPM). The
abnormal) return is the profit from holding the risky asset. The fair game prope
that on average the abnormal return is zero™. Thus an investor may experience
and losses (relative to the equilibrium return RT+1) in specific periods but the
out to zero over a series of ˜bets™.

Stock Prices and Martingales
Let us assume a simple model of returns, namely that the equilibrium or requ
is a constant = k . The fair game property implies that the conditional expec
return is zero:
E[(R,+l - k)lQ,] = 0
Given the definition of R,+l we have

where we have used the logarithmic approximation for the proportionate price c
ln(P,+l/P,) = ln[l (P,+1 - P,)/P,] % AP,+l/Pr). Since, in general, the div
ratio is non-zero and varies over time then, in general, the (log of the) price le
be a martingale in this class of model. In fact any increase in the expected c
must be exactly offset by a lower expected dividend yield. Hence in the efficie
literature when it is said that stock prices follow a martingale, it should be
that this refers to stock prices including dividends. In fact we could define a
price variable q, such that
ln(qr+l /qr 1 = Rr+l
Then the logarithm of qr is a martingale (under the assumption that expected
constant ).

Fair Game and the Rational Valuation Formula
If we let equilibrium or required returns by investors = k, then the fair gam
E [(RI+l - k,+l)lQ,] = 0
for returns also implies that the price of a stock equals the DPV of future
(Note, however, that in the presence of rational bubbles - discussed in Chap
fair game property holds but the RVF does not). The apparent paradox tha
returns can be unforecastable (i.e. a fair game), yet prices are determined by
fundamentals, is resolved.
A straightforward test of whether returns violate the fair game property
assumption of constant equilibrium returns is to see if returns can be predicte
data, Q r . Assuming a linear regression:

then if # ™ 0 (or E f + l is serially correlated), the fair game property is violate
test of the fair game property is equivalent to the orthogonality test for RE.

Economic Models and the Fair Game Property
Samuelson (1965) points out that the fair game model with constant requir
that is E,(Rt+l - k) = 0, can be derived under (restrictive) assumptions abo
preferences. All investors would have to have a common and constant time
rate, have homogeneous expectations and be risk neutral. Investors then pre
whichever asset has the highest expected return, regardless of risk. All ret
therefore be equalised and the required (real) rate of return equals the real i
which in turn equals the constant rate of time preference.

Martingales and Random Walks
A stochastic variable X , is said to follow a random walk with drift parameter

where is an identically and independently distributed random variable w

As the are independent random variables then the joint density function
f ( & , ) f ( ˜ ˜ for m # s and this rules out any dependence between E, and
linear or non-linear. The first way in which the martingale model is less rest
the random walk is that for a martingale and E t need only be uncorrelat
1inea r ly re 1ated) ,
A random walk without drift has 6 = 0. Clearly X f + l is a martingale an
Xr+l - Xr is a fair game (for 6 = 0). However, the random walk is more rest
a martingale since a martingale does not restrict the higher conditional mo
02) be statistically independent. For example, if the price of a stock (inc
dividend payments) is a martingale then successive price changes are unpred
it allows the conditional variance of the price changes E ( E : + ˜ ( Xto )be predi
Suppose that at any point in time all relevant (current and past) information fo
the return on an asset is denoted Qf while market participants, p , have an i
set Q/ which is assumed to be available without cost. In an efficient market,
assumed to know all relevant information (i.e. Qf = Q f ) and they know th
(true) probability density function of the possible outcomes for returns

Hence in an efficient market, investors know the true (stochastic) economic
generates future returns and use all relevant information to form their ˜best™
the expected return. This is the rational expectations element of the EMH.
Ex post, agents will see that they have made forecast errors and this w
ex-post profits or losses
- W&+IIQf)
= &+I

where the superscript ˜p™ indicates that the expectations and forecast errors are
on the equilibrium model of returns used by investors2. The expected or e
return will include an element to compensate for any (systematic) risk in the
to enable investors to earn normal profits. (Exactly what determines this ris
depends on the valuation model assumed.) The EMH assumes that excess
forecast errors) only change in response to news so that r&l are innovations w
to the information available (i.e. the orthogonality property of RE holds).
For empirical testing a definition is needed of what constitutes ˜relevant in
and three broad types have been distinguished.

Weak Form: the current price (return) is considered to incorporate all the i

in past prices (returns).
Semi-strong Form: the current price (return) incorporates all publicly avail

mation (including past prices or returns).
Strong Form: prices reflect all information that can possibly be known

˜insider information™ (e.g. such as an impending announcement of a t
In empirical testing and general usage, tests of the EMH are usually considered
of the semi-strong form. We can now sum up the basic ideas that constitute th

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