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equilibrium â€˜pricing modelâ€™ 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

gameâ€™ 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 â€˜tru

5.2 IMPLICATIONS OF THE EMH

The view that the return on shares is determined by the actions ofâ€™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

â€˜buy and holdâ€™ policy. He should spread his risks and hold the market portfo

20 or so shares that mimicethe market portfolio). Andrew Carnegieâ€™s advice

your eggs in one basket and watch the basketâ€™ 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) â€˜shopping aroundâ€™ 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 â€˜lowâ€™, 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 â€˜efficientâ€™. But their use of â€˜efficientâ€™ 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 â€˜packagesâ€™ of mutual funds. Also with this strategy

house will usually have at least one fund that it can boast has â€˜beaten the ma

Takeovers, Conglomerates and Financial Institutions

Let us turn now to some public policy issues. The stock market is supposed to

â€˜correctâ€™ 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 firmâ€™s new

project are expected to be high the existing share price will be â€˜highâ€™ and th

obtain its funds by issuing fewer shares. However, if the share price does

firndamentals but is influenced by whim or fads of â€˜irrationalâ€™ 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

termismâ€™. 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 managerâ€™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 â€˜badâ€™ incumbent managers to be replac

scenario, the hostile bidder recognises that the incumbent â€˜badâ€™ manageme

shareholders to mark down the firmâ€™s share price. The hostile bidder pays

excess of the existing share price. After replacing the â€˜badâ€™ 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. â€˜asset strippingâ€™).

erate mergers were sometimes justified on the grounds that the acquisition o

firms by â€˜firm Aâ€™ reduced risk to the shareholder who held Aâ€™s shares since the

erateâ€™ 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 â€˜newsâ€™ occurs freque

expected to make a substantial impact on a firmâ€™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 â€˜externalityâ€™) - for example, that a stock market crash led to insolvencies

institutions, a â€˜credit crunchâ€™ 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 â€˜cooling-off periodâ€™). Also, where systemic risk is invo

â€˜runâ€™ on one bank causes a run on other banks) one might be prepared to proh

institutions from holding â€˜highly volatileâ€™ assets such as shares (e.g. banks in

By definition, â€˜newsâ€™ is random around zero. Hence â€˜newsâ€™ will not influen

of share prices over a long horizon. Therefore except in exceptional circumsta

would be unlikely to cause panics leading to a â€˜runâ€™ on banks or financial ins

their depositors. However, if the market is inefficient and prices are subject to

â€˜irrational swingsâ€™ 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 â€˜financing conditions will improveâ€™ (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 firmâ€™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

â€˜rolling overâ€™ 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

â€˜active managerâ€™ either as regards the choice over the appropriate â€˜mixâ€™ 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 â€˜beat the marketâ€™ 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.

53 EXPECTATIONS, MARTINGALES AND FAIR GA

.

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 â€˜efficient markets hypothesisâ€™.

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

i=l

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,)

&r+l

This (mathematical) conditional forecast error can be shown (always) to b

average:

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

=0

E(E:+lQrIQr)

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

)I

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

ErEt+lEt+2

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 â€˜Muth-REâ€™ (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 â€˜tastesâ€™ on the demand side

based products such as cookies). The actual equilibrium price depends in p

actual value of such â€˜shocksâ€™ 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 â€˜errorâ€™ 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;â€™ 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 agentsâ€™ forecast errors for the actual equilib

To test whether agentsâ€™ 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 â€˜noisyâ€™ proxy v

each agentâ€™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 â€˜economistâ€™s shelfâ€™ (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 â€˜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 â€˜wrongâ€™ 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 â€˜returnâ€™ 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 zeroâ€™. 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 â€˜betsâ€™.

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

implies

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 # â€™ 0 (or E f + l is serially correlated), the fair game property is violate

l

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

Er+l

As the are independent random variables then the joint density function

El

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

to

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 â€˜bestâ€™

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

7),+1

P

where the superscript â€˜pâ€™ 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 â€˜relevant in

and three broad types have been distinguished.

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

0

in past prices (returns).

Semi-strong Form: the current price (return) incorporates all publicly avail

0

mation (including past prices or returns).

Strong Form: prices reflect all information that can possibly be known

0

â€˜insider informationâ€™ (e.g. such as an impending announcement of a t

merger).

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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