Old information and market efficiency

Economics Letters 0165.1765/93/$06.00

43 (1993) 955101 0 1993 Elscvier

95 Science

Old information Bernard

B.V. All rights

reserved

and market

efficiency

Gauci”

Lkpar~mentof Economics.

Thomas

Box 9732. Hollins College. VA 24020. USA

Baumgartner

Itrsritut fiir iikologische Received Accepted

Publishers

12 April 29 June

Wirtschuftsfi,rschung,

Heidelberg.

Germany

1993 1993

Abstract In an efficient market, agents will reuse old information However. this does not create exploitable profit opportunities. subjective model, and the implications for the random walk.

in the aftermath This paper studies

of corrections the difference

in the subjective model. between the objective and

1. Introduction The efficient market theory (EMT) asserts that a market price at ail times fully reflects all available information. This theory is conventionally interpreted as asserting that it is not possible for agents at time t to profitably exploit information that was already available in the past, at t - j. This paper will explore the circumstances under which such an event would occur without breaching the requirements of efficiency. In instances where agents make corrections in their model, efficiency will in fact require the reuse of past information, but such events do not create exploitable profit opportunities. This suggests that the EMT does not preclude chartist claims of the discovery of past correlation between prices, but the EMT still excludes that such correlations provide future exploitable opportunities.

2. Efficiency The process of generating information for the purposes of trading in, for example, a financial market, can be seen as being in two parts. One is the more straightforward, and consists of plugging new data into an existing theoretical structure, or model, for the purpose of generating predictions. The other, which logically and chronologically comes earlier, has to do with the development of the theoretical structure or the model itself. A model states which predictors are significant, and also specifies the relationship between these predictors and the predicted

* Corresponding

author

96

B. Gauci.

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Letters 43 (1993) 9S-101

variable. ’ This, then, is the initial development of the model. In turn, this is followed by the continuing estimation of the parameters of the model, and revisions in such estimates, as fresh supplies of data are plugged into the model. The efficient market theory, like the rational expectations theory, is a reasonable and widely held description of how information is used within a subjective model of a stationary objective situation. A long unchanging relation between an endogenous variable and a set of predetermined variables allows agents to fully exploit knowledge of such a stable relationship in their speculation about the future path of the endogenous variable. As the subjective model structure which specifies this relationship becomes widely accepted, new data are used by forecasters simply to update the set of predetermined variables as well as the parameter estimates. But a changing objective situation or a potential change in an objective situation limits the relevance, and challenges the continuing validity of past relationships. In such an environment, prediction failures may prompt a revision of the model structure itself. Lucas (1976) argues that expectations will be rational in an environment characterized by Knightian riskiness, situations in which the probabilities of interest concern a fairly well defined recurrent event. [Here,] the hypothesis of rational behaviour on the part of agents will have usable content in terms of economic theory. In such situations, expectations are rational in Muth’s sense. In cases of uncertainty, economic reasoning will be of no value. Bray (1983)

adds that,

if the rational expectations hypothesis is to be taken to mean that subjective beliefs correspond to objective probabilities, it lacks content unless the stochastic process generating outcomes and expectations is in some rather general sense stationary. In a completely nonstationary world, it is impossible to identify probabilities with observed frequencies. A Knight-risky situation is one that occurs at a time of objective-model stability, where the need for corrections of the subjective (or agent’s) model is not manifest, and where the agent can therefore confidently draw on past experience to assess his future risk. On the other hand, a situation of Knight-uncertainty is one where the past fails to provide the relevant experience, and this can include also a situation where the validity of the established subjective model has been cast into doubt by, for example, anomalies in the evidence. The latter would therefore be a time of subjective and objective model instability. Empirical tests of the EMT typically ascertain whether the market has absorbed all available information and arbitraged away excess returns. It is appropriate for such tests to deal with situations of model stability, that is of instances when market participants have reason to anticipate a continuation of the relationships between the same pertinent variables as in the past. An example of such long-term objective model stability can be found in the case of the so-called January effect. The evidence, particularly dealing with small firms, suggests a possible failure in stock market efficiency and a possible failure by agents to align their subjective model to the objective one. The first few days of January apparently manifest consistently higher than usual rates of return. The cause seems to be the same on each occasion: certain selling habits at the end of the previous year, as financial managers reposition their portfolios at year’s end. Managers’ I These two characteristics p. 148).

of a model

are equivalent,

respectively.

to decomposition

and specification

in Simon

(1981,

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97

9.5-101

behavior, which can be said to be an important component of the objective model, appear to be stable. An efficient market is one where other traders, prompted by their subjective model, would arbitrage away excess January returns. Whether in the case of this so-called anomaly the markets are in fact inefficient is a matter of dispute. For example, Malkiel (1989, p. 129), finds such departures from randomness to be weak, if still disturbing. Haugen and Lakonishok (1988) find stronger evidence. As another example, we shall choose one involving long-horizon returns, rather than shorthorizon returns * as in the previous example. Fama and French (1988) report evidence of substantial negative correlation between consecutive five-year rates of returns. This suggests the possibility that traders, motivated by an incorrect subjective model, repeatedly over-react to exogenous events. As in the first example, such anomaly is allegedly of long standing and therefore may occur when the objective model has been stable. It suggests that in the aftermath of their over-reaction traders repeatedly revise prices in an ad hoc fashion, but they fail to correct the subjective model itself. The EMT does not permit repeated similar price revisions, and instead requires immediate correction of the subjective model. However, while objective model stability enables the subjective model to coincide with the objective model, it does not guarantee such an outcome. This paper will proceed to elaborate on the distinction between the objective and subjective models, and will later investigate the implications of changes in subjective models for one of the corollaries of the EMT, namely the random walk.

3. Different

models and the random walk

Consider the following efficient market. price P is fully reflected in P,_, , such that

At time t - 1 all available

information

pertinent

to a

EP,=P,_,(l+r)+e,

where

e is a stochastic

term with E(e) = 0, and Y is the rate of discount:

E[EP,-P,_,(l-tr)]=E,(e)=O.

(1)

The best guess of where P will go to after f - 1 is summed up in the statement that P will follow a random walk with a discount-rate drift. Now add the appropriate subscript to the expectations operator E in (1) to identify the time at which the expectations are formed. Equation (1) becomes E,_,[P,

Now,

a question

- P,_,(l

+ Y)] = E,(e) = 0.

can be posed:

E,[P, - P,_,(l

(2)

if (2) is correct,

is it then

also true that

+ r)] = O?

(3)

This question is significant for the following reason. Equation (2) and the random walk. Speculation at t - 1 as to the value normal return, r, and therefore market participants are assured of at t - 1 of where P will be at t is that it will follow a random walk.

game

’ Fama

(1991)

contains

a literature

survey

that draws

a distinction

between

displays the concepts of the fair of P at t will generate only the a fair game. The best prediction But now shift the vantage point

long and short

horizon

returns

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Letters 43 (1993) 95-101

to t. The issue of whether the game was fair can be set aside. We now ask: If at t we look back in time, shall we find that P will have followed a random walk? Or, is (3) also true? The answer depends on whether the subjective model at t - 1 governing expectations E,_lP, is the same as the objective model which at t actually determines P,. It depends on whether the subjective density function f(P) in EP =

I

Pf(P) dP

at t - 1 is the same as the objective one at t. Indeed, constancy of f(P) is a condition for the random walk. 3 If the two models are the same, then (2) implies (3). Before presenting the preceding argument more formally, we shall introduce two definitions. Consider two consecutive price changes, P,_, and p,. One can distinguish between an a priori or prospective random walk between t - 1 and t, as anticipated at t - 1, and an ex post or retrospective random walk, as viewed at t. The fair game is synonymous only with the prospective random walk: if p,_, cannot be predicted at t, then the game is fair. But whether, once we are at t, P,_~ and p, will turn out to be uncorrelated depends on whether the path followed by P follows a retrospective random walk. A formal statement of the preceding goes as follows. Define a price change p,=P,-P,-,(l+r).

If the game is fair, for any two periods separated E(P,lP,p,)

= E(P,P,-,)I%‘-,)

by interval T, 4 (4)

= 0.

The point made in the previous section was that the density functions behind the expectations terms in (4) may change between t - T and t. Therefore even if E,_,(p,_,p,) = E,_,(p,lp,_,) = 0, it does not necessarily follow that E,(p,-,p,) = E,(p,(p,_,) = 0. Equation (4) is equivalent to E(P,lPJ

= [(J-

P,F,P,dP,-,P,)

dP,-T dP,

l/U

P,h(PJ

dP,

1 .

Let g be the joint density function and h the marginal density function. We now distinguish between the subjective and objective model. Let hs_, and gs_, represent, respectively, the marginal and joint subjective functions at t - T, while hp and gp represent respectively the marginal and joint objective functions at t. We now distinguish between the two versions of E(p,lp,_,). One, defined at t -‘T, is equal to E,dP,lP,-7) =

[II

P,~TP,L(P,-,PJ

3 See Fama (1970, p. 386, footnote 4). ’ A joint expectation is defined as the product E(P,+,P,)

= E(P,IP,~,)E(P,+,)

E(P,~P,.,)

= E(P,P,-,YE(P,~,)

Or

CL

dP,

dpt-, 1. l/Ip,-,h:-,b-,I

of the conditional

expectation

and the marginal

expectation:

B. Gab,

The other,

at t, is equal

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43 (1993) 95-101

to

E,_,(p,lp,_,) = E,(p,lp,-,) only if the subjective and objective functions are identical. If this is the case, then a fair game, i.e. an a priori random walk E,P,(p,lp,_,) = 0, would imply an ex post random walk, E,(p,p,-,) = 0. A non-financial analogy is along these lines: a drunk is gently dropped by helicopter in a large wooded field. The helicopter lifts off and returns an hour later to fetch the drunk. What is his and will have followed a likely position at that time? A priori, the drunk may be anywhere random walk. However, it is in the realm of the possible that the drunk will actually, ex post, be at the outside gate, having been guided there by some kind soul, whose presence in the neighborhood was previously unknown. An ex post objective model describing the path followed by the drunk will differ from the a priori subjective model of the helicopter pilot, who will promptly correct his subjective model to include this new information.

4. Model corrections

and the random

walk

Even when the objective situation is stationary, traders may find the subjective model to be incorrect, and they then would proceed to correct it. Consider a case where price P varies only with a vector X, which consists of variables X,, i=l,. .) n. Let each X, itself be efficiently determined on a financial market, such that X, always incorporates also all information pertinent to its own expected future value. To simplify, we shall remove the drift from X as well as from P by correcting for the normal rate of return. Define X*l-7 = Xfmr(l + Y)~, and define drift-adjusted P as P*. Differences between the various P”‘s and between the various X*‘s are for reasons other than the normal rate of return over time. Furthermore, assume that P* and X* are related in the following manner in successive time periods [. . , t - 2, t - 1, t]: = Px:

p::

>

p,*_, =/3x,*_,

1

P,?, = PX,*_? , ... ...

(5)

Equations (5) are statements of market efficiency, where the market has at each time taken into account all available information. As already indicated, a lower case p represents the change in price level, P*. The market is continually satisfied with its measurement of the relationship /3: p,=P;*-P,*_, = p
the determination E,_,p,

= px,* -XT-,>

-

px;-,

.

of X is efficient,

= PEl_,(X::

-X,*_,)

= 0.

E,_, (XT - XT_,) = 0, and therefore

(6)

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Letters 43 (1993) 95-101

Equation (6) is a statement of a prospective random walk. At t - 1, prices are seen as being about to follow a random walk with drift. (6) is also a statement of a fair game, in that at t - 1 the expected rate of return from holding the asset is the normal rate of return. Retrospectively, at t, p will have followed a random walk if the subjective distribution, at t - 1, of
pE,(X:

-x,*_,>= 0.

(7)

Now consider an alternative scenario. At t, market agents discover an error in their measure of p, which they promptly replace with p. Let us reconstruct the successive steps. Before the correction, that is for periods prior to t, (5) still holds, but at t, there is a model revision, such that

PX,*-2> P;*_, =/3x;*_,, P,* = /Lx:: . p,*_, =

At t - 1, p is unknown,

and therefore

E,_,p,=PE,_,(X;* But the determination

(8)

(6), which is defined

-x;*_,>=o.

as at t - 1, remains

true: (6)

of E,p, in (7) becomes

p, = P: - P,*_, = #ux: - px,*_, = /Ax,* - /-Lx,*-, + pxr*_, -

px;-,

=lu(x,*-x,*-l)+(~--Px,*~* E,P,=PE,V:

-X,*_,)+(P-PM,*-,.

(9)

If the objective distribution of (X;* -X;“_,) at t is unchanged from its subjective distribution at t - 1, the first term on the right-side of (9) is zero. Even then, the second term tells us that in the event of model correction P, varies with past data X,*_, Since (6) remains true, (9) does not invalidate the fairness of the game in that p, could not have been predicted prior to t. In fact, p which is necessary for the prediction of P, is discovered at t, and is unknown before. Unlike (7), however, (9) shows the reuse at t of information which was already available at t - 1.

5. Conclusion Proponents of market efficiency assert that past information is irrelevant to all later prediction. It is true that efficiency requires complete use of all information. But the full exploitation of information - a characteristic of efficient markets - does not exclude its reuse. In fact, X,? 1, which consists of information available at t - 1, is used in (8) in the determination of P* at t - 1, and is then reused in (9) in the determination of the change in P* between t - 1 and t. If a price change is correlated with information that was available earlier, it does not necessarily follow that the price is inefficiently determined. Indeed, model changes require the revision of prices, which in turn demands the reuse of past data. What the EMT precludes, however, is that model changes themselves become an exploitable regularity. Efficiency requires that the event of the model corrections be itself unpredictable. Therefore the chartist (or other) claim at t, of

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correlation between P, and information set Z,_, is not ruled out by efficiency. out that such correlation can be exploited profitably after t.

But efficiency

rules

References in: R. Fryman and E.S. Phelps, eds. Individual Bray, M., 1983, Convergence to rational expectations equilibrium, forecasting and aggregate outcomes: Rational expectations examined (Cambridge University Press, Cambridge) ch. 6. Fama, E.F., 1970, Efficient capital markets: A review of theory and empirical work. Journal of Finance 25, 383-417. Fama, E.F.. 1991. Efficient capital markets: II, Journal of Finance 46, 1575-1617. Fama, E.F. and K.R. French, 1988, Permanent and temporary components of stock prices, Journal of Political Economy 96, 246-273. Haughen R.A. and J. Lakonishok, 1988. The incredible January effect: The stock market’s unsolved mystery (Dow Jones-Irwin, Homewood. IL). Lucas, R.E., 1976, Econometric policy evaluation: A critique, in: K. Brunner and A.M. Meltzer, eds. The Phillips curve and labor market (North-Holland, Amsterdam) 19-46. Malkiel, B.C., 1989. Efficient markets hypothesis, in: J. Eatwell, M. Milgate and P. Newman, The New Palgrave: Finance (Macmillan, London). Simon,

H.A.,

1981, The sciences

of the artificial,

2nd edn.

(MIT Press.

Cambridge,

MA).