On the impossibility of weak-form efficient markets
Steve L Slezak. Journal of Financial and Quantitative Analysis. Seattle: Sep
2003.Vol.38, Iss. 3; pg. 523
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Abstract (Document Summary)
Recent theoretical models show that irrational expectations can generate return
predictability consistent with apparent violations of weak-form market efficiency
documented in the empirical literature. These behavioral models constrain rational
investors' ability to exploit inter-temporal predictability by assuming that rational agents
face high transactions costs, are myopic, or are non-existent. This paper presents a
model in which there are two types of irrational expectations, one that causes
momentum and another that creates reversals. I investigate whether these types of
predictability will persist in the presence of fully rational agents who face no transactions
costs, are long lived, and trade dynamically to optimally exploit any predictability due to
irrational mispricings. I show that weak-form market efficiency will be violated under two
very weak conditions: rational investors are risk averse and the fundamental value of the
asset is risky. The paper also investigates the accumulation of wealth by trader type and
shows that irrational agents will survive under a large set of parameters. [PUBLICATION
ABSTRACT]
Full Text (7699 words)
Copyright %Washington% Sep 2003[Headnote]
Abstract
Recent theoretical models show that irrational expectations can generate return
predictability consistent with apparent violations of weak-form market efficiency
documented in the empirical literature. These behavioral models constrain rational
investors' ability to exploit inter-temporal predictability by assuming that rational agents
face high transactions costs, are myopic, or are non-existent. This paper presents a
model in which there are two types of irrational expectations, one that causes
momentum and another that creates reversals. I investigate whether these types of
predictability will persist in the presence of fully rational agents who face no transactions
costs, are long lived, and trade dynamically to optimally exploit any predictability due to
irrational mispricings. I show that weak-form market efficiency will be violated under two
very weak conditions: rational investors are risk averse and the fundamental value of the
asset is risky. The paper also investigates the accumulation of wealth by trader type and
shows that irrational agents will survive under a large set of parameters.
I. Introduction
The weak form of the efficient market hypothesis states that the price of a security at any
point in time should reflect all of the relevant information about the security's future value
that can be gleaned from past market information on price and volume. In risk-neutral
markets, this implies that prices must follow a random walk, with future returns being
completely unpredictable given past returns. In markets with risk-averse investors who
require risk premia, the weak form implies that future returns in excess of predictable risk
premia cannot be predicted given past price movements.
Recent empirical research, however, has documented dynamic return phenomena that
appear to be inconsistent with the weak form of the efficient market hypothesis. For
example, studies have documented momentum, whereby stock returns (over intervals of
specific length) are positively correlated, and reversals, whereby stock returns are
negatively correlated. Both phenomena suggest inter-temporal predictability in stock
returns based on past price information. In addition, there are studies that document
predictable post-event price behavior, such as post-earnings drift (Bernard and Thomas
(1989), (1990)) and long-term predictability after seasoned equity offerings (see
Loughran and Ritter (1997), for example). However, there is considerable debate (both
in the printed literature and in spoken conversation) regarding whether the documented
phenomena constitute profitable trade opportunities (see, for example, De Bondt and
Thaler (1985) and Jegadeesh and Titman (1993)), statistical aberrations (see
Richardson (1993) and Lo and MacKinlay (1990)), or compensation for risk (see Grundy
and Martin (2001)).1
Recent behavioral models have been developed to explain these phenomena by
assuming investors make specific types of information processing errors that have been
documented in the psychology literature. For example, Daniel, Hirshleifer, and
Subrahmanyam (1998) show that overconfidence and biased self-attribution (the
phenomenon that agents excessively attribute success to ability while excessively
discounting low ability as the possible source of failure) can generate short horizon
momentum followed by reversals at longer horizons. In Barberis, Shleifer, and Vishny
(1998), investors behave in a manner consistent with representativeness, in which
agents view events as typical or representative, in violation of the probabilistic laws
governing the event process. Hong and Stein (1999) also generate momentum followed
by reversals by considering the interaction of two types of boundedly rational agents,
one type that conditions on information about the future but ignores current and past
price and another type that forms overly simple forecasts based solely on lagged
information. Kyle and Wang (1997), Odean (1998), and Gervais and Odean (2001)
consider models in which the agents are overconfident about the quality of the
information they trade on, while De Long, Shleifer, Summers, and Waldmann (1990a),
(1990b) consider a model in which some investors make forecast errors that are
correlated with their information sets.
In all of the behavioral models discussed above, rational traders are limited in their ability
to exploit the mispricings generated by irrational investors, either by their absence or via
their myopia.2 In either case, it is a priori unclear whether the types of dynamic
phenomena that obtain in these models will persist if long-lived dynamically rational
agents are active in the market. Furthermore, it is also unclear whether it would ever pay
an investor to learn to become rational. That is, even if agents start with these
psychological biases, might they learn to overcome their natural tendencies when faced
with the prospect of speculative profits? It seems natural to inquire whether an agent
who learns to be rational will be rewarded and whether the trades he/she uses to exploit
mispricings will make the irrational phenomena disappear in equilibrium. If so, then these
behavioral explanations may not be very robust or convincing.
Rather than investigate the robustness of the above-cited behavioral models by
examining the equilibria that obtain when fully rational agents are added to each model, I
chose to generally examine robustness by taking the common feature of their equilibria,
namely inter-temporal predictability, and to see if it survives the presence of rational
investors.3 In particular, I examine two types of irrational expectations that can
potentially generate momentum or reversals. The main result of the paper is that intertemporal predictability generated by irrational expectations is robust to the inclusion of
dynamic rational agents under very weak conditions. I purposefully define an economy in
which it is most likely that rational trade will correct any mispricings and then show that,
even in this economy, irrational trades affect the equilibrium price process. By showing
that behavioral phenomena persist in my setting, it is clear that they will also persist in
less frictionless settings (as in Shleifer and Vishny (1997), who examine limits to
arbitrage due to institutional constraints). As frictions lessen over time with the advent of
better institutional features and order processing, my results indicate that irrational
behavior will continue to matter.
To define an economy in which rational traders are most able to correct irrational
behavior, I employ the following assumptions. First, I consider an infinite agent economy
in which rational agents are competitive price takers. In this setting, competing rational
agents have no incentive to let deviations from fundamental value persist (perhaps to
generate a continued flow of profitable trade opportunities). Second, I assume that the
rational agents are long lived and optimally choose demands to maximize expected
utility of terminal wealth, exploiting any predictable variation in returns that the irrational
agents generate. That is, I assume that rational agents solve a dynamic programming
problem to maximize their expected utility of terminal wealth given all the profitable trade
opportunities provided by irrational agents. I provide closed-form solutions for optimal
dynamic demand in the presence of correlated normally distributed returns without
relying on approximations. Third, I abstract away from the specific predictable patterns
documented in the empirical literature and consider only a simple predictable deviation
from fundamental value. Thus, there are no informational issues, such as
structural/parameter uncertainty (as in Brav and Heaton (2002)), that prevent rational
agents from recognizing deviations from fundamental value. Furthermore, I assume no
transactions costs, short-sale constraints, borrowing constraints, or asymmetric
information and abstract away from consumption/production effects that might generate
time-varying risk premia. That is, I allow rational agents to trade in a perfectly frictionless
market in which the source of predictability is well known to be due to irrational
mispricings.
I show that even in this frictionless market, optimal dynamic rational trade will not drive
prices back to fundamental value. While there are no exogenous barriers that prevent
competition from rational agents in the model I develop, I assume that rational agents
are risk averse and face fundamental risk. However, I prove that only in the limit as
either fundamental risk or rational risk aversion goes to 0 will irrational behavior have no
effect on equilibrium prices and markets be weak-form efficient.4 This result is in
contrast to the result in De Long, Shleifer, Summers, and Waldmann (DSSW) (1990b)
that irrational traders cause deviations from fundamental value even when there is no
fundamental risk.
Unlike my model, DSSW considers an economy in which rational investors are myopic.
Given myopia, unpredictable future irrational expectations impose risk, which prevents
the risk-averse myopic rational investors from forcing price to equal fundamental value in
equilibrium. Although DSSW state that the impact of irrational trade will diminish as the
horizon of the rational agents increases, they do not consider situations in which the
rational investors can trade multiple times during that longer horizon. Rather, they
consider only rational agents that must maintain a given position over their investment
horizon. Many of the empirical phenomena on inter-temporal predictability, however,
concern behavior over fairly short horizons.5 Thus, the fixed-position strategies of
DSSW will not be effective at exploiting the types of inter-temporal predictability in the
data. In my model, when rational investors can trade frequently, are not myopic, and
have investment horizons that extend beyond the predictability, rational trades eliminate
any predictability in equilibrium when there is no fundamental risk. If fundamental value
is risky, however, then inter-temporal predictability caused by irrational expectations will
exist in equilibrium and markets will not be weak-form efficient.
The paper also examines the wealth accumulation of rational and irrational investors and
shows that although the rational agents have greater returns on average, irrational
agents can still enjoy positive growth in their wealth. Numerical analysis shows that both
momentum and reversals are consistent with the positive wealth growth for irrational
agents in equilibrium. The paper also shows that the return to being rational is bounded
and falls as the proportion of rational investors increases. In the limit as all investors
become rational, the return to being rational falls to 0. Thus, if irrational agents can
choose to become rational on the basis of expected return, a mass of irrational investors
will persist in equilibrium as long as there is a positive cost associated with overcoming
an irrational tendency.
The paper is organized as follows. Section II describes the economy: fundamental value
is defined and the structure of irrational expectations and irrational demands are
specified. In Section III, the solution to the dynamic programming problem of the
dynamically rational agents is specified and discussed. In Section IV, the equilibrium
price function is derived from the market-clearing condition. In Section V, various
properties of the equilibrium price function under specific special cases are investigated.
This section also derives the conditions under which irrational traders affect price.
Section VI describes the accumulation of wealth by rational and irrational traders and
discusses survivorship and the incentive to be rational. Section VII provides concluding
comments.
II. The Model
A. Fundamental Value and the Arrival of Information
A single risky security and riskless money are traded by rational and irrational investors
in a market that is open at dates t - 1, 2, . . . , T - 1. Without a loss of generality, the
riskless asset generates a zero net return and, as numeraire, has a price equal to 1 at
each trade date t. The price of the risky asset, however, fluctuates given the arrival of
information. At trade date t, the price of the risky security is P^sub t^. The information
that affects the price is with respect to a single liquidating dividend that is paid at the
terminal date T. The firm does not pay dividends in any other periods and the per-share
liquidating dividend is denoted by v. The per-capita supply of the risky security is X.
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B. Rational and Irrational Expectations
Here I make a distinction between a publicly announced objective value implication and
publicly available information that must first be processed to yield a subjective value
implication. I assume that the public announcement of [eta]^sub t^ at t is the first type of
information; once [eta]^sub t^ is publicly announced, all investors agree that it is
objectively correct. I will henceforth refer to the public announcement of [eta]^sub t^ as
objective information at t. An example of the second type of information includes
newspaper articles on market conditions in a particular industry or on the discovery of a
new technology. Because this type of information must be processed by an individual to
yield a value implication, I refer to this information as subjective information. Even
though investors may possess common subjective information (they all read the same
newspapers), they may have different expectations on the objective value implication
that will be publicly announced in the future.
At date t, all agents (rational and irrational) possess the same information set
[Omega]^sub t^, where [Omega]^sub t^ includes all past and current objective
information (including past prices, past public announcements of [eta]^sub t^ (for [tau] <
t), and the current objective announcement of [eta]^sub t^), and current subjective
information.6 That is, [Omega]^sub t^ = {O^sub t^, S^sub t^}, where O^sub t^ =
{[eta]^sub 1^,[eta]^sub 2^,...,[eta]^sub 1^, P^sub 1^, P^sub 2^,..., P^sub t^} denotes the
objective information at t, and S^sub t^ denotes the subjective information at t. For
concreteness, I assume that the subjective information received ar t allows investors to
predict only the next period's objective public announcement of [eta]^sub t^+1. Although
the various types of investors have the same information set [Omega]^sub t^, they differ
in the way in which they process the subjective information in forming their forecasts of
[eta]^sub t^+1.7 I denote the conditional forecast of an investor of type i in period t by
E^sup i^[[eta]^sub t+1^/[Omega]^sub t^].
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I assume that there are two different types of irrational investors. Their expectations of
[eta]^sub t+1^ given [Omega]^sub t^, differ from the forecast of rational investors, with
the differences having a specific form for each type. So that I can assess the robustness
of behavioral explanations of both momentum and reversals, I allow irrationality to take
two forms, one form that, if dominant, will generate positive serial correlation (as in
momentum), and another form that, if dominant, will generate negative serial correlation
(as in reversals).
One type of irrational investor (denoted C) makes a mistake that is correlated with the
subjective information contained in his/her information set [Omega]^sub t^. Specifically,
the expectation of the next period news announcement for type C investors takes the
following specific form,
(3) E^sup C^[[eta]^sub t+1^/[Omega]^sub t^] = [eta]^sub t+1^ + [epsilon]^sub t^,
where [epsilon]^sub t^ is normally distributed with mean 0 and variance [sigma]^sup
2^^sub [epsilon]^sub t^^ and is independent of [eta]^sub t+1^. As I will show below, this
type of non-rational forecast creates price behavior that makes a contrarian trade
strategy profitable. Thus, I denote investors who have this type of irrational expectation
by C.8 The expectations of the type C agents are irrational since the forecast error,
[eta]^sub t+1^ - E^sup C^[[eta]^sub t+1^\[Omega]^sub t^] = [epsilon]^sub t^, is correlated
with the forecast, E^sup C^[[eta]^sub t+1^\[Omega]^sub t^] = [eta]^sub t+1^ +
[epsilon]^sub t^. As a result of this correlation, type C forecasts are biased. For ease of
exposition, I will refer to [epsilon]^sub t^ as the bias shock at t.
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In contrast to some of the behavioral models, I assume that there is no asymmetric
information across the different investor types. By adopting this assumption, however, I
subject myself to the following critique in the event that inter-temporal predictability does
not survive rational trade. If inter-temporal predictability does not survive rational trade,
then my result may be erroneous due to an internal inconsistency. For example, in some
of the behavioral models, the irrational agents possess asymmetric (superior)
information. If my model ignores asymmetric information (by focusing on the intertemporal predictability of returns and not its source), then the rational agents in my
model may trade more aggressively than they would if they were trading against (albeit
irrational) better informed agents; they may trade so much more aggressively that they
eliminate any inter-temporal predictability. However, if my model incorporated
informational asymmetry, then rational agents may trade less aggressively, allowing
predictability to persist. Thus, if inter-temporal predictability did not survive rational trade,
the result implied by my approach would be misleading. My approach is not misleading,
however, if I find that inter-temporal predictability persists. If rational investors cannot
eliminate the behavior in a frictionless world without asymmetric information, then they
will not be able to eliminate it in markets with frictions, whatever the source.
Alternatively, if the rational agents would trade less in the face of asymmetric
information, then the results from my approach are biased in favor of finding no
predictability. As I show below, however, my approach shows that predictability persists
and, as a consequence, my results are informative.
C. The Objective Functions of Investors
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I assume that even if [Delta][pi]^sup t+1^^sub 0^ changes through time, the
econometrician knows its deterministic path. Given this knowledge, the covariance of
adjacent price changes is simply cov([Delta]P^sub t+1^, [Delta]P^sub t^) = (1 - [pi]^sup
t^^sub 1^)[pi]^sup t^^sub 1^[sigma]^sup 2^^sub [eta]t+1^ - ([pi]^sup t^^sub 2^)^sup
2^[sigma]^sup 2^^sub [epsilon]t^ while the covariance of non-adjacent price changes is
0 (i.e., cov([Delta]P^sub t+1+k^, [Delta]P^sub t^) = 0, for k > 0). I define any non-zero
covariance between excess price changes as a violation of weak-form market efficiency.
A non-zero covariance implies that, once the predictable part of the price change that is
due to risk premia is taken out, the past excess price change can (at least partially)
predict the subsequent excess price change.
The first part of the covariance of adjacent price changes is due to the correction of nonrationally reflected subjective information on [eta]^sub t+1^ at t getting rationally reflected
once [eta]^sub t+1^ is objectively announced at t + 1. If [pi]^sup t^^sub 1^ < 1, then a
component of price displays momentum since a component of adjacent price changes is
positively correlated. The more current prices reflect the rational expectation of [eta]^sub
t+1^ (i.e., the closer [pi]^sup t^^sub 1^ is to 1), the less the amount of momentum or
positive correlation that exists. The second part of the covariance is negative as the
price in period t + 1 bounces back from the temporary deviation from fundamental value
caused by the irrational component of irrational expectations at t (the bias shock
[epsilon]^sub t^) once the price reflects the objective announcement of [eta]^sub t+1^ at t
+ 1. Thus, when the effect of the bias shock at t is reversed at t + 1, a portion of the price
change is negatively correlated.
Whether the serial covariance is positive or negative depends upon the relative strength
of the momentum and reversion effects discussed above. The covariance is 0 only when
i) the effects completely offset one another or ii) [pi]^sup t^^sub 1^ = 1 and [pi]^sup
t^^sub 2^ = 0. This latter condition holds when only the rational expectation is accurately
reflected and the irrational component of irrational expectations has no effect on
equilibrium price. In all other cases, there is unconditional short horizon predictability in
returns. The issue is whether the trades of rational agents, designed to exploit this
predictability, will, in equilibrium, make the predictability disappear.
III. Optimal Demands
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In contrast to irrational investors, the rational investors are long lived and solve a
dynamic programming problem that maximizes their expected utility of terminal wealth
given expected future profitable trading opportunities created by irrational expectations.
Note that the price function in equation (8) above admits a wide variety of price
dynamics depending upon the series of values for [pi]^sup t^^sub 0^, [pi]^sup t^^sub 1^,
and [pi]^sup t^^sub 2^. Since I do not want to a priori restrict the dynamic behavior of
returns that might result, I must derive optimal demands for rational agents for arbitrary
price processes (with arbitrary dynamics). In my model, the only restriction that is
imposed (for tractability) is that current and future prices are jointly normally distributed.
The following theorem (which is proven in Slezak (1994)) provides the optimal demands
for a long-lived dynamically rational investor in that setting.
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IV. The Equilibrium Price Process
This section derives the market-clearing equilibrium price process and examines
whether the irrational component of irrational investors' expectations impacts price in
equilibrium. More specifically, I next examine whether [pi]^sup t^^sub 2^ is non-zero and
whether [pi]^sup t^^sub 1^, differs from 1. If [pi]^sup t^^sub 2^ is non-zero, then type C
investors have an effect on prices and if [pi]^sup t^^sub 1^, is not equal to 1, type M
investors affect price. In addition, irrational trade can have an effect on prices via the risk
premium or discount [pi]^sup t^^sub 0^, which is also examined below.
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For the remainder of the analysis, I assume that the shock variances are constant for all
periods. That is, [sigma]^sup 2^^sub [eta]t^ = [sigma]^sup 2^^sub [eta]^ and [sigma]^sup
2^^sub [epsilon]t^ = [sigma]^sup 2^^sub [epsilon]t^. Given constant variances
[sigma]^sup 2^^sub [eta]^ and [sigma]^sup 2^^sub [epsilon]t^, I examine the resulting
stationary equilibria in which the price coefficients are constant across periods (i.e.,
[pi]^sup t^^sub 1^ = [pi]^sup t+1^^sub 1^, [pi]^sup t^^sub 2^ = [pi]^sup t+1^^sub 2^, and
[Delta][pi]^sup t+1^^sub 0^ = [Delta][pi]^sup t+2^^sub 0^). That is, I consider equilibria in
which the values of the future coefficients imply (via the relationships in Lemma 1) the
same values for the current price coefficients. As in Slezak (1994), I refer to these
equilibria as steady response equilibria (SRE) since, in equilibrium, the responsiveness
to a particular type of shock is steady (or the same) across periods. Lemma 2 uses the
results in Lemma 1 to characterize the equilibrium price function coefficients in an SRE.
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In the next section, I use this system of equations to characterize when irrational
investors affect equilibrium prices. From the system of equations (18)-(25), the price
coefficient at t depends upon the whole future sequence of price coefficients. By
imposing a transversality condition that the price coefficients equal a particular set of
values at a particular future date (say T), one can determine the coefficients at t by
recursively working backward. (The transversality condition acts as a seed to the
dynamic system (18)-(25).) An SRE price function (when it exists) may obtain in the limit
as T goes to infinity and the parameters are constant. Alternatively, without the
imposition of a transversality condition, SRE price functions will obtain as long as the
conditions in Lemma 2 hold. That is, as long as the future price functions are an SRE
price function, then the current price function will be that SRE function and an SRE will
have been achieved. Thus, SREs are self-sustaining. Furthermore, an SRE does not
require transversality conditions. As a consequence, since all of the properties of price I
analyze derive from the conditions stated in Lemma 2, the results do not rely on price
equaling fundamental value at some future point.10
V. The Effect of Irrational Expectations on Equilibrium Price
A. Conditions under Which Irrational Expectations Do Not Affect Equilibrium Price
Given the characterization of SRE from Lemma 2 in the previous section, I now
investigate the conditions under which irrational investors will have no effect on price
and market will be weak-form efficient. Theorem 2 is the primary result of this section.
Theorem. 2. Under each of the following sufficient conditions, the market is weak-form
efficient. If any one of these conditions holds, then irrational expectations have no effect
on equilibrium prices (i.e., [pi]^sub 1^ = 1 and [pi]^sub 2^ = 0 in equilibrium).
1. [gamma]^sub R^ = 0
2. [sigma]^sup 2^^sub [eta]^ = 0
3. N^sub R^ = 1
When all three of the above conditions are not satisfied, irrational trades affect price and
prices will not be weak-form efficient. In particular:
If 0 < N^sub C^ < or = 1 and 0 < [sigma]^sup 2^^sub [epsilon]^ < [infinity], then [pi]2 > 0.
If 0 < N^sub M^ < or = 1 and 0 < or = [delta] < 1, then [pi]^sub 1^ < 1.
Proof. See Appendix B.
The first part of Theorem 2 states that as long as rational investors are either risk neutral
or face no fundamental risk, then the presence of irrational investors (regardless of their
mass) will have no effect upon the equilibrium price. When rational agents are risk
neutral, they are willing to take infinite positions to exploit even tiny deviations of price
from expected fundamental value generated by irrational traders. As a result, for market
clearing to hold, the equilibrium price must fully reflect the rational expectation of
[eta]^sub t+1^ (and current and past objective news via [theta]^sub t^) and nothing else
(i.e., [pi]^sub 1^ = 1 and [pi]^sub 2^ = 0). Similarly, when there is no fundamental risk
(i.e., when [sigma]^sup 2^^sub [eta]^ = 0), even risk-averse rational investors are willing
to take infinite positions on the basis of an irrational deviation.
Notice, however, that the future random irrational errors made by type C investors are
unpredictable (to rational investors) and, in principle, might cause future prices to
fluctuate, generating risk to the rational investors (as in DSSW (1990b)). The above
theorem states, however, that this is not true. Given that rational agents are long lived
and dynamic, potential fluctuations due to type C irrational errors do not generate risk
when the underlying fundamental value of the firm is constant over time. Thus, rational
agents are not averse to taking large positions on the basis of deviations created by
[epsilon]^sub t^ while facing future volatility created by these shocks, knowing that the
future fluctuations do not represent risk but rather generate profitable future trading
opportunities. Competition among the rational agents for these trade opportunities,
however, assures that these deviations go to 0 in equilibrium.
The second part of Theorem 2 states that as long as there is a non-negligible mass of at
least one type of irrational trader, and rational agents are risk averse and face
fundamental risk, the price function will differ from the fully rational price function. When
there are M type traders present, the price will only partially reflect the rational
expectation [eta]^sub t+1^; when there are C type traders present, the price will have
fluctuations due to the irrational component of their expectations, [epsilon]^sub t^.
Furthermore, it is never the case that there exists a combination of parameters for which
the effects of the two types of irrational traders cancel, yielding a rational price function
in equilibrium. Given risk aversion and fundamental risk, rational agents are unwilling to
take positions that are large enough to make the irrational deviations disappear in
equilibrium. In this case, when a rational investor trades to exploit a deviation from
fundamental value created by [epsilon]^sub t^, he exposes himself to fundamental risk.
B. How Irrationality Affects Equilibrium Price
In this section, I examine the features of the equilibrium price function. Because
analytical solutions for the non-linear system in Lemma 2 do not exist, I provide
numerical solutions for the price function under a variety of parameters to show how the
parameters affect equilibrium. Although I provide solutions for a limited set of parameters
here, the solutions I provide are representative of the behavior that I obtained under
numerous other parameter assumptions. Below, I summarize my findings with respect to
the price coefficients and the intertemporal predictability of returns.
1. Price Function Coefficients
Figures 1-3 show how the price function coefficients ([pi]^sub 1^2, [pi]^sub 2^ and
[Delta][pi]^sub 0^, respectively) vary with the variance of news ([sigma]^sup 2^^sub
[eta]^) and the variance of type C forecast errors ([sigma]^sup 2^^sub [epsilon]^) for
various values of [delta] and N^sub R^. Each individual graph shows how the
coefficients vary with [sigma]^sup 2^^sub [eta]^ and [sigma]^sup 2^^sub [epsilon]^. By
comparing across the rows of individual graphs in each figure, one can see how the
coefficients vary with [delta] while holding the percentage of rational investors fixed. By
comparing down the columns, one can see how the coefficients vary with the percent of
rational traders in the market while holding [delta] fixed. For all of these graphs, I divide
the irrational population equally between M and C type investors. That is, I assume
N^sub C^/(N^sub C^ + N^sub M^) = 0.5, or, equivalently, N^sub C^ = N^sub M^ = (1 N^sub R^)/2.
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FIGURE 1
Equilibrium [pi]^sub 1^ as a Function of the Exogenous Parameters
Figure 1 shows that [pi]^sub 1^ has the following properties.
Property 1. [pi]^sub 1^ is increasing in [delta] and N^sub R^.
For lower values of [delta], price reflects less of the unbiased forecast of the future public
announcement [eta]^sub t^. Recall that [pi]^sub 1^ is the weighted average of 1 and
[delta] and, as a consequence, the lower [delta], the lower [pi]^sub 1^. In addition, as the
percentage of rational investors increases (or the percentage of agents that are type M
decreases), greater weight is placed on 1, thus increasing [pi]^sub 1^.
Property 2. [pi]^sub 1^ is both increasing and decreasing in [sigma]^sup 2^^sub
[epsilon]^ and [sigma]^sup 2^^sub [eta]^.
As [sigma]^sup 2^^sub [eta]^ goes to 0, [pi]^sub 1^ goes to 1. As [sigma]^sup 2^^sub
[eta]^ gets larger, [pi]^sub 1^ falls toward [delta]. For very low values of [sigma]^sup
2^^sub [epsilon]^ however, [pi]^sub 1^ will fall and then rise as [sigma]^sup 2^^sub [eta]^
increases. [pi]^sub 1^ is increasing in [sigma]^sup 2^^sub [epsilon]^ for low values of
[sigma]^sup 2^^sub [eta]^ but can be decreasing for higher values of [sigma]^sup 2^^sub
[eta]^. Thus, the [pi]^sub 1^ surface twists as [sigma]^sup 2^^sub [eta]^ increases.
Whether [pi]^sub 1^ increases or decreases in [sigma]^sup 2^^sub [epsilon]^ depends
upon which of two effects dominates. The first effect is due to the effect of increasing
[sigma]^sup 2^^sub [epsilon]^ on type C trades. For low levels of [sigma]^sup 2^^sub
[epsilon]^, type C agents are less irrational and trade more like rational agents. That is,
the variance of their bias falls and, as a result, their expectations get closer to rational
expectations. As a result, there is a larger mass of traders with expectations that are
close to fully reflecting [eta]^sub t+1^, which pushes [pi]^sub 1^ closer to 1. The second
effect is from the effect of [sigma]^sup 2^^sub [epsilon]^ on type M demand. For smaller
values of [sigma]^sup 2^^sub [epsilon]^, the M type traders trade more aggressively
since the variance of future price, the appropriate measure of risk for type M traders, is
smaller. As [sigma]^sup 2^^sub [epsilon]^ increases, M type investors trade less
aggressively and, thus, do not push [pi]^sub 1^ as far down toward [delta]. When
[sigma]^sup 2^^sub [epsilon]^ is small, the first effect (due to C type investors)
dominates and [pi]^sub 1^ increases in [sigma]^sup 2^^sub [epsilon]^. When
[sigma]^sup 2^^sub [epsilon]^ is large, then the second effect (due to M type investors)
dominates and [pi]^sub 1^ decreases in [sigma]^sup 2^^sub [epsilon]^.
Figure 2 provides the following properties for [pi]^sub 2^.
Property 3. [pi]^sub 2^ is decreasing in N^sub R^.
As N^sub R^ increases, there is more competition to exploit the mispricing generated by
type C investors and, as a result, the type C irrational errors have less impact on price.
Property 4. [pi]2 is increasing in [sigma]^sup 2^^sub [eta]^ and decreasing in
[sigma]^sup 2^^sub [epsilon]^.
As [sigma]^sup 2^^sub [eta]^ increases, fundamental risk increases and rational agents
are willing to take less extreme positions to exploit the mispricings caused by type C
irrational forecast errors. As such, less of the mispricing is traded away and [pi]^sub 2^ is
larger. As [sigma]^sup 2^^sub [epsilon]^ increases, the return per unit of risk of trading
on mispricings increases and competition among rational investors makes the bias
shock affect price less.
Figure 3 provides the following relationships for mean returns.
Property 5. The change in the risk discount is increasing in both [sigma]^sup 2^^sub
[eta]^ and [sigma]^sup 2^^sub [epsilon]^.
That the risk premium is increasing in [sigma]^sup 2^^sub [eta]^ is not surprising; as the
variance of news increases, there is more fundamental risk and both rational and
irrational agents require larger mean returns if they are to want to hold the fixed supply
of the risky asset. Why the risk premium is increasing in [sigma]^sup 2^^sub [epsilon]^ is
less obvious since, for rational investors, [sigma]^sup 2^^sub [epsilon]^ does not
generate risk, but rather represents trading opportunities. However, the irrational agents
are myopic and as such they care about the variance of the