Belief Persistence and the Disposition Effect

†

Jay Lu

November 2013

Abstract

We consider a model where agents have heterogeneous beliefs about state persis-

tence. Equilibrium trading behavior is ordered if agents can be ranked according to

the degree of disposition effect (i.e. they buy when prices fall and sell when prices rise)

that they exhibit. We show that trading behavior is ordered if and only if beliefs in

the population can be ordered via a single parameter measuring persistence. Agents

who believe that states are less persistent exhibit the disposition effect while those who

believe that states are more persistent exhibit the opposite behavior (i.e. a form of the

house-money effect).

† I am deeply indebted to both Faruk Gul and Wolfgang Pesendorfer for their continuous advice, guidanceand encouragement. I would also like to thank Roland Bénabou, Sylvain Chassang, Stephen Morris and WeiXiong for their helpful suggestions and constructive input.

1

1 Introduction

One of the most robust findings in behavioral finance is the disposition effect. This refers

to the tendency of individual investors to oversell stocks that have gone up in value (i.e.

price) and to undersell stocks that gone down. Although suboptimal, this behavior has been

well-documented in many situations and in various markets around the world.1 Moreover,

similar observations have also been recorded in other contexts, such as the housing market

or in the exercise of executive stock options.2

In this paper, we consider a belief-based explanation for the disposition effect. In par-

ticular, we study a model of heterogeneous beliefs where beliefs differ only along a single

dimension measuring persistence. For example, some agents may believe that earnings infor-

mation from this quarter will be highly correlated with earnings information from the next

quarter (high persistence). On the other hand, others may believe that the earnings informa-

tion from both quarters are completely uncorrelated (no persistence). In equilibrium, agents

who believe in the least persistence decrease their holdings of the stock when prices rise and

increase their holdings when prices fall; in essence, they exhibit the disposition effect. On

the other hand, those who believe in the most persistence exhibit the opposite behavior;

they employ a trading strategy based on stock price momentum and exhibit a form of the

house-money effect.3

To be concrete, suppose that institutional investors pay close attention to certain financial

indicators after each earnings release which they believe to be highly correlated with future

earnings information. Individual investors on the other hand, believe these indicators to be

noisy and ignore them when placing their trades. In equilibrium, our model predicts that

individual investors will exhibit the disposition effect while institutional investors will engage

in momentum trading.4

Much of the existing research has focused on providing a preference-based explanation to1 Studies by Odean [15] in the U.S., Grinblatt and Keloharju [10] in Finland and Feng and Seasholes [8]

in China confirm that the disposition effect is a global phenomenon that transcends both institutional andcultural divisions.

2 See Genesove and Mayer [9] and Heath, Huddart and Lang [12] respectively.3 The house-money effect refers to the tendency of gamblers to increase their bets after a gain and to

decrease their bets after a loss (see Thaler and Johnson [20]).4 This is somewhat consistent with the empirical evidence suggesting that the disposition effect is much

more pronounced for investors who are financially less sophisticated (see Dhar and Zhu [6] for example).

1

rationalize the disposition effect.5 Our results show that by introducing belief heterogeneity

in a model with equilibrium trading, we can obtain behavior that exhibits the disposition

effect while still retaining the standard assumption of expected utility preferences. While our

approach does not insist that a belief-based equilibrium model is the only explanation for the

disposition effect, it does suggest that in many cases, a careful study of the disposition effect

should also take into account investor beliefs and how they interact with equilibrium forces.

We leave the more practical exercises of testing different explanations of the disposition effect

as avenues for future research.

Formally, we consider a simple two-period model where agents form beliefs about the

state space S in each period. For example, S could represent all the financial information

available after each earnings release. We assume that agents share the same correct prior

about S, but have different beliefs about the transition probabilities between the two time

periods. We model these beliefs as Markov kernels on S. For example, some agents may

believe that state realizations in the two periods are highly correlated (i.e. high persistence)

while others may believe that there is little correlation (i.e. low persistence). We say that

the distribution of beliefs in a population has a persistence representation iff each agent’s

belief corresponds to a Markov kernel

K = �K + (1� �)K

where K is the constant kernel, K is some other fixed kernel and � 2 [0, 1]. Thus, all agents

in the population can be parametrized by some � 2 [0, 1] measuring each agent’s belief of

state persistence. For example, an agent with � = 0 believes in zero persistence. In the case

where S is finite, K is simply the �-mixture of the two matrices K and K. Note that this is

not a model of asymmetric information; after the first period, all agents observe the realized

state s 2 S but form their own individual beliefs about what that means for second period

state realizations.

We consider an equilibrium where agents trade claims to two securities: a risky “stock”

that yields a payoff contingent on the realization of the state s 2 S and a risk-free “bond”

that yields the same payoff regardless of which state is realized. To simplify the exposition,5 See Barberis and Xiong [2], Strahilevitz, Odean and Barber [19] and Barberis and Xiong [3] for expla-

nations based on prospect theory, emotional regret and realization utility respectively.

2

we assume that all agents have the same endowment in each state and that they share the

same CARA utility index. An equilibrium is ordered iff all agents can be ranked according to

the degree of disposition effect they exhibit. Our first main result shows that any equilibrium

under a persistence representation is ordered. In this case, those who believe in the least

persistence (i.e. smallest �) exhibit the disposition effect while those who believe in the most

persistence (i.e. largest �) exhibit the opposite behavior (i.e. momentum trading or a form

of the house-money effect). Thus, in our model, the disposition effect emerges as equilibrium

behavior induced by belief heterogeneity.

We then study a special case where beliefs are Gaussian. There are two distinct groups

in the population: the first group believes that states are independent and identically dis-

tributed (i.i.d.) while the second believes that states are correlated. This CARA-Gaussian

setup is easily tractable and allows for simple expressions for both prices and trading strate-

gies. In equilibrium, a simple inequality characterizes when stock prices will increase (or

decrease) and when agents in the first group (i.e. those who believe in no persistence) will

decrease (or increase) their stock holdings. We then proceed to analyze some of the com-

parative statics. We show that increasing the proportion of agents in the second group (i.e.

those who believe in persistence) will inflate prices in “good” states (i.e. states where prices

rise) and deflate prices in “bad” states (i.e. states where prices fall). Thus, introducing belief

heterogeneity in our model does not uniformly increase prices and may result in greater

dispersion (i.e. higher volatility) of stock prices.

Lastly, we consider the uniqueness properties of the persistence representation. In general,

beliefs in a population are not uniquely specified given equilibrium prices and strategies.

However, by varying the payoffs of the stock and observing the resulting equilibrium behavior,

we can completely identify the distribution of beliefs in a population.6 We show that without

loss of generality, beliefs in a population has a persistence representation if and only if any

equilibrium is ordered by the same disposition ranking over agents. In other words, for any

other representation, there exists some stock where the disposition ranking is reversed in

some states. One agent exhibits a greater disposition effect than another agent under one

equilibrium but not in another. Thus, the persistence representation is the only distribution

of beliefs that permits a consistent disposition ranking of agents in all equilibria. This is6 This approach is similar in spirit to that of Savage [16] where one varies the state-contingent payoffs

(i.e. “acts”) to uniquely identify beliefs under individual decision-making.

3

a complete characterization of the persistence representation. It also allows us to equate

observable equilibrium prices and trading strategies with the unobservable beliefs of agents

in a population.

This paper is related to a long literature on the disposition effect. Shefrin and Statman

[18] first used the term the “disposition effect” to describe this behavior. Odean [15] provided

the first comprehensive study of the disposition effect and ruled out various explanations

including portfolio re-balancing, trading costs and tax considerations. Other studies include

Grinblatt and Keloharju [10] in Finland and Feng and Seasholes [8] in China. Weber and

Camerer [21] demonstrated that the effect is robust even in experimental studies where

subjects behave in a manner inconsistent with Bayesian updating. Barberis and Xiong [2]

illustrated that under certain parametric assumptions, the prospect theory of Kahneman

and Tversky [13] could explain the disposition effect although under other assumptions the

theory predicts the opposite effect. Other preference-based explanations include emotional

regret by Strahilevitz, Odean and Barber [19] and realization utility by Barberis and Xiong

[3]. In contrast, our belief-based model can be viewed as the “dual” approach to addressing

the disposition effect, analogous to how the dual theory of Yaari [22] addresses violations of

expected utility under individual decision-making.

This paper also fits in the large literature on heterogeneous beliefs. Harrison and Kreps

[11] introduce belief heterogeneity to obtain speculative pricing. Morris [14] relates belief

heterogeneity with short-term IPO overpricing while Scheinkman and Xiong [17] study het-

erogeneous beliefs with Gaussian learning. Eyster and Piccione [7] consider a model where

agents have incomplete knowledge but are all convinced about their own “theories” about

how states transition. In all these models, risk-neutrality and the absence of short-selling of

the risky security imply that belief heterogeneity generates inflated prices. In contrast, our

model allows for both risk-aversion and short-selling. As a result, the implications on prices

are more subtle as demonstrated in our Gaussian special case.

4

2 The Persistence Representation

Let S be a Polish space and let ⇧ be the set of all probability measures on (S,F). Consider a

simple two-period model S⇥S. We assume that all agents share the same prior belief p 2 ⇧on S at time 0. However, after the realization of some s 2 S at time 1, agents update their

beliefs differently and may have various posterior beliefs about the realization of s 2 S at

time 2. For example, S could represent earnings information about a company where agents

differ in their beliefs about how correlated earnings are over time.

Formally, we model these conditional beliefs as Markov kernels K : S ⇥ F ! [0, 1] that

have p as the invariant measure.

Definition. The Markov kernel K : S ⇥ F ! [0, 1] is p-invariant iff Ks is absolutely

continuous with respect to p and for all A 2 F

p (A) =

Z

S

p (ds)Ks (A)

If K is p-invariant, then each Ks has a density s with respect to p. In what follows,

almost surely (a.s.) always mean almost surely with respect to the measure p.7 We say that

a p-invariant Markov kernel exhibits persistence iff the conditional density of the same state

occurring is greater than unity.

Definition. K exhibits persistence iff s (s) � 1 a.s.

For example, if S is finite, then persistence implies that Ks {s} � p {s} for all s 2 S.

In other words, after observing the realization of s 2 S at time 1, all agents increase their

beliefs about s 2 S occurring again at time 2.

Let K denote the set of all p-invariant kernels that exhibit persistence. Since we are

interested in beliefs that differ only in the degree of persistence they exhibit, we only consider

kernels in K. One extreme example is the constant kernel K 2 K such that Ks = p for all

s 2 S. Thus, K corresponds to believing that realizations of s 2 S in both time periods

are completely independent and identically distributed (i.i.d.). This is an example of zero

persistence.7 This is without loss of generality since we only consider measures that are absolutely continuous with

respect to p.

5

Consider a population with heterogeneous beliefs about persistence. Formally, we model

this as a probability measure µ on K with finite support. Each K 2 K such that µ {K} > 0

represents the belief of an agent (or group of agents with the same belief) in the population.

A persistence representation is a linear one-dimensional parametrization of the degree of

persistence in the population.

Definition. µ has a persistence representation iff there is some K 2 K such that for all

µ {K} > 0, there is some � 2 [0, 1] where a.s.

K = �K + (1� �)K

Under a persistence representation, each agent’s belief K 2 K is characterized by a

parameter � 2 [0, 1] which specifies the level of persistence of S over time. For example, if � =

0, then K = K is the constant kernel and there is no persistence. Note that µ is completely

characterized by a scalar distribution on [0, 1] representing persistence. Hence, beliefs in

the population are heterogeneous only along this dimension measuring the persistence level

of realizations of S. Note that the model is completely agnostic as to what beliefs should

be, that is, the true distribution of states are irrelevant. Moreover, there is no asymmetric

information. For example, an agent with belief K also observes realizations of S in time 1

but believes they are irrelevant and chooses to ignore them.

We end this section with two examples of persistence representations.

Example 1. Let S = {s1, s2, s3}, p =

⇥13 ,

13 ,

13

⇤and K be the identity matrix. Note that

pK = p so K is p-invariant. Also, for all s 2 S,

Ks {s}p {s} = 3 � 1

so K 2 K. In fact, K is the identity kernel representing full persistence. Let K :=

12K +

12K

and

µ : =

1

3

�K +

1

3

�K +

1

3

�K

Thus, µ is a persistence representation with equal masses on three beliefs of increasing

persistence: K, K and K.

6

Example 2 (Gaussian Case). Let S = R and suppose that an agent believes that the joint

distribution on S ⇥ S is Gaussian (or normal). Let p be a Gaussian distribution with mean

m and variance �2. If we let ⌧ be the correlation coefficient, then for s 2 S, Ks is Gaussian

with mean m (1� ⌧) + ⌧s and variance (1� ⌧ 2) �2. Thus,

p (ds0)s (s0) = Ks (ds

0) = ds0

1

�p2⇡ (1� ⌧ 2)

e� (s0�m(1�⌧)�⌧s)2

2(1�⌧2)�2

Note that

p (ds0) = ds01

�p2⇡

e�(s0�m)2

2�2

Hence, for s = s0, we have

s (s) =1p

1� ⌧ 2e(

s�m� )

2 ⌧1+⌧ � 1

iff ⌧ � 0. Thus, K 2 K. Let K = K and µ :=

12�K +

12�K . This represents a population

where half of the agents believe that realizations of S over time are completely uncorrelated

while the other half of agents believe that realizations of S over time are correlated with

correlation ⌧ � 0.

3 Trading Equilibrium

We now consider the trading equilibrium for a population with beliefs distributed according

to µ. Let I ⇢ N be finite and for each i 2 I, let Ki 2 K be such that µ {Ki} > 0. Thus,

each i 2 I represents an agent (or group of agents) with belief Ki 2 K about the realization

of S in both time periods.

Let Z0 denote the set of all bounded and measurable z : S ! R+. We interpret each

z 2 Z0 as a security that gives payoff z (s) if s 2 S is realized. For example, 1 2 Z0 and

we call this risk-free security that pays one unit in every s 2 S a “bond”. Let Z denote the

set of all securities that have non-zero variance.8 We call any z 2 Z a “stock” and note that

1 62 Z.8 That is, the variance of z with respect to the measure p. Note that this is equivalent to requiring that

z is not constant a.s..

7

Fix some stock z 2 Z. Consider a trading equilibrium where at time 0, agents trade

claims to both the stock and bond. We let s0 62 S represent the initial state at time 0 and

S0 := {s0}[S. The price density be given by a measurable function : S0 ! R2. For s 2 S0,

we interpret 0 (s) and 1 (s) as the price densities for the bond and stock respectively. Note

that (s0) is the price vector for both the stock and the bond at time 0.

Let ⇥ denote the set of all measurable functions ✓ : S0 ! R2. Each ✓ 2 ⇥ represents a

trading strategy. For s 2 S0, we interpret ✓0 (s) and ✓1 (s) as the holdings for the bond and

stock respectively. As before, ✓ (s0) represents the initial holdings for both the stock and the

bond at time 0.

Given a price , the pricing functional : ⇥! R is given by

(✓) := (0) · ✓ (0) +Z

S

p (ds) (s) · ✓ (s)

for all ✓ 2 ⇥. Hence, (✓) specifies the total price for executing the trading strategy ✓ 2 ⇥.

We assume that agents have constant endowment 1 in all states. Thus, the budget set given

price is

B ( ) := {✓ 2 ⇥ | (✓) (1)}

Let u be a CARA utility index with constant risk aversion ⇢ > 0 and let � 2 (0, 1) denote

the discount rate.9 We let z := (1, z) denote the asset vector. The utility of an agent with

belief K 2 K is given by the function UK : ⇥! R where

UK (✓) :=

Z

S

p (ds) u (✓ (s0) · z (s)) + �

Z

S

p (ds)

Z

S

Ks (ds0) u (✓ (s) · z (s0))

Thus, an agent with belief K 2 K and chooses trading strategy ✓ 2 ⇥ obtains utility UK (✓).

We say ✓ 2 B ( ) is optimal for K 2 K iff UK (✓) � UK (✓0) for all ✓0 2 B ( ). For i 2 I, we

let ✓i 2 B ( ) be optimal for Ki 2 K.

An allocation is a probability ⌫ on ⇥. We let ( , ⌫) denote the price and allocation pair.

We now define an equilibrium given a stock z and a distribution of beliefs µ as follows.9 The CARA utility index is given by u (x) = �e

�⇢x for some ⇢ > 0.

8

Definition. ( , ⌫) is an equilibrium for (z, µ) iff ⌫ {✓i} = µ {Ki} for all i 2 I and a.s.

X

i2I

⌫�✓i ✓i (s) = 1

Thus, in an equilibrium, all agents elect their optimal strategies and all claim markets

clear. Note that since µ has finite support, ⌫ must also have finite support on ⇥.

Given a price , we can consider the normalized price density ¯ : S0 ! R such that for

all s 2 S0

¯ (s) := 1 (s)

0 (s)

The normalized price density gives the relative price of the stock to the bond in every

realization of s 2 S.10 We interpret this as the interest-adjusted price of the stock.

We now address the behavioral characteristics of an equilibrium. Let ⌫ be a complete

binary relation on I.

Definition. An equilibrium ( , ⌫) is ordered by ⌫ iff a.s. the following are equivalent:

(1) i ⌫ j

(2) ¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s)

We say an equilibrium is ordered iff it is ordered by some ⌫. To illustrate this definition,

consider two agents i and j such that i ⌫ j. Now, in all states where the normalized price

decreases (increases), agent i has greater (less) stock holdings than agent j. In other words,

agent i exhibits a greater disposition effect than agent j. If an equilibrium is ordered, then

this ranking on I is complete. In other words, all agents can be ranked according to the

degree of disposition effect that they exhibit. Note that this is a behavioral characterization

of the equilibrium that is completely observable.

Theorem 1 below asserts that every µ with a persistence representation has an ordered

equilibrium.

Theorem 1. If µ has a persistence representation, then any equilibrium of (z, µ) is ordered

by some ⌫. Moreover, �i �j iff i ⌫ j for all {i, j} ⇢ I.

Proof. See Appendix.10 Lemma A1 in the Appendix ensures that we can always define without loss of generality.

9

In an equilibrium where µ has a persistence representation, agents who exhibit the great-

est disposition effect are exactly those who have beliefs that exhibit the least persistence.

This is because at time 0, all agents share the same prior on S so they all hold the same

amount of the stock. In states where the price of the stock increases (decreases), the agent

with the least persistent belief holds the most (least) amount of stock in equilibrium. Market

clearing ensures that this agent exhibits the disposition effect.

On the other hand, agents with beliefs that have the greatest persistence exhibit the

opposite of the disposition effect. They increase stock holdings when prices rise and de-

crease holdings when prices fall. Thus, they trade based on stock price momentum. Note

that this is exactly the house-money effect if we consider the limit case where prices are

constant.11 Agents who believe in any state persistence increase holdings of the stock after

“good” realizations of s 2 S and decrease their holdings after “bad” realizations.

4 Special Case: Gaussian Beliefs

In this section, we consider a special case where beliefs are Gaussian. Let S = R and assume

that p is Gaussian with mean m and variance �2 > 0. Recall from Example 2 that if we let

⌧ 2 (�1, 1) be the correlation coefficient between the two periods, then for every s 2 S, Ks

is Gaussian with mean m (1� ⌧) + ⌧s and variance (1� ⌧ 2) �2.

Definition. K 2 K is Gaussian iff there is some ⌧ � 0 such that Ks is a Gaussian distribu-

tion with mean m (1� ⌧) + ⌧s and variance (1� ⌧ 2) �2.

Note that the constant kernel K 2 K is Gaussian with ⌧ = 0. Suppose that the measure

µ only puts strictly positive mass on the constant kernel K and some other Gaussian kernel

K 2 K. We call such a µ simple Gaussian.

Definition. µ is simple Gaussian iff there is some Gaussian K 2 K and ↵ 2 [0, 1] such that

µ = (1� ↵) �K + ↵�K

11 For example, by assuming that µ {K} ! 1.

10

Thus, a population with a simple Gaussian µ consists of agents (↵ proportion) who

believe in that S is correlated with coefficient ⌧ while the rest (1�↵ proportion) believe that

there is no correlation. Note that µ is trivially a persistence representation. Moreover, since

mixtures of Gaussian distributions are in general not Gaussian, any µ that has a persistence

representation and only puts weight on Gaussian beliefs must be simple Gaussian.

Now, suppose the stock is given by z (s) = s. In other words, payoffs are increasing in

s 2 S. The Proposition below characterizes the equilibrium prices and strategies for the

stock.

Proposition 1. Let µ be simple Gaussian and ✓ be the equilibrium strategy for K 2 K.

Then for all s 2 S,

¯ (s) =↵⌧ (s� (1� ⌧)m) + (1� ⌧ 2) (m� ⇢�2

)

1� (1� ↵) ⌧ 2

✓1 (s) =⇢�2

+ (s�m) (1� ↵) ⌧

⇢ (1� (1� ↵) ⌧ 2) �2

Proof. See Appendix.

This immediately implies the following corollary below.

Corollary 1. Let µ be simple Gaussian and ✓ be the equilibrium strategy for K 2 K. Then

the following are equivalent for all s 2 S.

(1) s � m� ⌧⇢�2

(2) ¯ (s) � ¯ (s0)

(3) ✓1 (s) ✓1 (s0)

Proof. See Appendix.

Corollary 1 provides a very precise illustration of Theorem 1. A realization s 2 S at time

1 is considered “good” for the stock iff s � m� ⌧⇢�2 and the price density increases. In this

case, agents who believe that is no persistence (⌧ = 0) decrease their holdings of the stock.

Vice-versa, a realization s 2 S at time 1 is considered “bad” for the stock iff s m�⌧⇢�2, the

price density decreases and the agents who believe in no persistence increase their holdings.

This is exactly the disposition effect in this special case with Gaussian beliefs.

Corollary 2 below summarizes some comparative statistics of pricing in this model.

11

Corollary 2. If µ is simple Gaussian, then

(1) ¯ is increasing in ↵ iff s � m� ⌧⇢�2

(2) ¯ is decreasing in ⇢

Proof. Follows directly from Proposition 1.

Thus, as more agents believe in persistence, the stock prices increase under “good” states

and decrease under “bad” states. In other words, increasing the relative mass of agents in

the population who believe in persistence results in prices that are more dispersed. This is in

contrast to many other models of heterogeneous beliefs where introducing belief heterogeneity

uniformly increases prices. In our model, risk aversion and the absence of any short-sale

constraints result in a more subtle interaction between belief heterogeneity and prices. Note

that on the other hand, increasing risk aversion uniformly lowers stock prices.

5 Characterization and Uniqueness

In this section, we consider the uniqueness properties of persistence representations. First,

note that we can also view each kernel K 2 K as an operator K : ⇧ ! ⇧ where for any

q 2 ⇧, K (q) 2 ⇧ is the measure that satisfies

(K (q)) (A) =

Z

S

q (ds)Ks (A)

for all A 2 F . Since K is p-invariant, p is a fixed point of this operator. We say K is generic

iff the operator is injective. If S is finite, then this is equivalent to requiring that all Ks are

linearly independent. Thus, in this case, an agent with a generic kernel possesses conditional

beliefs with enough persistence that they span the entire probability simplex. For example,

the matrix corresponding to the kernel K in Example 1 is invertible so K is generic. We say

µ is generic iff it puts strictly positive mass on some generic kernel.

Definition. µ is generic iff µ {K} > 0 for some generic K 2 K.

12

If µ is generic, then there is at least some non-trivial proportion of agents in the population

we exhibit enough variation and persistence in beliefs. The following is an example of a non-

generic µ.

Example 3. Let S = {s1, s2, s3}, p =

⇥13 ,

13 ,

13

⇤and

K :=

2

664

23

16

16

13

13

13

0

12

12

3

775

Note that pK = p so K is p-invariant. Also, for all s 2 S,

Ks {s}p {s} 2

⇢2, 1,

3

2

�

so K is persistent and K 2 K. However, note that K is not an invertible matrix so it is not

generic. This is because an agent with belief K does not update her beliefs if s2 occurs. If

we let

µ : =

1

2

�K +

1

2

�K

then µ is not generic.

Fix some z 2 Z and consider the equilibrium ( , ⌫) for some (z, µ). Clearly, there are

cases where we can find some other µ0 such that ( , ⌫) is also the equilibrium for (z, µ0). In

other words, µ is not unique given its equilibrium. However, suppose we were able to vary

the security z and observe the equilibria for (z, µ) for each z 2 Z. We say all the equilibria

of µ are ordered by some ⌫ iff ⌫ ranks all agents by the degree of disposition effect they

exhibit for all z 2 Z.

Definition. The equilibria of µ are ordered by ⌫ iff a.s. for all z 2 Z, the following are

equivalent:

(1) i ⌫ j

(2) ¯ (0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s)

13

As before, we say that equilibria of µ are ordered iff it is ordered by some ⌫. Theorem 2

below asserts that persistence representations are the only representations that ensure that

all equilibria of µ are ordered by the same ⌫.

Theorem 2. The equilibria of a generic µ are ordered iff µ has a persistence representation.

Proof. See Appendix.

Thus, if µ is generic, then observing a consistent disposition ranking ⌫ for all equilibria

completely characterizes persistence representations. For any other representation of µ,

we can find some z 2 Z such that the disposition ranking is violated. This allows us to

completely identify the unobservable µ with observable characteristics of equilibrium prices

and strategies. By varying the stock payoffs z 2 Z, we can uniquely pin down the distribution

of beliefs in the population.

6 Conclusion

We introduce a model where beliefs are heterogeneous along a single dimension measuring

persistence. In equilibrium, agents can all be ordered by the degree of disposition effect they

exhibit. In particular, those who hold the most persistent beliefs exhibit the disposition

effect while those who hold the least persistent beliefs engage in momentum trading (a form

of the house-money effect). Although we only consider a simple two-period trading model,

our results could be generalized to a dynamic infinite-period setup with Markov trading

strategies.

14

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15

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16

Appendix A

In this appendix, we prove the main results for the model.

Lemma (A1). If ( , ⌫) is an equilibrium for (z, µ), then 0 (s0) > 0 and 0 (s) > 0 a.s.

Proof. Consider K 2 K such that µ {K} > 0 and let ✓ 2 ⇥ be optimal for K. First,suppose 0 (s0) 0 and consider ˆ✓ 2 ⇥ such that ˆ✓ (s) = ✓ (s) for all s 2 S and ˆ✓ (s0) =

(✓0 (s0) + ", ✓1 (s0)) for some " > 0. Now,

⇣ˆ✓⌘= 0 (s0) "+ (✓) (1)

so ˆ✓ 2 B ( ). Since u is CARA,

UK

⇣ˆ✓⌘� UK (✓) =

Z

S

p (ds) [u (✓ (s0) · z (s) + ")� u (✓ (s0) · z (s))]

=

�e�⇢" � 1

� Z

S

p (ds) u (✓ (s0) · z (s))

Since z is bounded, UK

⇣ˆ✓⌘

> UK (✓) contradicting the fact that ✓ is optimal. Hence, 0 (s0) > 0.

DefineE := {s 2 S | 0 (s) 0}

and suppose p (E) > 0. Let ˆ✓ 2 B ( ) be such that ˆ✓ (s) = ✓ (s) if s 2 {s0} [ (S\E) andˆ✓ (s) = (✓0 (s) + ", ✓1 (s)) if s 2 E for some " > 0. Now,

⇣ˆ✓⌘= (✓) + "

Z

E

p (ds) 0 (s) (1)

so ˆ✓ 2 B ( ). Again, as u is CARA,

UK

⇣ˆ✓⌘� UK (✓) = �

�e�⇢" � 1

� Z

E

p (ds)

Z

S

Ks (ds0) u (✓ (s) · z (s0))

Let⇣ (s) :=

Z

S

Ks (ds0) u (✓ (s) · z (s0))

and note that since z is bounded, ⇣ (s) < 0 for all s 2 S. For ⌘ > 0, let

E⌘ := {s 2 E | ⇣ (s) < �⌘}

17

Now, for all ⌘ > 0, Z

E

p (ds) ⇣ (s) �p (E⌘) ⌘

Suppose p (E⌘) = 0 for all ⌘ > 0. As ⌘ ! 0, E⌘ % E so p (E⌘) ! p (E) > 0 a contradiction.Thus, 9⌘ > 0 such that p (E⌘) > 0 so

REp (ds) ⇣ (s) < 0. Hence, UK (b) > UK (a) again

contradicting the optimality of ✓. We thus have 0 (s) > 0 a.s..

Lemma A1 ensures that we can define the normalized price density ¯ : S0 ! R such that

¯ (s) := 1 (s)

0 (s)

For ease of notation, we let Eq denote the expectation operator with respect to the measure

q 2 ⇧. For q = Kis, we let Ei

s := EKis . Let ⇧0 be the set of probability measures on S

absolutely continuous with respect to p. Note that by definition, K 2 K implies Ks 2 ⇧0

for all s 2 S. Fix z 2 Z and let ⇠ : ⇧0 ⇥ R ! R be such that

⇠ (q, a) :=Eq

[u (az) z]

Eq[u (az)]

Note that since z is bounded and u is CARA, Eq[u (az)] < 0 so ⇠ is well-defined.

Lemma (A2). Fix z 2 Z.

(1) ⇠ is strictly decreasing in a

(2) lima!1 ⇠ (q, a) = sups2S z (s) and lima!�1 ⇠ (q, a) = infs2S z (s)

(3) If ⇠ (q, a) = ⇠ (r, a), then ⇠ (�q + (1� �) r, a) = ⇠ (q, a) for all � 2 [0, 1].

Proof. Fix z 2 Z. We prove the lemma in order.

(1) Since u is CARA,@u (az)

@a= u0

(az) z = �⇢u (az) z

As z is bounded, by Theorem 16.8 of Billingsley [4], ⇠ is differentiable in a and

@⇠

@a= �⇢E

q[u (az) z2]

Eq[u (az)]

+ ⇢

✓Eq

[u (az) z]

Eq[u (az)]

◆2

For a 2 R, define qa 2 ⇧ such that

qa (A) :=Eq

[1Au (az)]

Eq[u (az)]

18

for all A 2 F . Now,

@⇠

@a= ⇢ (Eqa

[z])2 � ⇢Eqa⇥z2⇤= �⇢Eqa

⇥(z � Eqa

[z])2⇤ 0

If the last inequality is an equality, then z = Eqa[z] a constant qa-a.s.. Note that since

q 2 ⇧0, p (A) = 0 implies q (A) = 0 which implies qa (A) = 0. Hence, qa 2 ⇧0 for alla 2 R so z is non-constant qa-a.s.. Thus, ⇠ must be strictly decreasing in a.

(2) Since z is measurable, we can approximate z by a sequence of increasing simple func-tions. In other words, z = limn zn where the zn are increasing and

zn =

X

t

cnt 1Ant

for Ant 2 F . Now,

Eq[u (azn) zn]

Eq[u (azn)]

=

Pt q (A

nt ) u (ac

nt ) c

ntP

t q (Ant ) u (ac

nt )

=

Pt q (A

nt ) e

�⇢acnt cntPt q (A

nt ) e

�⇢acnt

Clearly, if cnt = sups2S zn (s), then

lim

a!1

Eq[u (azn) zn]

Eq[u (azn)]

= cnt = sup

s2Szn (s)

By dominated convergence (see Theorem I.4.16 of Cinlar [5]),

⇠ (q, a) =Eq

[u (az) z]

Eq[u (az)]

= lim

n

Eq[u (azn) zn]

Eq[u (azn)]

Hence,

lim

a!1⇠ (q, a) = lim

nlim

a!1

Eq[u (azn) zn]

Eq[u (azn)]

= lim

nsup

s2Szn (s)

= sup

s2Slim

nzn (s) = sup

s2Sz (s)

The case for lim↵!�1 ⇠ (q,↵) = infs2S z (s) is symmetric.

(3) Suppose ⇠ (q, a) = ⇠ (r, a) = ⇠⇤ for {r, q} ⇢ ⇧0 and a 2 R. Thus,

Eq[u (az) (z � ⇠⇤)] = Er

[u (az) (z � ⇠⇤)] = 0

19

If we let q� := �q + (1� �) r for � 2 [0, 1], then

Eq�[u (az) (z � ⇠⇤)] = 0

which implies

⇠⇤ =Eq�

[u (az) z]

Eq�[u (az)]

= ⇠ (q�, a)

Lemma (A3). Let qj = �qi + (1� �) qk for some � 2 (0, 1), and suppose

⇠�qk, ak

�= ⇠

�qj, aj

�= ⇠

�qi, ai

�

Then ak = ai implies ak = aj = ai and ak > ai implies ak > aj > ai.

Proof. Let⇠⇤ := ⇠

�qk, ak

�= ⇠

�qj, aj

�= ⇠

�qi, ai

�

First, suppose a := ai = ak. Thus, from Lemma A2, ⇠ (qj, a) = ⇠⇤ and a = aj.Now, assume ak > ai. Again, by Lemma A2,

⇠�qk, ai

�> ⇠⇤ > ⇠

�qi, ak

�

Suppose aj > ak so ⇠�qj, ak

�> ⇠⇤. By continuity, we can find some � 2 (0, 1) such that

⇠⇤ = ⇠��qj + (1� �) qi, ak

�= ⇠

�qk, ak

�

Now, if we let � :=

��+(1��)(1��) , then

���qj + (1� �) qi

�+ (1� �) qk = �qi + (1� �) qk = qj

Thus, by Lemma A2, ⇠⇤ = ⇠�qj, ak

�which implies aj = ak a contradiction. The case for

ai > aj is symmetric, so we have ak � aj � ai. Lastly, suppose a := ak = ↵j so by LemmaA2,

⇠⇤ = ⇠�qi, ai

�> ⇠

�qi, a

�=

Eqi[u (az) z]

Eqi[u (az)]

20

Thus,

Eqi[u (az) (z � ⇠⇤)] > 0 = Eqk

[u (az) (z � ⇠⇤)]

= Eqj[u (az) (z � ⇠⇤)]

However, qj = �qi + (1� �) qk for � 2 (0, 1) yielding a contradiction. The case for ai = aj

is symmetric, so ak > aj > ai.

Theorem (A4). If µ has a persistence representation, then any equilibrium of (z, µ) isordered by some ⌫. Moreover, �i �j iff i ⌫ j for all {i, j} ⇢ I.

Proof. Let µ have a persistence representation and ( , ⌫) be an equilibrium for (z, µ). Fromthe optimality conditions and the fact that u is CARA, we have for all i 2 I,

¯ (s0) = 1 (s0)

0 (s0)=

Ep[u0

(✓i (s0) · z) z]Ep

[u0(✓i (s0) · z)]

=

Ep[u (✓i1 (s0) z) z]

Ep[u (✓i1 (s0) z)]

= ⇠�p, ✓i1 (s0)

�

Thus, by Lemma A2, ✓i1 (s0) is the same for all i 2 I so by market clearing, ✓i1 (s0) = 1 forall i 2 I. Note that by similar reasoning, we have a.s.

¯ (s) = 1 (s)

0 (s)=

Eis [u (✓

i1 (s) z) z]

Eis [u (✓

i1 (s) z)]

= ⇠�Ki

s, ✓i1 (s)

�

Define i ⌫ j iff �i �j iff i j for all {i, j} ⇢ I. We show that (z, µ) is ordered by ⌫.Let {1, n} ⇢ I be such that 1 i n for all i 2 I. Let

E :=

�s 2 S

�� ¯ (s0) � ¯ (s) and ✓11 (s) < 1

and suppose p (E) > 0. For s 2 E, the Lemma A2 ensures that we can always find someas 2 R such that ⇠ (p, as) =

¯ (s). Note that we can assume ¯ (s) = ⇠ (Kis, ✓

i1 (s)) for all

i 2 I without loss of generality, so

⇠�p, ✓11 (s)

�> ⇠ (p, 1) = ¯ (0) � ¯ (s) = ⇠ (p, as) = ⇠

�K1

s , ✓11 (s)

�= ⇠ (Kn

s , ✓n1 (s))

By Lemma A2 and A3, we have as � 1 > ✓11 (s) so as > ✓11 (s) > ✓n1 (s) as K1s = �1p +

(1� �1)Kns . Since this is true for all s 2 E, we have 1 > ✓i1 (s) for all i 2 I on a set

of strictly positive measure, contradicting market clearing. Thus, ¯ (s0) � ¯ (s) implies

21

✓11 (s) � 1 which implies ✓i1 (s) � ✓j1 (s) for j � i a.s.. By symmetric argument, we have¯ (s0) ¯ (s) implies ✓i1 (s) ✓j1 (s) for j � i a.s.. Thus, ¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s) forj � i a.s. so (z, µ) is ordered by ⌫.

Lemma (A5). Suppose Ls = �p + (1� �)Ks a.s. for {K,L} ⇢ K and � 2 (0, 1). Then K

is generic iff L is generic.

Proof. First, suppose K is generic. Let {r, q} ⇢ ⇧ be such thatZ

S

q (ds)Ls (A) =

Z

S

r (ds)Ls (A)

for all A 2 F . Now,Z

S

q (ds) (�p (A) + (1� �)Ks (A)) =

Z

S

r (ds) (�p (A) + (1� �)Ks (A))Z

S

q (ds)Ks (A) =

Z

S

r (ds)Ks (A)

implying q = r so L is generic. If L is generic, then let {r, q} ⇢ ⇧ be such thatZ

S

q (ds)Ks (A) =

Z

S

r (ds)Ks (A)Z

S

q (ds) (�p (A) + (1� �)Ks (A)) =

Z

S

r (ds) (�p (A) + (1� �)Ks (A))

for all A 2 F . Thus, r = q and K is generic.

For {q, r} ⇢ Rd where d 2 N, define

[q, r] :=�q�+ (1� �) r 2 Rd

�� � 2 [0, 1]

Similarly, for {q, r} ⇢ ⇧0, define

[q, r] := {q�+ (1� �) r 2 ⇧0 | � 2 [0, 1]}

Lemma (A6). If µ has a persistence representation, then its equilibria are ordered.

Proof. Suppose µ has a persistence representation, and for any z 2 Z, define Ez ⇢ S suchthat s 2 Ez iff i ⌫ j is equivalent to ¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s). By Theorem A4,p (Ez) = 1. Now, let ¯S ⇢ S be the set such that s 2 ¯S iff for all z 2 Z, i ⌫ j is equivalent

22

to ¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s). Note that

¯S =

\

z2Z

Ez

Let Z⇤ ⇢ Z be some dense countable subset of Z so

\

z2Z

Ez ⇢\

z2Z⇤

Ez

Now, let s 2 Tz2Z⇤ Ez. Thus, for all z 2 Z⇤, i ⌫ j is equivalent to ¯ (s0) � ¯ (s) iff

✓i1 (s) � ✓j1 (s). By the continuity of prices and holdings, we have i ⌫ j is equivalent to¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s) for all z 2 Z. Thus, s 2 ¯S so ¯S is measurable. Hence,

p�¯S�= p

\

z2Z⇤

Ez

!= 1

Theorem (A7). The equilibria of a generic µ are ordered iff µ has a persistence represen-tation.

Proof. Note that necessity follows from Lemma A6, so we prove sufficiency. Let µ be genericwith equilibria ordered by ⌫. Without loss of generality, let 1 ⌫ i � i+ 1 ⌫ n for all i 2 I.Define ¯S ⇢ S to be the a.s. set such that s 2 ¯S iff for all z 2 Z, i ⌫ j is equivalent to¯ (s0) � ¯ (s) iff ✓i1 (s) � ✓j1 (s).

Fix s 2 ¯S and first consider i � j � k for {i, j, k} ⇢ I. Let {A1, . . . , Ad} be a measurablepartition of S for some d 2 N and c 2 Rd

+. Let z =

Pt 1Atct for t 2 {1, . . . , d}. Note

that ✓i1 (s) � ✓j1 (s) � ✓k1 (s) or ✓i1 (s) ✓j1 (s) ✓k1 (s). Assume the former without loss ofgenerality. Let qi := Ki

s for all i 2 I and note that also without loss of generality

¯ (s) = ⇠�qi, ✓i1 (s)

�= ⇠

�qj, ✓j1 (s)

�= ⇠

�qk, ✓k1 (s)

�

If we let ac := ✓j1 (s) and ⇠c := ¯ (s), then by the lemma above,

⇠�qi, ac

� ⇠c = ⇠�qj, ac

� ⇠�qk, ac

�

Thus,

Eqi[u (acz) (z � ⇠c)] � 0 = Eqj

[u (acz) (z � ⇠c)] � Eqk[u (acz) (z � ⇠c)]

23

If we let vc (t) = u (acct) (ct � ⇠c) for all t 2 {1, . . . ,m}, then qi · vc � qj · vc � qk · vc where{qi, vc} ⇢ Rm.

For c = 1t,qi · vc = qi (t) u (ac) (1� ⇠z) +

�1� qi (t)

�u (0) (�⇠z)

If qi (t) = qk (t), then qj (t) = qi (t). Otherwise, we can find a t0 2 {1, . . . , d} such thatqi (t) > qk (t) and qi (t0) < qk (t0) without loss of generality. By continuity, there is somec = �1t + (1� �)1t0 such that qi · vc = qj · vc = qk · vc. Note that we can always find m� 2

such c where vc are linearly independent. SinceP

t qi(t) = 1, we must have qj 2 ⇥qi, qk⇤.

Since this is true for all {i, j, k} ⇢ I, we must have qi 2 [q1, qn] for all i 2 I. If p 62 [q1, qn],we can find some c such that ⇠c � ¯ (s0) and Eqi

[u (az) (z � ⇠c)] � Ep[u (az) (z � ⇠c)] for all

i 2 I where ⇠ (p, a) = ⇠c. Hence,

⇠�qi, a

� ⇠ (p, a) = ⇠�qi, ✓i1 (s)

�= ⇠c � ¯ (s0) = ⇠ (p, 1)

Thus, ✓i1 (s) a 1 contradicting market clearing. Hence, p 2 [q1, qn].Suppose qj 62 ⇥qi, qk⇤. By a standard Separating Hyperplane Theorem (see Theorem 5.61

of Aliprantis and Border [1]), we can find some measurable ⇣ : S ! R such that

Eqj[⇣] 62

hEqi

[⇣] , Eqk[⇣]i

Since ⇣ is measurable, it is the limit of a sequence of increasing simple functions. Hence,⇣ = liml ⇣l where

⇣l =X

t

blt1Alt

Now, for each ⇣l, we have

Eqj[⇣l] =

X

t

bltqj�Al

t

�= �

X

t

bltqi�Al

t

�+ (1� �)

X

t

bltqk�Al

t

�

so Eqj[⇣l] 2

hEqi

[⇣l] , Eqk[⇣l]i

for all ⇣l. By monotone convergence, we must have Eqj[⇣] 2

hEqi

[⇣] , Eqk[⇣]i

a contradiction. By similar argument, p 2 [q1, qn].Thus, we have p 2 [K1

s , Kns ] a.s.. Hence, we have a.s.

p (ds0) = �sK1s (ds

0) + �sK

ns (ds0)

= p (ds0)1s (s0)�s + p (ds0)ns (s

0) (1� �s)

24

Hence, we have 1s (s)�s + ns (s) (1� �s) = 1 a.s.. Since {K1, Kn} ⇢ K we have 1s (s) � 1

and ns (s) � 1 so �s 2 {0, 1} a.s.Now, consider Ki 2 [p,Kn

] so for all A 2 F ,

p (A) =

Z

S

p (ds)Kis (A) =

Z

S

p (ds)��isp (A) +

�1� �is

�Kn

s (A)�

= p (A)

Z

S

p (ds)�is + p (A)�Z

S

p (ds)Kns (A)�is

Note that if �is = 0 a.s. then Kis = p a.s.. Hence, suppose the �is > 0 on some set of strictly

positive p-measure. We can now define pi 2 ⇧ such that

pi (A) :=

Z

A

p (ds)�isRSp (ds)�is

sop (A) =

Z

S

pi (ds)Kns (A) =

Z

S

p (ds)Kns (A)

Since µ is generic, by Lemma A5, Kn is generic. Hence, p = pi so by the Radon-NikodymTheorem (Theorem I.5.11 of Cinlar [5]), we have

1 =

dpi

dp=

�isRSp (ds)�is

a.s. so �is = �i a.s.. The case for Ki 2 [p,K1] is symmetric, so we have �is = �i for all i 2 I.

Thus, µ has a persistence representation.

Appendix B

In this appendix, we prove the results for Gaussian model. Let z 2 Z be such that z (s) = s.

Lemma (B1). Let q 2 ⇧ be Gaussian distributed with mean m and variance �2 > 0. Then

Eq[u (az)] = u

✓ma� ⇢

�2

2

a2◆

Eq[u (az) z] = u

✓ma� ⇢

�2

2

a2◆�

m� a⇢�2�

25

Proof. A simple computation yields

Eq[u (az)] = �e

�⇢a⇣m�a⇢�2

2

⌘

Eq[u (az) z] = �e

�⇢a⇣m�a⇢�2

2

⌘ �m� a⇢�2

�

The result then follows from the definition of u.

Proposition (B2). Let µ be simple Gaussian and ✓ be the equilibrium strategy for K 2 K.Then for all s 2 S,

¯ (s) =↵⌧ (s� (1� ⌧)m) + (1� ⌧ 2) (m� ⇢�2

)

1� (1� ↵) ⌧ 2

✓1 (s) =⇢�2

+ (s�m) (1� ↵) ⌧

⇢ (1� (1� ↵) ⌧ 2) �2

Proof. Let K 2 K be Gaussian and ✓ be optimal for K. Fix s 2 S and let a = ✓1 (s). Now,by Lemma B1,

¯ (s) = ⇠ (Ks, a) =EKs

[u (az) z]

EKs[u (az)]

= m (1� ⌧) + ⌧s� a⇢�1� ⌧ 2

��2

Thus, we have

✓1 (s) = a =

m (1� ⌧) + ⌧s� ¯ (s)

⇢ (1� ⌧ 2) �2

Note that if ⌧ = 0, then ✓1 (s) = m� (s)⇢�2 . Since ✓1 (s0) = 1 by market clearing, we have

¯ (s0) = m� ⇢�2

Also by market clearing,

1 = ↵✓⌧1 (s) + (1� ↵) ✓01 (s) = ↵m (1� ⌧) + ⌧s� ¯ (s)

⇢ (1� ⌧ 2) �2+ (1� ↵)

m� ¯ (s)

⇢�2

Hence¯ (s) =

↵⌧ (s� (1� ⌧)m) + (1� ⌧ 2) (m� ⇢�2)

1� (1� ↵) ⌧ 2

Substituting the formula for ¯ yields

✓1 (s) =⇢�2

+ (s�m) (1� ↵) ⌧

⇢ (1� (1� ↵) ⌧ 2) �2

26

Corollary (B3). Let µ be simple Gaussian and ✓ be the equilibrium strategy for K 2 K.Then the following are equivalent for all s 2 S

(1) s � m� ⌧⇢�2

(2) ¯ (s) � ¯ (s0)

(3) ✓1 (s) ✓1 (s0)

Proof. From Proposition B2, we have ¯ (s) � ¯ (s0) iff

↵⌧ (s� (1� ⌧)m) +

�1� ⌧ 2

� �m� ⇢�2

� � �1� (1� ↵) ⌧ 2� �

m� ⇢�2�

s � m� ⌧⇢�2

so (1) and (2) are equivalent. Since it also follows readily that ✓1 (s) � 1 iff s � m � ⌧⇢�2

we have (1), (2) and (3) are all equivalent.

27