The Survival and Extinction of Overconfident Agents

Steven D. Bakersdbaker@cmu.edu

August 31, 2009

Abstract

I study the survival of an irrationally overconfident agent, who competes with a rational agent in astochastic growth economy subject to a short-sale constraint. I find that the constraint increases theoverconfident agent’s chance of survival, often providing an infinite relative improvement asymptotically.The magnitude of the effect increases with the level of overconfidence, and becomes stronger as the sourceof economic risk shifts from the growth process to direct innovations in the dividend process. As theoverconfident agent loses his endowment, the constraint binds more often, altering the interest rate andhis price of risk. A survival index is constructed, which links overconfidence with survival effects due toimpatience and risk aversion.

1 Introduction

This paper studies the survival of an irrationally overconfident agent who competes with a rational agent.The overconfident agent is certain that a signal carries information about future economic growth, when infact it does not. He is irrational in the sense that he never learns that the signal is meaningless. Both agentsare otherwise fully rational Bayesian optimizers of expected intertemporal utility.

Trading takes place in continuous time in a pure exchange economy with a single consumption good. Agents,who have heterogeneous CRRA utility and impatience, optimize over an infinite horizon. The aggregatedividend process and spurious signal are exogenous. Prices, the interest rate, and other aspects of marketbehavior are determined endogenously via market clearing and the characteristics of the agents.

Survival is investigated in markets where short-sales are allowed, and in those where they are not. Tradersoften face impediments to short-selling, for a variety of reasons. Regulatory bodies such as the Federal Reserveand the SEC have at times restricted short-sales, and individual funds may be contractually prohibited fromshort-selling. In some cases it may be difficult to sell short even absent restrictions. My implementation ofthe short-sale constraint is based upon Gallmeyer and Hollifield [2008], which also provides a more detailedreview of the topic. In the constrained setting, analysis is conducted for logarithmic utility only.

Of the many behavioral phenomena invoked as possible explanations for anomalies in financial markets, over-confidence has the advantage of being a widespread and well documented psychological trait, with possibleevolutionary advantages explaining its apparent prevalence and persistence. For a survey of overconfidencein connection with finance, see for example Daniel and Titman [2000]. Overconfidence is also able to ex-plain several well known financial phenomena at a stroke. Harrison and Kreps [1978] provide an importantearly study in a discrete setting without short-sales, where agents with heterogeneous beliefs equivalent to

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Steven D. Baker

overconfidence are shown to speculate, creating a price bubble. Scheinkman and Xiong [2003] formulatethe problem in continuous time, and argue that high trading volume and volatility may also be explainedby overconfidence. Their work examines the effects of trading costs upon these phenomena, but as budgetconstraints and fully rational traders are absent, the survival of overconfident traders is not considered. Mywork builds upon that of Dumas et al. [2009], who include both fully rational and overconfident agents, withCRRA power utility and budget constraints. However, survival of overconfident agents is only examinedunder one set of parameter assumptions, with homogeneous preferences and impatience. In their param-eterization, overconfident agents remain a significant fraction of the market for long periods (hundreds ofyears).

The survival of the overconfident agents is of critical importance to the validity of the overall theory, sinceotherwise it could be argued that they would quickly be forced from the markets, reducing their priceimpact and perhaps invalidating overconfidence as an explanation for observed speculative phenomena. Ifocus on the survival question under realistic and previously unexamined conditions. Important analysisof survival includes that of Kogan et al. [2006], who examine agents with heterogeneous beliefs and CRRApreferences who consume only in a terminal period. They conclude that agents with incorrect beliefs maydominate the market given appropriate preferences. More surprising, agents with negligible wealth canimpact prices significantly. This effect vanishes when agents have intermediate consumption, as reported byYan [2008]. However, Yan also concludes that disadvantages due to incorrect beliefs are easily overcome bythe effects of heterogeneous preferences. He also introduces a “survival index”, providing an elegant methodof characterizing the asymptotic survival of heterogeneous agents. I extend the analysis of Yan [2008] toinclude stochastic growth and overconfidence, and generalize his survival index.

2 The Model

The model is identical to that of Dumas et al. [2009]. The exogenous aggregate output of the economy δ(t)follows a geometric Brownian motion with dynamics

dδ(t) = δ(t)[f(t)dt + σδdZδ(t)], δ(0) > 0,

⇒ δ(t) = δ(0) exp∫ t

0

(f(s)− 1/2σ2δ )ds + σδZδ(t)

(1)

where Zδ(t) is a standard Brownian motion defined on (Ω,F , Ft,P) with σδ > 0. f , the mean rate ofeconomic growth, follows a mean reverting Ornstein-Uhlenbeck process:

df(t) = −ζ(f(t)− f)dt + σfdZf (t), ζ > 0

⇒ f(t) = e−ζt[f(0)− f + σf

∫ t

0

eζsdZf (s)] + f(2)

In addition, there is a signal that follows

ds(t) = σsdZs(t), (3)

where Zs(t) is a standard Brownian motion independent of Zδ(t), Zf (t). Assume there are two agents(investors), 1 and 2, who have different beliefs regarding future economic growth. Agent 1 correctly perceivess as independent of Zf (t), whereas agent 2 is irrationally “overconfident” that s is correlated with Zf (t) asfollows:

ds(t) = σsφdZf (t) + σs

√1− φ2dZs(t). (4)

Agent 2 always believes φ > 0 and constant. Neither agent observes f , neither are the innovations directlyobservable, but all other aspects of the economic model are assumed common knowledge unless otherwise

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Steven D. Baker

stated. From [Liptser and Shiryaev, 1977, Thm. 12.7, p.33], based on their respective beliefs, the agentsestimate the growth rate as

df1(t) = −ζ(f1(t)− f)dt +ϕ1

σ2δ

(dδ(t)δ(t)

− f1(t)dt

)(5)

⇒ f1(t) = e−ζt[f1(0)− f +ϕ1

σδ

∫ t

0

eζsdW 1δ (s)] + f , (6)

df2(t) = −ζ(f2(t)− f)dt +ϕ2

σ2δ

(dδ(t)δ(t)

− f2(t)dt

)+

φσf

σsds (7)

⇒ f2(t) = e−ζt[f2(0)− f +ϕ2

σδ

∫ t

0

eζsdW 2δ (s) + φσf

∫ t

0

eζsdZs(s)] + f (8)

Above, dW 1δ (t) = 1

σδ(dδ(t)

δ(t) −f1(t)dt) and dW 2δ (t) = 1

σδ(dδ(t)

δ(t) −f2(t)dt) are standard one dimensional Brownianmotions with respect to the beliefs of agents 1 and 2, respectively. The estimated steady state errors of thegrowth rate approximations are

ϕ1 = σ2δ

√ζ2 +

σ2f

σ2δ

− ζ

(9)

ϕ2 = σ2δ

√ζ2 + (1− φ2)

σ2f

σ2δ

− ζ

(10)

Although the real economy is described in terms of three independent Brownian motions, only two randomprocesses are actually observed by either agent. It is convenient to conduct equilibrium analysis in the spaceof observable random processes. See Riedel [2001] for a theoretical justification of this approach , and Dumaset al. [2009] for relevant derivations. Under agent 1’s beliefs, the exogenous aggregate output of the economyand the signal s follow

dδ(t) = δ(t)[f1(t)dt + σδdW 1δ (t)], δ(0) > 0

df1(t) = −ζ(f1(t)− f)dt +ϕ1

σfdW 1

δ (t)

ds(t) = σsdW 1s (t)

(11)

where W 1s is a one dimensional Brownian motion independent of W 1

δ .

3 Equilibrium

Assume there are three linearly independent assets such that markets are dynamically complete (in the sensethat the attainable consumption streams span the space of observable random processes). In particular, Iassume that shares of aggregate dividends are in net supply 1 and tradeable as a stock, whereas other assetsnecessary for market completeness are in net supply 0. Each agent i is endowed at time t = 0 with a fractionof the dividend stream θi, and has time additive CRRA preferences given by

ui(ci(t)) =ci(t)1−γi

1− γi, γi > 0, (12)

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Steven D. Baker

with γi = 1 corresponding to ui(ci(t)) = log ci(t). The first derivative of the utility function and its inverseare given by

u′i(ci(t)) = ci(t)−γi (13)

J(v) = v−1/γi (14)

As shown in Karatzas et al. [1990], an agent’s optimal consumption can be written as ci(t) = Ji(eρityiξi(t)),where ξi is the agent’s state price density, ρi > 0 is a constant reflecting impatience and yi is a constant thatsatisfies

Ei[∫ ∞

0

ξi(t)Ji(eρityiξi(t))dt] = Ei[∫ ∞

0

ξi(t)θiδ(t)dt] = xi. (15)

Above, Ei is expectation with respect to agent i’s beliefs and xi is initial wealth. I will now construct autility function for the representative agent following the approach of Basak [2005]. Let

U(δ(t); λ1,2(t)) = maxc1(t)+c2(t)≤δ(t)

λ1(t)e−ρ1tu1(c1(t)) + λ2(t)e−ρ2tu2(c2(t)) (16)

or equivalently we can normalize the maximization problem by λ1(t) and write

u(δ(t); λ(t)) = maxc1(t)+c2(t)≤δ(t)

e−ρ1tu1(c1(t)) + λ(t)e−ρ2tu2(c2(t)) (17)

where λ(t) = λ2(t)λ1(t)

. Dropping the time parameter for notational simplicity, we have the following Lagrangianand its derivatives

L(c1, c2, δ; λ) = e−ρ1tu1(c1) + λe−ρ2tu2(c2)− µ(c1 + c2 − δ)

µ = e−ρ1tu′1(c1)

µ = λe−ρ2tu′2(c2)

⇒ λ =e−ρ1tu′1(c1)e−ρ2tu

′2(c2)

(18)

Application of the envelope theorem yields

u′(δ; λ) =∂L

∂δ

= e−ρ1tu′1(c1)∂c1

∂δ+ λe−ρ2tu′2(c2)

∂c2

∂δ+ µ

= µ = e−ρ1tu′1(c1) = λe−ρ2tu′2(c2)

(19)

To summarize, we have

λ(t) =e−ρ1tu′1(c1(t))e−ρ2tu′2(c2(t))

=y1ξ1(t)y2ξ2(t)

(20)

Since it is the ratio y1y2

that matters in determining λ(t) and the resulting state price densities, I can choose

y1 = u′(δ(0)) ⇒ ξ1(t) =u′(δ(t))u′(δ(0))

. (21)

Consider time t = 0. Then ξ1(0) = 1 and

λ(0) =u′(δ(0))y2ξ2(0)

⇒ y2 =u′(δ(0))λ(0)ξ2(0)

⇒ ξ2(t) =ξ2(0)λ(0)u′(δ(t))

λ(t)u′(δ(0)).

(22)

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Steven D. Baker

Setting the initial value ξ2(0) = 1 satisfies the above equation to give

ξ2(t) =λ(0)λ(t)

ξ1(t) (23)

It remains to determine the process λ(t). Define the difference of opinion between agents as

g(t) =f1(t)− f2(t)

σδ(24)

The relationship W 2δ (t) = W 1

δ (t) +∫ t

0g(s)ds suggests an application of Girsanov’s theorem to construct a

change of measure between the probability spaces P1,P2 of the two agents. Suppose

η(t) =λ(t)λ(0)

= exp−∫ t

0

g(s)dW 1δ (t)− 1/2

∫ t

0

g2(s)ds

dλ(t) = −λ(t)g(s)dW 1δ (t).

(25)

Then η = η(∞) implements a change of measure from P2 to P1, such that for any event A

E2t [ξ2(t)1A] = E1

t [ηξ2(t)1A] = η(t)ξ2(t)P1(A)

= E1t [ξ1(t)1A] = ξ1(t)P1(A)

⇒ ξ2(t) =1

η(t)ξ1(t) =

λ(0)λ(t)

ξ1(t)

(26)

which establishes that the process for λ(t) is as posited. λ(0) is selected to satisfy the budget constraint (15).Although the preceding discussion characterizes equilibrium, it does not guarantee existence or uniqueness.Indeed, this is difficult to show in the general case; Karatzas et al. [1990] and Yan [2008] provide proofsunder certain conditions. Henceforth I assume the existence of a unique equilibrium.

It is possible to characterize λ(t) in a different way, using the market price of risk (i.e., the instantaneousSharpe ratio). This is defined as

κi(t) =fi(t)− r(t)

σδ(27)

where r(t) is the instantaneous risk free interest rate (excluding the term for impatience). Then the evolutionof lambda is governed by

dλ(t) = −λ(t)σλ(t)dW 1δ (t),

σλ(t) = κ1(t)− κ2(t)(28)

Likewise, the change of measure between agent i’s perceptions and the true economy is given by

dλi(t) = −λi(t)hi(t)dZδ(t)

hi(t) = κ(t)− κi(t), κ(t) =f(t)− r(t)

σδ

⇒ λi(t) = λi(0) exp−1/2∫ t

0

hi(s)2ds−∫ t

0

hi(s)dZδ(s)

(29)

If at present the two definitions of λ(t) seem to contribute nothing more than a surplus of notation, theirusefulness will become apparent when short-sale constraints are introduced, as they will not remain equiva-lent.

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Steven D. Baker

4 Survival

Survival may be defined either in terms of an agent’s asymptotic share of wealth, or his asymptotic shareof consumption. Dumas et al. [2009] argues that consumption share is the more informative statistic in thecontext of this model. Therefore we have the following definition.

Definition 1. An agent’s share of consumption is given by

ωi(t) =ci(t)δ(t)

(30)

Agent i becomes “extinct” iflim

t→∞ωi(t) = 0, a.s., (31)

and “survives” if this does not occur. He “dominates” the market if

limt→∞

ωi(t) = 1, a.s. (32)

In later sections I assume that all agents have log utility. In this case share of wealth and share of consumptionare equivalent, so results for survival may be compared directly with prior work in terms of either wealth orconsumption.

Yan [2008] introduces the concept of a “survival index” that determines which agent survives in the limit.As in this paper, his agents may have heterogeneous patience, risk aversion, and beliefs. However, the focusof Yan’s analysis is an economy with constant growth and constant errors in the agents’ beliefs, with someanalysis of learning given heterogeneous priors. As far as I am aware, the extension of the survival index tooverconfident agents in a stochastic growth economy is novel.

Survival is analyzed with respect to the true economy governed by Zδ(t), such that dWi(t) = dZδ(t) + hi(t).In the absence of a short-sale constraint, hi(t) = f(t)−fi(t)

σδ. Note that both the rational and the overconfident

agent estimate growth correctly on average, i.e., E[hi(t)] = 0. 1 The definition and proposition below helpto characterize survival in this economy.

Definition 2. The survival index for agent i at time t is

Ii(t) =12t

∫ t

0

hi(s)2ds + ρi +γi

t

∫ t

0

f(s)− 1/2σ2δds. (33)

The long-run survival index for agent i is

Ii = limt→∞

Ii = ψi + ρi + γi(f − 1/2σ2δ ) (34)

where ψi is half the steady-state error in agent i’s estimate of economic growth normalized by volatility σδ.For agents with varying degrees of overconfidence (whose unconditional estimates are unbiased), this reducesto half the steady state variance of hi(t) or, equivalently, the volatility of λi(t). 2 I refer to ψi as the beliefpenalty.

1In the case of agent 1, this results from the construction of f1(t), which follows Theorem 12.1 in Liptser and Shiryaev[1977]. While the condition E[f2(t)] = f is intuitively satisfied due to the symmetry of the spurious signal, the fact that f2(t)is misspecified relative to the true model makes a formal proof more difficult. However, the property has been confirmed viasimulation.

2Once again, convergence of V ar(h1) follows from Liptser and Shiryaev [1977] and the resulting process ϕ1(t), whereasconvergence and estimation of V ar(h2) has so far only been established numerically.

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Steven D. Baker

Proposition 1. The agent with the smallest long-run survival index survives asymptotically.

Proof. The proof is adapted from that of Proposition 2 in Yan [2008]. Although it is given with the case ofone rational agent and one overconfident agent in mind, the result is readily generalized to the case of Nagents with more general hi; only the definition of Ii would change. From equation 18,

λ2(t)λ1(t)

=e−ρ1tc1(t)−γ1

e−ρ2tc2(t)−γ2

⇒ c2(t)γ2

c1(t)γ1= e(ρ1−ρ2)t

λ2(t)λ1(t)

⇒ ωγ22 (t)

ωγ11 (t)

= e(ρ1−ρ2)tλ2(t)λ1(t)

δ(t)γ1−γ2

(35)

Substituting the solutions for λi(t) and δ(t) leads to

ωγ22 (t)

ωγ11 (t)

=λ2(0)λ1(0)

δ(0)γ1−γ2 exp(ρ1 − ρ2)t + 1/2∫ t

0

(h1(s)2 − h2(s)2)ds +∫ t

0

(h1(s)− h2(s))dZδ(s)

+ (γ1 − γ2)[∫ t

0

(f(s)− 1/2σ2δ )ds + σδZδ(t)]

=λ2(0)λ1(0)

δ(0)γ1−γ2 exp(I1 − I2)t + (∫ t

0

(h1(s)− h2(s))dZδ(s) + (γ1 − γ2)σδZδ(t))

=λ2(0)λ1(0)

δ(0)γ1−γ2 exp[(I1 − I2) +1t(∫ t

0

(h1(s)− h2(s))dZδ(s) + (γ1 − γ2)σδZδ(t))]t

(36)

Recall that the a law of large numbers implies limt→∞Zδ(t)

t = 0, a.s. (see for example [Karatzas and Shreve,1991, p. 104]). If we assume a bound |hi(t)| < h, ∀i, t on the error agents can reasonably make in estimatingthe growth rate, then

limt→∞

1t|∫ t

0

(h1(s)− h2(s))dZδ(s) + (γ1 − γ2)σδZδ(t)| ≤ (2h + |(γ1 − γ2)σδ|)|Zδ(t)t

| = 0, a.s. (37)

It follows that

limt→∞

(I1 − I2) +1t(∫ t

0

(h1(s)− h2(s))dZδ(s) + (γ1 − γ2)σδZδ(t)) = I1 − I2, a.s. (38)

Therefore if I1 > I2,

limt→∞

ωγ22 (t)

ωγ11 (t)

= limt→∞

λ2(0)λ1(0)

δ(0)γ1−γ2 exp(I1 − I2)t = ∞, a.s. (39)

Recalling that 0 ≤ ωi(t) ≤ 1, the above implies

limt→∞

ω1(t) = 0, a.s. (40)

The equivalent result applies for I1 < I2, implying that the agent with the smallest limiting survival indexsurvives 3, whereas the other agent becomes extinct. If I1 = I2, then both agents survive in the limit.

3In fact, since there are only two agents, the survivor dominates

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Steven D. Baker

4.1 Discussion

The survival index provides a convenient way to evaluate the asymptotic survival of heterogeneous agents.The interaction of heterogeneous impatience and risk aversion with incorrect beliefs is characterized by Yan[2008], and the basic characterization remains unchanged. However, the heterogeneous beliefs arising dueto overconfidence and stochastic growth lead to different finite-horizon dynamics. In addition, the relativemagnitude of the belief effects changes with the introduction of stochastic growth, as does the definition ofrational beliefs: no agent perfectly estimates growth asymptotically if σf > 0. Accordingly, the followingproposition summarizes optimal beliefs in this economy.

Proposition 2. The optimal (minimum mean-square error) estimate of the growth rate f(t) is given byf1(t), with asymptotic error in beliefs characterized by

ψ1 =ϕ1

2σ2δ

=12

√ζ2 +

σ2f

σ2δ

− ζ

, (41)

where ϕ1 is the steady-state error of the optimal filter. Assume agents have identical impatience and riskaversion. Then agent 1 survives. If in addition agents have identical initial growth estimate fi(0) = E[f(0)],then the optimal filter f1(t) is unique, and agent 1 dominates.

Proof. Survival follows from Theorem 12.1 in Liptser and Shiryaev [1977], which establishes optimality off1(t), and Proposition 1, which shows the connection between minimum error growth estimates and survival.Dominance follows from Theorem 12.3 in Liptser and Shiryaev [1977], which guarantees uniqueness of theoptimal filter up to initial conditions.

Although I focus on competition between the rational agent 1 (using the optimal growth estimate) and anoverconfident agent (who uses a misspecified filter), the proposition above establishes survival and dominanceof agent 1 versus any other investor, including (for example) an investor who simply estimates growth asits long run mean f . Also note that the initial growth estimate fi(0) - the source of heterogeneous beliefsconsidered in Yan [2008] and Gallmeyer and Hollifield [2008] - is of secondary importance here. In fact,agent 1 will dominate any agent who does not estimate growth in the form of f1(t); the assumption ofidentical initial conditions merely contains the case where another agent uses the optimal filter up to anirrational estimate of the initial condition. (Knowledge of the distribution of f(0) is a necessary assumptionfor construction of the optimal filter.)

5 Survival Under Short-sale Constraint

To simplify implementation of the short-sale constraint and focus upon its interactions with overconfidenceand stochastic growth, some standing assumptions are made. In particular, all agents are assumed tohave logarithmic utility (γi = 1). Agents also have uniform impatience ρi = 0.1, matching the parameterfrom Dumas et al. [2009], although heterogeneous impatience is explored at the end of this section. Theseassumptions make results easier to interpret, as several quantities become available in closed form.

Suppose we characterize the agents according to their current relative beliefs, calling them o (optimistic) andp (pessimistic), fo(t) ≥ fp(t). Define the difference of opinion g(t) = fo(t)−fp(t)

σδ. Then the representative

agent has the utility maximization problem

u(δ(t); λ(t)) = maxco(t)+cp(t)≤δ(t)

e−ρotuo(co(t)) + λ(t)e−ρptup(cp(t)) (42)

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Steven D. Baker

Gallmeyer and Hollifield [2008] show that, under these assumptions,

dλ(t) = −λ(t)σλ(t)dW oδ , σλ(t) = κo(t)− κp(t), (43)

and κi is given by the table below

Constraint Binds κo(t) κp(t)Yes (1 + λ(t))σδ 0No σδ + λ(t)g(t)

1+λ(t)σδ − g(t)

1+λ(t)

The constraint binds when κp(t) would otherwise become negative. Note that this is more likely to occurwhen agent p’s consumption share is small.

In the previously described economy with one rational and one overconfident agent, numbers 1 and 2,respectively, the optimistic and pessimistic agent fluctuates with the sign of g(t). Suppose f1(t) ≥ f2(t).Then the representative agent’s utility maximization problem corresponds to equation (43) with λ(t) =λ2(t)λ1(t)

= λ(t), o = 1, p = 2. Similarly, if f1(t) ≤ f2(t), then λ(t) = λ1(t)λ2(t)

= 1λ(t) , o = 2, p = 1. When

f1(t) = f2(t), the two problems are equivalent, as the constraint will not bind either agent. Prices of riskand the interest rate (also adapted from Gallmeyer and Hollifield [2008]) are in Table 1.

Constraint Binds κ1(t) κ2(t) r(t)

Agent 1 0(1 + 1

λ(t)

)σδ ρ + f2(t)−

(1 + 1

λ(t)

)σ2

δ

Agent 2 (1 + λ(t))σδ 0 ρ + f1(t)− (1 + λ(t))σ2δ

Neither agent σδ + λ(t)g(t)1+λ(t) σδ − g(t)

1+λ(t) ρ + f1(t)+λ(t)f2(t)λ(t)+1 − σ2

δ

Table 1: Prices of risk and interest rate under short-sale constraint

Note that the last row of Table 1 is identical to the case of the unconstrained economy.

Because the hypothetical omniscient agent (who observes f(t)) does not participate in the economy, he hasno impact upon the interest rate. Hence his price of risk is simply κ(t) = max( f(t)−r(t)

σδ, 0). Recalling that

hi(t) = κ(t)−κi(t) (which does not reduce to the difference of growth rates when short-sales are restricted),we have everything necessary to interpret the survival index under the constraint.

Definition 3. When short-sales are disallowed, the survival index for agent i at time t remains

Ii(t) =12t

∫ t

0

hi(s)2ds + ρi +γi

t

∫ t

0

f(s)− 1/2σ2δds. (44)

The long-run survival index for agent i is

Ii = limt→∞

Ii = ψi(λ) + ρi + γi(f − 1/2σ2δ ) (45)

where ψi(λ) is half the steady-state error of agent i’s estimated price of risk versus the true price of risk. λsignifies the asymptotic distribution of the weighting factor λ(t).

Thus the survival index may in fact depend on whether the agent survives! This conundrum will be examinedin the context of an experimental simulation. To simplify notation, the λ argument is omitted in futurediscussion.

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Steven D. Baker

5.1 Simulation and Baseline Parameters

Monte Carlo simulation is used to evaluate survival in a variety of scenarios. Antithetic variates were foundto reduce error somewhat, and are used in all experiments. Unless otherwise noted, results are generated bysimulating the economy 400,000 times. For terminal period T = 100, integrals are approximated with timestep dt = 0.02. For T = 1000, dt = 0.05. All times are in years. Further details accompany each figure.

Name Symbol ValueParameters from Dumas et al. [2009]Mean growth rate of aggregate endowment f 0.015Volatility of endowment growth rate σf 0.03Volatility of aggregate endowment σδ 0.13Mean reversion parameter ζ 0.2Agent 2’s correlation between signal and growth rate φ 0.95Agent’s initial shares of endowment θ1 = θ2 0.5Initial aggregate dividend δ(0) 1Initial growth expectations f(0) = f1(0) = f2(0) f

Parameters from Yan [2008]Constant growth rate f 0.018Volatility of aggregate endowment σδ 0.032

Table 2: Baseline parameter values and state variables

Table 2 gives the baseline parameters used to assess survival in this paper. They match those used byDumas et al. [2009], which were in turn based on estimates from Brennan and Xia [2001]. For comparison,I have also included the core economic parameters from the constant growth economy of Yan [2008]. Hisconstant growth rate f is most equivalent to my f , to which my experiments are generally insensitive. σδ

may be compared to σδ when considering instantaneous risk, but this comparison fails in the context of theforecast risk that applies over longer periods. Section 5.3 examines the effect of reducing (increasing) σδ

while increasing (reducing) risk from other sources.

5.2 Direct Effects of Overconfidence

What level of overconfidence is “reasonable” is open to speculation. The relatively large baseline valueφ = 0.95 was chosen primarily for consistency with prior work, and because it is large enough to providea convincing “worst case scenario”: if very overconfident agents survive for long periods, then clearly mod-erately overconfident agents will! But how important is the level of overconfidence when short-sales aredisallowed?

It turns out to be less important. Figure 1 shows how the consumption share of the irrational agent evolvesover time. The effect of the short-sale constraint can be dramatic. Under the baseline parameters (φ = 0.95),the unconstrained overconfident agent is nearly extinct after 300 years, but still consumes around 5% of thedividend under the constraint. The impact of the constraint declines as overconfidence is reduced to φ = 0.5,but it is still significant, offering a survival advantage through the terminal period. When agent 2 is onlyslightly overconfident his unconstrained wealth share is nearly indistinguishable from his constrained wealthshare.

The accompanying bar chart of belief penalties (ψ) provides intuition for the results. The leftmost barscorrespond to the rational agent, who has φ = 0 but still has ψ > 0, because perfect estimation of the

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Steven D. Baker

0 0.25 0.5 0.950

0.02

0.04

0.06

0.08

0.1

0.12

0.14

ψ

φ

Belief Penalty vs. Level of Overconfidence

UnconstrainedConstrained

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E[ω2] vs. Time for Varying φ

Time (years)

Sha

re o

f div

iden

ds

φ=0.25 (U)φ=0.5 (U)φ=0.95 (U)φ=0.25 (C)φ=0.5 (C)φ=0.95 (C)

Figure 1: Impact of overconfidence parameter φ.Maximum standard errors over the interval and all parameter combinations are as follows: E[ω2] : 4.23×10−4, ψ : 5.49×10−4.

growth rate is impossible given observable information. As all other components of the survival index areassumed identical for both agents, the agent with the smallest ψ survives. Further, because the wealth ratiois exponential in the survival index, it is the absolute difference ψ1−ψ2 that determines the rate of agent 2’sextinction, not the relative magnitude of the belief penalties. When modification of the baseline parametersis restricted to φ only, introduction of the short-sale constraint reduces a given belief penalty approximatelylinearly, by around 50%. Hence the absolute reduction in ψ is largest when the unconstrained ψ is large,e.g., for φ = 0.95, when agent 2 is very overconfident. When ψ1 − ψ2 is small to begin with (φ = 0.25), theconstraint causes a very small absolute reduction in the penalty difference, and E[ω2] is almost unaffected.

It is worth noting that imposing the short-sale constraint provides a relative increase in agent 2’s consumptionthat grows over time. Figure 2 illustrates this point: the ratio of expected consumptions converges to zerowith time, indicating that the overconfident agent’s consumption is infinitely larger with the constraint thanwithout it. However, when the benefit of the constraint is slight, the ratio may remain close to one for longperiods.

Unconstrained ConstrainedE[u1] -5.27 -5.80E[u2] -11.90 -8.43

Table 3: Expected Discounted Utility through T = 100

To further illustrate who benefits from imposition of the constraint, Table 3 presents the cumulative expecteddiscounted utility for each agent for the first 100 years under the baseline parameters. 4 Although hisattempts to trade are more often frustrated by the constraint than those of agent 1, agent 2 has significantlyhigher expected utility under the constraint. Therefore if agent 2 could be “convinced” of his irrationality(although he remained powerless to alter his behavior), he would prefer to trade in a market with theshort-sale constraint than without it.

4The estimate was computed using 100,000 samples with dt = 0.02. Because the discount factor is substantially larger thanthe growth rate, contributions to cumulative utility for t > 100 are slight.

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Steven D. Baker

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

Ratio of E[ω2] unconstrained/constrained

Time (years)

Unc

onst

rain

ed/C

onst

rain

ed

φ=0.25

φ=0.5

φ=0.95

Figure 2: Relative increase in E[ω2] due to short-sale constraint.Maximum standard error over the interval and all parameter combinations is 4.23× 10−4.

5.3 Sensitivity to Economic Parameters

Reasonable economic parameters are more precisely determined than the level of overconfidence. However itis important to evaluate the impact upon survival of modest changes to economic fundamentals. I attemptthis in a controlled fashion by simultaneously adjusting growth volatility σf and dividend volatility σδ. Usingequation (46) below, we can increase or decrease σf while holding ψ1 constant. Hence any impact upon thesurvival of agent 2 is due entirely to changes in his survival index; the parameter changes leave the index ofagent 1 unaltered.

σδ =σf√

(2ψ1 + ζ)2 − ζ2(46)

An interesting side effect of this normalization is that the overconfident agent’s perceived belief penalty(equation (47)) also remains unchanged.

ψ2 =12

√ζ2 + (1− φ2)

σ2f

σ2δ

− ζ

(47)

This is important, as an examination of agent 2’s growth rate estimate f2(t) reveals that his deficienciesstem from two sources. First, he underestimates the error of his estimate, which he perceives as ϕ2 < ϕ1.Because of this foolishness, agent 2 adjusts his growth estimate too little in response to observed changesto the dividend δ(t). It is ϕ2 that governs the overconfident agent’s perceived penalty ψ2. Since ψ2 remainsfixed, the aforementioned parameter changes affect survival only through agent 2’s second source of error:the pointless adjustments to his growth estimate in response to the signal. These adjustments are amplifiedby σf .

In Figure 3, σf is adjusted plus and minus 50% from the baseline parameter, while σδ is scaled to leaveψ1 fixed, as described. Three interesting results emerge from this experiment. In the unconstrained case,it appears that leaving the overconfident agent’s perceived belief penalty unchanged also leaves his actualpenalty unchanged. Whether this relationship holds when other parameter combinations are adjusted in asimilar way has yet to be investigated.

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0.015 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

ψ

σf

Belief Penalty vs. Normalized σf

A1 (U)

A2 (U)

A1 (C)

A2 (C)

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

E[ω2] vs. Time for Varying σ

f

Time (years)

Sha

re o

f div

iden

ds

σ

f=0.015 (U)

σf=0.06 (U)

σf=0.015 (C)

σf=0.06 (C)

Figure 3: Normalized adjustment of σf .Maximum standard errors over the interval and all parameter combinations are as follows: E[ω2] : 4.24×10−4, ψ : 5.52×10−4.

Matters are different when the short-sale constraint is imposed: the impact of the constraint is much moresignificant when growth volatility is low (but dividend volatility is high). Examination of the ψ values inthe bar chart suggests that the constraint has a big effect on both agents when the dividend is noisy butthe growth rate is not. The reason why this benefits agent 2 disproportionately is the same as discussed forvarying φ: a big proportional reduction in ψ1 and ψ2 leads to a big absolute reduction in ψ1 − ψ2.

The third point emerging from Figure 3 relates to the direct linkage between the short-sale constraint andthe consumption share: it is apparent from equation (43) that the short-sale constraint is more likely tobind an agent with a small consumption share. However in this experiment E[ω2] would decline at the samerate in the unconstrained case under either set of parameters. Since the effect of the constraint is not feltidentically, we can dispense with the idea that the constraint’s impact is merely more significant when theoverconfident agent’s decline is more severe.

5.4 The Belief Penalty and Consumption Share

The top plot in Figure 4 explores the interaction between the short-sale constraint, consumption share, andbelief penalty in more detail. Under the baseline parameterization, f(t), f1(t), and f2(t) are simulated untilthey achieve a steady state, in which each agent’s estimation of f(t) is (on average) unchanged from oneperiod to the next. Consumption share is then adjusted, from dominance of agent 2 at the left, to dominanceof agent 1 at the right. As an agent loses wealth, the constraint binds him more frequently. This is becausehis reduced consumption share leads to a smaller impact upon the equilibrium interest rate: the interestrate declines less in response to a poor agent’s low growth estimate, since he owns a small share of theendowment. The effect upon the belief penalties is more complicated. ψi captures the difference betweenagent i’s perceived market price of risk and that of an omniscient agent trading under the same economysubject to the same interest rate. Interestingly, this difference is minimized when the two agents have thesame consumption share.

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0.04

0.06

0.08

0.1

0.12

0.14Interaction of Consumption Share with Belief Penalty and Constraint

ψ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

% C

onst

rain

t Bin

ding

ω1

Agent 1Agent 2

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5Instances where Constraint Binds vs. Time

Time (years)

Per

cent

age

(0,1

)

Agent 1Agent 2

Figure 4: Frequency and effect of constraint binding.Maximum standard error for ψ over the interval and all parameter combinations is 4.20× 10−4.

The bottom plot in Figure 4 illustrates the evolution of the constraint’s effects over time under the baselineparameters. As the overconfident agent loses his endowment, he is more likely to be bound by the constraint.The rational agent is correspondingly less likely to be bound, and so the total percentage of paths for whichthe constraint binds one agent or the other remains roughly unchanged.

When short-sales are restricted, the interaction between ωi and ψi is sufficiently complex that an agent’ssteady state survival index Ii cannot be determined in isolation. Rather, it is only established when anagent’s equilibrium ωi is attained. Hence it is no longer possible to “rank” agents by survival index withoutsolving for the long run equilibrium, reducing the index’s usefulness as a “forecasting” tool. However, itremains useful for interpreting observed changes in consumption share.

5.5 Interest Rates and Prices of Risk

When agents are allowed to sell short, E[κi(t)] = E[

fi(t)−r(t)σδ

]. Because all agents agree on r(t) and

E[fi(t)] = f , ∀i, E[κi(t)] is unaffected by the level of overconfidence. With the introduction of the short-saleconstraint, this relationship no longer holds. Accordingly, I focus my analysis on the ratios of the constrainedto unconstrained E[κi(t)] and E[r(t)]. Figure 5 illustrates the evolution of these ratios. The terminal periodis set at 100 years, since the most dramatic changes occur relatively early on, as the highly overconfidentagent loses wealth rapidly.

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Steven D. Baker

1

1.02

1.04

1.06

1.08Ratio of E[r] constrained/unconstrained

Con

stra

ined

/Unc

onst

rain

ed

φ=0.5

φ=0.95

1.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62Ratio of E[κ] constrained/unconstrained

Con

stra

ined

/Unc

onst

rain

ed

φ=0.5

φ=0.95

0.975

0.98

0.985

0.99

0.995

1

1.005

Ratio of E[κ1] constrained/unconstrained

Con

stra

ined

/Unc

onst

rain

ed

φ=0.5

φ=0.95

0 10 20 30 40 50 60 70 80 90 1001

1.1

1.2

1.3

1.4

1.5

Ratio of E[κ2] constrained/unconstrained

Time (years)

Con

stra

ined

/Unc

onst

rain

ed

φ=0.5

φ=0.95

Figure 5: Interaction of constraint with interest rate and price of risk.Unconstrained expectations are constant at E[r] = 0.098, E[κ] = E[κ1] = E[κ2] = 0.13 for all values of φ. Note that the scales

along the y-axes of the charts for κi vary dramatically. Maximum standard errors over the interval and all parametercombinations are as follows: E[r] : 4.47× 10−5, E[κ] : 5.91× 10−4, E[κ1] : 2.89× 10−4, E[κ2] : 5.52× 10−4.

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Steven D. Baker

0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Sur

viva

l Ind

ex (

I)

ρ2

Survival Index vs. ρ2

A1 (U)

A2 (U)

A1 (C)

A2 (C)

0 100 200 300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

E[ω2] vs. Time for Varying ρ

2

Time (years)

Sha

re o

f div

iden

ds

ρ2=0.04 (U)

ρ2=0.06 (U)

ρ2=0.04 (C)

ρ2=0.06 (C)

Figure 6: Survival with overconfidence and impatience.Maximum standard errors over the interval and all parameter combinations are as follows: E[ω2] : 7.10× 10−4, ψ : 5.52× 10−4

Consistent with discussion in Gallmeyer and Hollifield [2008], imposing the constraint causes the interest rateto increase, significantly for φ = 0.95, but only slightly with φ = 0.5. As agent 1 increases his endowmentshare, his beliefs have a growing influence upon the equilibrium interest rate, and he is bound less frequentlyby the constraint. As a result, E[r(t)] gradually declines towards its unconstrained level.

Interpreting the price of risk is somewhat more complicated. As the overconfident agent loses his endowment,his beliefs have reduced impact upon r(t), and κ2(t) is more likely to be bound by the constraint. Effectively,the variance of κ2(t) increases as ω2(t) declines, but all realizations of κ2(t) below zero are truncated. Thiscauses E[κ2] to rise. κ1(t) is subject to two competing effects. As ω1(t) increases, the constraint is morelikely to bind when agent 1 is optimistic (i.e., it binds agent 2). In this case, κ1(t) will decrease as r(t)increases. As the overconfident agent’s influence wanes, however, r(t) will change little when agent 2 isunable to sell short. We see for φ = 0.95 that the first effect dominates initally, with E[κ1(t)] decreasing inthe first few years. Later the second effect dominates, and E[κ1(t)] increases.

5.6 The Relative Importance of Overconfidence

In a his constant growth economy, Yan [2008] observed that even significant errors in an agent’s beliefabout the growth rate can be dwarfed by relatively small differences in impatience or risk aversion, and thatestimates of the latter two parameters are widely dispersed. In his setup, a rational agent who knows thecorrect constant growth rate of f would be dominated by an irrational agent who perennially assumes growth1.25f but has impatience just 0.01 smaller than the rational agent’s. However, stochastic growth arguablyincreases what error in beliefs can be considered reasonable, since even the rational agent will never cease tomake errors.

In Figure 6, the relative importance of errors in belief is assessed by introducing heterogeneous impatience.Agent 1 retains the baseline value ρ1 = 0.1, while agent 2’s impatience is decreased. So long as agents areable to sell short, the magnitude of ψ1 − ψ2 is sufficient that agent 2 has a larger survival index despite his

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Steven D. Baker

patient nature. When short-sales are disallowed, however, belief penalties are reduced, and the overconfidentagent’s superior (lower) patience parameter gives him the smaller survival index. Although ψ1 < ψ2 evenunder the constraint, agent 2 dominates, since I1 > I2. If agents are likely to face obstacles to selling short,then overconfidence is arguably less significant than other factors in determining survival. This supportsYan’s suggestion that erroneous beliefs may be of secondary importance to survival, given the wide dispersionof estimates for impatience and risk aversion.

Finally, the path of agent 2’s consumption share under the constraint illustrates the connection between ω2

and ψ2 that was discussed earlier. The belief penalty takes a minimum around ω2(t) = 0.5, i.e., at timezero. As his consumption share increases, his penalty also increases, and I2(t) approaches I1(t), slowing thegrowth of agent 2’s endowment. The effect is particularly noticeable for ρ2 = 0.6.

6 Conclusions

When short-sales are constrained, even very overconfident investors may survive for over a thousand years.Overconfident agents who are modestly more patient than their rational competitors may even come todominate the market. This bolsters the argument that overconfident agents can survive long enough ina competitive environment to affect stock prices and volatility, offering possible explanations for observedphenomena in asset prices.

In addition, I have examined the dynamic effects of a short-sale constraint when agents are not staticallyoptimistic or pessimistic, but change their minds about the economy from time to time. Complex interactionsbetween wealth, beliefs, and the constraint are distilled somewhat.

The economy and the overconfident agent offer several opportunities for future work. In particular, thereare many kinds of market frictions. My work could be combined with that of Longstaff [2009] on illiquidity,or perhaps transaction costs could be modeled. Analytical solutions for the growth forecast error of theoverconfident agent could shed more light on his behavior under various economic parameters. Finally, thereare opportunities for empirical work in estimating whether (and how) overconfident agents could explainobserved asset pricing anomalies.

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M. Brennan and Y. Xia. Stock price volatility and equity premium. Journal of Monetary Economics, 47(2):249–283, 2001.

K. Daniel and S. Titman. Market Efficiency in an Irrational World. NBER Working Paper, 2000.

B. Dumas, A. Kurshev, and R. Uppal. Equilibrium Portfolio Strategies in the Presence of Sentiment Riskand Excess Volatility. The Journal of Finance, 64(2):579–629, 2009.

M. Gallmeyer and B. Hollifield. An Examination of Heterogeneous Beliefs with a Short-Sale Constraint ina Dynamic Economy. Review of Finance, 12(2):323–364, 2008.

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F. Longstaff. Portfolio claustrophobia: asset pricing in markets with illiquid assets. American EconomicReview, 2009.

F. Riedel. Existence of Arrow–Radner Equilibrium with Endogenously Complete Markets under IncompleteInformation. Journal of Economic Theory, 97(1):109–122, 2001.

J. Scheinkman and W. Xiong. Overconfidence and speculative bubbles. Journal of Political Economy, 111(6):1183–1220, 2003.

H. Yan. Natural Selection in Financial Markets: Does It Work? Management Science, 54(11):1935, 2008.

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