heterogeneous beliefs and asset pricing in discrete...

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Journal of Economic Dynamics & Control 30 (2006) 1233–1260

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Heterogeneous beliefs and asset pricing in discretetime: An analysis of pessimism and doubt

Elyes Jouinia,�, Clotilde Nappb

aInstitut Universitaire de France and CEREMADE, Universite de Paris Dauphine,

Place du Marechal de Lattre de Tassigny, 75116 Paris, FrancebCEREMADE, Universite de Paris Dauphine, Place du Marechal de Lattre de Tassigny,

75116 Paris and CREST, France

Received 21 May 2004; accepted 20 May 2005

Available online 19 September 2005

Abstract

The aim of the paper is to analyze the impact of heterogeneous beliefs in an otherwise

standard competitive complete markets discrete time economy. The construction of a

consensus belief, as well as a consensus consumer are shown to be valid modulo a predictable

aggregation bias, which takes the form of a discount factor. We use our construction of a

consensus consumer to investigate the impact of beliefs heterogeneity on the CCAPM and on

the expression of the risk free rate. We focus on the pessimism/doubt of the consensus

consumer and we study their impact on the equilibrium characteristics (market price of risk,

risk free rate). We finally analyze how pessimism and doubt at the aggregate level result from

pessimism and doubt at the individual level.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: G12; D80

Keywords: Beliefs; Heterogeneity; Consensus; Pessimism; Asset pricing

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.05.008

nding author. Tel.: +331 44 05 46 75; fax: +33 1 44 05 45 99.

dresses: [email protected] (E. Jouini), [email protected] (C. Napp).

www.elsevier.com/locate/jedc

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E. Jouini, C. Napp / Journal of Economic Dynamics & Control 30 (2006) 1233–12601234

1. Introduction

The aim of the paper is to analyze the impact of heterogeneous beliefs in anotherwise standard competitive complete markets discrete time economy.

We start from a given equilibrium with heterogeneous beliefs in a fairly generalsetting and we propose to decompose our analysis into three steps.

The first issue deals with the aggregation of these heterogeneous beliefs. The mainquestion is to know if it is possible to analyze the heterogeneous beliefs model interms of a classical homogeneous beliefs model, in which the common belief may bedifferent from the objective one. In other words, can we define a subjective consensusbelief, i.e. a belief which, if held by all individuals, would generate the sameequilibrium prices as in the actual heterogeneous economy?

We show that the answer to the previous question is positive. We are then ledto analyze, in a standard homogeneous model, the impact of a subjective beliefon the equilibrium characteristics (state price density, risk premium, risk free rate,etc.). This question has been explored by Abel (2002) in the specific setting ofpower utility functions and i.id asset returns. In particular, Abel introduces conceptsof pessimism and doubt, and analyzes their impact on the equilibrium character-istics. The concepts introduced by Abel (2002) are not suited for our framework andwe introduce alternative concepts that have an unambiguous impact on theequilibrium characteristics in a more general setting (in particular, without specificrestrictions on the utility functions nor on the dynamics and distribution of the assetreturns).

The last question is to understand how doubt and pessimism at the individual levelare captured at the aggregate level, i.e. by the consensus belief. Our definitions ofdoubt and pessimism appear as particularly adapted to this question. Indeed,roughly speaking, we obtain that the level of pessimism (resp. doubt) at theaggregate level is a weighted average of the level of pessimism (resp. doubt) at theindividual level.

In this paper, the different subjective beliefs are considered as given. As in Varian(1985, 1989), Abel (1989) or Harris and Raviv (1993), they reflect difference ofopinion among the agents rather than difference of information; indeed, ‘‘we assumethat investors receive common information, but differ in the way they interpret thisinformation’’ (Harris and Raviv, 1993). The different subjective beliefs might comefrom a Bayesian updating of the investors predictive distribution over the uncertainreturns on risky securities as in, e.g. Williams (1977), Detemple and Murthy (1994),Zapatero (1998), Gallmeyer (2000), Gallmeyer and Hollifield (2002), but we do notmake such an assumption; we only impose that the subjective probabilities beequivalent to the initial one. Notice that the above-mentioned models with learningare not ‘more endogenous’, since the investors’ updating rule and the correspondingprobabilities can be determined separately from his/her optimization problem (seee.g. Genotte, 1986).

In a companion paper, Jouini and Napp (2004) considered the same problem incontinuous time. Even if the approach developed therein is similar to the onedeveloped here, the techniques, the concepts and the results are quite different and

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E. Jouini, C. Napp / Journal of Economic Dynamics & Control 30 (2006) 1233–1260 1235

the contribution of this paper goes much further than a technical adaptation of thecontinuous time results. In particular, if we analyze the subjective beliefs in terms ofdistortions of the objective one, the discrete time setting permits us a much larger setof possible distortions than the continuous time setting. For instance, in continuoustime, the volatility of a given process is the same under the objective and the differentsubjective beliefs (by Girsanov’s Theorem) whereas, in discrete time some agentsmight overestimate or underestimate this volatility leading to doubtful oroverconfident behavior. More generally, in continuous time (again by Girsanov’sTheorem) agents can only disagree on the instantaneous rates of returns of theassets. Therefore, there is only one degree of possible divergence between agents andeach individual belief is measured by a one-dimensional parameter. The situation ismuch more complicated in discrete time since there are many ways of constructing aprobability, which is equivalent to a given initial one, which leads to a much richertypology of possible beliefs.

The main contributions of this paper are as follows:

�
To construct a representative agent in the heterogeneous beliefs framework suchthat utility functions are aggregated into a representative utility function, beliefsare aggregated into a consensus belief and time discount factors are aggregatedinto a time discount factor. The representative utility function is an average of theindividual ones and constructed as in the standard case, the consensus belief is anaverage of the individual ones and there is only a bias due to beliefs heterogeneityin the discount factor. � To provide in a very general setting a simple way to analyze the impact of beliefsheterogeneity on equilibrium characteristics. Through our construction, theheterogeneous beliefs framework can be analyzed in terms of subjective beliefs ina homogeneous beliefs framework. � To introduce concepts of pessimism and doubt at the individual level that havegood aggregation properties and that have a clear impact on the equilibrium riskpremium and risk free rate in a fairly general setting. � To decompose into three distinct effects the impact of beliefs aggregation intoa consensus belief, namely an average effect, a cautiousness effect and arelative weights effect and to show that a potential important source ofpessimism/doubt at the aggregate level is the correlation between individual risktolerance and the individual level of pessimism/doubt. In light of the risk premiumand risk free rate puzzles (Mehra and Prescott, 1985; Weil, 1989), we show that ifagents are on average pessimistic, there is a bias towards a higher market price ofrisk and a lower risk free rate than in the standard setting. Furthermore, if risktolerance and pessimism are positively correlated, this induces a higher marketprice of risk.
The paper is organized as follows. We present in Section 2 the aggregationprocedure that transforms the study of a model under heterogeneous subjectivebeliefs into the study of a model under homogeneous subjective beliefs and permitsus to derive an adjusted CCAPM formula.

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The next step consists in the study of the impact of a subjective belief on theequilibrium risk premium and risk free rate, which is the aim of Section 3. Moreprecisely, we try to determine characteristics of the subjective belief that have anunambiguous impact on the market price of risk and on the risk free rate, and, inrelation with the risk premium puzzle (see Mehra and Prescott, 1985 orKocherlakota, 1996 for a survey on the risk premium puzzle) and the risk freerate puzzle (Weil, 1989), which preferably lead to an increase of the market price ofrisk and a decrease of the risk free rate.

The aim of Section 4 is to analyze how pessimism and doubt at the aggregate levelresult from pessimism and doubt at the individual level.

All proofs are in the Appendix.

2. Consensus belief, adjusted CCAPM and risk free rate

In the classical representative agent approach, all investors are assumed toknow the true probability distribution and in this section, we analyze to whichextent this approach can be extended to heterogeneous subjective beliefs. Moreprecisely, we start from a given equilibrium with heterogeneous beliefs in anotherwise standard complete discrete time market model, and we explore to whichextent it is possible (1) to define a consensus belief, i.e. a belief, which, if held by allindividuals would generate the same equilibrium prices as in the actualheterogeneous economy and (2) to define a representative agent (or a consensusconsumer). We shall then analyze the impact of heterogeneity of beliefs on assetpricing, and more precisely on the CCAPM formula and on the expression of the riskfree rate.

The model is standard, except that we allow the agents to have distinct subjectiveprobabilities. We fix a finite time horizon T on which we are going to treat ourproblem. We consider a filtered probability space ðO; ðFtÞt2f0;...;Tg;PÞ, where thefiltration ðFtÞt2f0;...;Tg satisfies the usual conditions. Each investor indexed by i ¼

1; . . . ;N solves a standard dynamic utility maximization problem. He has a currentincome at date t denoted by e�

i

t and a von Neumann–Morgenstern utility functionfor consumption of the form EQi

½PT

t¼0uiðt; ctÞ�, where Qi is a probability measureequivalent to P which corresponds to the subjective belief of individual i. If wedenote by ðMi

tÞt2f0;...;Tg the positive density process of Qi with respect to P, then theutility function can be rewritten as EP½

PTt¼0Mi

tuiðt; ctÞ�.The different subjective beliefs might result from differences of opinions or from

incomplete information and Bayesian updating. The origin of beliefs heterogeneitydoes not impact our analysis. In order to take into account Bayesian updatingmodels, it suffices in such models to determine the individual beliefs (since, asunderlined by Genotte (1986), the investor’s updating rules and the correspondingprobabilities can be determined separately from his/her optimization problem) andto conduct our analysis.

We make the following classical assumptions on the utility functions and on thesubjective beliefs.

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Assumption.

�

1

for all t ¼ 1; . . . ;T ; uiðt; �Þ : ½ki;1Þ ! R [ f�1g is of class C1 on ðki;1Þ, strictlyincreasing and strictly concave,1

�
uið:; cÞ and u0iðt; :Þ ¼ qui=qcðt; :Þ are continuous on ½0;T �, � for i ¼ 1; . . . ;N, Pfei
t4kig40 and PfeitXkig ¼ 1,P P

�
there exists e40 such that e� � ei4 Ni¼1ki þ e, P a.s.,P
�
EP½Tt¼0e�t �o1,
�
the density process Mi is uniformly bounded for i ¼ 1; . . . ;N.
The third condition can be seen as a survival assumption at the individual level
and the fourth one as a survival assumption at the aggregate level. All the remainingconditions are very classical ones.
We do not specify the utility functions ui, although we shall focus on the classicalcases of linear risk tolerance utility functions (which include logarithmic, power aswell as exponential utility functions).

2.1. Consensus belief

We start from an equilibrium ðq�; ðy�iÞÞ relative to the beliefs ðMiÞ and the income

processes ei. We recall that an equilibrium relative to the beliefs ðMiÞ and the incomeprocesses ðeiÞ is defined by a positive, uniformly bounded price process q� and a familyof optimal admissible consumption plans ðy�

iÞ such that markets clear i.e. a family of

adapted ½ki;1Þ-valued processes y�isuch that EP½

PT0 jy�i

t j�o1 and satisfying

y�i¼ yiðq�;Mi; eiÞ;PN

i¼1

y�i¼PNi¼1

ei � e�;

8>><>>:where

yiðq;M ; eÞ � arg maxEP½PT

0qtðy

it�etÞ�p0

EPXT

0

Mtuiðt; ctÞ

" #.

Such an equilibrium, when it exists, can be characterized by the first order necessaryconditions for individual optimality and the market clearing condition. These conditionscan be written as follows:

Mitu0iðt; y

�i

t Þpliq�t on fy�

i¼ kig;

Mitu0iðt; y

�i

t Þ ¼ liq�t on fy�

i4kig;

EP½PT

0 q�t ðy�i

t � eitÞ� ¼ 0;PN

i¼1y�i ¼ e�

8>>>>><>>>>>:(2.1)

for some set of positive Lagrange multipliers ðliÞ.

Note that we could easily generalize the results of this paper to the case where ki is a function of t.

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In the next, we will say that ðq�; ðy�iÞÞ is an interior equilibrium relative to the

beliefs ðMiÞ and the income processes ðeiÞ if y�i4ki, P a.s. for i ¼ 1; . . . ;N.2

Our first aim is to find an ‘equivalent equilibrium’ in which the heterogeneoussubjective beliefs would be aggregated into a common characteristic M. We shalldefine an ‘equivalent equilibrium’ by two requirements. First, the ‘equivalentequilibrium’ should generate the same equilibrium trading volumes ðy�

i� eiÞ and

price process q� as in the original equilibrium with heterogeneous beliefs. Second,every investor should be indifferent at the margin between investing one additionalunit of income in the original equilibrium with heterogeneous beliefs and in the‘equivalent equilibrium’, so that each asset gets the same marginal valuation by eachinvestor (in terms of his marginal utility) in both equilibria.3 We give credit to Calvetet al. (2004) for the idea of constructing an equivalent equilibrium satisfying the twomentioned requirements. However, Calvet et al. (2004) allow an adjustment of theaggregate wealth while, in our construction, we keep the aggregate wealthunchanged.

The existence of such an ‘equivalent equilibrium’ is given by the followingproposition.

Proposition 2.1. Consider an interior equilibrium ðq�; ðy�iÞÞ relative to the beliefs ðMiÞ

and the income processes ðeiÞ. There exist a unique positive and adapted process

ðMtÞt2f0;...;Tg with M0 ¼ 1, a unique family of income processes ðeiÞ withPN

i¼1 ei ¼ e�

and a unique family of individual consumption processes ðyiÞ such that ðq�; ðyiÞÞ is an

equilibrium relative to the common characteristic M and the income processes ðeiÞ and

such that trading volumes and individual marginal valuation remain the same, i.e.

y�i� ei ¼ yi � ei i ¼ 1; . . . ;N

Mitu0iðt; y

�i Þ ¼Mtu0iðt; y

iÞ; t 2 f0; . . . ;Tg; i ¼ 1; . . . ;N.

This means that ðq�; ðyiÞÞ is an equilibrium with income transfers relative to thecommon characteristic M and the income processes ðeiÞ such that individualmarginal valuation is the same as in the original equilibrium with heterogeneousbeliefs. In other words, we proved that modulo a feasible modification of theindividual incomes (i.e.

PNi¼1ei ¼

PNi¼1eiÞ the initial equilibrium price process and

trading volumes remain equilibrium price process and trading volumes in ahomogeneous beliefs setting. The positive process M can then be interpreted as aconsensus characteristic. In particular, if there is no heterogeneity, i.e. if all theinvestors have the same belief represented by Mi ¼ ~M for all i, we obtain M ¼ ~M

2Note that under the following additional condition

u0iðt; kiÞ ¼ 1 for t 2 f0; . . . ;Tg and i ¼ 1; . . . ;N,

all the equilibria are interior ones.3This condition was introduced by Calvet et al. (2004). It would be equivalent to impose that the

Lagrange multipliers are the same for each investor in both equilibria. As a consequence, the

representative agent utility function will also be the same in both equilibria.

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and there is no transfer nor optimal allocations modification (i.e. ei ¼ ei and yi ¼ y�i

for all i).Once the result on beliefs aggregation is achieved, it is easy to construct as in the

standard case, a representative agent, i.e. an expected utility maximizing aggregateinvestor, representing the economy in equilibrium. As in the standard case, fora 2 ðR�þÞ

N , we introduce the function

uaðt;xÞ ¼ maxPN

i¼1xipx

XN

i¼1

1

ai

uiðt;xiÞ.

Proposition 2.2. Consider an interior equilibrium ðq�; ðy�iÞÞ relative to the beliefs ðMiÞ

and the income processes ðeiÞ and let ðliÞ denote the Lagrange multipliers associated to

this equilibrium. There exists a consensus investor defined by the normalized von

Neumann– Morgenstern utility function ul and the consensus characteristic M of

Proposition 2.1, in the sense that the portfolio e� maximizes his expected utility

EP½PT

t¼0 MtulðctÞ� under the market budget constraint EP½PT

t¼0q�t ðct � e�t Þ�p0.

Remark that the consensus investor utility function is constructed as usual withthe Lagrange multipliers of the initial equilibrium. In the specific case of linear risktolerance utility functions we obtain the following explicit expressions for theconsensus characteristic. Recall that the risk tolerance of agent i is defined byTiðt; yÞ ¼ �ðu0i=u00i Þðt; yÞ.

Example 2.3. 1. If the individual utility functions are of exponential type, i.e. if�ðu0iðt; xÞ=u00i ðt; xÞÞ ¼ yi40, then u0ðt;xÞ ¼ ae�x=y for a ¼ ee�

0=y and4

M ¼YNi¼1

ðMiÞyi=y ¼ Ey�

0 ðM�Þ.

2. If the individual utility functions are of logarithmic or power type, i.e. such that�ðu0iðt; xÞ=u00i ðt; xÞÞ ¼ yi þ Zx for Za0, then u0ðeÞ ¼ bðyþ ZeÞ�1=Z for b ¼ ðyþ Ze0Þ

1=Z

and

M ¼XN

i¼1

giðMiÞZ

" #1=Z¼ El�Z�

Z ðM�Þ

¼XN

i¼1

tiðMiÞ�Z

" #�1=Z¼ ET �ðt;y�

�Þ

�Z ðM �Þ

for gi ¼ l�Zi =PN

j¼1l�Zj and ti ¼ Tiðt; y�

iÞ=PN

j¼1Tjðt; y�iÞ.

Notice that the consensus characteristic M is a martingale (i.e. the density processof a given probability) only when Z ¼ 1 (logarithmic case). It is a supermartingalewhen Zo1, and a submartingale when Z41.

4For given families of positive real numbers ðxiÞi¼1;...;n and ðyiÞi¼1;...;n, we denote by y the sumPn

i¼1yi , by

Ey�k ðx�Þ the weighted k-average defined for k 2 Znf0g by Ey�

k ðx�Þ � ½Pn

i¼1ðyi=yÞxki �

1=k and we let Ey�0 ðx�Þ �Qn

i¼1xyi=yi denote the geometric average. We denote by 1 the vector of Rn whose coordinates are all equal

to 1. With these notations, E1kðx�Þ is then the equally weighted k-average of the xi.

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It is interesting to notice in Example 2.3 that for all utility functions in the classicalclass of linear risk tolerance utility functions, the consensus characteristic is obtainedas a weighted average of the individual subjective beliefs, the weights being given bythe individual risk tolerances. We assumed a linear risk tolerance and the samecoefficient of risk aversion for all the agents and it would be interesting to havesimilar formulas when the agents have different cautiousness parameters Zi. In thiscase, and even for more general utility functions, it is easy to see that the consensuscharacteristic can still be considered as an average of the individual beliefs, howeverthe formulas are intractable.

The process M represents a consensus characteristic, however, except in thelogarithmic case, it fails to be a martingale. Consequently, it cannot be interpreted asa belief, i.e. the density process of a given probability measure. It is easy to see that itis not possible in general to recover the consensus characteristic as a martingale, assoon as we want the equilibrium price to remain the same and the optimalallocations in the equivalent equilibrium to be feasible, in the sense that they still addup to e� (even if we do not impose the invariance of individual marginal valuation).

This means that in the general case, there is a bias induced by the aggregation ofthe individual probabilities into a consensus probability.

Proposition 2.4. Consider an interior equilibrium price process q� relative to the beliefs

ðMiÞ, and the income processes ðeiÞ. There exists a positive martingale process M with

M0 ¼ 1, and a predictable positive process B with B0 ¼ 1 such that

MtBtu0ðt; e�t Þ ¼ q�.

The process B is given by

Bt ¼ Bt�1Et�1½Mt�

Mt�1; B0 ¼ 1

and can be thought of as a discount factor.

The process B represents an aggregation bias. It satisfies Bt=Bt�1 ¼ ðEt�1½Mt�=Mt�1Þ

and measures the ‘distance’ from martingality of the consensus characteristic M. If allinvestors share the same beliefs, then B � 1. Otherwise, the process B leads to a(possibly negative) discount of utility from future consumption. We shall interpret thepresence and the properties of this ‘discount factor’ below, after Example 2.5.

As mentioned before, our construction is different from Calvet et al. (2004). Inboth constructions, a degree of freedom is needed. In the representative agentconstruction of Calvet et al. (2004), the degree of freedom is given by an adjustmentof the aggregate wealth while our degree of freedom is given by the introduction of adiscount factor, that is easy to interpret and that will disappear in the CCAPMformula. Furthermore, the introduced discount factor does not impact as in Calvetet al. (2004) the construction of the consensus belief.

We shall see now that the process M corresponds to a (weighted) average of theindividual beliefs and that the discount factor B is directly related to the dispersion ofthe individual beliefs. We first consider two simple and enlightening examples.

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Suppose that logðe�tþ1=e�t Þ has a normal conditional distribution under P andunder Qi, more precisely suppose that, under P, logðe�tþ1=e�t Þ�FtNðm; sÞ and thatunder Qi, logðe

�tþ1=e�t Þ�FtNðmi;sÞ and that all the utility functions are exponential,

i.e. u0iðt;xÞ ¼ e�x=yi . We have

Mitþ1

Mit

¼ exp�m2i �m2 þ 2 logðe�tþ1=e�t Þðm� miÞ

2s2

and consequently

Mtþ1

Mt

¼ exp�

PNi¼1ðyi=yÞm2i �m2 þ 2 logðe�tþ1=e�t Þ m�

PNi¼1ðyi=yÞmi

� �2s2

¼Mtþ1

Mt

Btþ1

Bt

with5

Mtþ1

Mt

¼ exp�½Ey�

1 ðm�Þ�2 �m2 þ 2 logðe�tþ1=e�t Þðm� Ey�

1 ðm�ÞÞ2s2

,

Btþ1

Bt

¼ exp�

PNi¼1ðyi=yÞm2i � ð

PNi¼1 ðyi=yÞmiÞ

2

2s2

¼ exp�Vary� ðm�Þ

2s2.

If we let Q denote the probability measure with density process M with respectto P, we have under Q, logðe�tþ1=e�t Þ�FtNðE

y�1 ðm�Þ;sÞ. The aggregate belief M

corresponds then clearly to a weighted average belief, the weights being given by theindividual risk tolerances. The discount factor Bðtþ 1Þ=BðtÞ is directly related toVary� ðm�Þ the weighted variance of the mi with the same weights and the process B canbe interpreted as a measure of beliefs dispersion.

With the same utility functions and if we now assume that under P,logðe�tþ1=e�t Þ�FtNðm;sÞ and that under Qi, logðe

�tþ1=e�t Þ�FtNðm;siÞ, we obtain with

the same notations that under Q, logðe�tþ1=e�t Þ�FtNðm;Ey��2ðs�ÞÞ which can still be

interpreted as an average belief (the variance under Q is an average of the variancesunder Qi). Furthermore, ½Bðtþ 1Þ�=BðtÞ ¼ ½Ey

�2ðsÞ�=Ey0ðsÞ and a simple second order

Taylor6 expansion gives us ½Bðtþ 1Þ�=BðtÞ�1� ½Vary� ðs�Þ�=½Ey�1 ðs�Þ

2� when the

dispersion of the si is sufficiently small. Once again, the process B is directly relatedto beliefs dispersion and its distance from 1 increases with beliefs dispersion.

More generally, with HARA utility functions, it is easy to see using similar Taylorapproximations that when beliefs dispersion is sufficiently small, M is equal, to the

5For given families of positive real numbers ðxiÞi¼1;...;n and ðyiÞi¼1;...;n, we denote by Vary: ðx:Þ the weighted

variance Vary: ðx:Þ ¼Pn

i¼1ðyi=yÞ½xi � Ey1ðxiÞ�

2.6Note that straightforward expansions give us for Vary� ðx�Þ small enough, the following approximations

Ey�k ðx�Þ�E

y�1 ðx�Þ þ ððk � 1Þ=2ÞðVary� ðx�Þ=E

y1ðx�ÞÞ.

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first order, to a risk tolerance weighted arithmetic average of the Mi and that thegrowth rate of B is, at the second order, proportional to the dispersion of the Mi.

We are now able to explicitly compute the process B and to determine whether it isincreasing, decreasing, smaller or greater than 1.

Example 2.5. For general power utility functions, the process B satisfies

Bt

Bt�1¼

Et�1½Eg�Z ðM

�tÞ�

Eg�Z ðM

�t�1Þ

so that by Minkowski’s Lemma, B is nondecreasing, greater than 1 (resp.nonincreasing, lower than 1) if ZX1 (resp. Zp1Þ.

For logarithmic utility functions, B � 1.In the exponential case, the process B satisfies

Bt

Bt�1¼

Et�1½Ey�0 ðM

�tÞ�

Ey�0 ðM

�t�1Þ

so that by Holder’s Lemma, B is nondecreasing, greater than 1.

The interpretation is the following. The parameter Z is a cautiousness parameter.When there is more risk involved, depending on whether the investor is cautious ornot, that is to say, depending on whether Zo1 or Z41, it can be shown that theinvestor will reduce or increase current consumption with respect to futureconsumption. For instance, for Zo1, the investor is cautious and increases currentconsumption acting as if his/her utility was discounted by a positive discount rate.The converse reasoning leads to a negative discount rate if Z41. Now in our contextwith heterogeneous beliefs, a possible interpretation consists in considering thedispersion of beliefs as a source of risk thereby leading for the representative agent toa discount factor associated to a positive or negative discount rate depending onwhether Zo1 or Z41. This interpretation of beliefs heterogeneity as a source of riskis compatible with the empirical findings of Cragg and Malkiel (1982) who studiedthe relationship between ex post returns and various measures of risk and found thatthe measure that performed best was a measure of divergence of opinion about theasset returns.

To summarize, we have pointed out through previous propositions two distincteffects of the introduction of beliefs heterogeneity on the equilibrium price.

There is first, a change of probability effect from P to the new common probabilityQ, whose density is given by M. This aggregate probability can be seen as a weightedaverage of the individual subjective probabilities. The weights of this average, in thecase of linear risk tolerance utility functions, are given by the individual risktolerances. The second effect is represented by an ‘aggregation bias’, which ispredictable and takes the form of a discount factor. We have seen on simpleexamples that this discount factor is directly related to beliefs dispersion. We areable, for linear risk tolerance utility functions, to determine if it is associated to apositive or negative discount rate. Modulo this discount factor, the equilibrium(state) price (density) is the same as in an economy in which all agents (or the

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representative agent) would share the same subjective belief Q. Remark that theinterpretation of beliefs heterogeneity as a source of risk is only related to the secondeffect. The relative importance of these effects is discussed in detail in Section 4.

We shall now use our construction of a representative consumer to analyze theimpact of heterogeneity of beliefs on the equilibrium properties.

2.2. Adjusted CCAPM, risk premium and risk free rate

In this subsection, we wish to compare the equilibrium characteristics (CCAPMformula, risk premium, risk free rate) in the heterogeneous beliefs setting and in thestandard setting. We shall consider as the standard setting an equilibrium, where thebeliefs are homogeneous and given by the objective probability P, and where therepresentative agent utility function is given by7 ul with the same ðliÞ as in ourheterogeneous beliefs setting.8

We suppose the existence of a riskless asset with price process S0 such that S00 ¼ 1

and S0t ¼

Qts¼1ð1þ rf

s Þ for some predictable risk free rate process rf . We consider arisky asset with positive price process S and associated rate of return between date t

and ðtþ 1Þ denoted by Rtþ1 � ðStþ1=StÞ � 1. In such a context, since q�S is a P�-martingale, we obtain, as in the classical case (see the Appendix),

EPt ½Rtþ1� � r

ftþ1 ¼ �covPt

q�tþ1

EPt ½q�tþ1�

;Rtþ1

" #. (2.2)

Now, since q�t ¼MtBtu0ðt; e�t Þ, with B predictable, Eq. (2.2) can be written


ftþ1 ¼ � covPt

Mtþ1Btþ1u0ðtþ 1; e�tþ1Þ

EPt ½Mtþ1Btþ1u0ðtþ 1; e�tþ1Þ�

;Rtþ1

" #

¼ � covPtMtþ1u

0ðtþ 1; e�tþ1Þ

EPt ½Mtþ1u0ðtþ 1; e�tþ1Þ�

;Rtþ1

" #, ð2:3Þ

so that the adjustment process B plays no role in the CCAPM formula withheterogeneous beliefs. This adjusted CCAPM formula differs from the followingstandard formula:


ftþ1 ¼ �covPt

u0ðtþ 1; e�tþ1Þ

EPt ½u0ðtþ 1; e�tþ1Þ�

;Rtþ1

" #(2.4)

only through the change of probability from P to the consensus probability Q. Thisimplies that only the change of probability has an impact on the difference of the riskpremium in the standard and in the heterogeneous beliefs setting.

7We recall that for a 2 ðR�þÞN , uaðt; xÞ � maxPN

i¼1xipx

PNi¼1 ð1=aiÞuiðt; xiÞ.

8This is in particular the case when the standard setting equilibrium has the same Lagrange multipliers

(or equivalently the same marginal valuations at date 0 or the same initial individual equilibrium

consumptions) as in the heterogeneous beliefs framework, or without restriction on the Lagrange

multipliers when investors have linear risk tolerance utility functions.

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The risk free rate rf is given, as in the classical case, by (see the Appendix)

1þ rftþ1 ¼

q�tEP

t ½q�tþ1�

.

Now, since q�t ¼MtBtu0ðt; e�t Þ, with B predictable, we obtain

1þ rftþ1½het.� ¼

Bt

Btþ1

� �Mtu

0ðt; e�t Þ

EPt ½Mtþ1u0ðtþ 1; e�tþ1Þ�

ð2:5Þ

¼Bt

Btþ1

� �1þ r

ftþ1½homogeneous under Q�

h i, ð2:6Þ

where rf ½homogeneous under Q� denotes the equilibrium risk free rate in a modelwhere all investors share the same subjective probability belief Q. Both the change ofprobability and the discount factor have an impact on the risk free rate. The impactof the discount factor leads to a decrease (resp. increase) of the riskfree rate if B isnondecreasing (resp. nonincreasing). This effect has a clear interpretation. Anonincreasing (resp. nondecreasing) discount factor B corresponds to the fact thatthe representative agent ‘discounts’ future consumption, which means that futureconsumption is less (resp. more) important for the representative agent, it is thennatural to obtain a higher (resp. lower) equilibrium risk free rate. For instance, forpower utility functions with Zp1, as well as for exponential utility functions, we haveseen that the ‘discount factor’ B is nonincreasing and therefore contributes to ahigher risk free rate.

The impact of the change of probability effect from P to the consensus probabilityQ depends on the sign of E

Qt ½u0ðtþ 1; e�tþ1Þ=u0ðt; e�t Þ� � EP

t ½u0ðtþ 1; e�tþ1Þ=u0ðt; e�t Þ�.

Through our aggregation procedure, we have then transformed the study of amodel under heterogeneous subjective beliefs into the study of a model underhomogeneous subjective beliefs. Indeed, the equilibrium risk premium formula underheterogeneous beliefs (Eq. (2.3)) is the same as in an economy where all investorswould share the same probability belief, namely the consensus belief obtainedthrough the aggregation procedure. The same is true of the risk free rate, modulo the‘discount factor’, and we have seen that we are able to determine the impact of thediscount factor for classical utility functions. The next step consists in the study ofthe impact of a subjective belief on the equilibrium risk premium and risk free rate,which is the aim of the next section.

3. Pessimism/doubt of the consensus belief

We consider in this section a homogeneous beliefs model, the agents commonsubjective belief bP being possibly different from the objective probability P. Wesuppose that bP is equivalent to the objective probability P and we denote by bM itsdensity. We want to determine characteristics of the subjective belief bP that have anunambiguous impact on the risk premium and on the risk free rate, and, in relationwith the risk premium puzzle (see Mehra and Prescott, 1985 or Kocherlakota, 1996

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for a survey on the risk premium puzzle) and the risk free rate puzzle (Weil, 1989),which preferably lead to an increase of the risk premium and a decrease of the riskfree rate. In particular, we wish to analyze how pessimism/doubt of the subjectiveprobability may affect the equilibrium risk premium and risk free rate.

In order to analyze the impact of the subjective belief on the value of the riskpremium, we need to be aware of the fact that the possible returns R ¼ ðRtÞ

Tt¼1

obtained in equilibrium are not the same in the standard (objective belief) setting andin the subjective belief settings. What exactly is to be compared?

As in the classical CCAPM, we consider an asset, whose returns Rtþ1 between datet and date tþ 1 are perfectly correlated with e�tþ1 from date t point of view. Moreprecisely, we consider a return process such that Rtþ1 ¼ kte

�tþ1 þ bt for some F t-

measurable positive random variables kt and bt. Notice that such an asset exists bymarket (dynamic) completeness. The risk premium for this asset in the standardobjective belief setting is given by


ftþ1 ¼ �covPt

u0ðe�tþ1Þ

EPt ½u0ðe�tþ1Þ�

; kte�tþ1

" #.

For any positively homogeneous risk measure r, the market price of risk i.e., theratio between the risk premium and the ‘level of risk’ is given by

MPRobjt ðRtþ1Þ ¼ �

1

rðe�tþ1ÞcovPt

u0ðe�tþ1Þ


; e�tþ1

" #.

Since this quantity does not depend upon kt and bt, we shall denote it byMPR

objt ðe

�tþ1Þ. We obtain analogously in the subjective belief setting under bP that

MPRsubjðbPÞt ðe�tþ1Þ ¼ �

1

rðe�tþ1ÞcovPt

bMtþ1u0ðe�tþ1Þ

EPt ½bMtþ1u0ðe

�tþ1Þ�

; e�tþ1

" #.

In order to compare the market price of risk in both settings, we are then led tocompare �covPt ½u

0ðe�tþ1Þ=EPt ½u0ðe�tþ1Þ�; e

�tþ1� with �covPt ½

bMtþ1u0ðe�tþ1Þ=EP

t ½bMtþ1u

0ðe�tþ1Þ�;e�tþ1�. The subjective belief setting leads to a higher market price of risk if and only if

EbPt ½u0ðe�tþ1Þe

�tþ1�

EbPt ½u0ðe�tþ1Þ�

pEP

t ½u0ðe�tþ1Þe

�tþ1�


. (3.1)

Abel (2002) studies a similar problem. He considers a return process Rt generatedby an asset whose dividends are given by e�t . In Abel’s framework (e�tþ1=e�t i.id. andisoelastic utility functions), such a return process satisfies our condition Rtþ1 ¼

kte�tþ1 þ bt for some Ft-measurable positive random variables kt and bt. Abel’s

definition of the risk premium is slightly different, since it is equal to the ratiobetween the expected return of the considered asset and the riskfree return, insteadof the difference between these two quantities. However, the probability mea-sures leading to a higher risk premium (in Abel’s sense) are those satisfyingInequality (3.1).

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To summarize what we have just seen, the subjective probability measures bPleading to an increase of the market price of risk (compared to the standard objectivebelief setting) are characterized by Inequality (3.1) or equivalently by the fact that

covPut ðbMtþ1; e

�tþ1Þp0, (3.2)

where dPu=dP is given (up to a constant) by u0ðtþ 1; e�tþ1Þ and bMtþ1 ¼ E½dbP=dP j Ftþ1�.The subjective belief setting leads to a higher market price of risk when the subjectiveprobability is negatively correlated with the total wealth under some probability and it isnatural to interpret this property as a form of pessimism. Moreover, as seen in theprevious section, the subjective probability measures bP leading to a decrease of the riskfree rate are characterized by

EbPt


u0ðt; e�t Þ

� �XEP

t


u0ðt; e�t Þ

� �or equivalently by

covPt ½bMtþ1; u

0ðe�tþ1Þ�X0,

which also corresponds to some form of pessimism. We shall introduce more formallydifferent notions of pessimism in the next section. Note that even if this section is in thesame spirit as Abel (2002), our framework and definitions are different. Indeed, Abel(2002) first characterizes pessimism by the first-order stochastic dominance and thenintroduces the concept of uniform pessimism characterized by a stronger form ofdominance.9 Abel (2002) proves then that uniform pessimism increases the risk premium,when agents have power utility functions. Jouini and Napp (2005) showed that, this is nolonger the case when uniform pessimism is replaced by pessimism (in Abel’s sense) orwhen the class of utility functions is enlarged to the whole class of concave andnondecreasing functions. In the next section we consider a dynamic concept of pessimism,that is particularly well adapted to the beliefs aggregation problem. Abel (2002) alsointroduces the concept of doubt based on second-order stochastic dominance. In the nextsection we shall propose another definition that will appear in Section 4 as much moretractable in order to analyze how individual doubt is captured at the aggregate level.

3.1. Doubt and first-order pessimism

Before introducing our concept of pessimism, we recall the following concept ofconditional comonotonicity, that generalizes the classical unconditional comonoto-nicity concept. Two random variables x and z are said to be comonotonicconditionally to Ft if the conditional law of ðx; zÞ with respect to F t has acomonotonic support.10 It can be shown that it is equivalent to assuming that x andz are conditionally to Ft nonincreasing functions of some third random variable.

9Abel (2002) defines a pessimistic (resp. uniformly pessimistic) probability measure (with respect to e) as

a probability measure bP such that, for all x, bP½epx�XP½epx� (resp. bP½epx� ¼ P½epkx� for some k41).10A subset A � R2 is said to be comonotonic if for all ððx; yÞ; ðx0; y0ÞÞ 2 A A we have

ðx� yÞðx0 � y0ÞX0.

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Definition 3.1. We say that a probability bP on ðO;F ;PÞ equivalent to P, with densityprocess ð bMtÞ is pessimistic (with respect to e�) if for all t, bMtþ1 and �e�tþ1 arecomonotonic conditionally to Ft.

A pessimistic probability is such that its density bMtþ1 at date ðtþ 1Þ decreases withe�tþ1 conditionally to Ft for all t; this means that conditionally to date t-information,it puts more weight on states of nature where e�tþ1 is low, which clearly correspondsto a notion of pessimism.

For instance, if the information structure is described by a tree and if, for a givennode$, the transition probability of bP (resp. P) between$ and its immediate successorsis given by bp$ (resp. p$), then bP is pessimistic (with respect to e�) if, for each t and eachdate-t node $, the transition density bp$=p$ is comonotonic with �e�tþ1 on the set ofsuccessors of $. In particular, this will be the case if bp$=p$ decreases with e�tþ1.

If F tþ1 is generated by e�tþ1 and e�tþ1=e�t�FtNðm;sÞ (resp. Nðbm;bsÞÞ under P (resp.bP) then bP is pessimistic (with respect to e�) between t and ðtþ 1Þ if and only if mXbmand s ¼ bs. The same result holds for lognormal distributions. This result remainsvalid if m and s are F t-measurable random variables and if the distributions arereplaced by conditional distributions.

If ðe�t Þ follows a Cox–Ross–Rubinstein binomial process with ‘returns’ at eachperiod denoted by u and d and associated transition probabilities pu and pd ¼ 1� pu

(resp. bpu and bpd ¼ 1� bpuÞ under P (resp. bP), then bP is pessimistic if and only ifpuXbpu. As previously, the result remains valid if we introduce a time and statedependence for u, d, pu and bpu as long as uðtÞ and dðtÞ, the returns between t andðtþ 1Þ are Ft-measurable.

In a static setting, Jouini and Napp (2005) introduced a concept of second-orderpessimism, characterized by the fact that Inequality (3.1) is satisfied for allnondecreasing and concave utility functions.

In particular, it is shown that if bP is second-order pessimistic with respect to e� then

EbP½e�je�px�pEP½e�je�px� for all x, which means that the notion of second order

pessimism is another way of expressing the fact that bP ‘puts more weight’ than P onthe bad states of the world and corresponds then naturally to a concept of pessimism.

It is also shown that if a second-order pessimistic probability measure bP satisfies

EbP½e�� ¼ EP½e��, then Var

bP½e��XVarP½e��. This permits one to relate this concept ofpessimism to the concept of doubt introduced by Abel (2002).

This notion of second order pessimism appears as hard to characterize from apractical point of view. In a static setting, Jouini and Napp (2005) introduced asubset of the second order pessimistic probability measures that are simple todescribe and that clearly exhibit doubt.11 A natural generalization of this class ofmeasures in a dynamic framework leads to the following definition in the case wherethe conditional distribution of e�tþ1 is symmetric.

11Suppose that e� has a symmetric distribution under P with respect to EP½e��. A probability P0 on ðO;F Þequivalent to P, with density M 0 is said to exhibit doubt with respect to e� if M 0 is a function of e�,

symmetric with respect to EP½e�� nonincreasing before EP½e�� (and nondecreasing after).

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Definition 3.2. Suppose that for all t, the Ft-conditional distribution of e�tþ1 under Pis symmetric with respect to Et½e

�tþ1�. We say that a probability bP on ðO;F ;PÞ

equivalent to P, with density process ðcMtÞ exhibits doubt (resp. overconfidence)between date t and ðtþ 1Þ (with respect to e�) if for all t, dMtþ1ðoÞ ¼ f tðo; e

�tþ1

ðoÞ � Et½e�tþ1�ðoÞÞ, where f t is Ft-measurable with respect to its first variable,

even and nondecreasing (resp. nonincreasing) on Rþ with respect to its secondvariable.

This means that a probability measure, equivalent to P, exhibits doubt (resp.overconfidence) between date t and ðtþ 1Þ if conditionally to date t information, itsdensity puts more (resp. less) weight on the tails and less (resp. more) weight on thecenter of the distribution. If F tþ1 is generated by e�tþ1 and e�tþ1=e�t�FtNðm;sÞ (resp.Nðbm;bsÞÞ under P (resp. bP), then bP exhibits doubt (with respect to e�) between t andðtþ 1Þ if and only if m ¼ bm and spbs.

If ðe�t Þ follows a trinomial process with ‘returns’ at each period u;m and d andassociated transition probabilities p; 1� 2p, and p (resp. bp, 1� 2bp and bpÞ under P(resp. bP), then bP exhibits doubt if and only if ppbp.
3.2. Impact on the market price of risk and on the interest rate
We prove that pessimism and doubt lead to a higher market price of risk.

Proposition 3.3. If the probability bP is pessimistic or exhibits doubt in the sense of the

previous definitions, then for all t,

MPRsubjðbPÞt ðe�tþ1ÞXMPR

objt ðe

�tþ1Þ

i.e., the market price of risk between t and ðtþ 1Þ in the subjective belief setting under bPis greater than in the standard objective belief setting.

The interpretation is the following. In fact, the market price of risk subjectivelyexpected is not modified by some pessimism. The reason why pessimism increases theobjective expectation of the market price of risk is not that, a pessimisticrepresentative agent requires a higher market price of risk. He/she requires thesame market price of risk but his/her pessimism leads him/her to underestimate theaverage rate of return of equity (leaving unchanged his/her estimation of the risk freerate). Thus the objective expectation of the equilibrium market price of risk is greaterthan the representative agent’s subjective expectation, hence is greater than thestandard market price of risk. The same interpretation holds for doubt. These resultsare consistent with the empirical findings of Cecchetti et al. (2000) and Giordani andSoderlind (2003). In the first reference the authors prove that a model in whichconsumers exhibit pessimistic beliefs can better match sample moments of assetreturns than can a rational expectations model. In the second reference the authorsprovide evidence of pessimism in investors forecasts.

As far as the risk free rate is concerned, we obtain the following result.

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Proposition 3.4. (1) If the subjective probability belief bP exhibits pessimism and if the

representative agent’s utility function is concave and nondecreasing, then the

equilibrium risk free rate is lower in the subjective belief setting than in the standard

objective belief setting.(2) If the subjective probability belief bP exhibits doubt and if the representative

agent’s utility function is nondecreasing, concave, with a convex derivative, then the

equilibrium risk free rate is lower in the subjective belief setting than in the standard

objective belief setting.

Hence, for a large class of utility functions including HARA utility functions, ifthere is doubt, the change of probability decreases the risk free rate. This result holdstrue if there is pessimism without restriction on the utility functions.

3.3. Application to the heterogeneous beliefs model

We adopt the same notations as above and we denote by MPRhett ðe

�tþ1Þ the market

price of risk in the heterogeneous beliefs setting given by

MPRhett ðe

�tþ1Þ ¼ �

1

rðe�tþ1ÞcovPt

Mtþ1u0ðe�tþ1Þ

EPt ½Mtþ1u0ðe

�tþ1Þ�

; e�tþ1

" #.

We want to compare this quantity to the market price of risk in the standardsetting, as well as the risk free rate in both settings. We have seen in the pre-ceding section that the heterogeneous beliefs model can be analyzed in termsof an equivalent homogeneous beliefs model, the homogeneous belief beinggiven by the consensus probability belief. In particular, we have seen that

MPRhett ðe

�tþ1Þ ¼MPR

subjðQÞt ðe�tþ1Þ, where Q denotes the consensus probability belief

and 1þ rftþ1½heterogeneous� ¼ ðBt=Btþ1Þ½1þ r

ftþ1½homogeneous under Q��. An im-

mediate application of our results on pessimism and doubt leads then to thefollowing proposition.

Proposition 3.5. (1) If the consensus probability belief Q exhibits pessimism and/or

doubt then the market price of risk in the heterogeneous beliefs setting is higher than in

the standard objective belief setting.(2) If the representative agent utility function is in the HARA class (i.e.

�u0ðt;xÞ=u00ðt; xÞ ¼ yþ ZxÞ with a cautiousness parameter Z41 and if Q exhibits

pessimism and/or doubt then the equilibrium interest rate in the heterogeneous beliefs

setting is lower than in the standard objective belief setting.

The last step of our analysis consists now in determining how pessimism anddoubt at the aggregate level (i.e. on the consensus belief Q) can be inherited from thepessimism and doubt at the individual level (i.e. on the Qi).

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4. From individual to aggregate pessimism/doubt

The aim of this subsection is to analyze how the properties of the individual beliefsare transferred to the consensus belief. In particular, since we have seen in thepreceding section that a pessimistic consensus probability as well as a consensusprobability which exhibits doubt leads to a higher market price of risk and lowers therisk free rate, we wish to explore which properties of the individual beliefs lead to apessimistic consensus belief or to a consensus belief which exhibits doubt.

We start with the analysis of the notion of pessimism.

4.1. From individual to aggregate pessimism

Notice that there is no need for all investors to be pessimistic in order to obtain apessimistic consensus belief, hence an increase of the market price of risk and adecrease of the risk free rate. Indeed, since the consensus probability corresponds tosome average of the individual beliefs, it suffices that some average of the individualbeliefs be pessimistic.

If all the agents have exponential utility functions, we have seen that M ¼ Ey�0 ðM

�Þ.The aggregate characteristic M overweights the beliefs Mi for which yi is greaterthan the average and underweights the beliefs Mi for which yi is smaller than theaverage.

In order to enlighten our analysis, let us consider the case where

e�tþ1=e�t�FtNðm;sÞ under P, and e�tþ1=e�t�FtNðmi;sÞ under Qi. As in Section 2, we

easily obtain that e�tþ1=e�t�Ft NðEy�1 ðm�Þ;sÞ under Q.

If all yi are equal, then M ¼ E10ðM

�Þ, which means that the consensus characteristicM is an equally weighted geometric average of the individual beliefs and the expectedgrowth rate under the consensus characteristic appears as the equal weighted averageof the individual subjective expected growth rates. The impact on the market price ofrisk is then simply given by the pessimism/optimism of the ‘equal-weighted average’investor. If the investors are on average pessimistic i.e. if E1

1ðm�Þpm (resp. optimistic,i.e. if E1

1ðm�ÞXm), then the market price of risk is higher (resp. lower) than in thestandard setting – and the risk free rate is lower (resp. higher).

If the yi are different, the expected return under the consensus belief can be writtenas Ey�

1 ðm�Þ ¼ E11ðm�Þ þ

PNi¼1ðmi � E1

1ðm�ÞÞðyi=y� 1=NÞ. There are then two effects ofbeliefs heterogeneity on the equilibrium characteristics. The first effect is given as inthe previous case by the average level of optimism/pessimism and the second effect isgiven by the ‘correlation’ between risk tolerance/aversion and optimism/pessimism.If we assume for instance that the more risk tolerant investors are pessimistic (resp.optimistic) and the less risk tolerant investors are optimistic (resp. pessimistic), thenthe second effect increases (resp. reduces) the market price of risk. The intuitionleads us to think that risk tolerance and optimism are positively correlated, whichwould induce a lower risk premium for assets with a higher dispersion. It remains toprove that this intuition is correct (or not). This could be done through behavioral orpsychological empirical studies, and to our knowledge, this question is still open.

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This could also be done through the introduction in our model of a specific learningprocess which would lead to such a correlation, and this is left for future research.

Let us now assume that the utility functions are no longer exponential but of thepower form u0ðxÞ ¼ ðyi þ ZxÞ�1=Z. The consensus characteristic M is now given byEg�Z ðM

�Þ. Our aim is to exhibit a third effect and for this purpose, let us assume thatall the agents are equally weighted in the definition of M (no weights effect) and thatthere is no systematic bias in the average returns individual estimations (i.e.E11ðm�Þ ¼ m). In order to simplify the analysis, we will even assume that the mi are

symmetrically distributed around m. The equally weighted geometric average belief isthen equal to the objective belief and the consensus belief is an equally weightedaverage of the individual beliefs. However, this last average is an Z-average and not ageometric average and, contrarily to the exponential case, the consensus belief is notequal to the objective one. The consensus characteristic M is such that

Mtþ1

Mt

¼1

N

XN

i¼1

expZ2s2�ðmi � mÞ2 � 2ðm� miÞ

e�tþ1e�t� m

� �� !1=Z

.

This function is clearly symmetric with respect to m, increasing after m. Then theconsensus probability Q exhibits doubt and the market price of risk is higher than inthe standard setting. It is easy to check that this doubt effect increases with thecautiousness parameter Z. More precisely, if we denote by QZ (resp. QZ0) theconsensus probability when all the agents have a cautiousness parameter Z (resp. Z0)then, for Z0XZ, the density of QZ0 with respect to QZ is symmetric with respect to m,and increases after m. The probability QZ0 exhibits more doubt than QZ and themarket price of risk is higher in the Z0 cautiousness framework than in the Zcautiousness framework.

We have shown in these examples that there are three possible effects when weaggregate individual heterogeneous beliefs into a consensus belief:

�

1

an average effect: if the (equally weighted) average belief is pessimistic oroptimistic, then the consensus belief will be influenced accordingly;
� a cautiousness effect: when the consensus belief is an Z0-average with given
weights12 and the objective probability is an Z-average with the same weights, thenthere is a bias towards doubt if Z0XZ and a bias towards overconfidence if ZXZ0;
� a relative weights effect: if the more risk tolerant investors are pessimistic (resp.
optimistic) and the less risk tolerant investors are optimistic (resp. pessimistic),then the consensus belief will present a bias toward pessimism (resp. optimism).

The first effect is natural and does not necessitate specific developments. Thesecond effect has been studied in a different framework by Gollier (2003) where theauthor analyzes the social impact of different exogeneous aggregation of beliefsprocedures. With our notations, Gollier (2003) states in particular for some specificvalues of the cautiousness parameter Z that the Z0-average aggregation procedure is

2That is when we have CRRA utility functions with a cautiousness parameter Z.

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socially efficient if and only if Z0 ¼ Z. When this condition is not satisfied, Gollier(2003) analyzes the impact of a disagreement increase depending on the location ofthis increase (in the tails or in the center of the distribution).

However, the Taylor expansions provided in Section 2 permit us to show thatthese distinct averages differ only by second order terms, when the beliefs dispersionis sufficiently small. Hence the second effect deserves specific attention only when thetwo other effects cancel out.

Abel (1989) studied a problem that is similar to ours in a specific framework. Heimposes a normal distribution for the aggregate wealth and considers exponentialutility functions with the same risk tolerance parameters yi. Furthermore, it is assumedthat the ‘average investor is rational’, i.e. the geometric average belief does not exhibitany bias and is equal to the objective probability. Under these assumptions our threeeffects vanish and there is no impact on the market price of risk as defined in thispaper. Abel (1989) is not interested in the impact of beliefs heterogeneity on themarket price of risk but on the risk premium. The impact of beliefs heterogeneity onthe risk premium is directly related to the impact on the risk free rate and Abel findsthat beliefs heterogeneity leads to an increase of the risk premium.

In the following, we focus on the third and last effect and, for this purpose, wesuppose that the equally weighted average of the individual beliefs corresponds tothe objective probability P. We obtain the following result, without any specificassumption on the distributions of the growth rate of aggregate wealth.

Proposition 4.1. Assume that �u0iðt;xÞ=u00i ðt;xÞ ¼ yi þ Zx40, and that

E1ZðM

�tþ1Þ

Et½E1ZðM

�tþ1Þ�¼ 1 t ¼ 0; . . . ;T .

If the pessimistic (resp. optimistic) agents have a risk tolerance higher (resp. lower) than

the average, then the market price of risk in the heterogeneous beliefs setting is greater

than in the standard setting.

Remark that the condition on the ðMiÞ still means that the equally weightedaverage of the individual beliefs corresponds to the objective probability P. However,the geometric average is replaced by the power Z average. It is possible to replace thiscondition by a geometric average condition (or an Z0-average with Z0oZ) modulo anadditional ‘cautiousness’ effect. As mentioned previously, this effect is of the secondorder for small dispersions. Furthermore, this effect should, as in the examplesabove, lead to an additional increase of the market price of risk.

We can adopt the same approach to analyze how doubt/overconfidence at theaggregate level can be inherited from the doubt/overconfidence at the individuallevel.

4.2. From individual to aggregate doubt

In the case of exponential utility functions, we have seen that M ¼ Eh�0 ðM

�Þ and M

overweights the beliefs Mi for which yi is greater than the average and underweights

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the beliefs Mi for which yi is smaller than the average. For instance, suppose thatunder P, e�tþ1=e�t�FtNðm;sÞ and that under Qi, e�tþ1=e�t�Ft Nðm;siÞ. It is then easy tosee that, as in Section 2, e�tþ1=e�t�FtNðm;E

y��2ðs�ÞÞ under Q. There are then two effects

of beliefs heterogeneity on the market price of risk. The first effect (the averageeffect) is given by the average level of doubt/overconfidence, measured by an equallyweighted average of the individual levels of overconfidence 1=s2i . The second effect(relative weights effect) is given by the covariance between individual risk tolerance/aversion and the individual level of overconfidence. If we assume for instance thatthe more risk tolerant investors exhibit doubt and that the less risk tolerant investorsexhibit overconfidence, then we get a higher (resp. lower) market price of risk. Wecould, as in the analysis of pessimism, exhibit a third ‘cautiousness’ effect if weconsider power utility functions.

As above, if we focus on the ‘relative weights effect’ and suppose that the equallyweighted average of the individual beliefs corresponds to the objective probability P,we obtain the analog of Proposition 4.1.

Proposition 4.2. Assume that �½u0iðt; xÞ�=½u00i ðt;xÞ� ¼ yi þ Zx (with Z possibly equal to

zero) and for all t, ½E1ZðM

�tþ1Þ�=fEt½E

1ZðM

�tþ1Þ�g ¼ 1.

If the agents that exhibit doubt (resp. overconfidence) have a risk tolerance higher

(resp. lower) than the average, then the market price of risk in the heterogeneous beliefs

setting is greater than in the standard setting.

Pessimism and doubt can then be seen as possible explanations for the riskpremium puzzle as well as for the risk free rate puzzle as underlined by Abel (1989).However, it is not necessary to assume pessimism or doubt at the individual level noron (equally weighted) average. A potential important source of pessimism or doubtat the aggregate level is the correlation between the individual risk tolerance and theindividual level of pessimism/doubt.

5. Conclusion

In this paper, we provided an aggregation procedure which permits one to rewritein a simple way the equilibrium characteristics (state price density, market price ofrisk, risk premium, risk free rate) in a heterogeneous beliefs framework and tocompare them with an otherwise similar standard setting. This procedure permitsone to analyze in detail the impact of beliefs heterogeneity on the equilibriumcharacteristics.

In particular, we introduced concepts of pessimism and doubt and we proved, in afairly general setting, that pessimism and doubt at the aggregate level lead to anincrease of the market price of risk and to a decrease of the risk free rate. We alsohave shown how pessimism and doubt are transmitted from the individual to theaggregate level.

It appears that pessimism and doubt lead to an increase of the market price of riskand, under some additional conditions on the cautiousness level, to a decrease of therisk free rate. Furthermore, for utility functions in the HARA class, the aggregate

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level of pessimism and/or doubt is a weighted average of the individual levels ofpessimism and doubt. These weights are proportional to the individual risktolerances and, at the equilibrium, there is a bias toward the beliefs of the more risktolerant agents. The intuition leads us to think that risk tolerant agents are moreoptimistic. It remains to prove that this intuition is correct and this could be donethrough behavioral experimental studies or through the introduction of a theoreticalmodel of learning and beliefs construction. This is left for future research.

Appendix

Proof of Proposition 2.1. Since q� is an interior equilibrium price process relative tothe beliefs ðMiÞ, and the income processes ei, we know that

PNi¼1y

�i ¼ e� and thatthere exist positive Lagrange multipliers ðliÞ such that for all i and for all t,

Mitu0iðt; y

�i

t Þ ¼ liq�t .

We consider the maximization problem

ðPlÞ : maxXN

i¼1

1

li

UiðyiÞ under the constraint

XN

i¼1

yipe�,

where UiðcÞ ¼ E½PT

t¼0uiðt; ctÞ�. Denoting the solution by ðyi;lÞi, we get thatPNi¼1y

i;ðlÞ ¼ e� and the process ðð1=liÞu0iðt; y

i;ðlÞt ÞÞt is independent from i. We denote

this process by pðlÞ. Letting MðlÞ � q�=pðlÞ, we then have for all i and for all t

MðlÞt u0iðt; y

i;ðlÞt Þ ¼Mi

tu0iðt; y

�i

t Þ.

The process MðlÞ is adapted and positive. Moreover, at date t ¼ 0, we have for all i,

Mi0 ¼ 1, and

PNyi;ðlÞ0

i

i¼1 ¼PN

i¼1y�i

0 ¼ e�0, so that MðlÞ0 ¼ 1. Then, it suffices to take

M ¼MðlÞ and yi ¼ yi;ðlÞ.As far as uniqueness is concerned, notice that any process yi such that

PNi¼1y

i ¼ e�

and

Mtu0iðt; y

itÞ ¼Mi

tu0iðt; y

�i

t Þ

for some positive process M is a solution of the maximization problem ðPðlÞÞ. Theuniqueness follows from the strict concavity of this maximization program. &

Proof of Proposition 2.2. Similar to the proof of the analogous result in a standardsetting. &

Proof of Example 2.3. Since the representative utility function u is given by

ulðt;xÞ ¼ maxPN

i¼1xipx

XN

i¼1

1

li

uiðt;xiÞ

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the expression of ul in the specific setting of linear risk tolerance utility functions isobtained as in the standard case (see e.g. Huang and Litzenberger, 1988).

The expression of M is obtained by using Mtu0iðt; y

itÞ ¼Mi

tu0iðt; y

�i

t Þ, as well asXN

i¼1

y�i¼XN

i¼1

yi ¼ e�.

Indeed, in the case of exponential utility functions, we have for all i,

Mi exp �y�

i

yi

!¼M exp �

yi

yi

� �,

henceYNi¼1

ðMiÞyi exp �

XN

i¼1

y�i

!¼My exp �

XN

i¼1

yi

!,

or equivalently

M ¼YNi¼1

ðMiÞyi=y.

In the case of power utility functions, we get for all i,

Miðyi þ Zy�iÞ�1=Z¼Mðyi þ ZyiÞ

�1=Z¼ liMðyþ Ze�Þ�1=Zb,

hence

ðMiÞZl�Zi ¼MZðyþ Ze�Þ�1bZ

ðyi þ Zy�iÞ

and

M ¼XN

i¼1

l�ZiPNi¼1l

�Zi

ðMiÞZ

" #1=Z: &

Proof of Proposition 2.4. Immediate defining B and M by

Bt ¼ Bt�1Et�1½Mt�

Mt�1; B0 ¼ 1

and

Mt ¼Mt

Bt

: &

Proof of Example 2.5. By Example (2.3) and Proposition (2.4), we immediately getthe expressions for Bt. Holder ’s inequality permits us to conclude in the exponentialcase. In the case of power utility functions, for 0oZo1, we get by Minkowski’sinequality that Et�1½½

PNi¼1giðM

itÞZ�1=Z�p½

PNi¼1giðM

it�1Þ

Z�1=Z so that B is nonincreas-

ing. The case Z41 can be treated similarly. &

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Proof of the CCAPM formula and the risk free rate expression. We know that for anyasset with associated price process ðStÞ

Tt¼0, we must have

q�t St ¼ EPt ½q�tþ1Stþ1� for all t ¼ 0; . . . ;T � 1. (A.1)

or equivalently, since bMtBtu0ðt; e�t Þ ¼ q�t ,

St ¼ Ept

bMtþ1Btþ1u0ðtþ 1; e�tþ1ÞbMtBtu0ðt; e�t ÞStþ1

" #or by definition of Rtþ1

1 ¼ Ept

bMtþ1Btþ1u0ðtþ 1; e�tþ1ÞbMtBtu0ðt; e�t Þ

ð1þ Rtþ1Þ

" #.

This leads to

1 ¼ Ept


" #ð1þ EP

t ½Rtþ1�Þ þ covPt

bMtþ1Btþ1u0ðtþ 1; e�tþ1ÞbMtBtu0ðt; e�t Þ;Rtþ1

!.

Now, applying Eq. (A.1) to the riskless asset yields

Ept ½bMtþ1Btþ1u0ðtþ 1; e�tþ1ÞbMtBtu0ðt; e�t Þ

� ¼1

1þ rftþ1

,

hence

1þ rftþ1 ¼ ð1þ EP

t ½Rtþ1�Þ þ ð1þ rftþ1ÞcovPt


;Rtþ1

!and


ftþ1 ¼ �

covPtbMtþ1Btþ1u0 tþ1;e�

tþ1

� �bMtBtu0ðt;e�t Þ

;Rtþ1

� �E

pt

bMtþ1Btþ1u0 tþ1;e�tþ1

� �bMtBtu0ðt;e�t Þ

� �or equivalently


ftþ1 ¼ � covPt

Mtþ1Btþ1u0ðtþ 1; e�tþ1Þ

EPt ½Mtþ1Btþ1u0ðtþ 1; e�tþ1Þ�

;Rtþ1

" #

¼ � covPt

bMtþ1u0ðtþ 1; e�tþ1Þ

EPt ½bMtþ1u0ðtþ 1; e�tþ1Þ�

;Rtþ1

" #: &

Proof of Proposition 3.3. (1) Consider first the case of a pessimistic probabilitymeasure bP. We have seen (Inequality 3.2) that the probability measures leading to anincrease of the market price of risk are characterized by covPu

t ðbMtþ1; e�tþ1Þp0 where

dPu=dP is given (up to a constant) by u0ðtþ 1; e�tþ1Þ. By definition, since bP is

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pessimistic with respect to e�, the random variables bMtþ1 and �e�tþ1 are comonotonicconditionally to Ft. For positive random variables X and Z such that X and Z arecomonotonic conditionally to Ft, we have cov

Qt ðX ;ZÞX0 for any probability

measure Q absolutely continuous with respect to P. Indeed, we have

covQt ðX ;ZÞ ¼ E

Qt ½XZ�ðoÞ � E

Qt ½X �ðoÞE

Qt ½Z�ðoÞ

¼

Zðx� x0Þðy� y0ÞMðo; dðx; yÞÞ Mðo; dðx0; y0ÞÞ,

where Mðo; :Þ is the conditional law of ðx; yÞ with respect to Ft. It is clear then thatcov

Qt ðX ;ZÞX0 and we easily deduce from there that covPu

t ðbMtþ1; e�tþ1Þp0.

(2) Consider now the case of a probability measure bP which exhibits doubt. Wehave obtained (Inequality 3.1) that bP leads to a higher market price of risk if andonly if

EbPt ½u0ðe�tþ1Þe

�tþ1�

EbPt ½u0ðe�tþ1Þ�

pEP

t ½u0ðe�tþ1Þe

�tþ1�


.

We have

EbPt ðe�tþ1u0ðe�tþ1ÞÞ

EbPt ðu0ðe�tþ1ÞÞ

¼EP

t ½ðe�tþ1 �mtÞ bMtþ1u

0ðe�tþ1Þ�

EPt ½bMtþ1u0ðe

�tþ1Þ�

þmt,

where mt ¼ Et½e�tþ1�. Since the random vector ðe�tþ1 �mt; bMtþ1Þ is distributed like

ðmt � e�tþ1; bMtþ1Þ conditionally to F t, we have

EPt ½ðe�tþ1 �mtÞ bMtþ1u0ðe�tþ1Þ1e�

tþ1pmt � ¼ EP

t ½ðmt � e�tþ1ÞbMtþ1u0ð2mt � e�tþ1Þ1e�

tþ1Xmt �,

hence

EPt ½ðe�tþ1 �mtÞ bMtþ1u

0ðe�tþ1Þ� ¼ EPt ½ðmt � e�tþ1Þ

bMtþ1½u0ð2mt � e�tþ1Þ

� u0ðe�tþ1Þ�1e�tþ1

Xmt �.

Then

EbPt ðe�tþ1u0ðe�tþ1ÞÞ

EbPt ðu0ðe�tþ1ÞÞ

¼EP

t ½ðmt � e�tþ1ÞbMtþ1½u

0ð2mt � e�tþ1Þ � u0ðe�tþ1Þ�1e�tþ1

Xmt �

EPt ½bMtþ1½u0ð2mt � e�tþ1Þ þ u0ðe�tþ1Þ�1e�

tþ1Xmt �

þmt.

We want to compare this quantity with

EPt ½ðmt � e�tþ1Þ½u

0ð2mt � e�tþ1Þ � u0ðe�tþ1Þ�1e�tþ1

Xmt �

EPt ½½u0ð2mt � e�tþ1Þ þ u0ðe�tþ1Þ�1e�

tþ1Xmt �

þmt.

Letting gðmt; e�tþ1Þ � ðmt � e�tþ1Þ½u0ð2mt � e�tþ1Þ � u0ðe�tþ1Þ� and hðmt; e�tþ1Þ � u0ð2mt�

e�tþ1Þ þ u0ðe�tþ1Þ, we are led to compare EPt ½bMtþ1gðmt; e�tþ1Þ1e�

tþ1Xmt �=EP

t ½bMtþ1hðmt;

e�tþ1Þ1e�tþ1

Xmt � with EPt ½gðmt; e�tþ1Þ1e�

tþ1Xmt �=EP

t ½hðmt; e�tþ1Þ1e�tþ1

Xmt �. Let us now define

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the probability measure Pg by dPg=dP ¼ gðmt; e�tþ1Þ1e�tþ1

Xmt=EP½gðmt; e�tþ1Þ1e�tþ1

Xmt �.

We are led to compare EPg

t ½bMtþ1�=EPg

t ½bMtþ1ðh=gÞðmt; e�tþ1Þ�with 1=EPg

t ½ðh=gÞðmt; e�tþ1Þ�.It is easy to check that the function ðh=gÞðx; �Þ : y 7!ðu0ðxþ yÞ þ u0ðx� yÞÞ=yðu0ðx�

yÞ � u0ðxþ yÞÞ is decreasing on Rþ, so that bMtþ11e�tþ1

Xmt and �ðh=gÞðmt; e�tþ1Þ1e�tþ1

Xmt

are comonotonic conditionally to Ft. We have then, as seen in (1),covP

g

t ððh=gÞðmt; e�tþ1Þ; bMtþ1Þp0 or equivalently

EPg

tbMtþ1

h

gðmt; e

�tþ1Þ

� �pEPg

t ½bMtþ1�E

Pg

t

h

gðmt; e

�tþ1Þ

� �which concludes the proof. &

Proof of Proposition 3.4. (1) If bP exhibits pessimism with respect to e�, then bydefinition of pessimism and since u is concave, we easily obtain that for all t, Mtþ1

and u0ðe�tþ1Þ are comonotonic conditionally to Ft. As seen in the proof of the previousproposition, we deduce that for all probability measure bP absolutely continuous with

respect to P, we have for all t, covbPt ðMtþ1; u0ðe�tþ1ÞÞX0. In particular, we have for all t,

covPt ðMtþ1; u0ðe�tþ1ÞÞX0 and EbPt ½u0ðe�tþ1Þ�XEP

t ½u0ðe�tþ1Þ�.

(2) If bP exhibits doubt with respect to e�, then we know that

EbPt ½u0ðe�tþ1Þ� ¼ E

bPt ½½u0ð2mt � e�tþ1Þ þ u0ðe�tþ1Þ�1e�

tþ1Xmt �

and

EPt ½u0ðe�tþ1Þ� ¼ EP

t ½½u0ð2mt � e�tþ1Þ þ u0ðe�tþ1Þ�1e�

tþ1Xmt �.

Since u has a convex derivative, u0ð2m� xÞ þ u0ðxÞ is nondecreasing on xXm and weconclude as in the previous case. &

Proof of Proposition 3.5. (1) Immediate consequence of Proposition 3.3 and the factthat the market price of risk under heterogeneous beliefs is the same as the marketprice of risk under the subjective belief Q.

(2) Immediate consequence of Equality 2.6, Proposition 3.4 and Example 2.5. &

Proof of Proposition 4.1. In the exponential utility case, we have E10ðM

�tþ1Þ=

Et½E10ðM

�tþ1Þ� ¼ 1 and

Ey�0 ðM

�tþ1Þ ¼

YNi¼1

ðMitþ1Þððyi=yÞ�ð1=NÞÞE1

0ðM�tþ1Þ

¼YNi¼1

ðMitþ1Þððyi=yÞ�ð1=NÞÞEt½E

10ðM

�tþ1Þ�.

Consequently, M ¼QN

i¼1ðMitþ1Þððyi=yÞ�ð1=NÞÞNt, where Nt is some F t-measurable

random variable. It is easy to see that under our assumptions, the random variable

M ¼QN

i¼1ðMitþ1Þðyi�ðy=NÞÞ=y is comonotonic with �e�tþ1 conditionally to Ft, hence the

consensus probability is pessimistic. Proposition 3.3 concludes.

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For power utility functions, let us remark that ½Eg�Z ðM

�tþ1Þ�

Z ¼PN

i¼1ðgi�

ð1=NÞÞðMitþ1Þ

Zþ ½E1

ZðM�tþ1Þ�

Z. Under our assumptions, since ½E1ZðM

�tþ1Þ�

Z is F t-measurable, this expression is comonotonic with �e�tþ1 conditionally to F t and again,Proposition 3.3 concludes. &

Proof of Proposition 4.2. For Z ¼ 0, as in the proof of Proposition 4.1, we haveM ¼

QNi¼1ðM

itþ1Þððyi=yÞ�ð1=NÞÞNt where Nt is some F t-measurable random variable.

Now, if the more risk tolerant agents (i.e. those for which yi4y=N) exhibit doubt(i.e. have a density Mi

tþ1 that is symmetric with respect to Et½e�tþ1� nondecreasing

after Et½e�tþ1� conditionally to FtÞ and if the less risk tolerant agents (i.e. those for

which yioy=N) are overconfident (i.e. have a density Mitþ1 that is nonincreasing

with e�tþ1 after Et½e�tþ1� conditionally to F tÞ, we clearly obtain that M exhibits doubt.

The case Za0 can be similarly treated using the arguments of the proof ofPropositions 4.1. &

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