# Risk Weighted Utility Theory as a Solution to the Equity Premium Puzzle

Thierry Chauveau, Nicolas Nalpas

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Thierry Chauveau, Nicolas Nalpas. Risk Weighted Utility Theory as a Solution to the Equity Premium Puzzle. 1999. halshs-03591443

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UNIVERSITÉ PARIS 1
-PANTHÉON - SORBONNE -

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Cahiers de la MSE

Risk Weighted Utility Theory as a Solution
to the Equity Premium Puzzle

Thierry CHAUVEAU

Nicolas NALPAS

1999.20

CENTRE NATIONAL
DE LA RECHERCHE
SCIENTIFIQUE

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13

ISSN : 1624-0340
Risk Weighted Utility Theory as a Solution
to the Equity Premium Puzzle*

Thierry Chauveau

Nicolas Nalpas†

TEAM-CNRS and CDC-Marchés

TEAM-CNRS

First draft : June 1998

This draft : September 1999

<footnote>*The authors thank Michèle Cohen, Mark Machina, Bernard Walliser, and partici-
pants at seminars at Université Paris I, Université Libre de Bruxelles, The 1998 Journées
de Microéconomie appliquée (Point-à-Pitre), The 1999 Association Française de Finance
(Aix-en-Provence), and The 1999 Macro Money Finance Research Group (University Col-
lege, Oxford) for useful comments.</footnote>

<footnote>†Corresponding author: TEAM-CNRS ESA 8059, Maison des Sciences Economiques, 106-112 Boulevard de l'Hôpital 75647 Paris cedex 13. E-mail: nalpas@univ-paris1.fr</footnote>
Abstract

In this paper, we formulate a restatement of the theory of choice under un-
certainty. As an alternative to the rank-dependent expected utility model, we
develop a probability-altering theory in which the transformation of probabili-
ties is weighted by the centered outcome of the lottery which may be viewed as
"pure" risk. Using a weak restriction on the changes of probability measure,
we avoid stochastic dominance inconsistency. We examine the main effects
of this new approach on financial market equilibrium, especially in terms of
Euler stochastic equations. Using such an approach allows for accounting
for both high equity risk premia and low risk free rates without unrealistic
assumptions upon the values of parameters characterizing the behavior of the
representative consumer.

Résumé

Nous cherchons à apporter une solution satisfaisante à l'énigme de la
prime de risque. Notre solution passe par une reformulation du comporte-
ment du consommateur en univers risqué. Nous proposons une théorie de
l'altération des probabilités alternative à celle de l'espérance de l'utilité dépen-
dant du rang (RDEU), où la déformation des probabilités est pondérée par le
revenu centré de la loterie, considéré comme le risque "pur" de cette loterie.
L'introduction d'une condition peu restrictive sur les changements de mesure
de probabilité utilisés nous permet d'obtenir un critère où le principe de la
dominance stochastique est respecté. Après avoir examiné les principales con-
séquences que cette nouvelle représentation des préférences implique en terme
d'équilibre d'un marché financier et en particulier, en matière d'équations
d'Euler stochastiques, nous sommes en mesure de rendre compte, sans hy-
pothèse irréaliste sur la valeur des paramètres caractérisant le comportement
du consommateur représentatif, du niveau de la prime de risque historique
observé sur données américaines.

Keywords: probability transformation function, CAPM, equity premium puzzle.

J.E.L. Classification: D81, D91, G12.
# 1 Introduction

In this article, we go back to the most commonly used assumption in economics and finance, namely the behaviour of a rational individual in an uncertain environment: the expected utility rule. Such an assumption, which was suggested as early as 1738 by D. Bernoulli, was justified by Von Neumann and Morgenstern [1954], using an axiomatics of the behaviour of a rational individual in a probabilized environment. This vision has been largely successful in that the criterion of expected utility remains the classical criterion of individual decisions in a risky environment.

However, two main criticisms have been addressed at Von Neumann and Morgenstern (VNM) theory. The first one is the rigid link the theory imposes between two notions: that of risk aversion, as defined by Pratt [1964] and Arrow [1965] and that of decreasing marginal utility<fnref n="1" />. It is also well known that the criterion of expected utility can lead to experimental inconsistency. Indeed, Allais [1953] was the first to criticize the American school axioms, especially the independence axiom. He related experiments in which people when filling out questionnaires behaved in ways contradictory to the search for the maximum of the expected utility. In addition, the work of Cohen [1992] and Jaffray [1988]<fnref n="2" /> pointed out that the deficiencies of the expected utility theory essentially depend on the effect of changes in behaviour upon the appearance of a “certainty effect”.

Faced with this difficulty, economists have had two reactions. The first was to try and “save” the VNM axiomatics. In view of the importance of the expected utility theory in the development of economics, Machina [1982,1989] was able to demonstrate that many results based on this theory still locally hold when the independence axiom is withdrawn.

However, many economists, psychologists, and mathematicians have followed Allais' point of view and have suggested behaviours for the rational individual that are in better agreement with experimental results. One trend favors a so-called deformed probabilities approach, in which the decision-makers handle the probabilities in a nonlinear fashion. Ground-breaking

<footnote n="1">¹For more details, see Munier [1995] or Rousseau [1995]. It could also be said, as pointed out by Gayant [1997], that it is impossible to distinguish between risk aversion and increasing risk aversion, as defined by Rothschild et Stiglitz [1970].</footnote>

<footnote n="2">²In the interest of brevity, we skip mention of studies in this area (see Cohen and Jaffray [1988]).</footnote>
work by Quiggin [1982]<fnref n="3" />, Allais [1986], Yaari [1987], and Segal [1989], led to the rank-dependent expected utility model (RDEU model<fnref n="4" />) and its reaxiomatization by Chateauneuf [1990] and Wakker [1994]. This model has had a few practical applications in finance and insurance; it distinguishes the concept of risk aversion from the concept of diminishing marginal utility. However, it is analytically intractable in an intertemporal framework and the interpretation of its results is not straightforward -i.e. risk premia do not express very easily, as in the VNM universe-.

In this study, we look again at the concept of subjective alteration of probabilities. We assume that any consumer who behaves according to the VNM theory is, in fact risk-neutral. This is not a new idea since it has been often argued (see e.g. Gayant [1995]) that, when RDEU theory is used, the elementary utility function characterizes the behaviour of the consumer in a certain world and that, on the contrary, the transformation of probabilities characterizes his “true” behaviour towards risk. An extreme point of view is that of the “dual” theory by Yaari [1987] in which agents have a constant marginal utility.

To obtain a “true” behaviour towards risk, we develop a theory in which the transformation of probabilities depends on the centered outcome of the lottery, which may be viewed as the risk perceived by an individual facing a lottery. The probability measure used by an individual to evaluate such a lottery will be a probability measure equivalent to that of the lottery under review, and whose Radon-Nykodim derivative with respect to the original probability is a random variable, which depends on both of the risk perceived by an individual facing a lottery -i.e. the corresponding centered outcome-and risk aversion<fnref n="5" />.

Given the transformation of the probabilities, our individual will consider the expected utility of the outcomes of the lottery he faces. Any risk premium will then be the sum of two elements:

(i) A concavity premium, which also may be called dispersion premium: it is the usual Arrow-Pratt premium. We emphasize that such a premium

<footnote n="3">³The concept of subjective alteration of probabilities can also be found in Allais [1953], Edwards [1954, 1962], Bernard [1964], and Tversky [1967].</footnote>

<footnote n="4">⁴As Gayant [1995] points out, this model can be viewed as a particular case, in a probabilized environment, of the one developed by Schmeidler [1989] in an uncertain environment.</footnote>

<footnote n="5">⁵called pure risk aversion or P-aversion to distinguish it from the usual Arrow-Pratt definition. We shall later define more precisely these notions.</footnote>
exists only because consumers are supposed to maximize a non linear function
of their terminal wealth. In other words, Arrow-Pratt risk aversion is but
the reverse of decreasing marginal utility. Consequently, the behaviour which
can be inferred from observing concavity premia is a behaviour towards the
dispersion of the results and we shall often call it, from now, "dispersion-
risk" aversion (hereafter D-risk aversion). Such a behaviour may be observed
either in a certain world, when a consumer has to determine an optimal
intertemporal set of consumptions given a sequence of incomes or, in an
uncertain one, when he faces a random experiment.

(ii) A “pure” risk premium (P-premium). The P-premium corresponds to the fact that the empirical frequencies of the values of a random variable usually deviate from the corresponding probabilities. If we consider a single experiment, we can only observe one outcome. The “empirical frequency” of this outcome is just one and risk can be viewed as the difference between such an outcome and its expected value. Then, the premium will depend on the distribution of the centered outcome and on the tastes of our individual, i.e. on his aversion towards “pure” risk.

To sum up, we propose a risk weighted utility theory which, as in the
RDEU model, distinguishes between a behaviour towards wealth in a certain
world (through the elementary utility function which will be denoted *u* and a
behaviour towards risk in an uncertain world (through his aversion towards
P-risk *A*<fnref n="6" />). However, this is as far as the analogy with the RDEU model
goes.

Indeed, although our analysis, which, from now, we shall call Risk Weighted
Utility theory (RWU) is based on the transformation of original probabili-
ties, it can be viewed neither as a variant of the subjectively weighted utility
theory (SWU) nor as one of RDEU analysis.

Actually, the idea underlying non-expected utility theories which trans-
form the original probabilities, is that the individual mis-evaluates low prob-
abilities, and that he is led to over-weight them. This kind of argument
appears to be normative. In our theory, we provide positive arguments in
order to explain the transformation of the original probabilities: our individ-
ual does it because he faces the risk described above, which vanishes when
the law of large numbers plays its role.

Moreover, in the RDEU models, the transformed cumulative distribution
function depends on the original one through a probability transformation

<footnote n="6">⁶As it will be proved later in the paper.</footnote>
function which is independent of the lottery considered. On the contrary, we,
here, assume that the transformation of probabilities depends on the pure
risk of the lottery and on the P-risk aversion of our individual and, conse-
quently on the characteristics of the lottery. Hence, the transformation of
probabilities used by an investor to take into account the P-risk will generally
differ from one lottery to another.

This does not guarantee that the criticism traditionally addressed at the
proponents of probabilities deformation<sup>7</sup>, namely the possible violation of
the “dominance property”<sup>8</sup>, will never occur. Hence, we have been led to
impose on the probabilities alteration function a restrictive condition which
eliminates stochastic dominance inconsistency. Such a condition is nice and
little restrictive (See Section 1).

All these results have been obtained using an extension of the axiomatics
of Von Neumann and Morgenstern. We still assume that the preferences on
the lotteries can be represented by a total ordering; however such an ordering
now induces an equivalent ordering on the transformed lotteries and the two
other axioms of VNM (continuity and independence) are supposed to hold
on the set of the transformed lotteries. This allows for levying paradoxes
such as Allais' and Kahneman and Tversky's (See Appendix 7).

Having thus defined and axiomatized a new behaviour for a rational in-
dividual in an uncertain environment, we examine how our new assumptions
may modify financial theory. Starting with the assumption of no arbitrage
opportunities on a complete market, we derive the link between risk neutral
probability measure and the subjective probability measure of a consumer at
equilibrium.

We then address the issue of the existence of a static market-equilibrium.
We establish that an appealing justification of using mean and variance as
choice criteria (or the use of CAPM for valuation) is the existence of decision-
makers with constant marginal utility and positive P-risk aversion. Finally,
we show that the Euler equations derived from the intertemporal equilibrium
model take two effects into account<sup>9</sup>.

<footnote n="7"><sup>7</sup>Except, of course, for those who developed the RDEU model precisely to escape this
criticism.</footnote>

<footnote n="8"><sup>8</sup>The dominance principle consists in the following statement: if a lottery *L* first-order stochastically dominates a lottery *L*′, then any rational individual will prefer *L* to *L*′.</footnote>

<footnote n="9">⁹The first effect is related to the covariance between the P-risk aversion of the decision-
maker and the product of the utility of wealth and the price of an asset. The second effect
relates to the covariance between the change in probability measure and the product of</footnote>
The paper is organized as follows: In Section 2, we present the set of the assumptions which are necessary to building deformed probabilities, and, we examine the consequences of shifting from the original to the new probabilities on the definition and the properties of risk aversion. In Section 3, we examine the characteristics of equilibrium on financial markets when agents “cautiously” alter objective probabilities; static and intertemporal equilibria are successively considered. In Section 4, we apply the Mehra and Prescott [1985] methodology to our new Euler equations in order to evaluate the ability of our theory to solve the equity premium puzzle. Section 5 concludes.

# 2 A Risk Weighted Utility Theory as an alternative to Non-Expected Utility Theories

## 2.1 Intuition underlying the RWU theory

The intuition underlying the paper can be grasped from the following example: consider an individual whose horizon is $T$ and whose intertemporal utility function is the sum of $T$ identical elementary utility functions $u(\cdot)^{10}$. Assume that he faces a $T$-repeated lottery $\mathcal{L}_T$, giving him at time 0, the right of earning $x_t$ at time $t$ ($t = 1, ..., T$), and whose elementary outcomes are $x_1$ and $x_2$ ($x_1 < x_2$) with probabilities ($p = 1/2, (1-p) = 1/2$).

The total expected utility of our individual will be $(u(x_1) + u(x_2))T/2$. If $T$ is large enough, his total expected utility will not differ significantly from the utility obtained through earning the following certain intertemporal income: $x_t = x_1$ ($x_t = x_2$) when $t$ is even (odd). This is due to the law of large numbers.

Hence, using the EU criterion, when uncertainty occurs, may be interpreted as equivalent to using the utility criterion in a certain world, if uncertain outcomes are weighted using probabilities as proxies to empirical frequencies. As has been already said, such a way of doing may be interpreted as a lack of specific behaviour towards risk.

Nevertheless risk premia can be defined. They are due to the fact that the utility of an expected outcome differs from expected utility of the outcomes if the utility function $u$ is concave; in other words, such premia, i.e.

<footnote>the marginal utility of wealth and the price of an asset.</footnote>

<footnote n="10">¹⁰For the make of simplicity, the discount factor is unitary.</footnote>
Arrow-Pratt premia, only derive from the assumption of diminishing mar-
ginal utility, and not from one characterizing a genuine behaviour towards
risk.

On the contrary, if our individual is assumed to use the EU criterion for
a single-trial lottery L₁ whose other caracteristics are identical to that of L<sub>T</sub>,
we are led to the odd result that he considers that the fair price of L₁ is 1/T
times the price of L<sub>T</sub>.

We claim that our individual is unlikely to use the EU criterion and that,
on the contrary, he will take into account what we have called the P-risk of
the lottery, which is due to the fact that, here, the empirical frequencies will
be either (1,0) or (0,1) which is very different from (1/2,1/2). Consider, now,
a second lottery L* whose payoffs are (x<sub>1</sub>*, x<sub>2</sub>*) with probabilities p* = 1/2 and
(1 - p*) = 1/2 and assume that:

(i) the payoffs rank as follows : x₁* < x1 < x2 < x₂*

(ii) the expected utilities $U(L) = (u(x_1) + u(x_2))/2$ and

$U(L^*) = (u(x_1^*) + u(x_2^*)) / 2$ are equal.

Notwithstanding the last equality, which holds by assumption, we claim
that our individual is likely to prefer lottery *L* to lottery *L*\* since *x*<sub>1</sub>, the
lower outcome of *L*, is higher than *x*<sub>1</sub>\*, the lower outcome of *L*\*. He may be
thus more disappointed earning *x*<sub>1</sub> instead of *x*<sub>1</sub>\* than he will be more satisfied
earning *x*<sub>2</sub> instead of *x*<sub>2</sub>\*. Hence he will care more for the unfavorable event
when facing lottery *L*\* than when facing lottery *L*. There is an incentive for
distorting probabilities which increases as *x*<sub>1</sub>(*x*<sub>2</sub>) decreases (increases) *i.e.*
when the P-risk increases; on the contrary, there will be no incentive for
distorting probabilities if the lottery is degenerate (*x*<sub>1</sub> = *x*<sub>2</sub>).

Finally, we assume that the utility got from a single trial lottery is the
difference between, on the one hand, the utility of the expected gain, and,
on the other one, the sum of the usual Markowitz premium and of a P-risk
premium. Our individual will still use as a criterion the expected utility of
the outcomes of the lottery, but the weights used for the calculation of the
expected utility will be no more the probabilities characterizing the lottery
under review; they will be transformed probabilities. In our example the
change of probabilities would be:

$$
\frac{q(x_i)}{p(x_i)} = 1 - A(E_P[\tilde{x}]) (x_i - E_P[\tilde{x}]) \text{ with } i = 1, 2.
$$

And the criterion used would express as

$$
E_q[u(x)] = q(x_1)u(x_1) + q(x_2)u(x_2).
$$

Note that, in the preceding formulas, we have taken into account the fact
that P-risk aversion generally depends on $E_P[\tilde{x}]$.
## 2.2 Notations

Consider a set $\mathbb{L}$ of lotteries that we can describe as follows: a lottery $L \in \mathbb{L}$ is a discrete or continuous probability distribution on outcomes defined on a set $W \subset \mathbb{R}$; in the first case, we assume that the number of events is finite and that the outcomes are ordered. Hence, we can denote the lottery $L$ like

$$L = \{p_1^L, p_2^L, \dots, p_{K-1}^L, p_K^L; w_1, w_2, \dots, w_{K-1}, w_K\}$$

Thus we have the equivalence $k < l \Leftrightarrow w_k < w_l$, and the constraints: $\sum_{k=1}^{K} p_k^L = 1$ and $p_k^L \ge 0 \ \forall k$.

In the continuous case, we describe $L$ as a random variable $\tilde{x}$ defined on an interval $W = [a, b]$. We denote $F$ its cumulative distribution function. If $F$ is absolutely continuous, its density function $f$ is such that:

$$f(w) = \frac{dF}{dw}(w) \quad \forall w \in [a, b]$$

As usual in financial theory, we only consider the positive outcomes. Thus, we have $0 \le a < b$. Sometimes, we consider $b$ as infinite, and we therefore take the outcomes space over an interval $[0, +\infty[$.

## 2.3 Axiomatization

In this section, we develop a simple axiomatics for the intuition that agents maximize the expected utility of their terminal wealth but that the weights they use while doing so, are not the original probabilities but, deformed probabilities, with the understanding that the deformation should satisfy the following four conditions: i) no violation of the dominance principle may occur; ii) favorable events are under-weighted and unfavorable ones are over-weighted; iii) the degree of deformation only depends on the outcome of the lottery under review; iv) the deformation should be monotonic in wealth.

We assume the existence of an ordering over the lotteries which represents the preferences of the agent. It will be labelled $\prec_U$.

**Assumption 1:** The ordering obeys the first axiom of Von Neumann and Morgenstern [1944], i.e.:

**A1 (Total weak order):**
(i) Relation which is defined on the set $\mathbb{L}$ of the available lotteries is reflexive and transitive i.e., whatever the lotteries $L, L'$ and $L'',$ belonging to set $\mathbb{L}$, we have:

$$L \prec_U L' \qquad L \prec_U L' \text{ and } L' \prec_U L'' \Rightarrow L \prec_U L''$$

(ii) An equivalence relation can be defined as follows:

$L$ is said to be as prefered as $L'$ (and we shall denote $L \sim_U L'$) if and only if:

$$L \prec_U L' \text{ and } L' \prec_U L$$

(iii) Whatever the couple of lotteries $(L, L')$ belonging to $\mathbb{L} \times \mathbb{L}$,

$$\text{Either } L \prec_U L' \text{ or } L' \prec_U L$$

Recall that we seek for a change of probability measure which will depend on pure risk i.e. centered outcome. Note that any change of probability is characterized by the Radon-Nykodim derivative of the transformed cumulative distribution $\Phi_L$ with respect to the original one $F_L$ i.e.

$$\frac{d\Phi_L}{dF_L}(\tilde{w}) = \gamma_L(\tilde{w}) \quad \text{with} \quad E_L[\gamma_L(\tilde{w})] = 1$$

and:

$$\gamma_L(\tilde{w}) \geq 0$$

We thus consider changes of measure which depend on pure risk i.e. those which write:

$$\frac{d\Phi_L}{dF_L}(\tilde{w}) = \gamma_L(\tilde{w} - E_L[(\tilde{w})])$$
For the sake of simplicity we limit, in this paper, to take into account affine functions of centered outcome; hence, without loss of generality:

$$ \gamma_{L,A}(\tilde{w}) = 1 + A(E_L[\tilde{w}], E_L[\tilde{w}^2], E_L[\tilde{w}^3], \dots) (\tilde{w} - E_L[\tilde{w}]) $$

where A depends on the moments of the lottery. Again for the sake of simplicity, we restrict to the case when those changes of measure, called, from now, standard changes of measure, depend only on the first moment of the lottery under review; hence:

$$ \gamma_{L,A}(\tilde{w}) = 1 + A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}]) $$

and, without real loss of generality, we can impose:

$$ A(E_L[\tilde{w}]) \geq 0 $$

or, more conveniently:

$$ \forall w \in [a, b] \quad , \quad A(w) \geq 0 $$

and our standard change of measure will write:

$$ \gamma_{L,A}(\tilde{w}) = 1 \pm A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}]) $$

Note that $A(.)$ characterizes the sensitiveness of our individual to P-risk, $(\tilde{w} - E_L[\tilde{w}])$. Finally, we are led to consider three different cases:

**Definition 1** Given any positive function $A(.)$, one can define three standard changes of measure of probability as follows:

1) *cautious transformations*:

$$ \gamma_{L,A}(w) = 1 - A(E_L[\tilde{w}]) (w - E_L[\tilde{w}]) $$

with
$$ \mathrm{Min} (\gamma_{L,A}(\tilde{w})) = 1 - A(E_L[\tilde{w}]) (b - E_L[\tilde{w}]) \geq 0 \quad (1) $$

2) neutral transformations: $\gamma_{L,A}(\cdot) \equiv 1$

3) audacious transformations:

$$ \gamma_{L,A}(\tilde{w}) = 1 + A(E_L[\tilde{w}]) (w - E_L[\tilde{w}]) $$

with

$$ \mathrm{Min} (\gamma_{L,A}(\tilde{w})) = 1 + A(E_L[\tilde{w}]) (a - E_L[\tilde{w}]) \geq 0 \quad (2) $$

Here the terms “cautiously” and “audaciously” have their common meaning; they are not related to the concept of prudence as defined by Kimball [1990], which is the third derivative of the utility function. Nevertheless, common sense tells us that a prudent individual would tend to over-weight the probability of unfavorable events and to under-weight favorable ones.

To sum up we have, up to now, defined standard changes of measure of probability which can be easily interpreted in terms of P-risk and of behaviour towards pure risk (the sign and the importance of the sensitiveness of the individual to pure risk characterizing such a behaviour).

Instead of considering the original lotteries such as $L$, one can now consider altered lotteries i.e. using transformed probabilities instead of the original ones. We shall denote $T_A(L)$ the lottery whose cumulative distribution $\Phi_L$ is defined using a standard change of measure i.e. by the following formula:

$$ \frac{d\Phi_L}{dF_L}(\tilde{w}) = 1 \pm A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}]) $$

where $A(\cdot)$ is a positive function and satisfies condition (1) or (2). We can thus define a correspondence between the initial set of lotteries $\mathbb{L}$ and the set $T_A(\mathbb{L})$ of altered lotteries. It will be called the canonical risk weighted change of measure of probability. Moreover, we limit, from now on, to considering the case of cautious (or neutral) individuals; hence we restrict to changes of the following form:

$$ \frac{d\Phi_L}{dF_L}(\tilde{w}) = 1 - A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}]) \text{ and } A(E_L[\tilde{w}]) \geq 0 $$

and we now establish the following result:
**Proposition 1** A sufficient condition for the correspondence $T_A$ to be one to one is that the two following conditions hold:

(i) $A(.)$ is a decreasing function of $w$, i.e.

$$ \frac{dA}{dw}(w) = A'(w) \leq 0 \quad \forall w \in [a, b] \qquad (3) $$

(ii) $wA(.)$ is an increasing function of $w$, i.e.:

$$ \frac{d(wA)}{dw}(w) \geq 0 \quad \forall w \in [a, b] \qquad (4) $$

**Proof: See Appendix 1.**

It is interesting to note that condition (3) will be later interpreted as implying that pure risk aversion is a decreasing function of wealth and that condition (4) limits the importance of distortion.

We can now define a canonical correspondence between the initial set of lotteries $\mathbb{L}$ and the set $T_A(\mathbb{L})$ of altered lotteries. It will be called the canonical risk weighted change of measure of probability.

**Definition 2** Given any function $A$ satisfying the following conditions:

$$ \begin{array}{l} \forall w \in [a, b], \quad A(w) \geq 0; \quad A'(w) \leq 0; \quad \frac{d(wA)}{dw}(w) \geq 0 \\ \text{and} \\ 1 - A(E_L[\tilde{w}]) (b - E_L[\tilde{w}]) \geq 0 \end{array} $$

The canonical risk weighted change of measure associated to $A$ is given by:

$$ \frac{d\Phi_L}{dF_L}(\tilde{w}) = 1 - A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}]) $$

and the correspondence $T_A$ is said to be canonical.

**Proposition 2** If $T_A$ is canonical, the ordering $\prec_U$ which represents the preferences of the agent over the lotteries belonging to $\mathbb{L}$, induces an equivalent ordering $\prec_{T_A}$ over the transformed lotteries belonging to $T_A(\mathbb{L})$. It is defined using the following equivalences :
$$L \prec_U L' \Leftrightarrow T_A(L) \prec_{T_A} T_A(L')$$

$$L \sim_U L' \Leftrightarrow T_A(L) \sim_{T_A} T_A(L')$$

Instead of assuming the continuity and the independence axioms on the original set of probabilities we assume that they are valid on the set $T_A(\mathbb{L})$. This will lead us to the expected conclusion that our individual behaves as if he were an expected utility maximizer who would use the □'s instead of the p's. Hence:

**Assumption 2:** There exists an increasing function $A(\cdot)$ defining a canonical risk weighted change of measure of probability $T_A(\cdot)$ such that the ordering $\prec_{T_A}$ induced by $\prec_U$ obeys the last two axioms of Von Neumann and Morgenstern [1944], i.e.:

**A2 (Modified Continuity Axiom):**

Whatever the triple of lotteries $(L, L', L'')$ belonging to $\mathbb{L} \times \mathbb{L} \times \mathbb{L}$ and the triple of the corresponding transformed lotteries $(T_A(L), T_A(L'), T_A(L''))$, if we have:

$$L \prec_U L' \prec_U L'' \Leftrightarrow T_A(L) \prec_{T_A} T_A(L') \prec_{T_A} T_A(L'')$$

Then, there exists a scalar $\alpha \in [0, 1]$ such that:

$$\alpha T_A(L) + (1 - \alpha) T_A(L'') \sim_{T_A} T_A(L')$$

**A3 (Modified Independence axiom):**

Whatever the triple of lotteries $(L, L', L'')$ belonging to $\mathbb{L} \times \mathbb{L} \times \mathbb{L}$ and the triple of the corresponding transformed lotteries $(T_A(L), T_A(L'), T_A(L''))$, and whatever $\alpha$ belonging to $[0, 1]$,

$$(i) L \prec_U L' \Leftrightarrow T_A(L) \prec_{T_A} T_A(L') \\ \Leftrightarrow \alpha T_A(L) + (1 - \alpha) T_A(L'') \prec_{T_A} \alpha T_A(L') + (1 - \alpha) T_A(L'')$$

$$(ii) L \sim_U L' \Leftrightarrow T_A(L) \sim_{T_A} T_A(L') \Leftrightarrow \\ \alpha T_A(L) + (1 - \alpha) T_A(L'') \sim_{T_A} \alpha T_A(L') + (1 - \alpha) T_A(L'')$$
We can now sum up the preceding discussion in the following proposition.

**Representation Theorem:** Under the assumptions above, there exists a relation of preferences $\prec_U$, which admits a utility representation of the expected utility form with probabilities' alteration, that is:

for any two lotteries,

$$ 
\begin{align*}
L &\prec_U L' \Leftrightarrow \int_a^b u(\tilde{w}) d\Phi_L(\tilde{w}) \le \int_a^b u(\tilde{w}) d\Phi_{L'}(\tilde{w}) \Leftrightarrow \\
&\phantom{\prec_U L' \Leftrightarrow} \int_a^b u(\tilde{w}) (1 - A(E_L[\tilde{w}]) (\tilde{w} - E_L[\tilde{w}])) dF_L(\tilde{w}) \\
&\le \int_a^b u(\tilde{w}) (1 - A(E_{L'}[\tilde{w}]) (\tilde{w} - E_{L'}[\tilde{w}])) dF_{L'}(\tilde{w})
\end{align*}
$$

Note that if $A$ is a constant, i.e. if $A$ does not depend on the characteristics of the lottery, the transformation of probabilities can be viewed as a particular case of a utility function which is quadratic in the probabilities (e.g. see Machina [1982] and Chew, Epstein and Segal [1991]). However, note that this is not true in general since $A$ will usually depend on $E_P[\tilde{w}]$. Moreover, our representation cannot be assimilated to a RDEU's one since, in general, the change of measure cannot be written as

$$ \frac{d\Phi_L}{dF_L}(\tilde{w}) = h'(F_L) \quad \text{with:} \quad h'(.) \ge 0; \quad h(0) = 0 \quad \text{and} \quad h(1) = 1 $$

It is shown in Appendix 8, that, if the canonical change of measure satisfies the preceding equation, two lotteries exhibiting the same mean cannot differ. In other words, the transformation function $h(.)$ depends on the lottery considered and hence must be indexed by $L$.

## 2.4 Dominance property

We now address the issue of dominance property. We establish a sufficient condition for which the usetilization of distributions $\Phi_L$ instead of $F_L$ does not lead to the violation of the stochastic dominance principle.

**Proposition 3** Consider a rational cautious decision-maker and a set $\mathbb{L}$ of lotteries defined on interval $[a, b]$. A sufficient condition for the change
in measure to be compatible with the first-order stochastic dominance principle is:

$$ \forall L \in \mathbb{L}, \quad A(E_L[\tilde{w}]) \leq (1/(b - E_L[\tilde{w}])) + 1/\Box(b) $$

where $\Box(b) = u'(b)/u(b)$.

**Proof: See Appendix 2.**

Since $u(.)$ is defined up to an affine increasing transformation we can, without loss of generality, choose a utility function in such a way that the last term becomes negligible; we thus only consider the condition:

$$ \forall L \in \mathbb{L}, \quad A(E_L[\tilde{w}]) \leq 1/(b - E_L[\tilde{w}]) \quad (5) $$

or, equivalently:

$$ \forall L \in \mathbb{L}, \quad aE_L[\tilde{w}] \leq 1/(b - E_L[\tilde{w}]) \Leftrightarrow a \leq 1/(b/E_L[\tilde{w}] - 1) \quad (6) $$

with:

$$ a = A/E_L[\tilde{w}] \quad (7) $$

In Finance, condition (5) will appear very little restrictive. For a complete discussion, see Section 4.

## 2.5 Two definitions for Risk premia and risk aversion

We now turn to studying risk “in the small” i.e. infinitesimal risks. We define as usual the risk premium of an infinitesimal risk as the difference between the expected outcome of the lottery and its certainty equivalent. We are led to the following equality, which is established in Appendix 3:

$$ \underbrace{PR(\bar{w})}_{TRP} = \underbrace{A(\bar{w}) Var[\tilde{w}]}_{PRP} - \underbrace{\left(\frac{u'(\bar{w})}{2u''(\bar{w})}\right) Var(\tilde{w})}_{COP} $$

$$ \begin{aligned} & \text{TOTAL RISK PREMIUM} \\ &= \text{PURE RISK PREMIUM} + \\ & \text{CONCAVITY PREMIUM (or Arrow-Pratt Premium)} \end{aligned} $$
Each of the two premia is proportional to the variance of the lottery under review. When the first premium is considered, the coefficient of proportionality only depends on the tastes of the investor since it is equal to $A(\bar{w})$, which now characterizes P-risk aversion. Hence, we can write:

$$TRA(\bar{w}) = 2A(\bar{w}) + \underbrace{(-u''(\bar{w})/u'(\bar{w}))}_{DRA(\bar{w})}$$

$$ \begin{aligned} & \text{TOTAL RISK AVERSION} \\ = & \text{PURE RISK AVERSION} + \\ & \text{DISPERSION RISK AVERSION (or usual risk aversion)} \end{aligned} $$

The reason for coefficient 2 to appear is that, according to tradition, we have defined risk aversion as twice the ratio of the premium to the variance of the payoffs.

Up to now, we have only defined the notions of absolute risk aversion. The following equality define how we can deal with the relative notions:

$$ \begin{aligned} TRR(\bar{w}) &= \bar{w}TRA(\bar{w}) = \bar{w}(PRA(\bar{w}) + DRA(\bar{w})) \\ &= \bar{w}\left(2A(\bar{w}) - u''(\bar{w})/u'(\bar{w})\right) \equiv PRR(\bar{w}) + DRR(\bar{w}) \end{aligned} $$

All the preceding results hold “in the small”, i.e. for infinitesimal risks. We could, now generalize Pratt's theorem and cope with risk in the large.

To spare space we limit here, as far as finite risks are considered, to selecting Markowitz' point of view, leading to an alternative definition of a risk premium, i.e. the difference between the utility of the expected gain $E_p[u(\tilde{w})]$ and the utility $U(\tilde{w})$ of the uncertain wealth $\tilde{w}$ which is but the expected utility of wealth using the transformed probabilities $E_{\square}[u(\tilde{w})]$. Hence:

$$ \begin{aligned} U(\tilde{w}) &= E_{\square}[u(\tilde{w})] = E_p[(1 - A(\tilde{w} - E_p[\tilde{w}]))u(\tilde{w})] \\ &= E_p[u(\tilde{w})] - ACov_p[\tilde{w}, u(\tilde{w})] \end{aligned} $$

and, finally:
$$ \underbrace{u(E_p[\tilde{w}]) - E_{\square}[u(\tilde{w})]}_{\text{TOTAL PREMIUM}} = \underbrace{u(E_p[\tilde{w}]) - E_p[u(\tilde{w})]}_{\text{MARKOWITZ PREMIUM}} + \underbrace{A(E_p[\tilde{w}]) \text{Cov}_p[\tilde{w}, u(\tilde{w})]}_{\text{PURE RISK PREMIUM}} $$

The new premium is the sum of two terms: the first, which is the Markowitz' premium, would correspond to the case of a VNM individual. The second corresponds to the behaviour of probability alteration.

## 2.6 The mean-variance criterion as dual theory

A particular case arises when individuals have P-risk aversion and a constant marginal utility<fnref n="11" />. This case is the analogue of that of the dual theory of Yaari [1987] for the RDEU model. Although we cannot bring here some empirical evidence we think it is probably a very frequent situation in the daily life, since institutional investors are likely to have both a marginal utility close to one, and a “true” behaviour facing risk. Assuming a constant marginal utility, the difference between the utility of expected wealth and its expected utility is:

$$ u(E_p[\tilde{w}]) - E_{\square}[\tilde{w}] = A \text{Var}_p[\tilde{w}] $$

One can view this relation as a justification of the utilization of the mean-variance criterion since the equation above can be written as follows:

$$ U(\tilde{w}) = E_{\square}[\tilde{w}] = u(E_p[\tilde{w}]) - A \text{Var}_p[\tilde{w}] $$

It must be underlined that this conclusion does not depend on whether A is independent of the lottery characteristics or not. One can explicitly take into account the dependency of A on $E_p[\tilde{w}]$ writing:

$$ U(\tilde{w}) = E_{\square}[\tilde{w}] = u(E_p[\tilde{w}]) - A(E_p[\tilde{w}]) \text{Var}_p[\tilde{w}] $$

where A is a decreasing function of $E_p[\tilde{w}]$. Then, $U(\tilde{w})$ is well an increasing function of $E_p[\tilde{w}]$.

<footnote n="11">¹¹The VNM axiomatics cannot suitably deal with this case except for admitting the risk neutral assumption of all investors.</footnote>
# 3 Market Equilibrium

We now address the issue of market equilibrium with individuals altering “cautiously” probabilities. Our analysis is threefold. We first consider the behaviour of a rational consumer, and we study the implications of the link between risk-neutral probabilities and subjective probabilities, assuming no arbitrage opportunities on a complete market. This leads us to characterize static market equilibrium. We show that the CAPM is the model that perfectly corresponds to the dual case of the constant marginal utility. Finally, we deal with intertemporal models and we propose new Euler equations.

## 3.1 The link between risk-neutral probabilities and altered probabilities

We consider the very simple case of one single period securities market. We assume there exists a finite set of states of the world, with $K$ elements; the value of each state is revealed to the investors at time $t = 1$. Two kinds of assets can be traded on this market:

There is one risk free asset noticed by the subscript 0 and $N$ risked assets labelled with the subscript $n$ ($n = 1, ..., N$). $S_0^n$ will denote the time 0 price of the security $n$, and, $S_k^n$ ($k = 1, ..., K$) the time 1 price of the security $n$. We have:

$$S_0^0 = 1 \text{ and } S_k^0 = \beta = 1 + r_0, \forall k = 1, ..., K$$

Where $r_0$ is the rate of return of the risk free asset, and, $\beta$ the discount factor. Since, by assumption, there is no arbitrage opportunity and that the market is complete, there exists a unique vector $q$ of risk neutral probabilities allowing for the asset evaluation by the usual formula:

$$S_0^n = \beta^{-1} \left( \sum_{k=1}^{K} q_k S_k^n \right) \text{ with } q_k > 0 \text{ and } \sum_{k=1}^{K} q_k = 1$$

We now assume that the preferences of our consumer on this market are represented by an additive and separable utility function, using $\alpha^{-1}$ as discount factor. We restrict our analysis to the case of linear alteration of probabilities. His utility function can be written as follows:

$$ \begin{aligned} U(w_0, \tilde{w}) &= u(w_0) + \alpha^{-1} \sum_{k=1}^{K} p_k \gamma_k^p u(w_k) \\ &= u(w_0) + \alpha^{-1} E_p [(1 - A(\tilde{w} - E_p[\tilde{w}])) u(\tilde{w})] \end{aligned} $$
With, if $\delta_{jk}$ is the Kronecker symbol:

$$ \gamma_k^p = 1 - A(w_k - E_p[\tilde{w}]) = 1 - A\left(w_k - \sum_{j=1}^{K} p_j w_j\right) \Rightarrow \sum_{j=1}^{K} p_j \gamma_j^p = 1 $$

$$ \frac{\partial \gamma_j^p}{\partial w_k} = A(p_k - \delta_{jk}) + A' p_k (E_p[\tilde{w}] - w_j) \Rightarrow \sum_{j=1}^{K} p_j \frac{\partial \gamma_j^p}{\partial w_k} = 0 $$

From now on, we shall define $\Gamma(k)$ as follows:

$$ \Gamma(k) = \sum_{j=1}^{K} p_j u(w_j) \frac{\partial \gamma_j^p}{\partial w_k} = \sum_{j=1}^{K} p_j u(w_j) (A(p_k - \delta_{jk}) + A' p_k (E_p[\tilde{w}] - w_j)) \\ = p_k (A(E_p[u(\tilde{w})] - u(w_k)) - A' \mathrm{Cov}_p(\tilde{w}, u(\tilde{w}))) $$

and therefore we have:

$$ \sum_{k=1}^{K} \Gamma(k) = \sum_{k=1}^{K} \sum_{j=1}^{K} p_j u(w_j) \frac{\partial \gamma_j^p}{\partial w_k} = \sum_{k=1}^{K} A p_k (E_p[u(\tilde{w})] - u(w_k)) \\ = -A' \mathrm{Cov}_p(\tilde{w}, u(\tilde{w})) $$

If the consumer has an initial wealth $W_0$, trading in competitive markets, he maximizes his expected utility using the transformed probabilities:

$MAXU(w_0, \tilde{w})$

Subject to the budget constraints:

$$ \text{i) } w_0 + \theta^0 + \sum_{n=1}^{N} \theta^n S_0^n - W_0 = 0 \Rightarrow w_0 = W_0 - \left( \theta^0 + \sum_{n=1}^{N} \theta^n S_0^n \right) $$

$$ \text{ii) } w_k - \beta \theta^0 - \sum_{n=1}^{N} \theta^n S_k^n = 0 $$

$$ \Rightarrow w_k = \beta \theta^0 + \sum_{n=1}^{N} \theta^n S_k^n, \quad k = 1, \dots, K $$
where $\theta^n$ is the agent's asset $n$ holdings at date 0, and $\tilde{w}$ its final consumption. We now examine how the asset pricing formula is modified when altered probabilities are taken into account: A straightforward calculation, detailed in Appendix 4, gives:

$$S_0^n = \frac{1}{\alpha u'(w_0)} \left( E_p[u'(\tilde{w}) \tilde{S}^n] - A \left( \begin{array}{c} \mathrm{Cov}_p[u'(\tilde{w}) \tilde{S}^n, \tilde{w}] \\ + \mathrm{Cov}_p[u(\tilde{w}), \tilde{S}^n] \end{array} \right) - A'E_p[\tilde{S}^n] \mathrm{Cov}_p[u(\tilde{w}), \tilde{w}] \right) \quad (8)$$

and, for the risk-free asset:

$$S_0^0 = 1 = \frac{1}{\alpha u'(w_0)} \left( \beta \left( E_p[u'(\tilde{w})] - A \mathrm{Cov}_p[u'(\tilde{w}), \tilde{w}] - A' \mathrm{Cov}_p[u(\tilde{w}), \tilde{w}] \right) \right)$$

Equation (8) exhibits an unusual new term whose numerator is the product of a factor, which is proportional to the “pure” risk aversion, by the sum of two covariances characterizing the risk implied by our probabilities’ deformation.

## 3.2 Static market equilibrium

The existence and the uniqueness of a static equilibrium depend on the behaviour of the utility function of the representative agent. We, here, have to show that the function below is continuous, increasing and quasi-concave in each of its arguments $w_k$:

$$\begin{align}
U(w_0, \tilde{w}) &= u(w_0) + \alpha^{-1} \sum_{k=1}^{K} p_k \gamma_k^p u(w_k) \tag{9} \\
&= u(w_0) + \alpha^{-1} \sum_{k=1}^{K} p_k \lambda_p (1 - A(w_k - E_p[\tilde{w}])) u(w_k) \nonumber
\end{align}$$

**Proposition 4** Function $U(w_0, \tilde{w})$ defined by relation (9) is continuous, increasing and quasi-concave in each of its arguments $w_k$.

**Proof:**

- The continuity is trivial since we have supposed A and $u$ are continuous.
- The proof that $U(w_0, \tilde{w})$ is an increasing and quasi-concave function is given in Appendix 5.

The conditions above allow us to conclude that the existence and the uniqueness of the market equilibrium is established under the assumption of probabilities alteration as in the VNM case.

## 3.3 Constant marginal utility and constant risk aversion

We turn again to the very interesting case of constant marginal utility. The link between subjective and risk neutral probabilities can be written as follows:

$$q_k - \square_k = q_k - p_k \gamma_k^p = p_k A (E_p[\tilde{w}] - w_k)$$

**Proposition 5** The difference between the risk neutral probability and the subjective probability of a consumer, whose both marginal utility of wealth and risk aversion are constant, is equal to the product of three terms: his risk aversion, the original probability and the difference between the expected outcome and the current value of the outcome.

**Proof: See Appendix 4.**

Finally, note that if the risk aversion tends to zero, the subjective probability tends to the risk neutral probability. The limiting case is that of VNM with constant marginal utility that is the case of risk neutral agents (in the VNM sense). We now turn to this particular case. Analogous calculations to these undertaken in the general case allow us to write:

$$S_0^n = \sum_{k=1}^{K} \frac{1}{\alpha} p_k [(1 + 2A(E_p[\tilde{w}] - w_k))] S_k^n$$

hence,

$$S_0^n = \frac{1}{\alpha} \left( E_p[\tilde{S}^n] + 2AE_p[\tilde{S}^n] E_p[\tilde{w}] - 2AE_p[\tilde{S}^n \tilde{w}] \right) \\ = \frac{1}{\alpha} \left( E_p[\tilde{S}^n] - 2ACov_p[\tilde{S}^n, \tilde{w}] \right)$$

For the risk-free asset (n=0), we have:
$$1 = \frac{1}{\alpha} (1 + r_0) \Leftrightarrow \beta = \alpha$$

The expected return of a risked asset is done by:

$$E_P[\tilde{r}^n] = \beta - 1 + 2ACov[\tilde{w}, \tilde{r}^n] \quad (10)$$

At the equilibrium, the risk premium is equal to the product between the risk aversion and the covariance between the rate return of the asset and the income of the consumer. To go on with our analysis, we now suppose there exists a representative consumer. Let us denote $M_0$ and $M$ the values of the market portfolio in dates 0 and 1, then:

$$\tilde{w} = \beta \left( W_0 - w_0 - \sum_{j=1}^{N} \theta_k^j S_0^j \right) + \sum_{j=1}^{N} \theta_k^j \tilde{S}_k^j = M + \beta (W_0 - w_0 - M_0)$$

If $\tilde{r}^M$ designates the rate of return of the market portfolio, we have $M = M_0(1 + \tilde{r}^M)$, and, for the representative consumer:

$$Cov(\tilde{w}, \tilde{r}^n) = \sum_{j=1}^{N} \theta_k^j Cov(\tilde{S}^j, \tilde{r}^n) = Cov(M, \tilde{r}^n) = M_0 Cov(\tilde{r}^M, \tilde{r}^n)$$

Using this result and Equation (10), we can write for all assets:

$$E_P[\tilde{r}^n] = r_0 + 2AM_0Cov(\tilde{r}^M, \tilde{r}^n) \quad (11)$$

And, for the market portfolio:

$$E_P[\tilde{r}^M] = r_0 + 2AM_0Var[\tilde{r}^M] \quad (12)$$

Note that $Cov(\tilde{r}^M, \tilde{r}^n) = \beta_M^n Var[\tilde{r}^n]$. Using (11) and (12), leads to write the CAPM equation:

$$E_P[\tilde{r}^n] = r_0 + \beta_M^n (E_P[\tilde{r}^M] - r_0)$$

It is well known that; in the VNM universe, the CAPM relation can be established if and only if at least one of the two following conditions is verified:

- The asset returns are distributed following by laws fully defined from their two first moments;

- The investors utility functions are quadratic;
Nevertheless, it is not less known that the second assumption implies a result invalidated by empirical works: the absolute risk aversion in the Arrow-Pratt sense is not an increasing function of wealth. The first assumption leads to consider the normality of the asset returns. But it is contradicted by the empirical observations even if we consider the interesting Lévy-stable laws.

Hence, it is somewhat paradoxical that the CAPM has been so popular, that it remains the reference for any empirical work on the stock markets and that fund managers still continue to use the mean-variance criterion. Such a paradox can be levied, if one assumes that investors, especially institutional investors, who play the most important role on financial markets, have both a constant marginal utility and a P-risk aversion; if so, the market equilibrium must be in accordance with the CAPM. This justification of the use of the CAPM appears to us more convincing than that of the one traditionally made, in terms of quadratic utility functions or normal distributions.

Finally, to confirm the interest of our theory alternative to the RDEU model, we establish the Euler equations within the framework of an intertemporal asset pricing model with alteration of probabilities.

## 3.4 The Consumption Based Asset Pricing Model with probabilities alteration:

We only consider, for the sake of simplicity, the case of linear alteration of probabilities and we can write, using the same labelling as before, the maximization program of the representative consumer as follows:

$$MAX \quad E_{\square,0} \left[ \sum_{t=0}^{\infty} \alpha^{-t} u(\tilde{w}_t) \right]$$

subject to the budget constraints:

$$i) W_t = \sum_{n=1}^{N} \theta_t^n S_t^n + w_t$$

$$ii) W_{t+1} = \sum_{n=1}^{N} \theta_t^n S_{t+1}^n$$

Note that current wealth $Wt$ is a state variable and that current asset holdings $\theta_t^n$ are the control variables of this program; we can thus write the following Bellman equation:
$$U(W_t, \vec{S}_t) = \underbrace{MAX}_{\vec{\theta}_t} \left[ u \left( W_t - \sum_{n=1}^{N} \theta_t^n S_t^n \right) + \alpha^{-1} E_{p,t} \left[ \gamma^p(W_{t+1}) U(W_{t+1}, \vec{S}_{t+1}) \right] \right]$$

Appendix 6 provides the resolution of this program. The corresponding Euler equations are:

$$S_t^n = \frac{1}{\alpha u'(w_t)} \left[ \begin{array}{c} E_{p,t} \left[ \tilde{S}_{t+1}^n u'(W_{t+1}) \right] \\ -A' E_{p,t} [W_{t+1}] Cov_{p,t} (W_{t+1}, u(W_{t+1})) \\ -A \left( Cov_{p,t} (W_{t+1}, \tilde{S}_{t+1}^n u'(W_{t+1})) + Cov_{p,t} (u(W_{t+1}), \tilde{S}_{t+1}^n) \right) \end{array} \right] \quad \text{for } n = \{0, \dots, N\} \qquad (13)$$

# 4 The equity premium puzzle and the cautious alteration of probabilities

In this section we show the consistency of our new measure to deal with a well known anomaly in the financial literature: The Equity Premium Puzzle. In a seminal paper, Mehra and Prescott [1985] found that the Lucas model with an expected utility maximizer was not able to account for the average equity premium over the period 1889-1978. Actually, the model can only produce an average equity premium of 0.35% instead of 6.18% observed in the US data.

The model used is based on three main assumptions<fnref n="12" /> described in the recent survey of Kocherlakota [1996]. The different attempts to solve this puzzle had been made by relaxing one of these assumptions. Nevertheless, the main approach used in the literature consists in generalizing preferences of the representative consumer in order to introduce non-separabilities in the state space or in the temporal space. But, as Kocherlakota noticed, no one of these approaches have totally succeeded in the resolution of this puzzle.

<footnote n="12">¹²i) Expected Utility Maximizer</footnote>

<footnote>ii) Asset markets are complete</footnote>

<footnote>iii) Asset trading is costless</footnote>
To evaluate the pertinence of our new measure of probability, we cali-
brate<fnref n="13" /> our new Euler equations (13) in a Mehra and Prescott framework.
We simulate a theoretical average equity premium that we compare with that
observed on the period 1889-1978 in the United States.

The calibration needs the use of two kinds of parameters: (i) those defin-
ing the preferences of the representative agent<fnref n="14" />, and (ii) those defining the
technology of the economy<fnref n="15" />. The latter parameters are selected so that they
match the sample values for the US economy over the period 1889-1978.

To avoid complex calculations, we make the assumption that our decision
maker has a constant marginal utility and a constant relative risk aversion<fnref n="16" />.
Figure 1 gives the average simulated equity premia corresponding to the
Mehra and Prescott original model, in which, as the authors did, we have
introduced a leverage effect. Indeed we have allowed θ to vary in the interval
[0,0.975]. The maximal premium simulated by the model does not exceed
0.7%. This value is much less than the observed equity premium (6.18%)
and it is obtained with an induced risk free rate close to 4%, which is much
higher than that observed (0.8%).

In order to compare our results with those of Mehra and Prescott, we
have used similar specifications:

i) we impose a maximal value of 4% for the simulated risk free
rate, which does produce an interval of [0.96,1] for the subjective
discount factor;

ii) We limit the values of the simulated equity premia to those
which correspond to a relative risk aversion coefficient less than
10. If we neglect rare events, or if the lottery is not very risky<sup>17</sup>,
the condition (5) is not restritive and does not limit our analysis
since it authorizes a value of *a* up to 20. In our framework, we
consider a sequence of one period models where a risky asset (or a
group of risky assets such as a portfolio replicating a stock index)
plays the role of the lottery, and where an individual invests his
wealth partly in the risky asset and partly in the safe asset. For

<footnote n="13"><sup>13</sup>For details on the technical appendix, see Chauveau and Nalpas [1998].</footnote>

<footnote n="14"><sup>14</sup>His subjective discount factor and his relative risk aversion coefficient.</footnote>

<footnote n="15"><sup>15</sup>The average growth rate of the per capita consumption on non-durables and services, its standard deviation, and its first-order serial correlation.</footnote>

<footnote n="16"><sup>16</sup>As in the Merha and Prescott study.</footnote>

<footnote n="17"><sup>17</sup>When *b* is very close to *E*<sub>*L*</sub>[w̃].</footnote>
instance, if we consider an investor whose risky part of wealth is less than 10%, condition (5) implies a maximum value of 20 for the relative risk aversion coefficient, even if we take into account the extremal events which have occured on the stock market over the period 1889-1978¹⁸.

The values of the equity risk premia obtained with our model are plotted on the figure 2.

Thus, our model is able to produce both high equity premia and low risk free rates. It can be considered as a model which solves the equity premium puzzle. Details can be found in Chauveau and Nalpas [1998].

<footnote n="18">¹⁸We could observe annual return greater than 50%.</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 2: Simulated Equity Premia for The Risk Weighted Utility Theory</figcaption></figure>

<figure><img src="image_6.png" /></figure>
# 5 Conclusion

We have developed a theory of probabilities’ alteration, which is an alternative to the Rank Dependent Expected Utility model and where the deformation of probabilities only depends on the centered outcome of the lottery considered. We have distinguished the true behaviour towards risk, which is identified to a cautious, neutral, or audacious alteration of objective probabilities and the traditional so called behaviour towards risk, which can be identified to the importance of the concavity of the utility function. Three main results have been obtained:

- (i) the risk premium is the sum of the Arrow-Pratt premium that is, in reality, only a concavity premium, and a pure risk premium characterizing the intensity of the deformation of probabilities; such a result holds in the small and in the large<fnref n="19" />;

- (ii) an appealing justification for utilizing the mean-variance criterion, and, consequently for referring to the CAPM is the existence of investors having both a constant marginal utility and a pure risk aversion;

- (iii) the Euler equations of an intertemporal model are modified by taking account of, on the one hand the covariance between terminal wealth and its utility and, on the other one, the covariance between terminal wealth and the product of its marginal utility by the price of the asset. The use of these new Euler equations has enabled us to obtain results much more realistic than those of Mehra and Prescott [1985].

Three final remarks can be made: one could believe that assuming a prudent alteration of probabilities may challenge the validity of the CAPM, in the case of decreasing marginal utility. This is obviously not true since a Taylor development to the second (third, ...) order of the utility function allows for the 3CAPM (4CAPM, ...) to prevail. Second, the ideas that we have advanced in this study concerning cautious alteration may be generalized in several directions among which one appears us more particularly interesting. Relaxing the monotonicity assumption of the alteration function should allow taking account of observed behaviours as in the case of lotto games.

<footnote n="19">¹⁹However we have limited ourselves to considering Markowitz’ point of view and not Pratt’s, for risk in the large.</footnote>
## REFERENCES

ALLAIS M. [1953], “Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école américaine”, *Econometrica*, 21, p. 503-546.

ALLAIS M. [1986], “The General Theory of Random Choices in Relation to the Invariant Cardinal Utility Function and the Specific Probability Function (U,Q) Model a General Overview”, *Risk, Decision and Rationality, Dordrecht*, D. REIDEL, p. 231-289.

ARROW K. J. [1965], “Aspects of the Theory of Risk Bearing”, Helsinki.

BERNARD G. [1964], “Réduction du paradoxe de Saint Petersbourg par la théorie de l’utilité”, *Comptes rendus de l’Académie des Sciences*, PARIS, 259, p. 3168-3170.

BERNOULLI D. [1738], “Specimen theoriae novae de mensurasortis”, *Comentarii Academiae Scientarum Imperialis Petropolitanae* (vol. 5), translated in *Econometrica*, January 1954 under the title: “Exposition of a new theory on the new measure of risk”.

CHATEAUNEUF A. [1990], “On the Use of Comonotonicity in the Axiomatization of EURDP Theory for arbitrary Consequences”, *mimeo CERSEM*, Université de Paris I.

CHAUVEAU T., NALPAS N. [1998], “L’altération prudente des probabilités comme solution à l’énigme de la prime de risque”, *Working Paper of the Caisse des Dépôts et Consignations*, n°98-04.

COHEN M. [1992], “Expected utility, security level, potential level: a three criteria decision model under risk”, *Theory and Decision*, 33, p. 101-134.

COHEN M., JAFFRAY J.Y. [1998], “Certainty Effect versus probability distortion: an experimental Analysis of Decision Making under Risk”, *Journal of Experimental Psychology*, H.P.P., 14 (4), p.554-560.

DANA R.A., JEANBLANC-PICQUE M. [1994], “Marchés Financiers en Temps Continu”, *Economica*, Paris.

EDWARDS W. [1954], “The Theory of Decision Making”, *Psychological Bulletin*, 51, p. 380-417.

EDWARDS W. [1962], “Subjective Probabilities inferred from Decisions”, *Psychological Review*, 69, p. 109-135.

GAYANT J.M. [1995], “Généralisation de l’espérance d’utilité en univers risqué: représentation et estimation”, *Revue Economique*, 46, p. 1047-1061.
GAYANT J.M. [1997], “Décroissance de l’utilité marginale et aversion probabiliste pour le risque”, *Revue d’Economie Politique*, 107 (3), p. 331-342.

HANSEN L.P., SINGLETON K. [1982], “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models”, *Econometrica*, 50 (5), p. 1269-1286.

JAFFRAY J.Y. [1988], “Choice under risk and the security factor”, *Theory and Decision*, 24.

KAHNEMAN D., TVERSKY A. [1979], “Prospect Theory: an analysis of Decision under Risk”, *Econometrica*, 47, p. 263-291.

KIMBALL M.S. [1990], “Precautionary Saving in the Small and in the Large”, *Econometrica*, 58, p. 53-73.

KOCHERLAKOTA N. [1996], “The Equity Premium: It’s Still a Puzzle”, *Journal of Economic Literature*, 34, p. 43-71.

MACHINA M. J. [1982], “Expected Utility Analysis without the Independence Axiom”, *Econometrica*, 50, p. 277-323.

MACHINA M. J. [1989], “Comparative Statics and Non-expected Utility Preferences”, *Journal of Economic Theory*, 47, p. 393-405.

MEHRA R., PRESCOTT E.C. [1985], “The Equity Premium: A Puzzle”, *Journal of Monetary Economics*, 15, p. 145-161.

MUNIER B. [1995], “Entre rationalités instrumentale et cognitive: contributions de la dernière décennie à la modélisation du risque”, *Revue d’Economie politique*, 105, p. 5-70.

PRATT. J. [1964], “Risk Aversion in the Small and in the Large”, *Econometrica*, 49, p. 621-638.

QUIGGIN J. [1982], “A Theory of Anticipated Utility”, *Journal of Economic behaviour and Organization*, 3, p. 323-343.

ROTHSCHILD M., STIGLITZ J. [1970], “Increasing Risk: I. A definition”, *Journal of Economic Theory*, 2, p.225-243.

ROUSSEAU J.M. [1995], “Risque et décision financière dans le cadre de la théorie du portefeuille”, *Revue d’Economie politique*, 105, p. 91-131.

SCHMEIDLER D. [1989], “Subjective Probability and Expected Utility without Additivity”, *Econometrica*, 57, p. 571-587.

SEGAL U. [1989], “Anticipated Utility: A Measure Representation Approach”, *Annals of Operational Research*, 19, p. 359-373.

TVERSKY A. [1967], “Additivity, Utility and subjective Probability”, *Journal of Mathematical Psychology*, 4, p. 175-201.
VON NEUMANN J., MORGENSTERN O. [1944], “Theory of Games and Economic behaviour”, *Princeton University Press*.

WAKKER P. [1994], “Separating Marginal Utility and Probabilistic Risk Aversion”, *Theory and Decision*, 36, p. 1-44.

YAARI M. [1987], “The Dual Theory of Choice under Risk”, *Econometrica*, 55, p. 95-105.
# APPENDIX 1:

## Characterization of the canonical risk weighted changes of measure

Consider two lotteries $L_p$ and $L_q$ and the two canonical changes of measure:

$$ \frac{d\Phi_{L_j}(\tilde{w})}{dF_{L_j}} = 1 - A(E_{L_j}[\tilde{w}]) (\tilde{w} - E_{L_j}[\tilde{w}]); \quad j = p, q $$

We assume that the cumulative distributions are absolutely continuous; hence:

$$ \frac{d\Phi_{L_j}(\tilde{w})}{dw} = \varphi_{L_j}(w) \quad \text{and} \quad \frac{dF_{L_j}(\tilde{w})}{dw} = f_{L_j}(w); \quad j = p, q $$

and finally

$$ \varphi_{L_j}(w) = (1 - A(E_{L_j}[\tilde{w}]) (\tilde{w} - E_{L_j}[\tilde{w}])) f_{L_j}(w); \quad j = p, q $$

We shall establish that the correspondence $T_A$ is always injective (statement (a)) and that (statement (b)) a sufficient condition for $T_A$ to be sufficient is that A obeys the two following conditions:

$$ A'(w) \le 0 \quad \text{and} \quad wA'(w) \ge 0 $$

(a) we have to establish the following implication:

$$ L_p = L_q \Rightarrow T_A(L_p) = T_A(L_q) $$

To do so, note that we have, by assumption:

$$ \begin{align*} L_p = L_q &\Rightarrow f_{L_p}[\tilde{w}] = f_{L_q}[\tilde{w}] \\ &\Rightarrow E_{L_p}[\tilde{w}] = E_{L_q}[\tilde{w}] \Rightarrow A(E_{L_p}[\tilde{w}]) = A(E_{L_q}[\tilde{w}]) \end{align*} $$

hence:

$$ \varphi_{L_p}[\tilde{w}] = \varphi_{L_q}[\tilde{w}] $$

(b) We have to find a sufficient condition for the following implication:
$$T_A(L_p) = T_A(L_q) \Rightarrow L_p = L_q$$

If without loss of generality we assume that $E_p[w] \le E_q[w]$ then: if $A'(w) \le 0$ and $wA'(w) \ge 0$

we get:

$$A(E_p[w]) \ge A(E_q[w]) \text{ and } E_p[w] A(E_p[w]) \le E_q[w] A(E_q[w])$$

Hence:

$$(1 - A(E_{Lp}[\tilde{w}])) (\tilde{w} - E_{Lp}[\tilde{w}])) \le (1 - A(E_{Lq}[\tilde{w}])) (\tilde{w} - E_{Lq}[\tilde{w}])) \quad \forall w \in [a, b]$$

since, by assumption, we have:

$$\varphi_{Lp}[\tilde{w}] = \varphi_{Lq}[\tilde{w}]$$

we get:

$$\varphi_{Lp}(\tilde{w}) / \left( 1 - A(E_{Lp}[\tilde{w}]) \begin{pmatrix} \tilde{w} \\ -E_{Lp}[\tilde{w}] \end{pmatrix} \right) \\ \ge \varphi_{Lq}(\tilde{w}) / \left( 1 - A(E_{Lq}[\tilde{w}]) \begin{pmatrix} \tilde{w} \\ -E_{Lq}[\tilde{w}] \end{pmatrix} \right)$$

or:

$$f_{Lp}[\tilde{w}] \ge f_{Lq}[\tilde{w}]$$

but we have:

$$\int_a^b f_{Lp}(w) dw = \int_a^b f_{Lq}(w) dw = 1$$

⇔

$$\int_a^b (f_{Lp}(w) - f_{Lq}(w)) dw = 0$$
The difference between the two functions is positive and its integral is zero; consequently we must have:

$$f_{Lp}[\tilde{w}] - f_{Lq}[\tilde{w}] = 0$$

This establishes the sufficiency condition.

# APPENDIX 2:

## Stochastic Dominance

We have, by definition, for any lottery $L_p$:

$$U(L_p) = \int_a^b u(x) d\Phi_p(x) = \int_a^b u(x) (1 - A_p(x - x_p)) dF_p(x)$$

with:

$$x_q = E_q[x]; x_p = E_p[x]$$

$$A_p = A(E_p[x]); A_q = A(E_q[x])$$

Hence:

$$\begin{align*} U(L_p) - U(L_q) &= \int_a^b u(x) \left[ \begin{array}{l} (1 - A_p(x - x_p)) dF_p(x) \\ -(1 - A_q(x - x_q)) dF_q(x) \end{array} \right] \\ &= \int_a^b u(x) (dF_p(x) - dF_q(x)) \\ &\quad + \int_a^b xu(x) (A_q dF_q(x) - A_p dF_p(x)) \\ &\quad + \int_a^b u(x) (A_p x_p dF_p(x) - A_q x_q dF_q(x)) \end{align*}$$

or

$$\begin{align*} U(L_p) - U(L_q) &= \int_a^b u(x) (1 - A_q x + A_p x_p) (dF_p(x) - dF_q(x)) \quad (14) \\ &\quad + \int_a^b xu(x) (A_q - A_p) dF_p(x) \\ &\quad + \int_a^b u(x) (A_p x_p - A_q x_q) dF_q(x) \end{align*}$$
Suppose that $L_p$ dominates $L_q$ with respect to the first-order stochastic dominance property. Then, we have:

$$ \forall x \in [a, b], F_p(x) \leq F_q(x) \text{ and } x_q \leq x_p. $$

Since $A' \leq 0$ and $(xA)' \geq 0$ we have:

$$ A_p \leq A_q \text{ and } A_p x_p \geq A_q x_q $$

This establishes that the last two integrals are positive. Moreover, if $L_p$ dominates $L_q$, it implies that for any increasing function, $v_p(x)$, $\int_a^b v_p(x)$ $(dF_p(x) - dF_q(x))$ is positive; hence the first integral in Equation (14) is also positive if $v_p(x) = u(x)(1 - A_q x + A_p x_p)$ is increasing.

Note that if $w_p(x) = u(x)(1 - A_p x + A_p x_p)$ is increasing $v_p(x)$ is also increasing. Hence a sufficient condition for $U(L_p) - U(L_q)$ to be positive, given that $L_p$ dominates $L_q$ is that:

$u(x)(1 - A_p x + A_p x_p)$ is increasing i.e. that:

$$ F(x) = u'(x)(1 - A_p x + A_p x_p) - A_p u(x) \geq 0 $$

One can easily check that if $u''(x) \leq 0$ then

$$ F'(x) = u''(x)(1 - A_p x + A_p x_p) - 2A_p u'(x) \leq 0 $$

Therefore the minimum value for $F(x)$ is $F(b)$ i.e.:

$$ F(b) = u'(b)(1 - A_p b + A_p x_p) - A_p u(b) $$

and the condition

$$ F(b) \geq 0 $$

is equivalent to

$$ u'(b) \geq A_p((b - x_p)u'(b) + u(b)) $$

or

$$ A_p \leq 1/((b - x_p) + u(b)/u'(b)) $$

or equivalently

$$ A(E_p[\tilde{x}]) \leq 1/((b - x_p) + 1/\square(b)) $$

This condition will hold for all the lotteries if

$$ A(E_p[\tilde{x}]) \leq \frac{1}{((b - E_p[\tilde{x}]) 1/\square(b))} $$
# APPENDIX 3:

## Calculation of the risk premium

We write therefore, for an additive risk with usual notations:

$$ \tilde{x} = \bar{x} + \tilde{\varepsilon} \quad , \quad E_p[\tilde{\varepsilon}] = 0 \quad , \quad E_p[\tilde{x}] = \bar{x} \quad , \quad \mathrm{Var}_p[\tilde{x}] = \mathrm{Var}_p[\tilde{\varepsilon}] = \Box^2 $$

The utility of this lottery is:

$$ \begin{align*} U(\tilde{x}) &= E_p[\gamma_p(\tilde{x}) u(\tilde{x})] = E_p[(1 - A(\tilde{x} - E_p[\tilde{x}])) u(\tilde{x})] \\ &= E_p\left[(1 - A(\tilde{x} - E_p[\tilde{x}])) \begin{pmatrix} u(\bar{x}) \\ + (\tilde{x} - E_p[\tilde{x}]) u'(\bar{x}) \\ + \frac{1}{2} (\tilde{x} - E_p[\tilde{x}])^2 u''(\bar{x}) \\ + o((\tilde{x} - E_p[\tilde{x}])^2) \end{pmatrix} \right] \\ &= u(\bar{x}) + u'(\bar{x}) E_p[(\tilde{x} - E_p[\tilde{x}]) (1 - A(\tilde{x} - E_p[\tilde{x}]))] \\ &\quad + \frac{1}{2} u''(\bar{x}) E_p[(\tilde{x} - E_p[\tilde{x}])^2 (1 - A(\tilde{x} - E_p[\tilde{x}]))] \end{align*} $$

So, if we limit to second order terms<fnref n="20" />:

$$ U(\tilde{x}) = u(\bar{x}) - \left( Au'(\bar{x}) - \frac{1}{2} u''(\bar{x}) \right) \mathrm{Var}_p[\tilde{x}] $$

The certainty equivalent of this lottery is defined by:

$$ U(\tilde{x}) = u(EC(\tilde{x})) $$

And the risk premium is:

$$ PR(\tilde{x}) = \bar{x} - EC(\tilde{x}) $$

Combining the preceding equations, we get:

$$ U(\tilde{x}) = u(EC(\tilde{x})) = u(\bar{x} - PR(\bar{x})) = u(\bar{x}) - PR(\bar{x})u'(\bar{x}) + o(PR(\bar{x})) $$

And, we finally have:

$$ PR(\bar{x}) = AVar_p[\tilde{x}] - \frac{1}{2} \frac{u''(\bar{x})}{u'(\bar{x})} Var_p[\tilde{x}] $$

<footnote n="20">²⁰See Dana and Jeanblanc-Picque [1994].</footnote>
# APPENDIX 4:

## Static maximization program with probabilities alteration

$$ \underset{\square}{\operatorname{MAX}} E_{\square}[U(w_0, \tilde{w})] $$

subject to the contraints:

$$ \begin{array}{l} \text{i) } x_0 + \theta^0 + \sum_{n=1}^{N} \theta^n S_0^n - W_0 = 0 \\ \qquad \Rightarrow x_0 = W_0 - \left( \theta^0 + \sum_{n=1}^{N} \theta^n S_0^n \right) \end{array} $$

$$ \begin{array}{l} \text{ii) } x_k - \beta\theta^0 - \sum_{n=1}^{N} \theta^n S_k^n = 0 \\ \qquad \Rightarrow x_k = \beta\theta^0 + \sum_{n=1}^{N} \theta^n S_k^n \quad , k = 1, \dots, K \end{array} $$

Where $\theta^n$ is the date 0 asset n holding.

The Lagrangian function of this program is:

$$ \begin{aligned} L = & U(w_0, \tilde{w}) - \lambda_0 \left( w_0 + \theta^0 + \sum_{n=1}^{N} \theta^n S_0^n - W_0 \right) \\ & - \sum_{k=1}^{K} \lambda_k \left( w_k - \beta\theta^0 - \sum_{n=1}^{N} \theta^n S_k^n \right) \end{aligned} $$

The first order conditions are:

$$ \square \frac{\partial L}{\partial w_0} = u'(w_0) - \lambda_0 = 0 \Rightarrow \lambda_0 = u'(w_0) $$

$$ \begin{array}{l} \square \frac{\partial L}{\partial w_k} = \alpha^{-1} \left[ p_k \gamma_k^p u'(w_k) + \sum_{j=1}^{K} p_j u(w_j) \frac{\partial \gamma_j^p}{\partial w_k} \right] - \lambda_k = 0 \\ \qquad \Rightarrow \lambda_k = \alpha^{-1} \left[ p_k \gamma_k^p u'(w_k) + \Gamma(k) \right] \quad , k = 1, \dots, K \end{array} $$

$$ \square \frac{\partial L}{\partial \theta^0} = -\lambda_0 + \left( \sum_{k=1}^{K} \lambda_k \right) \beta = 0 \Rightarrow \sum_{k=1}^{K} \lambda_k = \beta^{-1} u'(w_0) $$
$$ \square \frac{\partial L}{\partial \theta^n} = -\lambda_0 S_0^n + \left( \sum_{k=1}^{K} \lambda_k S_k^n \right) = 0, \quad n = 1, \dots, N $$

Combining the third and last equations, we get:

$$ S_0^n = \frac{\sum_{k=1}^{K} \lambda_k S_k^n}{\alpha u'(w_0)} = \beta^{-1} \frac{\sum_{k=1}^{K} \lambda_k S_k^n}{\sum_{k=1}^{K} \lambda_k} = \beta^{-1} \sum_{k=1}^{K} \Box_k S_k^n $$

$$ \text{With } \Box_k = \frac{\lambda_k}{\sum_{j=1}^{K} \lambda_j} \ge 0 \quad \text{and} \quad \sum_{k=1}^{K} \Box_k = 1 $$

The equilibrium is defined by the condition that the $\Box_k$ are equal to the risk neutral probabilities $q_k$. Thus, we have the solution of our program:

$$ S_0^n = \frac{\sum_{k=1}^{K} \lambda_k S_k^n}{\alpha u'(w_0)} = \frac{1}{\alpha u'(w_0)} \sum_{k=1}^{K} \left( p_k \gamma_k^p u'(w_k) S_k^n + \Gamma(k) S_k^n \right) $$

Since,

$$ \begin{align*} & \sum_{k=1}^{K} \left( p_k u'(w_k) S_k^n \left( (1 - A(w_k - E_p[\tilde{w}])) \right) \right) \\ &= E_p[u'(\tilde{w}) \tilde{S}] - E_p[A(\tilde{w} - E_p[\tilde{w}]) u'(\tilde{w}) \tilde{S}] \\ &= E_p[u'(\tilde{w}) \tilde{S}] - A \mathrm{Cov}_p(\tilde{w}, u'(\tilde{w}) \tilde{S}) \end{align*} $$

And,

$$ \Gamma(k) = p_k (A(E_p[u(\tilde{w})] - u(w_k)) - A'\mathrm{Cov}_p(\tilde{w}, u(\tilde{w}))) $$

Therefore,

$$ \begin{align*} \sum_{k=1}^{K} \Gamma(k) S_k^n &= \sum_{k=1}^{K} S_k^n p_k (A(E_p[u(\tilde{w})] - u(w_k)) - A'\mathrm{Cov}_p(\tilde{w}, u(\tilde{w}))) \\ &= -A\mathrm{Cov}_p(u(\tilde{w}), \tilde{S}) - A'E_p[\tilde{S}] \mathrm{Cov}_p(\tilde{w}, u(\tilde{w})) \end{align*} $$
So, we get:

$$S_0^n = \frac{1}{\alpha u'(w_0)} \left( \begin{array}{c} E_p[u'(\tilde{w}) \tilde{S}^n] \\ -A(Cov_p[u'(\tilde{w}) \tilde{S}^n, \tilde{w}] + Cov_p[u(\tilde{w}), \tilde{S}^n]) \\ -A'E_p[\tilde{S}^n] Cov_p[u(\tilde{w}), \tilde{w}] \end{array} \right)$$

# APPENDIX 5: Proof of Proposition 3

## (a) Quasi-concavity property

To spare space we limit to the case when A is constant. We have to demonstrate that the alteration of probabilities leads to the strong quasi-concavity of U.

Let $w^1$ and $w^2$ be the income vectors of the individual in the different states of world: $w^i = (w_1^i, ..., w_k^i, ..., w_K^i)$ for $i = 1, 2$

Let w be a linear convex combination of $w^1$ and $w^2$,

$$w = \alpha w^1 + (1 - \alpha) w^2, \quad \alpha \in [0, 1]$$

We assume the inequality:

$$U(w^1) < U(w^2)$$

$U(\cdot)$ is quasi-concave if and only if $U(w^1) < U(w^2) \Rightarrow U(w^1) < U(w^2)$

Since:

$$\begin{align*} U(w) &= \sum_{j=1}^{K} p_j u(w_j) (1 + A(\bar{w} - w_j)) \\ &= \sum_{j=1}^{K} p_j u(\alpha w_j^1 + (1 - \alpha) w_j^2) (1 + A(\bar{w} - w_j)) \\ &= \sum_{j=1}^{K} p_j (\alpha u(w_j^1) + (1 - \alpha)u(w_j^2) + \delta_j) (1 + A(\bar{w} - w_j)) \end{align*}$$
Since $u$ is concave, $\delta$ is a positive random variable. Hence we have:

$$
\begin{align*}
U(w) ={}& \sum_{j=1}^{K} p_j \delta_j (1 + A(\bar{w} - w_j)) \\
& + \alpha \sum_{j=1}^{K} p_j u(w_j^1) (1 + A(\bar{w} - w_j)) \\
& + (1 - \alpha) \sum_{j=1}^{K} p_j u(w_j^2) (1 + A(\bar{w} - w_j))
\end{align*}
$$

So,

$$
\begin{align*}
U(w) ={}& \sum_{j=1}^{K} p_j \delta_j (1 + A(\bar{w} - w_j)) \\
& + \alpha \sum_{j=1}^{K} p_j u(w_j^1) (1 + A(\bar{w}^1 - w_j^1)) \\
& + \alpha \sum_{j=1}^{K} p_j u(w_j^1) A((\bar{w} - \bar{w}^1) - (w_j - w_j^1)) \\
& + (1 - \alpha) \sum_{j=1}^{K} p_j u(w_j^2) (1 + A(\bar{w}^2 - w_j^2)) \\
& + (1 - \alpha) \sum_{j=1}^{K} p_j u(w_j^2) A((\bar{w} - \bar{w}^2) - (w_j - w_j^2))
\end{align*}
$$

Hence:

$$
\begin{align*}
U(w) ={}& E_{\square}[\delta] + \alpha U(w^1) + (1-\alpha) U(w^2) \\
& + \alpha \sum_{j=1}^{K} p_j u(w_j^1) A(1-\alpha) ((\bar{w}^2 - \bar{w}^1) - (w_j^2 - w_j^1)) \\
& - (1-\alpha) \sum_{j=1}^{K} p_j u(w_j^2) A\alpha ((\bar{w}^2 - \bar{w}^1) - (w_j^2 - w_j^1))
\end{align*}
$$
$$
\begin{align*}
&= E_{\square}[\delta] + \alpha U(w^1) + (1 - \alpha) U(w^2) + \\
&\qquad A\alpha(1 - \alpha) \sum_{j=1}^{K} p_j (u(w_j^1) - u(w_j^2)) ((\bar{w}^2 - \bar{w}^1) - (w_j^2 - w_j^1))
\end{align*}
$$

Hence, we get:

$$
\begin{equation} \tag{15}
\begin{split}
U(w) - U(w^1) ={}& E_{\square}[\delta] + (1-\alpha)(U(w^2) - U(w^1)) \\
& + A\alpha(1-\alpha)\operatorname{Cov}_p(u(w^2) - u(w^1), w^2 - w^1)
\end{split}
\end{equation}
$$

Since the following identity prevails:

$$
\begin{align*}
& \sum_{j=1}^{K} p_j \left[ \begin{array}{@{}l@{}} \left( \bar{u} (w_j^1) - \bar{u} (w_j^2) \right) \\ \left( (\bar{w}^2 - \bar{w}^1) - (w_j^2 - w_j^1) \right) \end{array} \right] \\
&= \left( (\bar{u} (w_j^1) - \bar{u} (w_j^2)) \sum_{j=1}^{K} p_j \left( (\bar{w}^2 - \bar{w}^1) - (w_j^2 - w_j^1) \right) \right) \\
&= 0
\end{align*}
$$

The first term in the equation (15) is positive: it is the expectation of $\delta$
using the probabilities $\square_j = p_j (1 + A (\bar{w} - w_j))$. By assumption, the second
term is positive; the last one is positive because $u(\cdot)$ is an increasing function
of $w$.

(b) *U is increasing in w<sub>k</sub>*.

Left to the reader

APPENDIX 6:
Intertemporal maximization program with probabilities alteration

The consumer maximizes:

$$
\begin{gather*}
\text{MAX } E_{\square,0} \left[ \sum_{t=0}^{\infty} \alpha^{-t} u(\tilde{w}_t) \right] \\
\text{subject to the contraints:}
\end{gather*}
$$

$$
\text{i) } W_t = \sum_{n=0}^{N} \theta_t^n S_t^n + w_t
$$
ii) $$W_{t+1} = \sum_{n=0}^{N} \theta_{t}^{n} S_{t+1}^{n}$$

Note that current wealth $Wt$ is a state variable and that current asset holdings $\theta_t^n$ are the control variables of this program; we can thus write the following Bellman equation:

$$U(W_t, \vec{S}_t) = \underbrace{MAX}_{\vec{\theta}_t} \left[ u \left( W_t - \sum_{n=0}^{N} \theta_t^n S_t^n \right) + \alpha^{-1} E_{p,t} \left[ \gamma^p(W_{t+1}) U(W_{t+1}, \vec{S}_{t+1}) \right] \right]$$

The first order conditions are:

$$ \begin{aligned} \square \frac{\partial}{\partial \theta_t^n} \left\{ \begin{array}{l} u \left( \begin{array}{l} W_t \\ -\sum_{n=0}^N \theta_t^n S_t^n \end{array} \right) \\ + \alpha^{-1} E_{p,t} \left[ \begin{array}{l} \gamma^p \left( \sum_{n=0}^N \theta_t^n S_{t+1}^n \right) \\ U \left( \sum_{n=0}^N \theta_t^n S_{t+1}^n, S_{t+1} \right) \end{array} \right] \end{array} \right\} &= 0 \quad \text{for } n = \{0, \dots, N\} \\ \Leftrightarrow & -S_t^n u' \left( \begin{array}{l} W_t \\ -\sum_{n=0}^N \theta_t^n S_t^n \end{array} \right) \\ &+ \alpha^{-1} E_{p,t} \left[ S_{t+1}^n \left( \begin{array}{l} \gamma^{p'}(W_{t+1}) U(W_{t+1}, S_{t+1}) \\ + \frac{\partial U}{\partial W}(W_{t+1}, S_{t+1}) \gamma^p(W_{t+1}) \end{array} \right) \right] = 0 \end{aligned} $$

So:

$$1 = \frac{1}{\alpha u'(w_t) S_t^n} \left[ E_{p,t} \left[ \left( \begin{array}{l} \gamma^{p'}(W_{t+1}) U(W_{t+1}, \vec{S}_{t+1}) \\ + \frac{\partial U}{\partial W}(W_{t+1}, S_{t+1}) \gamma^p(W_{t+1}) \end{array} \right) S_{t+1}^n \right] \right]$$

We establish the Benveniste-Sheinkman condition:

$$\frac{\partial U}{\partial W}(W_t, S_t) = u'(w_t)$$

Hence, for date $t+1$:

$$\frac{\partial U}{\partial W}(W_{t+1}, S_{t+1}) = u'(w_{t+1}) \text{ and } U(W_{t+1}, S_{t+1}) = u(w_{t+1})$$

Combining the preceding equations, we get the Euler equations:
$$ S_t^n = \frac{1}{\alpha u'(w_t)} \left[ \begin{array}{c} E_{p,t} \left[ \tilde{S}_{t+1}^n u'(W_{t+1}) \right] \\ -A \left( \begin{array}{c} \mathrm{Cov}_p[u'(W_{t+1}) \tilde{S}_{t+1}^n, W_{t+1}] \\ +\mathrm{Cov}_p[u(W_{t+1}), \tilde{S}_{t+1}^n] \end{array} \right) \\ -A'E_{p,t} \left[ \tilde{S}_{t+1}^n \right] \mathrm{Cov}_{p,t}(W_{t+1}, u(W_{t+1})) \end{array} \right] \quad \text{for } n = \{0, \dots, N\} $$

# APPENDIX 7:

## Detailed discussion of Kahneman and Tversky example

Using our axiomatics we get:

$$ \square U(L) = u(3000) $$

$$ \square U(M) = 0.8(1 - 800A) $$

$$ \square U(L) = 0.25(1 - 2250A) $$

$$ \square U(M) = 0.2(1 - 3200A) $$

And the inequalities:

$$ U(M)/U(L) \le 1; U(L)/U(M) \le 1 $$

We can re-write the inequalities above as follows:

$$ 0.8(1 - 800A) \le 1 \le 0.2(1 - 3200A) / (0.25(1 - 2250A)) \quad (16) $$

Let us assume:

$$ f(A) = 0.8(1 - 800A) \text{ and } g(A) = (1 - 3200A)/(1 - 2250A) $$

Inequality (16) can be written: $f(A) < 1 < g(A)$, and an elementary study of $f(.)$ and $g(.)$ variations show that the double inequality is verified for values of $A$ greater than $(1/2250)$ (or for negative values).