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Temi di discussione

(Working papers)

April 2008

664

Number

Portfolio selection with monotone

mean-variance preferences

by Fabio Maccheroni, Massimo Marinacci, Aldo Rustichini

and Marco Taboga

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The purpose of the Temi di discussione series is to promote the circulation of working

papers prepared within the Bank of Italy or presented in Bank seminars by outside

economists with the aim of stimulating comments and suggestions.

The views expressed in the articles are those of the authors and do not involve the

responsibility of the Bank.

Editorial Board: Domenico J. Marchetti, Patrizio Pagano, Ugo Albertazzi, Michele

Caivano, Stefano Iezzi, Paolo Pinotti, Alessandro Secchi, Enrico Sette, Marco

Taboga, Pietro Tommasino.

Editorial Assistants: Roberto Marano, Nicoletta Olivanti.

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PORTFOLIO SELECTION

WITH MONOTONE MEAN-VARIANCE PREFERENCES

by Fabio Maccheroni*, Massimo Marinacci**,

Aldo Rustichini*** and Marco Taboga****

Abstract

We propose a portfolio selection model based on a class of monotone preferences that

coincide with mean-variance preferences on their domain of monotonicity, but differ where

mean-variance preferences fail to be monotone and are therefore not economically

meaningful. The functional associated with this new class of preferences is the best

approximation of the mean-variance functional among those which are monotonic. We solve

the portfolio selection problem and we derive a monotone version of the CAPM, which has

two main features: (i) it is, unlike the standard CAPM model, arbitrage free, (ii) it has

empirically testable CAPM-like relations. Therefore, the monotone CAPM has a sounder

theoretical foundation than the standard CAPM and comparable empirical tractability.

JEL Classification: C6, D8, G11.

Keywords: portfolio theory, CAPM, mean-variance, monotone preferences.

Contents

1. Introduction..........................................................................................................................3

2. Monotone Mean-Variance preferences................................................................................6

3. The portfolio selection problem.........................................................................................10

4. The optimal portfolio.........................................................................................................11

5. Monotone CAPM...............................................................................................................13

6. Some examples..................................................................................................................17

7. Conclusions........................................................................................................................19

8. Acknowledgements............................................................................................................20

Appendix A: Monotone Fenchel duality................................................................................20

Appendix B: Proofs................................................................................................................25

References..............................................................................................................................41

_______________________________________

* Istituto di Metodi Quantitativi and IGIER, Università Bocconi.

** Collegio Carlo Alberto and Università di Torino.

*** Department of Economics, University of Minnesota.

**** Bank of Italy, Economic Outlook and Monetary Policy Department.

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1Introduction

Since the seminal contributions of Markowitz [Ma] and Tobin [To], mean-

variance preferences have been extensively used to model the behavior of

economic agents choosing among uncertain prospects and have become one

of the workhorses of portfolio selection theory.1These preferences, denoted

by ?mv, assign to an uncertain prospect f the following utility score:

U?(f) = EP[f] ??

where P is a given probability measure and ? is an index of the agent’s

aversion to variance.

The success of this speci…cation of preferences is due to its analytical

tractability and clear intuitive meaning. Mean-variance preferences have,

however, a major theoretical drawback: they may fail to be monotone. It

may happen that an agent with mean-variance preferences strictly prefers

less to more, thus violating one of the most compelling principles of economic

rationality. This is especially troublesome in Finance because monotonicity

is the crucial assumption on preferences that arbitrage arguments require

(see Dybvig and Ross [DR] and Ross [R]).

2VarP[f];

The lack of monotonicity of mean-variance preferences is a well known

problem (see, e.g., Dybvig and Ingersoll [DI] and Jarrow and Madan [JM])

and not a minor one, since it can be (partly) bypassed only under very

restrictive assumptions about the statistical distribution of asset returns (see,

e.g., Bigelow [Bi]).

The non-monotonicity of mean-variance preferences can be illustrated

with a simple example. Consider a mean-variance agent with ? = 2. Suppose

she has to choose between the two following prospects f and g:

States of Nature

Probabilities

Payo¤ of f

Payo¤ of g

s1

0:25

1

1

s2

0:25

2

2

s3

0:25

3

3

s4

0:25

4

5

Prospect g yields a higher payo¤ than f in every state. Any rational agent

should prefer g to f. However, it turns out that our mean-variance agent

strictly prefers f to g. In fact:

U2(f) = 1:25 > 0:5625 = U2(g):

1See, e.g., Bodie, Kane, and Marcus [BKM], Britten-Jones [Br], Gibbons, Ross, and

Shanken [GRS], Kandel and Stambaugh [KS], and MacKinlay and Richardson [MR].

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The reason why monotonicity fails here is fairly intuitive. By choosing g

rather than f, the payo¤ in state s4increases by one unit. This additional

unit increases the mean payo¤, but it also makes the distribution of payo¤s

more spread out, thus increasing the variance. The increase in the mean is

more than compensated by the increase in the variance, and this makes our

mean-variance agent worse o¤.

In this paper we consider the minimal modi…cation of mean-variance pref-

erences needed to overcome their lack of monotonicity. This amended version,

based on the variational preferences of Maccheroni, Marinacci, and Rustichini

[MMR], is not only sounder from an economic rationality viewpoint, but, be-

ing as close as possible to the original, also maintains the basic intuition and

tractability of mean-variance preferences.

Speci…cally, we consider the variational preference ?mmvrepresented by

the choice functional

?

where Q ranges over all probability measures with square-integrable density

with respect to P, and C (QjjP) is the relative Gini concentration index (or

?2-distance), a concentration index that enjoys properties similar to those of

the relative entropy.

The preferences ?mmvhave the following key properties:

V?(f) = min

Q

EQ[f] +1

2?C (QjjP)

?

8f 2 L2(P);

? They are monotone and they agree with mean-variance preferences

where the latter are monotone, that is, economically meaningful.2

? Their choice functional V? is the minimal, and so the most cautious,

monotone functional that extends the mean-variance functional U?out-

side its domain of monotonicity.

? The functional V?is also the best possible monotone approximation of

U?: that is, if V0

?is any other monotone extension of U? outside its

domain of monotonicity, then

jV?(f) ? U?(f)j ? jV0

?(f) ? U?(f)j

for each prospect f.

Moreover:

2This is the set where rU?is positive, called domain of monotonicity of U?.

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? The functional relation between V?and U?can be explicitly formulated.

? The parameter ? retains the usual interpretation in terms of uncertainty

aversion.

? The functional V?preserves second order stochastic dominance.

All these features make the preferences ?mmva natural adjusted version of

mean-variance preferences that satis…es monotonicity. For this reason we call

them monotone mean-variance preferences.

In view of all this, it is natural to wonder what happens in a portfolio

problem à la Markowitz when we use monotone mean-variance preferences

in place of standard mean-variance preferences. This is the main subject

matter of this paper. Markowitz’s well-known optimal allocation rule under

mean-variance preferences is:

?VarP[X]?1EPh

where ??

mvis the optimal portfolio of risky assets, X is the vector of gross

returns on the risky assets, R is the gross return on the risk-free asset, and

~1 is a vector of 1s. We show that with monotone mean-variance preferences

the optimal allocation rule becomes:

?P (W ? ?)VarP[X jW ? ?]?1EPh

where W is future wealth and ? is a constant determined along with ??

by solving a suitable system of equations.

Except for a scaling factor, the di¤erence between Markowitz’s optimal

portfolio ??

moments of asset returns EP[?jW ? ?] and VarP[?jW ? ?] are used instead

of unconditional moments, so that the allocation ??

the distribution where wealth is higher than ?. As a result, a monotone

mean-variance agent does not take into account those high payo¤ states that

contribute to increase the mean return, but give an even greater contribution

to increase the variance. By doing so, this agent does not incur in violations

of monotonicity caused for mean-variance preferences by an exaggerate pe-

nalization of “positive deviations from the mean.” This is a key feature of

monotone mean-variance preferences. We further illustrate this point in Sec-

tion 6 by showing how this functional avoids some pathological situations in

which the more the payo¤ to an asset is increased in some states, the more a

??

mv=1

X ?~1R

i

;

??

mmv=

1

X ?~1RjW ? ?

i

;

mmv

mvand the above portfolio ??

mmvis that in the latter conditional

mmvignores the part of

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mean-variance agent reduces the quantity of it in her portfolio, until in the

limit she ends up holding none.

In the last part of the paper we derive a monotone CAPM model based on

the above portfolio analysis with our monotone mean-variance preferences.

We …rst show that in our model optimal portfolios satisfy the classic two

fund separation principle: (i) portfolios of risky assets optimally held by

agents with di¤erent degrees of uncertainty aversion are all proportional to

each other, (ii) at an optimum the only di¤erence between two agents is the

amount of wealth invested in the risk-free asset. This separation has signif-

icant theoretical implications because it allows to identify the equilibrium

market portfolio with the optimal portfolio of risky assets held by any agent

(as in [Sh], [Sh-2] and [To]), and it allows to derive a monotone version of

the classic CAPM.

In Section 5 we show that the monotone CAPM that we derive has the em-

pirical tractability of the standard CAPM. Moreover, thanks to monotonicity

of the preference functional V?, in the monotone CAPM there are no arbitrage

opportunities. This is a key property of the monotone CAPM and is in stark

contrast with what happens with the standard CAPM. In fact, as observed by

Dybvig and Ingersoll [DI], the lack of monotonicity of mean-variance pref-

erences generates arbitrage opportunities in the standard CAPM. In turn,

these arbitrage opportunities make impossible to have CAPM equilibrium

prices of all assets in a complete-markets economy. This is, instead, possible

in our arbitrage free monotone CAPM, which can thus be integrated in the

classic Arrow-Debreu complete-markets framework.

The paper is organized as follows. Section 2 illustrates in detail monotone

mean-variance preferences. Sections 3 and 4 state and solve the portfolio se-

lection problem under the proposed speci…cation of preferences. Section 5

contains the CAPM analysis. Section 6 presents some examples that illus-

trate the di¤erence between the optimal allocation rule proposed here and

Markowitz’s. Section 7 concludes. All proofs are collected in the appendices,

along with some general results on monotone approximations of concave func-

tionals.

2 Monotone Mean-Variance Preferences

We consider a measurable space (S;?) of states of nature. An uncertain

prospect is a ?-measurable real valued function f : S ! R, that is, a sto-

chastic monetary payo¤.

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The agent’s preferences are described by a binary relation ? on a set of

uncertain prospects. [MMR] provides a set of simple behavioral conditions

that guarantee the existence of an increasing utility function u : R ! R and

a convex uncertainty index c : ? ! [0;1] on the set ? of all probability

measures, such that

?EQ[u(f)] + c(Q)?? inf

for all (simple) prospects f;g.

Preferences having such a representation are called variational, and two

important special cases of variational preferences are the multiple priors pref-

erences of Gilboa and Schmeidler [GS], obtained when c only takes on values

0 and 1, and the multiplier preferences of Hansen and Sargent [HS], ob-

tained when c(Q) is proportional to the relative entropy of Q with respect

to a reference probability measure P.3

Variational preferences satisfy the basic tenets of economic rationality. In

particular, they are monotone, that is, given any two prospects f and g, we

have f ? g whenever f (s) ? g (s) for each s 2 S.4

For concreteness, given a probability measure P on (S;?), we consider

the set L2(P) of all square integrable uncertain prospects. A relation ?mv

on L2(P) is a mean-variance preference if it is represented by the choice

functional

U?(f) = EP[f] ??

for some ? > 0.

The subset G?of L2(P) where the Gateaux di¤erential of U?is positive

is called domain of monotonicity of U?. The preference ?mvis monotone on

the set G?, which has the following properties.

Lemma 1 The set G?is convex, closed, and

?

Moreover, for all f = 2 G? and every " > 0 there exists g 2 L2(P) that is

"-close to f (i.e., jf (s) ? g (s)j < " for all s 2 S), and such that g > f and

U?(g) < U?(f).

3The relative entropy of Q given P is EPh

and Kupper [FK] and Kupper and Cheridito [KC]) call monetary utility functions the

functionals representing variational preferences.

f ? g , inf

Q2?

Q2?

?EQ[u(g)] + c(Q)?

(1)

2VarP[f]

8f 2 L2(P);

G?=f 2 L2(P) : f ? EP[f] 61

?

?

.(2)

dQ

dPlndQ

dP

i

if Q ? P and 1 otherwise.

4In the special case in which u is linear, some recent …nance papers (e.g., Filipovic

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The domain of monotonicity has thus nice properties. More importantly,

the last part of the lemma shows why G?is where the mean-variance pref-

erence ?mvis economically meaningful. In fact, it says that if we take any

prospect f outside G?, then in every its neighborhood, however small, there

is at least a prospect g that is statewise better than f, but ranked by ?mv

below f.

The mean-variance preference ?mvthus exhibits irrational non-monotone

behavior in every neighborhood, however small, of prospects f outside G?.

For this reason ?mvhas no economic meaning outside G?.

It can be shown that the restriction of ?mvto G?is a variational prefer-

ence, and

?

where ?2(P) is the set of all probability measures with square-integrable

density with respect to P, and

(

+1

is the relative Gini concentration index (or ?2-distance).5

This suggests to call monotone mean-variance preference the relation

?mmvon L2(P) represented by the choice functional

?

Our …rst result is the following:6

U?(f) =min

Q2?2(P)

EQ[f] +1

2?C (QjjP)

?

8f 2 G?,

C (QjjP) =

EPh?dQ

dP

?2i

? 1

if Q ? P

otherwise

V?(f) =min

Q2?2(P)

EQ[f] +1

2?C (QjjP)

?

8f 2 L2(P):

(3)

Theorem 2 The functional V?: L2(P) ! R de…ned by (3) is the minimal

monotone functional on L2(P) such that V?(g) = U?(g) for all g 2 G?; that

is,

V?(f) = supfU?(g) : g 2 G?and g 6 fg

Moreover, V?(f) ? U?(f) for each f 2 L2(P).

5Along with the Shannon entropy, the Gini index is the most classic concentration

index. For discrete distributions it is given byPn

6The proof of this theorem builds on a general result on the minimal monotone func-

tional that dominates a concave functional on an ordered Banach space, which we prove

in Appendix A (Proposition 12).

8f 2 L2(P):

(4)

i=1Q2

i?1, and C (QjjP) is its continuous

and relative version. We refer to [LV] for a comprehensive study of concentration indices.

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The functional V?is concave, continuous, and in view of this theorem it

has the following fundamental properties:

(i) V?coincides with the mean-variance choice functional U?on its domain

of monotonicity G?.

(ii) V? is the minimal monotone extension of U? outside the domain of

monotonicity G?, and so it is the most cautious monotone adjustment

of the mean-variance choice functional.

(iii) V?is the best possible monotone approximation of U?: if V0

monotone extension of U?outside the domain of monotonicity G?, then

V0

?(f) ? V?(f) ? U?(f) and so

jV?(f) ? U?(f)j ? jV0

?is any other

?(f) ? U?(f)j8f 2 L2(P):

Next theorem shows explicitly the functional relation between V?and U?.

Theorem 3 Let f 2 L2(P). Then:

V?(f) =

?U?(f)

if f 2 G?;

else,

U?(f ^ ?f)

where

?f= maxft 2 R : f ^ t 2 G?g:

(5)

A monotone mean-variance agent can thus be regarded as still using the

mean-variance functional U?even in evaluating prospects outside the domain

of monotonicity G?. In this case, however, the agent no longer considers the

original prospects, but rather their truncations at ?f, the largest constant t

such that f ^ t belongs to G?.

Besides depending on the given prospect f, the constant ?falso depends

on the parameter ?. Corollary 16 in Appendix B shows that ?fdecreases as ?

increases, and it is the unique solution of the equation EP?(f ? ?)??= 1=?.

Given two preferences over uncertain prospects, we say that ?1is more

uncertainty averse than ?2if and only if

f ?1c =) f ?2c

for all f 2 L2(P) and c 2 R. That is, Agent 1 is more uncertainty averse

than Agent 2 if, whenever Agent 1 prefers the uncertain prospect f to a sure

payo¤ c, so does Agent 2.7

7We refer the interested reader to [MMR] for a discussion of this notion, and its inter-

pretation in terms of risk aversion and ambiguity aversion (not mentioned here in order

to keep the paper focused).

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A mean-variance preference ??

mean-variance preference ??

as an uncertainty aversion coe¢cient.

The next result, a variation of [MMR, Cor. 21], shows that the same is

true for monotone mean-variance preferences.

mvis more uncertainty averse than another

mvif and only if ? > ?. Thus, ? can be interpreted

Proposition 4 The preference ??

if and only if ? > ?.

mmvis more uncertainty averse than ??

mmv

We conclude this section by observing that, unlike U?, the preference

functional V?preserves second order stochastic dominance (?SSD).8This is

a further proof of the sounder economic meaning of monotone mean-variance

preferences relative to mean-variance ones.

Theorem 5 Let f;g 2 L2(P). If f ?SSDg, then V?(f) ? V?(g).

Summing up, the monotone choice functional V?provides a natural ad-

justment of the mean-variance choice functional. It also has the remarkable

feature of involving, like multiplier preferences ([HS]), a classic concentration

index. This ensures to V?a good analytical tractability, as the next sections

show.

3The Portfolio Selection Problem

We consider the one-period allocation problem of an agent who has to decide

how to invest a unit of wealth at time 0, dividing it among n+1 assets. The

…rst n assets are risky, while the (n + 1)-th is risk-free. The gross return

on the i-th asset after one period is denoted by Xi. The (n ? 1) vector of

the returns on the …rst n assets is denoted by X and the (n ? 1) vector of

portfolio weights, indicating the fraction of wealth invested in each of the

risky assets, is denoted by ?. The return on the (n + 1)-th asset is risk-free

and equal to a constant R.

The end-of-period wealth W?induced by a choice of ? is given by:

W?= R + ? ?

?

X ?~1R

?

:

We assume that there are no frictions of any kind: securities are perfectly

divisible; there are no transaction costs or taxes; agents are price-takers, in

8Recall that f ?SSD g i¤ EPh

Dana [Da], to which the proof of Theorem 5 is inspired, for references on second order

stochastic dominance.

(f ? t)?i

? EPh

(g ? t)?i

for all t 2 R. We refer to

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that they believe that their choices do not a¤ect the distribution of asset

returns; there are no institutional restrictions, so that agents are allowed to

buy, sell or short sell any desired amount of any security.9As a result, ? can

be chosen in Rn.

We adopt ?mmv as a speci…cation of preferences, and so portfolios are

ranked according to the preference functional:

?

where P is the reference probability measure. Hence, the portfolio problem

can be written as:

?

Notice that, if the agent’s initial wealth is w > 0, then her end-of-period

wealth is wW?, therefore she solves the problem

?

V?(W?) = min

Q2?2(P)

EQ[W?] +1

2?C (QjjP)

?

;

max

?2Rn

min

Q2?2(P)

EQ[W?] +1

2?C (QjjP)

?

:

(6)

max

?2Rn

min

Q2?2(P)

EQ[wW?] +1

2?C (QjjP)

?

which – dividing the argument by w – reduces to (6) up to replacement of ?

with ?w.

4The Optimal Portfolio

In this section we give a solution to the portfolio selection problem outlined

in the previous section. The characterization of the optimal portfolio is given

by the following theorem.10

Theorem 6 The vector ??2 Rnis a solution of the portfolio selection prob-

lem (6) if and only if there exists ??2 R such that (??;??) satis…es the

system of equations:

(

?P (W?? ?)VarP[X jW?? ?]? = EPh

X ?~1RjW?? ?

i

;

EP?(W?? ?)??= 1=?:

tional restrictions are not binding.

10EP[?jW?? ?] and VarP[?jW?? ?] are the expectation and variance conditional on

the event fW?? ?g. Note that VarP[?jW?? ?] is an (n ? n) matrix.

9This assumption can be weakened, by simply requiring that at an optimum institu-

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As observed in Section 2, the second displayed equation guarantees that ??

is the largest constant such that W??^??belongs to the domain of monotonic-

ity of the mean-variance functional U?. The optimal portfolio ??is thus de-

termined along with the threshold ??by solving a system of n+1 equations

in n + 1 unknowns.

Although it is not generally possible to …nd explicitly a solution of the

above system of equations, numerical calculation with a standard equation

solver is straightforward: given an initial guess (?;?), one is able to cal-

culate the …rst two moments of the conditional distribution of wealth; if

the moments thus calculated, together with the initial guess (?;?), satisfy

the system of equations, then (?;?) = (??;??) and numerical search stops;

otherwise, the search procedure continues with a new initial guess for the pa-

rameters.11In the next section we will solve in this way few simple portfolio

problems in order to illustrate some features of the model.

The optimal allocation rule of Theorem 6 is easily compared to the rule

that would obtain in a classic Markowitz’s setting. In the traditional mean-

variance model we would have:

??=1

?

The monotone mean-variance model yields:

?P (W?? ? ??)VarP[X jW?? ? ??]?1EPh

Relative to Markowitz’s optimal allocation (7), here the unconditional mean

and variance of the vector of returns X are replaced by a conditional mean

and a conditional variance, both calculated by conditioning on the event

fW?? ? ??g. Furthermore a scaling factor is introduced, which is inversely

proportional to the probability of not exceeding the threshold ??.

Roughly speaking, when computing the optimal portfolio we ignore that

part of the distribution where wealth is higher than ??. To see why it is

optimal to ignore the part of the distribution where one obtains the high-

est returns, recall the example of non-monotonicity of mean-variance illus-

trated in the Introduction. In that example, high payo¤s were increasing

the variance more than the mean, thus leading the mean-variance agent to

irrationally prefer a strictly smaller prospect. With monotone mean-variance

preferences, this kind of behavior is avoided by arti…cially setting the prob-

ability of some high payo¤ states equal to zero. In our portfolio selection

problem we set the probability of the event fW?? > ??g equal to zero.

11A R (S-Plus) routine to calculate the optimal portfolio in an economy with …nitely

many states of nature is available upon request.

?VarP[X]??1EPh

X ?~1R

i

:

(7)

??=

1

X ?~1RjW?? ? ??i

:

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When there is only one risky asset, the optimal quantities ??

prescribed, respectively, by our model and by the mean-variance model can

be compared by means of the following result.

mmvand ??

mv

Proposition 7 Suppose that S is …nite, with P (s) > 0 for all s 2 S, and

that there is only one risky asset. Then, either

??

mmv? ??

mv? 0

or

??

mmv? ??

mv? 0:

If, in addition, P?W??

mmv> ???> 0, then:

??

??

mmv> ??

mmv< ??

mv

if ??

if ??

mmv> 0;

mv mmv< 0:

That is, an investor with monotone mean-variance preferences always

holds a portfolio which is more leveraged than the portfolio held by a mean-

variance investor. If she buys a positive quantity of the risky asset, this

is greater than the quantity that would be bought by a mean-variance in-

vestor; on the contrary, if she sells the risky asset short, she sells more than

a mean-variance investor would do.

oughly illustrated by the examples in the next section: the intuition behind

it is that in some cases a favorable investment opportunity is discarded by a

mean-variance investor because of non-monotonicity of her preferences, while

a monotone mean-variance investor exploits the opportunity, thus taking a

more leveraged position.

This kind of behavior will be thor-

5 Monotone CAPM

In this section we show how the standard CAPM analysis can be carried out

in the monotone mean-variance setup.

We begin by establishing a two-funds separation result, which shows that

agents’ optimal investment choices can be done in two stages: …rst agents

decide the amount of wealth to invest in the risk-free asset; then, they decide

how to allocate the remaining wealth among the risky assets. The outcome

of this second decision is the same for all agents, regardless of their initial

wealth or aversion to uncertainty.

Proposition 8 Let ?;? > 0. If???;???solves the portfolio selection problem

(6) for an investor with uncertainty aversion ?, then

solves it for an investor with uncertainty aversion ?.

?

?

???;?

???+

?

1 ??

?

?

R

?

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Given a ? > 0 with ???~1 > 0, de…ne m = ????~1. Hence,

?m=

??

???~1

and Proposition 8 guarantees that m and ?mdo not depend on the choice of

?. The equality ?m?~1 = 1 implies that ?mis the portfolio held by an investor

who does not invest any of her wealth in the risk-free asset. Following the

majority of the literature, we call ?mthe market portfolio and denote by

Xm= ?m? X its payo¤. In particular, W?m = R + ?m?

In an economy consisting of monotone-mean variance agents, all investors

hold a portfolio of risky assets proportional to the market portfolio. Speci…-

cally, an investor with uncertainty aversion ? will invest m=? in the market

portfolio and the rest of her wealth in the risk free asset. Like in the standard

mean-variance setting, also here the amount of wealth invested in the market

portfolio only depends on the coe¢cient ? of the agent.

?

X ?~1R

?

= Xm.

All this has strong empirical implications. From market data – more

precisely, by observing the market values of the assets in the economy – it

is possible to determine the equilibrium composition of the market portfolio

?m. Once we know the equilibrium ?m, and so its equilibrium payo¤ Xm,

thanks to the next result we can …nd the values of m and ?mby solving a

system of equations with observable coe¢cients.12

Proposition 9 The pair (x?;y?) ? (m;?m) solves the following system of

equations

?P (Xm? y)VarP[XmjXm? y]x = EP[Xm? RjXm? y];

The knowledge of m and ?mmakes it possible to determine the equilibrium

pricing kernel rVm(Xm), which will become very important momentarily

when discussing the monotone CAPM. To see why this is the case, we need

the following lemma, which gives some properties of rVm(Xm).

EP?(Xm? y)??= 1=x:

Lemma 10 The quantity rVm(Xm) has the following properties:

(i) rVm(Xm) = m(Xm? ?m)?= rV?(W??) for all ? > 0:

12Notice that, like in the standard mean-variance setting, it can also be shown that the

uncertainty aversion coe¢cient m is a mean of the uncertainty aversion coe¢cients of the

?P

of agent j and ? is the market value of all assets.

agents. Speci…cally, m = ?

j

??j??1??1

where ?jis the uncertainty aversion coe¢cient

14