PORTFOLIO SELECTION WITH MONOTONE MEAN-VARIANCE PREFERENCES

Abstract

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.

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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].
3

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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?.
4