Temi di discussione
Portfolio selection with monotone
by Fabio Maccheroni, Massimo Marinacci, Aldo Rustichini
and Marco Taboga
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.
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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.
WITH MONOTONE MEAN-VARIANCE PREFERENCES
by Fabio Maccheroni*, Massimo Marinacci**,
Aldo Rustichini*** and Marco Taboga****
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.
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
Appendix A: Monotone Fenchel duality................................................................................20
Appendix B: Proofs................................................................................................................25
* 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.
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]).
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
Payo¤ of f
Payo¤ of g
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].
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
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.
2This is the set where rU?is positive, called domain of monotonicity of U?.
? The functional relation between V?and U?can be explicitly formulated.
? The parameter ? retains the usual interpretation in terms of uncertainty
? 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:
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
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
X ?~1RjW ? ?
mvand the above portfolio ??
mmvis that in the latter conditional
mmvignores the part of
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-
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¤.
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
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
?EQ[u(g)] + c(Q)?
8f 2 L2(P);
G?=f 2 L2(P) : f ? EP[f] 61
if Q ? P and 1 otherwise.
4In the special case in which u is linear, some recent …nance papers (e.g., Filipovic
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
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-
where ?2(P) is the set of all probability measures with square-integrable
density with respect to P, and
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
8f 2 G?,
C (QjjP) =
if Q ? P
8f 2 L2(P):
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
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):
i?1, and C (QjjP) is its continuous
and relative version. We refer to [LV] for a comprehensive study of concentration indices.
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
?(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:
if f 2 G?;
U?(f ^ ?f)
?f= maxft 2 R : f ^ t 2 G?g:
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).
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 ??
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
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 + ? ?
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
(f ? t)?i
(g ? t)?i
for all t 2 R. We refer to
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
which – dividing the argument by w – reduces to (6) up to replacement of ?
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?? ?
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-
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:
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.
X ?~1RjW?? ? ??i
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.
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
If, in addition, P?W??
mmv> ???> 0, then:
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 ?.
Given a ? > 0 with ???~1 > 0, de…ne m = ????~1. Hence,
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.
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
?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
of agent j and ? is the market value of all assets.
agents. Speci…cally, m = ?
where ?jis the uncertainty aversion coe¢cient