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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?.
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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.
2Monotone 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
7
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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.
3 The 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-
5Monotone 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
Page 16
(ii) EP[XirVm(Xm)] = R for all i = 1;:::;n.
(iii) EP[rVm(Xm)] = 1.
By property (i) of this lemma, once we know the values of m and ?m
we can determine the value of rVm(Xm) via the equation m(Xm? ?m)?.
The value of rVm(Xm) can thus be determined from market data. To ease
notation, in what follows we set rV ? rVm(Xm).
Lemma 10 also makes it possible to derive our monotone version of the
CAPM. In fact, by Lemma 10.(ii), EP[XmrV ] = ?m? EP[XrV ] = R.
Together with Lemma 10.(iii), this implies
CovP[Xi;rV ] = R ? EP[Xi] and CovP[Xm;rV ] = R ? EP[Xm];
which proves the following theorem, the main result of this section.
Theorem 11 (Monotone CAPM) Let Xmbe de…ned as above. Then,
?EP[Xm] ? R?;
where
?i=CovP[Xi;rV ]
CovP[Xm;rV ].
Theorem 11 gives our monotone CAPM, with security market line (8),
and shows its key theoretical and empirical features.
On the theoretical side, the pricing rule delivered by our CAPM is arbi-
trage free: there are no portfolios with strictly negative prices and positive
…nal payo¤s. In fact, let Yithe end-of-period payo¤ per share of asset i and
piits current price. Then, Xi= Yi=piand (8) becomes
EP[Xi] ? R = ?i
8i = 1;:::;n;
(8)
(9)
EP[Yi]
pi
? R =1
pi
CovP[Yi;rV ]
CovP[Xm;rV ]
?EP[Xm] ? R?
This delivers the pricing rule:
pi=1
REP[YirV ]:
This pricing rule is a positive linear functional. For, the price of a portfolio
consisting of qishares of asset i is
"
n
X
i=1
qipi=1
REP
n
X
i=1
qiYi
!
rV
#
;
15
Page 17
which is positive as long as
n
X
i=1
qiYiis positive (rV is positive because Vmis
monotone).
The absence of arbitrage opportunities in our monotone CAPM is a key
theoretical feature of our model. As observed in the Introduction, this is in
stark contrast with their presence in the standard CAPM model, caused by
the lack of monotonicity of mean-variance preferences. This was observed
by Dybvig and Ingersoll [DI], who show that if Xm = 2 Gmand the market
is complete, then the pricing rule obtained from standard CAPM is not a
positive linear functional, and it thus allows arbitrages.
Inter alia, this means that, di¤erently from the standard CAPM, our
monotone CAPM pricing rule can be integrated in a standard Arrow-Debreu
complete-markets economy, with all assets in such an economy priced in
equilibrium according to our CAPM.
On the empirical side, our monotone CAPM model can be fully analyzed
from market data. First observe that the values of the betas (9) can be
derived from market data because we just observed that, besides those of Xi,
also the values of Xmand rV can be determined from market data.
Second, Theorem 11 suggests that, by regressing the excess returns to the
single assets on the excess return to the market portfolio, the empirical betas
of the single assets can be estimated by instrumental variables, using rV as
an instrument. In fact, de…ne "i= Xi? R ? ?i(Xm? R), so that:
Xi= R + ?i(Xm? R) + "i.
By Lemma 10.(ii), rV is easily seen to be orthogonal to "i, and so rV can
be used as an instrument.
We cannot use, instead, ordinary least squares because, in general, Xm?R
is not orthogonal to "i. For:
EP["i(Xm? R)] = EP[(Xi? R)(Xm? R)] ? ?iEP?(Xm? R)2?
= EP[(Xi? R)(Xm? R)] ?CovP[rV;Xi]
CovP[rV;Xm]EP?(Xm? R)2?:
Summing up, the monotone CAPM is arbitrage free and its betas can be
inferred from market data. The monotone CAPM has thus a sounder theo-
retical foundation than the standard CAPM, while retaining its remarkable
empirical tractability.
16
Page 18
6Some Examples
In this section we present three simple examples to illustrate the optimal
portfolio rule we derived above. In every example there are …ve possible
states of Nature. Each of them obtains with a probability P (si) that remains
…xed throughout the examples. In all examples we also set ? = 10.
Example 1 is a case in which our model and the traditional mean-variance
model deliver the same optimal composition of the portfolio. This is not
interesting per se, but it serves as a benchmark and it helps to introduce
Example 2, where the two optimal portfolios di¤er. In Example 1 there is
only one risky asset, whose gross return is denoted by X1and is reported in
the next table, and a risk-free asset, whose gross return R is equal to 1 across
all states. In this example, the optimal portfolio ??
to our rule is equal to the mean-variance optimal portfolio ??
Wmmvrepresent the overall return to the two optimal portfolios for each state
of the world. The table also displays the value of the constant ??at which it
is optimal to truncate the distribution of the return to the portfolio of risky
assets.
mmvcalculated according
mv. Wmv and
P (si)
0:1
0:2
0:4
0:2
0:1
P (sijWmmv? ??)
0:1
0:2
0:4
0:2
0:1
R
1
1
1
1
1
X1
0:97
0:99
1:01
1:03
1:05
Wmv
0:9375
0:9791
1:0208
1:0620
1:1041
Wmmv
0:9375
0:9791
1:0208
1:0620
1:1041
s1
s2
s3
s4
s5
??
??
mv= 2:083
mmv= 2:083??= 1:1211
Example 2 is a slight modi…cation of Example 1. We increase the payo¤
to the risky asset in state s5from 1:05 to 1:10, leaving everything else un-
changed. The e¤ect of this change is an increase in both the mean and the
variance of X1, the payo¤ to the risky asset. The optimal behavior according
to the mean-variance model is to reduce the fraction of wealth invested in the
risky asset from 2:083 to 1:3574. According to our model it is also optimal
to decrease the position in the risky asset, but less, from 2:083 to 1:8382.
17
Page 19
P (si)
0:1
0:2
0:4
0:2
0:1
P (sijWmmv? ??)
0:1111
0:2222
0:4444
0:2222
0
R
1
1
1
1
1
X1
0:97
0:99
1:01
1:03
1:10
Wmv
0:9592
0:9864
1:0135
1:0407
1:1357
Wmmv
0:9448
0:9816
1:0183
1:0551
1:1838
s1
s2
s3
s4
s5
??
??
mv= 1:3574
mmv= 1:8382??= 1:1213
In both cases the optimal behavior might seem puzzling at a …rst sight:
when the payo¤ of an asset increases in one state, it is optimal to hold less of
that asset. This behavior can be understood by looking at the distributions
of the overall return in the two tables. By reducing the fraction of wealth
invested in the risky asset, the overall return increases in the states where
the risky asset pays less than the risk-free asset. On the contrary, the overall
return decreases in the states where the risky asset pays more than the risk-
free asset. In state s5, however, the e¤ect of this decrease is compensated
by the fact that we have raised the payo¤ to the risky asset from 1:05 to
1:10. Hence, by reducing the amount of wealth invested in the risky asset,
the investor gives up some of the extra payo¤ received in state s5in order
to guarantee himself a higher overall return in the states where the risky
asset has a low payo¤. The problem with this kind of behavior is that it can
become pathological with mean-variance preferences. The next table shows
what happens if we further increase the payo¤ in state s5.
X1(s5)
??
mv
??
mmv
1:05
2:0830
2:0830
1:10
1:3574
1:8382
1:15
0:9174
1:8382
1:20
0:6747
1:8382
1:50
0:2465
1:8382
2
0:1175
1:8382
3
0:0572
1:8382
The more we increase the payo¤ in state s5, the more the mean-variance
optimal fraction ??
mvof wealth invested in the risky asset decreases, until it
goes to zero when the payo¤ in state s5becomes very large. In our model this
does not happen. At …rst ??
mmvdecreases, but it then stops decreasing and
it remains …xed at the same value, though the payo¤ in state s5is further
increased. The reason why this happens is that, once probabilities have been
optimally reassigned to states and a zero probability has been assigned to
state s5, any further increases of the payo¤ in s5are disregarded and have
no in‡uence on the formation of the optimal portfolio.
18
Page 20
Example 3 is slightly more complicated. Everything is as in Example 2,
but a second risky asset is added. The payo¤ to this new asset, denoted by
X2, is high in the states where X1is low and low where X1is high.
P (si)
0:1
0:2
0:4
0:2
0:1
P (sijWmmv? ??)
0:1111
0:2222
0:4444
0:2222
0
R
1
1
1
1
1
X1
0:97
0:99
1:01
1:03
1:10
X2
1:05
1:00
0:99
0:99
0:99
Wmv
1:002
0:9833
1:0061
1:0393
1:1556
Wmmv
1:0231
0:9570
1:0125
1:0985
1:3994
s1
s2
s3
s4
s5
??
??
mv= (1:6613;1:0495)
mmv= (4:2989;3:0423)??= 1:1316
Also in this case the optimal portfolios suggested by our model and by the
traditional model are di¤erent. To get an intuitive idea of what is happening,
note that, although the market is still arbitrage-free, asset 2 allows to hedge
away almost completely the risks taken by investing in asset 1. Consider for
example a portfolio formed by 0:5 units of asset 1 and 0:5 units of asset 2.
Its payo¤s in the …ve states are collected in the following vector:
(1:01;0:995;1;1:01;1:045)
A qualitative inspection of this payo¤ vector reveals that in state s2 this
portfolio pays o¤ slightly less than the risk-free asset, while in all other states
it pays o¤ more and in some states considerably more. Roughly speaking, if
it was not for the slightly low payo¤ in state s2, there would be an arbitrage
opportunity because the portfolio would pay o¤ more than the risk-free asset
in every state. As a consequence, we would expect an optimal portfolio
rule to exploit this favorable con…guration of payo¤s by prescribing to take
a highly levered position. As reported in the last table, according to our
model it is optimal to take a highly levered position in the risky assets in
order to exploit this opportunity, at the cost of facing a low payo¤ in state
s2. In contrast, with the mean-variance model the optimal portfolio is much
less aggressive, and the investor is overly concerned with the unique state in
which the payo¤ is lower than the payo¤ to the risk-free asset.
7 Conclusions
We have derived a portfolio allocation rule using a corrected version of the
mean-variance principle, which avoids the problem of non-monotonicity. In
19
Page 21
the cases where mean-variance preferences are well-behaved (i.e., monotone)
the optimal portfolios suggested by our rule do not di¤er from standard
mean-variance e¢cient portfolios.
The important property of separability in two funds still holds in our set-
ting, and this allows to derive a monotone CAPM that retains the empirical
tractability of the standard CAPM, but, unlike the latter, is arbitrage free.
We close by observing that Maccheroni, Marinacci, and Rustichini [MMR-2]
recently extended [MMR] to a dynamic setting, and for this reason we expect
that also our analysis can be extended to an intertemporal framework. This
will be the subject of future research, along with an empirical analysis of the
monotone CAPM.
8Acknowledgements
We thank Erio Castagnoli, Rose-Anne Dana, Paolo Guasoni, Jianjun Miao,
Giovanna Nicodano, Francesco Sangiorgi, Giacomo Scandolo, Elena Vigna,
and, especially, Dilip Madan, an associate editor, and two anonymous refer-
ees for helpful suggestions. We are also grateful for stimulating discussions to
the participants at the VI Quantitative Finance Workshop (Milan, January
2005), SMAI 2005 (Evian, May 2005), RUD 2005 (Heidelberg, June 2005)
and to the seminar audiences at Banca d’Italia, Boston University, Colle-
gio Carlo Alberto, Università Bocconi, Università di Bologna, Università del
Piemonte Orientale, Università di Salerno, and Università di Udine. Part of
this research was done while the …rst two authors were visiting the Depart-
ment of Economics of Boston University, which they thank for its hospitality.
The authors also gratefully acknowledge the …nancial support of the Colle-
gio Carlo Alberto, National Science Foundation (grant SES-0452477), and
Università Bocconi.
AMonotone Fenchel Duality
In this section we recall some important de…nitions of Convex Analysis, and
we prove some properties of monotone extensions of concave functionals that
pave the way for the proof of Theorem 2.
Let (E;k?k;>) be an ordered Banach space, i.e., an ordered vector space
endowed with a Banach norm such that the positive cone E+is closed and
generates E. Denote by E0the norm dual of E, and by E0
positive, linear, and continuous functionals on E. (See [Ch].)
+the cone of all
20
Page 22
Let : E ! R be a concave and continuous functional. The directional
derivative of at x is
d+ (x)(v) = lim
t!0+
(x + tv) ? (x)
t
8v 2 E:
If the above limit exists for t ! 0 and all v 2 E, is Gateaux di¤erentiable at
x, and the functional r (x) : v 7! d+ (x)(v) is called Gateaux di¤erential.
(See [Ph]). The superdi¤erential of at x is the subset of E0de…ned by
@ (x) = fx02 E0: (y) ? (x) ? hy ? x;x0i
while its Fenchel conjugate ?: E0! [?1;1) is given by
?(x0) = inf
8y 2 Eg;
x2Efhx;x0i ? (x)g8x02 E0:
Finally, the domain of monotonicity of is the set G ? E given by
G =?x 2 E : @ (x) \ E0
Next proposition con…rms the intuition that the set G is where the func-
tional is monotone.
+6= ??:
Proposition 12 Let E be an ordered Banach space and : E ! R be a
concave and continuous functional with G 6= ?.
(i) The functional~ : E ! R, given by
~ (x) = min
x02E0
+
fhx;x0i ? ?(x0)g8x 2 E;
(10)
is the minimal monotone functional that dominates ; that is,
~ (x) = supf (y) : y 2 E and y 6 xg8x 2 E:
(11)
(ii) Let x 2 E, x 2 G , d+ (x)(v) ? 0 8v 6 0 ,~ (x) = (x):
(iii) Let x02 E0,
?~
(iv) If ?is strictly concave on the intersection of its domain with E0
~ is Gateaux di¤erentiable and r~ (x) = argminx02E0
for all x 2 E.
??
(x0) =
? ?(x0) if x02 E0
+,
?1 otherwise.
(12)
+, then
+fhx;x0i ? ?(x0)g
21
Page 23
Moreover, if G is convex, for all x 2 E there exists y 2 G such that
y 6 x, and there exists a linear subspace F ? G such that jF is linear and
?(x0) is attained for all x02 E0
monotone functional that extends jG from G to E. In this case,
+with x0
jF= jF; then,~ is the minimal
~ (x) = supf (y) : y 2 G and y 6 xg =min
+:x0
x02E0
jF= jFfhx;x0i ? ?(x0)g
(13)
for all x 2 E.
By (i) and (ii),~ is the minimal monotone extension of jG from G
to E that dominates on E; in particular,~ (x) ? (x) for each x 2 E
and~ (x) = (x) for all x 2 G . Moreover, (iii) shows that the Fenchel
conjugates of~ and coincide on the cone E0
property. The …nal part, shows that, under additional assumptions (satis…ed,
e.g., by the mean-variance functional), the extension~ has even stronger
minimality properties.13
+. (iv) is a useful di¤erentiability
Proof. By the Fenchel-Moreau Theorem, (x) = infx02E0 fhx;x0i ? ?(x0)g
for all x 2 E. Set
~ (x) = inf
x02E0
+
fhx;x0i ? ?(x0)g8x 2 E:
(14)
Let x02 G (6= ?) and x0
02 @ (x0) \ E0
+(6= ?), then
(x) ? (x0) + hx;x0
0i ? hx0;x0
0i8x 2 E:
Therefore, the functional x0
dominates . Notice that
0+ ( (x0) ? hx0;x0
0i) is a¢ne, monotone, and it
hx0;x0
0i ? (x0) ? hx;x0
0i ? (x) 8x 2 E and ?(x0
0) = hx0;x0
0i ? (x0):
(15)
In particular, x0
02 E0
+and ?(x0
0) 2 R, hence, for all x 2 E,
fhx;x0i ? ?(x0)g < 1:
?1 < (x) ? inf
x02E0
+
Then~ dominates and it takes …nite values (i.e. it is a functional). Obvi-
ously,~ is concave and monotone.
13Similar results can be obtained in greater generality, e.g., in the case of a proper
concave, and upper semicontinuous function : E ! [?1;1) on a ordered locally
convex space. Details are available upon request.
22
Page 24
(i) Let ' : E ! R be a concave and monotone functional such that
' ? . Then ' is continuous,14and '(x) = infx02E0
all x 2 E.15In addition, since ' ? , then '?? ?and
'(x) = inf
x02E0
+fhx;x0i ? '?(x0)g for
+
fhx;x0i ? '?(x0)g ? inf
x02E0
+
fhx;x0i ? ?(x0)g =~ (x)
8x 2 E:
This shows that~ is the minimal concave and monotone functional that
dominates . It is easy to check that the function de…ned by Eq. (11) is the
minimal monotone functional that dominates , and it is concave, hence it
coincides with~ .
(iii) Follows from the de…nition of~ (Eq. 14) and the observation that
the function de…ned by Eq. (12) is proper, concave, and weak* upper semi-
continuous.
Since~ is concave and continuous, for all x02 E,16(iii) delivers
x0
02 @~ (x0) , x0
02 arg min
, x0
x02E0
n
fhx0;x0i ? ?(x0)g:
hx0;x0i ?~ ?(x0)
o
02 arg min
x02E0
+
This shows that the in…mum in Eq. (14) is attained, and that (iv) holds.
(ii) If x 2 G , then there is x0
02 @ (x) \ E0
+, and for all v 6 0
D
d+ (x)(v) = min
x02@ (x)hv;x0i ?
v;x
0
0
E
? 0:
Conversely, assume that d+ (x)(v) ? 0 for all v 6 0 and, by contradiction,
that x = 2 G , that is @ (x)\E0
cone and @ (x) is weak* compact, convex, and nonempty, there exists v 2 E
such that
hv;y0i ? 0 < hv;x0i
Since E+ is closed, then E+ =
hand side of Eq. (16) amounts to say that v 6 0, the right hand side that
minx02@ (x)hv;x0i > 0, which is absurd.
14If ' : ? ! R is concave and monotone on an open subset ? of an ordered Banach
space, then it is continuous:
15If ' : E ! [?1;1] is monotone and not identically ?1, then '?(x0) = ?1 for all
x0= 2 E0
16If ' : E ! R is concave and continuous, then, for all x02 E;
@'(x0) = arg min
+= ?. Since E0
+is a weak* closed and convex
8y02 E0
?x 2 E : hx;y0i ? 0 8y02 E0
+;x02 @ (x):
(16)
+
?.The left
+:
x02E0fhx0;x0i ? '?(x0)g 6= ?:
23
Page 25
If x 2 G , then there is x0
x0
02 arg min
x0
02 arg min
02 @ (x) \ E0
+that is
x02E0fhx;x0i ? ?(x0)g and x0
fhx;x0i ? ?(x0)g:
02 E0
+, then
x02E0
+
The …rst line implies (x) = hx;x0
hx;x0
there exists x0
0i ? ?(x0
0), the second that~ (x) =
+fhx;x0i ? ?(x0)g, then
0i ? ?(x0
0). Conversely, if (x) = minx02E0
02 E0
0i ? ?(x0
x0
02 arg min
+such that
hx;x0
0) = (x) = min
x02E0fhx;x0i ? ?(x0)g, then
x02E0fhx;x0i ? ?(x0)g and x0
02 E0
+,
therefore, x0
Finally, assume G is convex, for all x 2 E there exists y 2 G such that
y 6 x, and there exists a linear subspace F ? G such that jFis linear and
?(x0) is attained for all x02 E0
^ (x) = supf (y) : y 2 G and y 6 xg
It is easy to check that^ : E ! R is the minimal monotone functional
that extends jG from G to E, and that convexity of G implies that it
is concave. Therefore,^ ?~ ,^ (x) = (x) for all x 2 G ,^ is concave,
monotone, and linear on F (where it coincides with ). Hence, for all x 2
E,17
n
=inf
x02E0
02 @ (x) \ E0
+and x 2 G .
+with x0
jF= jF. Set
8x 2 E:
^ (x) = inf
x02E0
hx;x0i ?^ ?(x0)
n
+such that x0
n
o
=inf
+:x0
o
x02E0
jF=^ jF
n
hx;x0i ?^ ?(x0)
o
+:x0
jF= jF
hx;x0i ?^ ?(x0):
For all x02 E0
jF= jF,
o
^ ?(x0) = inf
x2E
hx;x0i ?^ (x)
? inf
x2G
n
hx;x0i ?^ (x)
o
= inf
x2G fhx;x0i ? (x)g:
(17)
Analogously, since is linear on F then, for all x 2 E,
~ (x) =
x02E0
min
+:x0
jF= jFfhx;x0i ? ?(x0)g.(18)
17If ' : E ! [?1;1] is linear on a subspace F, then '?(x0) = ?1 if x0
jF6= 'jF:
24
Page 26
Since ?(x0) is attained for all x02 E0
these properties there exists x02 E such that
hx0;x0i ? (x0) ? hx;x0i ? (x)
Then x02 @ (x0)\E0
That is, for x02 E0
?(x0) = inf
+such that x0
jF= jF, for every x0with
8x 2 E:
+and x02 G , in particular, ?(x0) is attained in G .
+such that x0
jF= jF,
x2G fhx;x0i ? (x)g ?^ ?(x0)
where the last inequality descends from Eq. (17). We conclude that
n
for all x 2 E, as wanted.
These results have been recently extended in Filipovic and Kupper [FK-2].
^ (x) = inf
+:x0
x02E0
jF= jF
hx;x0i ?^ ?(x0)
o
?
inf
+:x0
x02E0
jF= jFfhx;x0i ? ?(x0)g =~ (x)
?
B Proofs
Let f 2 L(P), we denote by Ff(t) = P (f ? t) its cumulative distribution
function and by gf(t) =Rt
functions. We report the proofs for the sake of completeness.
?1Ff(z)dz its integral distribution function.
Next two lemmas regroup some useful properties of integrated distribution
Lemma 13 For all z 2 R,
gf(z) =
Z
(f ? z)?dP
Z
Z
z ? (f ^ z)dP
Zz
= zP (f ? z) ?
f1ff?zgdP
= zP (f < z) ?
f1ff<zgdP
=
Z
=
?1
P (f < t)dt:
Proof. Let z 2 R. (f ? z)?= (z ? f)1ff?zg, hence
Z
(f ? z)?dP = zP (f ? z) ?
Z
f1ff?zgdP:
25
Page 27
Moreover,
zP (f ? z) ?
Z
f1ff?zgdP = zP (f < z) + zP (f = z) ?
Z
Z
ff<zg
fdP ?
Z
ff=zg
fdP
= zP (f < z) + zP (f = z) ?
ff<zg
fdP ? zP (f = z)
= zP (f < z) ?
Z
f1ff<zgdP:
Since f1ff?zg= (f ^ z) ? z1ff>zg, then
Z
zP (f ? z) ?f1ff?zgdP = zP (f ? z) ?
Z?f ^ z ? z1ff>zg
(f ^ z)dP + z
?dP
1ff>zgdP= zP (f ? z) ?
ZZ
= zP (f ? z) + zP (f > z) ?
Z
Z
(f ^ z)dP
=
z ? (f ^ z)dP:
Observe that z ? (f ^ z) ? 0, and so
Z
On the other hand, fz ? (f ^ z) ? ug = ff ? z ? ug for all u > 0. In fact,
z ? (f ^ z) ? u ) (f ^ z) ? z ? u < z ) (f ^ z) = f ) f ? z ? u
and
z ? (f ^ z)dP =
Z1
0
P (z ? (f ^ z) ? u)du:
f ? z ? u ) f ^ z ? (z ? u) ^ z ) f ^ z ? z ? u ) z ? (f ^ z) ? u:
Hence,
Z
=
0
z ? (f ^ z)dP =
Z1
Z1
0
P (z ? (f ^ z) ? u)du
P (f ? z ? u)du =
Zz
?1
P (f ? t)dt = gf(z);
thus the …rst four equalities hold.
Finally, notice that P (f < t) = limu!t? P (f ? u) 6= P (f ? t) for at
most a countably many ts.
?
26
Page 28
Lemma 14 The function gf: R ! [0;1) is continuous and
?gf(z + ") ? gf(z)
That is, Ff is the right derivative of gf, and Ff(z) is the derivative of gf at
every point z at which Ffis continuous. Moreover, setting ? = essinf (f), gf
is strictly increasing on (?;1), gf ? 0 on (?1;?],18limz!?+ gf(z) = 0+,
and limz!1gf(z) = 1.
Proof. The Fundamental Theorem of Calculus guarantees the continuity
and derivability properties of gf. Recall that
Ff(z) = lim
"!0+
"
?
8z 2 R:
essinf (f) = supf? 2 R : P (f < ?) = 0g:
If z 2 R and z ? essinf (f), for all t < z there exists ? > t such that
P (f < ?) = 0. Then,
0 ? P (f < t) ? P (f < ?) = 0:
This implies gf(z) = 0 for all z 2 (?1;?].
On the other hand, if ? < z < z0, then
gf(z0) ? gf(z) =
Zz0
z
P (f < t)dt ? P (f < z)(z0? z):
But P (f < z) = 0 would imply z ? ?, a contradiction. Therefore, gf(z0) ?
gf(z) > 0. That is, gfis strictly increasing on (?;1).
Notice that limt!1Ff(t) = 1. Then, for all n > 1 there exists k ? 1
(n;k 2 N) such that Ff(t) > 1 ?1
Zk+n
? n
n
nfor all t ? k. Therefore,
Zk+n
1 ?1
gf(k + n) =
?1
?
Ff(t)dt ?
?
k
Ff(t)dt
= n ? 1:
Since gfis increasing on R, then limz!1gf(z) = 1.
If ? > ?1, gf(?) = 0, continuity and nonnegativity imply limz!?+ gf(z) =
0+. Let ? = ?1, for all n > 1,
Z
= ((?f) ? ((?f) ^ n))dP:
gf(?n) =(?n ? (f ^ (?n)))dP =
Z
(((?f) _ n) ? n)dP
Z
18With the convention (?1;?1] = ?.
27
Page 29
The Monotone Convergence Theorem guarantees that limn!1gf(?n) = 0:
Monotonicity and nonnegativity imply limz!?+ gf(z) = 0+.
?
For the rest of the Appendix we indi¤erently write EPor just E. Denoting
by > the relation ? P-a.s., (L2(P);k?k2;>) is an ordered Banach space,
and its norm dual can be identi…ed with L2(P), with the duality relation
hf;Y i = E[fY ]. Simple computation shows that, for all ? > 0,
?f 2 L2(P) : rU?(f) > 0?=
this set is denoted by G?.
Lemma 15 Let f 2 L2(P) ? G?and t 2 R. Then
f ^ t 2 G?, gf(t) ?1
?
f 2 L2(P) : f ? EP[f] 61
?
?
(19)
?:
(20)
Proof. Notice that
f ^ t ? E[f ^ t] = f1ff?tg+ t1ff>tg? tP (f > t) ? E?f1ff?tg
= (f ? t)1ff?tg+ gf(t):
Since (f ? t)1ff?tg6 0,
gf(t) ?1
?
= f1ff?tg+ t1ff>tg? t + tP (f ? t) ? E?f1ff?tg
?
?) f ^ t ? E[f ^ t] 61
?;
i.e., gf(t) ?1
For the converse implication, notice that we are assuming f = 2 G?. Then,
being f ^ t 2 G?, it cannot be f ^ t = f P-a.s.. Hence, essup(f ^ t) = t. It
follows that:
?) f ^ t 2 G?.
f ^ t ? E[f ^ t] 61
?) essup(f ^ t) ? tP (f > t) ? E?f1ff?tg
) t ? tP (f > t) ? E?f1ff?tg
) tP (f ? t) ? E?f1ff?tg
) gf(t) ?1
??1
?
??1
?
?
??1
?;
i.e., f ^ t 2 G?) gf(t) ?1
Lemmas 14 and 15 immediately yield the following:
?.
?
28
Page 30
Corollary 16 Let f 2 L2(P)?G?, then g?1
f
?1
?
?= maxft 2 R : f ^ t 2 G?g:
Lemma 17 For all ? > 0;
(i) G?is convex, closed, and for all f 2 L2(P) ? G?and every " > 0 there
exists g 2 L2(P) such that jf (s) ? g (s)j < " for all s 2 S, g > f, and
U?(g) < U?(f).
(ii) U?is linear on the subspace T ? G?of all P-a.s. constant functions.
(iii) For all Y 2 L2(P),
??1
n
attained and strictly concave on this set.
U?
?(Y ) =
2?(E[Y2] ? 1)
?1
if E[Y ] = 1;
otherwise.
(21)
(iv) fY 2 L2(P) : E[Y ] = 1g =Y 2 L2(P) : h?;Y ijT= U?jT
o
and U?
?is
Proof. (i) Convexity of G? is trivial. Next we show closure. If fn 2 G?
and fn! f in L2(P), then there exists a subsequence gnof fnsuch that
gn(s) ! f (s) for P-almost all s in S. Let A0= fs 2 S : gn(s) ! f (s)g, and
for all n ? 1, An= fs 2 S : gn(s) ? E[gn] ? 1=?g, then P?T
n?0An
?
= 1
and for all s 2T
n?0An,
f (s) ? E[f] = lim
n!1(gn(s) ? E[gn]) ?1
?
that is f ? E[f] 6 1=?.
Moreover, if f = 2 G?, then rU?(f) is not positive and there exists A with
P (A) > 0 such thatR1ArU?(f)dP < 0, then
lim
t!0
t
U?(f + t1A) ? U?(f)
=
Z
1ArU?(f)dP < 0
that is, there exists " > 0 such that
U?(f + t1A) ? U?(f)
t
< 0
for all t 2 (0;"). As a consequence for all such ts, f+t1A> f, U?(f + t1A) <
U?(f) (set g = f + ("=2)1A).
(ii) is trivial.
29
Page 31
For convenience, (iii) and (iv) are proved together. For all t 2 R, U?(t1S) =
t, then for all Y 2 L2(P)
n
and U?
?(Y ) = ?1 for all Y that does not belong to this set.
If E[Y ] = 1, the functional W?: L2(P) ! R given, for all f 2 L2(P), by
W?(f) = hf;Y i ? U?(f) = hf;Y i ? hf;1i +?
Y 2 L2(P) : h?;Y ijT= U?jT
o
=?Y 2 L2(P) : E[Y ] = 1?;
2hf ? E[f];f ? E[f]i
is well de…ned, convex, and Gateaux di¤erentiable. Its Gateaux di¤erential
is
rW?(f) = Y ? 1 + ?(f ? E[f]):
Notice that^f = ???1Y solves rW?
tains its minimumon L2(P) at^f. This implies that U?
is attained, and
?
= ?1
2
(22)
?^f
?
= 0 (since E[Y ] = 1), and W?at-
?(Y ) = inff2L2(P)W?(f)
U?
?(Y ) = min
f2L2(P)W?(f) = W?
?E?Y2?+1
?^f
?
=
?1
?Y;Y
?
?
?
?E?Y2?? 1?:
?1
?Y;1
?
+?
2Var
?
?1
?Y
?
?E[Y ] +?1
?2Var[Y ] = ?1
2?
This concludes the proof of Eq. (21). Strict concavity of U?
a straightforward application of the the Cauchy-Schwartz inequality.
?in its domain is
?
Proof of Lemma 1. It is an immediate consequence of Eq. (19) and part
(i) of the above Lemma 17.
?
Proof of Theorem 2. As observed, U? : L2(P) ! R is a concave and
continuous functional on an ordered Banach space, G? = GU?, Lemma 17
and Corollary 16, guarantee that all the hypotheses of Proposition 12 are
satis…ed. For all f 2 L2(P),
?
?
and V?has all the desired properties.
~U?(f) = min
+(P):E[Y ]=1
Y 2L2
E[fY ] +1
2?
?E?Y2?? 1??
?
=min
Q2?2(P)
EQ[f] +1
2?C (QjjP)= V?(f);
?
30
Page 32
Remark 18 Proposition 12.(iv) and Lemma 17.(iv) guarantee that V? is
Gateaux di¤erentiable, and
?
rV?(f) = arg min
+(P):E[Y ]=1
Y 2L2
E[fY ] +1
2?
?E?Y2?? 1??
8f 2 L2(P):
(23)
Theorem 19 Let f 2 L2(P). Then
V?(f) =
?E[f] ??
?: Moreover, the Gateaux di¤erential of V?at f is
rV?(f) = ?(? ? f)1ff??g:
Proof. For all f 2 L2(P), V?(f) = minQ2?2(P)
That is, V?(f) is the value of the problem:
8
:
Remark 18 guarantees that the solution of such problem exists, is unique,
and it coincides with the Gateaux derivative of V? at f. Notice that Y is
a solution of problem (24) if and only if it is a solution of the constrained
optimization problem:
8
:
The Lagrangian is
2Var[f]
if f 2 G?;
else,
E[f ^ ?] ??
2Var[f ^ ?]
where ? = g?1
f
?1
?
?EQ(f) +
?
1
2?C (QjjP)?.
<
min?E[fY ] +
E [Y ] = 1
1
2?E[Y2] ?
1
2?
Y > 0
:
(24)
<
min?E[fY ] +
E[Y ] = 1
1
2?E[Y2]?
Y > 0
:
(25)
L(Y;?;?) = E(fY ) +1
2?E?Y2?? E(?Y ) ? ?(E(Y ) ? 1);
with ? 2 L2
+(P), ? 2 R. The Kuhn-Tucker optimality conditions are:
f +1
?Y ? ? ? ? = 0
E(?Y ) = 0
Y > 0;? > 0
E(Y ) = 1
(P-a.s.)
31
Page 33
Since ?;Y > 0, they are equivalent to:
f +1
?Y ? ? ? ? = 0
?Y = 0
(P-a.s.)
Y > 0;? > 0
E(Y ) = 1
(P-a.s.)
that is,
f +1
?Y ? ? ? 0
f +1
?Y ? ?
P-a.s.(26)
?
?
Y = 0P-a.s.(27)
Y ? 0
E[Y ] = 1
P-a.s.(28)
(29)
It is su¢cient to …nd (Y?;??) that satisfy (26) - (29) everywhere (not
only P-a.s.).
If s 2 fY?> 0g, then by (27) f (s) +1
Y?(s) = ?(??? f (s)):
In particular, ???f (s) > 0, and s 2 ff < ??g. Conversely, if s 2 ff < ??g,
then by (26) Y?(s) ? ?(??? f (s)) > 0 and s 2 fY?> 0g. In sum,
fY?> 0g = ff < ??g and
Y?= ?(??? f)1ff<??g.
By (29),
1 = E[Y?] = E??(??? f)1ff<??g
?Y?(s) ? ??= 0 and
(30)
?
= ????P (f < ??) ? E?f1ff<??g
gf(??) = ??P (f < ??) ? E?f1ff<??g
??;
?=1
that is,
?:
In other words,
??= g?1
f
?1
?
?
? ?;
(31)
and ??is unique. A fortiori, Y?is unique and
Y?= ?(? ? f)1ff<?g:
(32)
32
Page 34
By construction, the pair (Y?;??) de…ned by (31) and (32) is a solution of
(26) - (29). Since the solution of (24) exists and it is unique, we conclude
that Y?de…ned as in Eq. (32) is the unique solution of (24).
Notice that Y?= ?(? ? f)1ff<?g+ ?(? ? f)1ff=?g= ?(? ? f)1ff??g,
(Y?)2= ?2?f21ff??g+ ?21ff??g? 2?f1ff??g
and
?Z
Moreover,
E[fY?] = E?f?(? ? f)1ff??g
= ??f1ff??gdP ? ?
?
E?(Y?)2?= ?2
f21ff??gdP + ?2P (f ? ?) ? 2?
Z
f1ff??gdP
?
:
?= E???f1ff??g? ?f21ff??g
f21ff??gdP:
?
ZZ
Therefore,
V?(f) = E[fY?] +1
2?E?(Y?)2??1
f1ff??gdP ? ?
2?
= ??
ZZ
f21ff??gdP+
+?
2
?Z
f21ff??gdP + ?2P (f ? k) ? 2?
f21ff??gdP +?
Z
f1ff??gdP
?
?1
2?
= ??
2
Z
2?2P (f ? k) ?1
2?
Also observe that f1ff??g+ ?1ff>?g= f ^ ?, whence
E[f ^ ?] = E?f1ff??g
?+ ?P (f > ?) = E?f1ff??g
?;
?? ?P (f ? ?) + ?
= ?gf(?) + ? = ? ?1
and
Var[f ^ ?] = E
h?f1ff??g+ ?1ff>?g
f21ff??gdP + ?2P (f > k) ? ?2?1
f21ff??gdP ? ?2P (f ? k) ?1
?2i
?
?
? ?1
?
?2
=
Z
Z
?2+ 2?
?
=
?2+ 2?
?:
33
Page 35
Finally,
E[f ^ ?] ??
2Var[f ^ ?] = ? ?1
???
2
??
?Z
f21ff??gdP ? ?2P (f ? k) ?1
f21ff??gdP ??
f21ff??gdP +?
?2+ 2?
?
?
?
= ? ?1
= ??
= V?(f):
??
Z
2
Z
2?2P (f ? k) ?1
2?2P (f ? k) ?1
2?+ ?
22?
?
Proof of Theorem 3.
Theorem 19.
It is now enough to combine Corollary 16 and
?
Remark 20 Inspection of the proof of Theorem 19 shows that:
? For all f 2 L2(P) (not only for f 2 L2(P)?G?), setting ? = g?1
have
f
?1
?
?we
V?(f) = E[f ^ ?] ??
rV?(f) = ?(? ? f)1ff??g= ?(f ? ?)?;
rV?(f) = argminQ2?2(P)
2Var[f ^ ?];
?
EQ(f) +1
2?C (QjjP)
?
:
? The properties of gf guarantee that ? exists, it is unique, and
?P (f ? ?) ?Rf1ff??gdP; therefore, P (f ? ?) > 0.
? Moreover,1
?, and so
?
=
P (f ? ?)? ?(f ? E[fjf ? ?])
1
?=
?= ?P (f ? ?)?Rf1ff??gdP implies
1
?P(f??)+RfdPff??g=
rV?(f) = ?(? ? f)1ff??g=
?
1
P (f ? ?)+ ?
Z
fdPff??g? ?f
?
1ff??g
1
?
1ff??g:
N
Proof of Theorem 5. By Theorem 13.8 of Chong and Rice [CR], for all
f;Y 2 L2(P), the set?RfY0dP : Y02 L2(P) and Y0?cY?coincides with
?Z1
the interval
0
F?1
f
(1 ? u)F?1
Y (u)du;
Z1
0
F?1
f
(u)F?1
Y (u)du
?
;
34
Page 36
where ?cis the concave order and F?1
sider the function ? : L2(P) ! [0;1] de…ned by
?
It is easy to check that Y0?cY implies ?(Y0) ? ?(Y ).20Let f 2 L2(P),
f
is the quantile function of f.19Con-
?(Y ) =
1
2?(E[Y2] ? 1)
1
if Y > 0 and E[Y ] = 1;
otherwise.
V?(f) =min
Y 2L2(P)fE[fY ] + ?(Y )g ?
Moreover, for all Y 2 L2(P) there exists Y02 L2(P) with Y0?cY such
thatR1
F?1
f
(1 ? u)F?1
inf
Y 2L2(P)
?Z1
0
F?1
f
(1 ? u)F?1
Y (u)du + ?(Y )
?
:
0F?1
f
(1 ? u)F?1
Y (u)du =RfY0dP. Since ?(Y0) ? ?(Y ), then
Y (u)du+?(Y ) = fY0dP +?(Y ) ?
Z1
hence
0
ZZ
fY0dP +?(Y0);
V?(f) = inf
Y 2L2(P)
?Z1
0
F?1
f
(1 ? u)F?1
Y (u)du + ?(Y )
?
:
If f ?cg, an inequality of Hardy (see, e.g., [CR, p. 57-58]) delivers
Z1
and V?(f) ? V?(g). Let f ?SSDg, Theorem 1.1 of [Cn] guarantees that
f ? h ?c g for some h 2 L2
V?(f ? h) ? V?(g).
Proof of Theorem 6. The maximization problem is
0
F?1
f
(1 ? u)F?1
Y (u)du ?
Z1
0
F?1
g
(1 ? u)F?1
Y (u)du
8Y 2 L2(P);
+(P). Since V? is monotone, then V?(f) ?
?
sup
?2RV?(W?)
(remember that ? 2 Rn, X 2 L2(P)n, and~1 iswhere W?= R+??
a vector of 1s). From Theorem 19 we know that V?is Gateaux di¤erentiable
and
?
19I.e.Y0?c Y i¤
and [Cn] denote this relation by Y0? Y and call it majorization), and F?1
inf fz 2 R : Ff(z) ? tg for all t 2 [0;1]:
20Let Y0?cY . If ?(Y ) = 1, then ?(Y0) ? ?(Y ). Else Y > 0 and E[Y ] = 1, then also
Y0> 0 and E[Y0] = 1 (see [CR, p. 62]). The function ?(t) = ?1
whence E[?(Y0)] ? E[?(Y )], and ?(Y ) = ?E[?(Y )] ? ?E[?(Y0)] = ?(Y0).
?
X ?~1R
?
rV?(W?) =
1
P (W?? ??)? ?(W?? E[W?jW?? ??])
R?(Y0)dP ?
?
1fW????g;
R?(Y )dP for all concave ? : R ! R ([CR]
f
(t) =
2?
?t2? 1?
is concave,
35
Page 37
where ??solves:
P (W?? ??)???? EP[W?jW?? ??]?=1
Since for all i = 1;:::;n
?:
(33)
@V?(W?)
@?i
= E[rV?(W?)(Xi? R)];
the …rst order conditions for an optimum are:
E[XrV?(W?)] =~1R:
(34)
Substituting rV?(W?):
?
set A = fW?? ??g to obtain
E[X jA] ? ?P (A)(E[(? ? X)X jA] ? E[X jA]E[? ? X jA]) =~1R
E
1fW????g
P (W?? ??)X ? ??(? ? X)1fW????gX ? E[? ? X jW?? ??]1fW????gX??
=~1R
the observation that E[(? ? X)X jA] ? E[X jA]E[? ? X jA] = Var[X jA]?
yields:
h
These are the …rst n equations. The (n + 1)-th is the equation which deter-
mines ??, that is (33) or (see Lemma 13)
EX ?~1RjW?? ??
i
= ?P (W?? ??)Var[X jW?? ??]?:
E?(W?? ??)??=1
?:
Concavity of V?guarantees the su¢ciency of …rst order conditions.
?
Proof of Proposition 7. Set ??= ??
solves is
mmv. The maximization problem it
max
?2Rmin
Y 2Y
?
E[(R + ?(X ? R))Y ] +1
+: E[Y ] = 1?. Clearly, Y is convex and compact, and
2?E?Y2??1
2?
?
(35)
where Y =?Y 2 RS
(??;Y?) is a solution of (35) if and only if it is a solution of
max
?2Rmin
Y 2YG(?;Y )
36
Page 38
where G(?;Y ) = E[(R + ?(X ? R))Y ] +
G : R ? Y ! R is continuous, it is a¢ne in ? (for each …xed Y ) and strictly
convex in Y (for each …xed ?). Set v = max?2RminY 2YG(?;Y ), by (a
version of) the Min-Max Theorem (e.g. [?, p. 134]) there exists?Y 2 Y such
that
v = sup
1
2?E[Y2]. Moreover, notice that
?2RG??;?Y?:
Moreover,
G???;?Y?? min
therefore, G???;?Y?
Y 2YG(??;Y ) = G(??;Y?) = v = sup
= minY 2YG(??;Y ), strict convexity implies?Y = Y?.
In turn, this yields sup?2RG(?;Y?) = v 6= 1 and it cannot be
?
therefore
E[Y?(X ? R)] = 0:
Y?is the solution of problem (25) in the proof of Theorem 19 with f =
R + ??(X ? R) = W??. Therefore, there exist ??2 R and ? 2 L2(P)
(= RS) such that Y?satis…es the following conditions:
?2RG??;?Y?? G???;?Y?;
sup
?2R
R + ?E[(X ? R)Y?] +1
2?E?(Y?)2??
= 1;
(36)
R + ??(X ? R) +1
E[Y?] = 1;
Y?> 0;? > 0;?Y?= 0:
?Y?? ? ? ??= 0;
(37)
(38)
(39)
Taking the expectation of both sides of (37) we obtain:
(1 ? ??)R + ??E[X] +1
?E[Y?] ? E[?] ? ??= 0
(40)
and, subtracting (40) from (37):
??(X ? E[X]) +1
?(Y?? E[Y?]) ? (? ? E[?]) = 0:
Rearranging and using (38):
Y?= 1 ? ???(X ? E[X]) + ?(? ? E[?]):
Multiply both sides by ?, take expectations and use (39) to get:
(41)
E[?] ? ???Cov[?;X] + ?Var[?] = 0
37
Page 39
and, rearranging terms:
???Cov[?;X] = E[?] + ?Var[?]:
(42)
Since ? > 0, then E[?] ? 0 thus:
??= 0 ) ? = 0 ) Cov[?;X] = 0;
??> 0 ) Cov[?;X] ? 0;
??< 0 ) Cov[?;X] ? 0:
(43)
(44)
(45)
Now, plugging (41) into (36) we obtain:
E[(1 ? ???(X ? E[X]) + ?(? ? E[?]))X] = R
or:
E[X] ? ???Var[X] + ?Cov[?;X] = R
which becomes:
??=1
?
E[X ? R]
Var[X]
+Cov[?;X]
Var[X]
Recalling that:
??
mv=1
?
E[X ? R]
Var[X]
we obtain:
??= ??
mv+Cov[?;X]
Var[X]
(46)
Using (43) - (45), it is now obvious that:
??= 0 ) ??= ??
??> 0 ) ??? ??
??< 0 ) ??? ??
mv= 0;
(47)
mv;
(48)
mv:
(49)
From the proof of Theorem 19 – Eq. (31) – we know that ??= g?1
Furthermore, if P (W?? > ??) > 0, since S is …nite, there exists s such that
W??
?1
?
?= ??.
R + ??(X (s) ? R) = W?? (s) > ??= ??;
that is R + ??(X (s) ? R) ? ??> 0. Since Y?(s) ? 0, by (37), we have
?(s) = R + ??(X (s) ? R) ? ??+1
?Y?> 0;
and E[?] > 0. Thus, in this case (42) implies ??Cov[?;X] > 0 and the
inequalities in (48) and (49) become strict.
38
Page 40
Finally, we want to show that ????
????
Suppose ??> 0 and ??
mv? 0. By contradiction, suppose
mv< 0 or ??< 0 and ??
mv< 0. Then, either ??> 0 and ??
mv< 0, since
mv> 0.
??
mv=1
?
E[X ? R]
Var[X]
;
it must be E[X ? R] < 0 and
??E[X ? R] < 0:
(50)
Clearly, if ??< 0 and ??
a saddle point for
mv> 0, (50) still holds. Remember that (??;Y?) is
G(?;Y ) = E[(R + ?(X ? R))Y ] +1
2?E?Y2?
and so:
G(??;Y?) ? G(??;1S) = R + ??E[X ? R] < R
where the last inequality follows from (50). But,
min
Y 2YG(0;Y ) = R + min
Y 2Y
1
2?E?Y2?? R
?2Rmin > G(??;Y?) = max
Y 2YG(?;Y ) ? min
Y 2YG(0;Y );
which is impossible.
?
Proof of Proposition 8. Assume
8
:
Set
<
?P?W?? ? ???Var?X??W?? ? ?????= E
h
X ?~1R??W?? ? ??i
;
E
h?W?? ? ????i
=1
?:
(51)
??=?
???and ??=?
?
???? R?+ R:
Then,
W?? = R + ???
= R ??
=?
?W?? + R
?
X ?~1R
?R +?
?
?
???? X ??
1 ??
?
= R +?
???? X ??
????~1R
????~1R
?R +?
?
;
39
Page 41
and so
W?? =?
?W?? + R
?
1 ??
?
?
and ??=?
???+ R
?
1 ??
?
?
:
(52)
Equation (51) becomes
8
:
that is
(
<
?P (W?? ? ??)Var[X jW?? ? ??]?
EPh??
???= E
h
X ?~1RjW?? ? ??i
;
?(W?? ? ??)??i
=1
?:
?P (W?? ? ??)Var[X jW?? ? ??]??= E
E?(W?? ? ??)??=1
so that (??;??) solves the portfolio selection problem for an investor with
uncertainty aversion ?.
h
X ?~1RjW?? ? ??i
;
?:
?
Proof of Lemma 9. Consider the portfolio selection problem (6) with
? = 1. Next we show that
max
?2RnV1(W?) = max
?2RV1(?Xm+ (1 ? ?)R)
and m 2 argmax?2RV1(?Xm+ (1 ? ?)R).
First observe that, for all ? 2 R,
?Xm+ (1 ? ?)R = ? (?m? X) +
?
1 ? ??m?~1
?
R = W??m
hence max?2RV1(?Xm+ (1 ? ?)R) ? max?2Rn V1(W?). Conversely, if ?12
argmax?2Rn V1(W?), then m = ?1?~1 and ?m= ?1=
Hence
?
?1?~1
?
or ?1= m?m.
max
?2RnV1(W?) = V1(W?1) = V1(Wm?m) = V1
= V1(mXm+ (1 ? m)R) ? max
? max
?
m?m? X +
?2RV1(?Xm+ (1 ? ?)R)
?
1 ? m?m?~1
?
R
?
?2RnV1(W?)
as wanted. Now, applying Theorem 6 with Xminstead of X, m must satisfy
the following conditions:
(
E = 1:
P (mXm+ (1 ? m)R ? ?)Var[XmjmXm+ (1 ? m)R ? ?]m = E[Xm? RjmXm+ (1 ? m)R ? ?];
h
(mXm+ (1 ? m)R ? ?)?i
(53)
40
Page 42
Moreover, since Xm is the (optimal) …nal wealth of an agent with initial
wealth 1 and uncertainty aversion coe¢cient m, again by Theorem 6 applied
to such an agent it must be the case that
E?(Xm? ?m)??= 1=m:
But the second equation of previous system is equivalent to
"?
(54)
EXm?
??
m?(1 ? m)
m
R
???#
=1
m
(55)
and since gXmis strictly increasing (see Lemma 14), then
?
m?(1 ? m)
m
R = ?m:
(56)
By (56), fmXm+ (1 ? m)R ? ?g =
and (55) is equivalent to (54), thus (53) amounts to
?P (Xm? ?m)Var[XmjXm? ?m]m = E[Xm? RjXm? ?m];
as wanted.
n
Xm?
?
m?(1?m)
mR
o
= fXm? ?mg
E?(Xm? ?m)??= 1=m;
?
Proof of Lemma 10. (i) First observe that by Remark 20 and Eq. (52) in
the proof of Proposition 8, for all ? > 0
rV?(W??) = ????? W???1fW?????g= ?
= m(?m? Xm)1fXm??mg= m(Xm? ?m)?= rVm(Xm):
(ii) The …rst order conditions for an optimum are E[XrVm(W?m)] =~1R
(see Eq. 34).
(iii) Descends from Remark 18.
?m
??m?m
?W?m
?
1fW?m??mg
?
References
[Bi] Bigelow, J. P., Consistency of mean-variance analysis and expected
utility analysis: a complete characterisation, Economic Letters, 43,
187-192, 1993.
[Br] Britten-Jones, M., The sampling error in estimates of mean-variance
e¢cient portfolio weights, Journal of Finance, 45, 655-671, 1999.
41
Page 43
[BKM] Bodie, Z., A. Kane, and A. J. Marcus, Investments, McGraw Hill,
Boston, 2002.
[CR]Chong, K. M. and N. M. Rice, Equimeasurable rearrangements of
functions, Queens Papers in Pure and Applied Mathematics, 28,
1971.
[Cn]Chong, K. M., Doubly stochastic operators and rearrangement the-
orems, Journal of Mathematical Analysis and Applications, 56, 309–
316, 1976.
[Ch]Choquet, G., Lectures on Analysis (vol. 2), Benjamin, New York,
1969.
[Da]Dana, R-A., A representation result for concave Schur concave func-
tions, Mathematical Finance, 15, 613-634, 2005.
[DI]Dybvig, P. H. and J. E. Ingersoll, Mean-Variance theory in complete
markets, The Journal of Business, 55, 233-251, 1982.
[DR]Dybvig, P. H. and S. A. Ross, Arbitrage, in New Palgrave: A Dictio-
nary of Economics, J. Eatwell, M. Milgate, and P. Newman (eds.),
MacMillan, London, 1987.
[FK] Filipovic, D. and M. Kupper, Equilibrium Prices for Monetary Util-
ity Functions, mimeo, 2006.
[FK-2] Filipovic, D. and M. Kupper, Monotone and cash-invariant convex
functions and hulls, mimeo, 2006.
[GS]Gilboa, I. and D. Schmeidler, Maxmin expected utility with non-
unique prior, Journal of Mathematical Economics, 18, 141-153, 1989.
[GRS] Gibbons, M. R., S. A. Ross and J. Shanken, A test of the e¢ciency
of a given portfolio, Econometrica, 57, 1121-1152, 1989.
[HS]Hansen, L. and T. Sargent, Robust control and model uncertainty,
American Economic Review, 91, 60-66, 2001.
[JM]Jarrow, R. A. and D. B. Madan, Is Mean-Variance analysis vacuous:
Or was Beta still born?, European Finance Review, 1, 15-30, 1997.
[KS]Kandel, S. and R. Stambaugh, A mean-variance framework for tests
of asset pricing models, Review of Financial Studies, 2, 125-156,
1987.
42
Page 44
[KC] Kupper, M. and P. Cheridito, Time-consistency of indi¤erence prices
and monetary utility functions, mimeo, 2006.
[LV]Liese, F. and I. Vajda, Convex Statistical Distances, Teubner,
Leipzig, 1987.
[MMR] Maccheroni, F., M. Marinacci and A. Rustichini, Ambiguity aver-
sion, robustness, and the variational representation of preferences,
Econometrica, 74, 1447-1498, 2006.
[MMR-2] Maccheroni, F., M. Marinacci, and A. Rustichini, Dynamic varia-
tional preferences, Journal of Economic Theory, 128, 4-44, 2006.
[MR]MacKinlay, A. C. and M. P. Richardson, Using generalized method
of moments to test mean-variance e¢ciency, Journal of Finance, 46,
511-527, 1991.
[Ma]Markowitz, H. M., Portfolio selection, Journal of Finance, 7, 77-91,
1952.
[Ph] Phelps, R. R., Convex functions, monotone operators and di¤eren-
tiability, Springer-Verlag, New York, 1992.
[R]Ross S. A., Neoclassical Finance, Princeton University Press, Prince-
ton, 2005.
[Sh] Sharpe, W. F., Capital asset prices: a theory of market equilibrium
under conditions of risk, Journal of Finance, 19, 425-442, 1964.
[Sh-2] Sharpe, W. F., Capital asset prices with and without negative hold-
ings, Journal of Finance, 46, 489-509,1991.
[To]Tobin, J., Liquidity preference as behavior towards risk, Review of
Economic Studies, 25, 65-86, 1958.
43
Page 45
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Page 46
"TEMI" LATER PUBLISHED ELSEWHERE
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L. DEDOLA and F. LIPPI, The monetary transmission mechanism: Evidence from the industries of 5 OECD
countries, European Economic Review, 2005, Vol. 49, 6, pp. 1543-1569, TD No. 389 (December
2000).
D. Jr. MARCHETTI and F. NUCCI, Price stickiness and the contractionary effects of technology shocks.
European Economic Review, Vol. 49, 5, pp. 1137-1164, TD No. 392 (February 2001).
G. CORSETTI, M. PERICOLI and M. SBRACIA, Some contagion, some interdependence: More pitfalls in tests
of financial contagion, Journal of International Money and Finance, Vol. 24, 8, pp. 1177-1199, TD
No. 408 (June 2001).
GUISO L., L. PISTAFERRI and F. SCHIVARDI, Insurance within the firm. Journal of Political Economy, Vol.
113, 5, pp. 1054-1087, TD No. 414 (August 2001)
R. CRISTADORO, M. FORNI, L. REICHLIN and G. VERONESE, A core inflation indicator for the euro area,
Journal of Money, Credit, and Banking, Vol. 37, 3, pp. 539-560, TD No. 435 (December 2001).
F. ALTISSIMO, E. GAIOTTI and A. LOCARNO, Is money informative? Evidence from a large model used for
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G. DE BLASIO and S. DI ADDARIO, Do workers benefit from industrial agglomeration? Journal of regional
Science, Vol. 45, (4), pp. 797-827, TD No. 453 (October 2002).
G. DE BLASIO and S. DI ADDARIO, Salari, imprenditorialità e mobilità nei distretti industriali italiani, in L.
F. Signorini, M. Omiccioli (eds.), Economie locali e competizione globale: il localismo industriale
italiano di fronte a nuove sfide, Bologna, il Mulino, TD No. 453 (October 2002).
R. TORRINI, Cross-country differences in self-employment rates: The role of institutions, Labour
Economics, Vol. 12, 5, pp. 661-683, TD No. 459 (December 2002).
A. CUKIERMAN and F. LIPPI, Endogenous monetary policy with unobserved potential output, Journal of
Economic Dynamics and Control, Vol. 29, 11, pp. 1951-1983, TD No. 493 (June 2004).
M. OMICCIOLI, Il credito commerciale: problemi e teorie, in L. Cannari, S. Chiri e M. Omiccioli (eds.),
Imprese o intermediari? Aspetti finanziari e commerciali del credito tra imprese in Italia,
Bologna, Il Mulino, TD No. 494 (June 2004).
L. CANNARI, S. CHIRI and M. OMICCIOLI, Condizioni di pagamento e differenziazione della clientela, in L.
Cannari, S. Chiri e M. Omiccioli (eds.), Imprese o intermediari? Aspetti finanziari e commerciali
del credito tra imprese in Italia, Bologna, Il Mulino, TD No. 495 (June 2004).
P. FINALDI RUSSO and L. LEVA, Il debito commerciale in Italia: quanto contano le motivazioni finanziarie?, in
L. Cannari, S. Chiri e M. Omiccioli (eds.), Imprese o intermediari? Aspetti finanziari e commerciali
del credito tra imprese in Italia, Bologna, Il Mulino, TD No. 496 (June 2004).
A. CARMIGNANI, Funzionamento della giustizia civile e struttura finanziaria delle imprese: il ruolo del
credito commerciale, in L. Cannari, S. Chiri e M. Omiccioli (eds.), Imprese o intermediari? Aspetti
finanziari e commerciali del credito tra imprese in Italia, Bologna, Il Mulino, TD No. 497 (June
2004).
G. DE BLASIO, Credito commerciale e politica monetaria: una verifica basata sull’investimento in scorte,
in L. Cannari, S. Chiri e M. Omiccioli (eds.), Imprese o intermediari? Aspetti finanziari e
commerciali del credito tra imprese in Italia, Bologna, Il Mulino, TD No. 498 (June 2004).
G. DE BLASIO, Does trade credit substitute bank credit? Evidence from firm-level data. Economic notes,
Vol. 34, 1, pp. 85-112, TD No. 498 (June 2004).
A. DI CESARE, Estimating expectations of shocks using option prices, The ICFAI Journal of Derivatives
Markets, Vol. 2, 1, pp. 42-53, TD No. 506 (July 2004).
M. BENVENUTI and M. GALLO, Il ricorso al "factoring" da parte delle imprese italiane, in L. Cannari, S.
Chiri e M. Omiccioli (eds.), Imprese o intermediari? Aspetti finanziari e commerciali del credito
tra imprese in Italia, Bologna, Il Mulino, TD No. 518 (October 2004).
L. CASOLARO and L. GAMBACORTA, Redditività bancaria e ciclo economico, Bancaria, Vol. 61, 3, pp. 19-
27, TD No. 519 (October 2004).
F. PANETTA, F. SCHIVARDI and M. SHUM, Do mergers improve information? Evidence from the loan
market, CEPR Discussion Paper, 4961, TD No. 521 (October 2004).
Page 47
P. DEL GIOVANE and R. SABBATINI, La divergenza tra inflazione rilevata e percepita in Italia, in P. Del
Giovane, F. Lippi e R. Sabbatini (eds.), L'euro e l'inflazione: percezioni, fatti e analisi, Bologna,
Il Mulino, TD No. 532 (December 2004).
R. TORRINI, Quota dei profitti e redditività del capitale in Italia: un tentativo di interpretazione, Politica
economica, Vol. 21, 1, pp. 7-41, TD No. 551 (June 2005).
M. OMICCIOLI, Il credito commerciale come “collateral”, in L. Cannari, S. Chiri, M. Omiccioli (eds.),
Imprese o intermediari? Aspetti finanziari e commerciali del credito tra imprese in Italia, Bologna,
il Mulino, TD No. 553 (June 2005).
L. CASOLARO, L. GAMBACORTA and L. GUISO, Regulation, formal and informal enforcement and the
development of the household loan market. Lessons from Italy, in Bertola G., Grant C. and Disney
R. (eds.) The Economics of Consumer Credit: European Experience and Lessons from the US,
Boston, MIT Press, TD No. 560 (September 2005).
S. DI ADDARIO and E. PATACCHINI, Lavorare in una grande città paga, ma poco, in Brucchi Luchino (ed.),
Per un’analisi critica del mercato del lavoro, Bologna , Il Mulino, TD No. 570 (January 2006).
P. ANGELINI and F. LIPPI, Did inflation really soar after the euro changeover? Indirect evidence from ATM
withdrawals, CEPR Discussion Paper, 4950, TD No. 581 (March 2006).
S. FEDERICO, Internazionalizzazione produttiva, distretti industriali e investimenti diretti all'estero, in L. F.
Signorini, M. Omiccioli (eds.), Economie locali e competizione globale: il localismo industriale
italiano di fronte a nuove sfide, Bologna, il Mulino, TD No. 592 (October 2002).
S. DI ADDARIO, Job search in thick markets: Evidence from Italy, Oxford Discussion Paper 235,
Department of Economics Series, TD No. 605 (December 2006).
2006
F. BUSETTI, Tests of seasonal integration and cointegration in multivariate unobserved component
models, Journal of Applied Econometrics, Vol. 21, 4, pp. 419-438, TD No. 476 (June 2003).
C. BIANCOTTI, A polarization of inequality? The distribution of national Gini coefficients 1970-1996,
Journal of Economic Inequality, Vol. 4, 1, pp. 1-32, TD No. 487 (March 2004).
L. CANNARI and S. CHIRI, La bilancia dei pagamenti di parte corrente Nord-Sud (1998-2000), in L.
Cannari, F. Panetta (a cura di), Il sistema finanziario e il Mezzogiorno: squilibri strutturali e divari
finanziari, Bari, Cacucci, TD No. 490 (March 2004).
M. BOFONDI and G. GOBBI, Information barriers to entry into credit markets, Review of Finance, Vol. 10,
1, pp. 39-67, TD No. 509 (July 2004).
FUCHS W. and LIPPI F., Monetary union with voluntary participation, Review of Economic Studies, Vol.
73, pp. 437-457 TD No. 512 (July 2004).
GAIOTTI E. and A. SECCHI, Is there a cost channel of monetary transmission? An investigation into the
pricing behaviour of 2000 firms, Journal of Money, Credit and Banking, Vol. 38, 8, pp. 2013-2038
TD No. 525 (December 2004).
A. BRANDOLINI, P. CIPOLLONE and E. VIVIANO, Does the ILO definition capture all unemployment?, Journal
of the European Economic Association, Vol. 4, 1, pp. 153-179, TD No. 529 (December 2004).
A. BRANDOLINI, L. CANNARI, G. D’ALESSIO and I. FAIELLA, Household wealth distribution in Italy in the
1990s, in E. N. Wolff (ed.) International Perspectives on Household Wealth, Cheltenham, Edward
Elgar, TD No. 530 (December 2004).
P. DEL GIOVANE and R. SABBATINI, Perceived and measured inflation after the launch of the Euro:
Explaining the gap in Italy, Giornale degli economisti e annali di economia, Vol. 65, 2 , pp. 155-
192, TD No. 532 (December 2004).
M. CARUSO, Monetary policy impulses, local output and the transmission mechanism, Giornale degli
economisti e annali di economia, Vol. 65, 1, pp. 1-30, TD No. 537 (December 2004).
A. NOBILI, Assessing the predictive power of financial spreads in the euro area: does parameters
instability matter?, Empirical Economics, Vol. 31, 1, pp. 177-195, TD No. 544 (February 2005).
L. GUISO and M. PAIELLA, The role of risk aversion in predicting individual behavior, In P. A. Chiappori e
C. Gollier (eds.) Competitive Failures in Insurance Markets: Theory and Policy Implications,
Monaco, CESifo, TD No. 546 (February 2005).
Page 48
G. M. TOMAT, Prices product differentiation and quality measurement: A comparison between hedonic
and matched model methods, Research in Economics, Vol. 60, 1, pp. 54-68, TD No. 547
(February 2005).
F. LOTTI, E. SANTARELLI and M. VIVARELLI, Gibrat's law in a medium-technology industry: Empirical
evidence for Italy, in E. Santarelli (ed.), Entrepreneurship, Growth, and Innovation: the Dynamics
of Firms and Industries, New York, Springer, TD No. 555 (June 2005).
F. BUSETTI, S. FABIANI and A. HARVEY, Convergence of prices and rates of inflation, Oxford Bulletin of
Economics and Statistics, Vol. 68, 1, pp. 863-878, TD No. 575 (February 2006).
M. CARUSO, Stock market fluctuations and money demand in Italy, 1913 - 2003, Economic Notes, Vol. 35,
1, pp. 1-47, TD No. 576 (February 2006).
S. IRANZO, F. SCHIVARDI and E. TOSETTI, Skill dispersion and productivity: An analysis with matched
data, CEPR Discussion Paper, 5539, TD No. 577 (February 2006).
R. BRONZINI and G. DE BLASIO, Evaluating the impact of investment incentives: The case of Italy’s Law
488/92. Journal of Urban Economics, Vol. 60, 2, pp. 327-349, TD No. 582 (March 2006).
R. BRONZINI and G. DE BLASIO, Una valutazione degli incentivi pubblici agli investimenti, Rivista Italiana
degli Economisti , Vol. 11, 3, pp. 331-362, TD No. 582 (March 2006).
A. DI CESARE, Do market-based indicators anticipate rating agencies? Evidence for international banks,
Economic Notes, Vol. 35, pp. 121-150, TD No. 593 (May 2006).
L. DEDOLA and S. NERI, What does a technology shock do? A VAR analysis with model-based sign
restrictions, Journal of Monetary Economics, Vol. 54, 2, pp. 512-549, TD No. 607 (December
2006).
R. GOLINELLI and S. MOMIGLIANO, Real-time determinants of fiscal policies in the euro area, Journal of
Policy Modeling, Vol. 28, 9, pp. 943-964, TD No. 609 (December 2006).
P. ANGELINI, S. GERLACH, G. GRANDE, A. LEVY, F. PANETTA, R. PERLI,S. RAMASWAMY, M. SCATIGNA
and P. YESIN, The recent behaviour of financial market volatility, BIS Papers, 29, QEF No. 2
(August 2006).
2007
L. CASOLARO. and G. GOBBI, Information technology and productivity changes in the banking industry,
Economic Notes, Vol. 36, 1, pp. 43-76, TD No. 489 (March 2004).
M. PAIELLA, Does wealth affect consumption? Evidence for Italy, Journal of Macroeconomics, Vol. 29, 1,
pp. 189-205, TD No. 510 (July 2004).
F. LIPPI. and S. NERI, Information variables for monetary policy in a small structural model of the euro
area, Journal of Monetary Economics, Vol. 54, 4, pp. 1256-1270, TD No. 511 (July 2004).
A. ANZUINI and A. LEVY, Monetary policy shocks in the new EU members: A VAR approach, Applied
Economics, Vol. 39, 9, pp. 1147-1161, TD No. 514 (July 2004).
R. BRONZINI, FDI Inflows, agglomeration and host country firms' size: Evidence from Italy, Regional
Studies, Vol. 41, 7, pp. 963-978, TD No. 526 (December 2004).
L. MONTEFORTE, Aggregation bias in macro models: Does it matter for the euro area?, Economic
Modelling, 24, pp. 236-261, TD No. 534 (December 2004).
A. DALMAZZO and G. DE BLASIO, Production and consumption externalities of human capital: An empirical
study for Italy, Journal of Population Economics, Vol. 20, 2, pp. 359-382, TD No. 554 (June 2005).
M. BUGAMELLI and R. TEDESCHI, Le strategie di prezzo delle imprese esportatrici italiane, Politica
Economica, v. 3, pp. 321-350, TD No. 563 (November 2005).
L. GAMBACORTA and S. IANNOTTI, Are there asymmetries in the response of bank interest rates to
monetary shocks?, Applied Economics, v. 39, 19, pp. 2503-2517, TD No. 566 (November 2005).
S. DI ADDARIO and E. PATACCHINI, Wages and the city. Evidence from Italy, Development Studies
Working Papers 231, Centro Studi Luca d’Agliano, TD No. 570 (January 2006).
P. ANGELINI and F. LIPPI, Did prices really soar after the euro cash changeover? Evidence from ATM
withdrawals, International Journal of Central Banking, Vol. 3, 4, pp. 1-22, TD No. 581 (March
2006).
A. LOCARNO, Imperfect knowledge, adaptive learning and the bias against activist monetary policies,
International Journal of Central Banking, v. 3, 3, pp. 47-85, TD No. 590 (May 2006).
Page 49
F. LOTTI and J. MARCUCCI, Revisiting the empirical evidence on firms' money demand, Journal of
Economics and Business, Vol. 59, 1, pp. 51-73, TD No. 595 (May 2006).
P. CIPOLLONE and A. ROSOLIA, Social interactions in high school: Lessons from an earthquake, American
Economic Review, Vol. 97, 3, pp. 948-965, TD No. 596 (September 2006).
A. BRANDOLINI, Measurement of income distribution in supranational entities: The case of the European
Union, in S. P. Jenkins e J. Micklewright (eds.), Inequality and Poverty Re-examined, Oxford,
Oxford University Press, TD No. 623 (April 2007).
M. PAIELLA, The foregone gains of incomplete portfolios, Review of Financial Studies, Vol. 20, 5, pp.
1623-1646, TD No. 625 (April 2007).
K. BEHRENS, A. R. LAMORGESE, G.I.P. OTTAVIANO and T. TABUCHI, Changes in transport and non
transport costs: local vs. global impacts in a spatial network, Regional Science and Urban
Economics, Vol. 37, 6, pp. 625-648, TD No. 628 (April 2007).
G. ASCARI and T. ROPELE, Optimal monetary policy under low trend inflation, Journal of Monetary
Economics, v. 54, 8, pp. 2568-2583, TD No. 647 (November 2007).
R. GIORDANO, S. MOMIGLIANO, S. NERI and R. PEROTTI, The Effects of Fiscal Policy in Italy: Evidence
from a VAR Model, European Journal of Political Economy, Vol. 23, 3, pp. 707-733, TD No. 656
(December 2007).
2008
S. MOMIGLIANO, J. Henry and P. Hernández de Cos, The impact of government budget on prices: Evidence
from macroeconometric models, Journal of Policy Modelling, v. 30, 1, pp. 123-143 TD No. 523
(October 2004).
P. DEL GIOVANE, S. FABIANI and R. SABATINI, What’s behind “inflation perceptions”? A survey-based
analysis of Italian consumers, in P. Del Giovane e R. Sabbatini (eds.), The Euro Inflation and
Consumers’ Perceptions. Lessons from Italy, Berlin-Heidelberg, Springer, TD No. 655 (January
2008).
FORTHCOMING
S. SIVIERO and D. TERLIZZESE, Macroeconomic forecasting: Debunking a few old wives’ tales, Journal of
Business Cycle Measurement and Analysis, TD No. 395 (February 2001).
P. ANGELINI, Liquidity and announcement effects in the euro area, Giornale degli economisti e annali di
economia, TD No. 451 (October 2002).
S. MAGRI, Italian households' debt: The participation to the debt market and the size of the loan, Empirical
Economics, TD No. 454 (October 2002).
P. ANGELINI, P. DEL GIOVANE, S. SIVIERO and D. TERLIZZESE, Monetary policy in a monetary union: What
role for regional information?, International Journal of Central Banking, TD No. 457 (December
2002).
L. MONTEFORTE and S. SIVIERO, The Economic Consequences of Euro Area Modelling Shortcuts, Applied
Economics, TD No. 458 (December 2002).
L. GUISO and M. PAIELLA,, Risk aversion, wealth and background risk, Journal of the European Economic
Association, TD No. 483 (September 2003).
G. FERRERO, Monetary policy, learning and the speed of convergence, Journal of Economic Dynamics and
Control, TD No. 499 (June 2004).
F. SCHIVARDI e R. TORRINI, Identifying the effects of firing restrictions through size-contingent Differences
in regulation, Labour Economics, TD No. 504 (giugno 2004).
C. BIANCOTTI, G. D'ALESSIO and A. NERI, Measurement errors in the Bank of Italy’s survey of household
income and wealth, Review of Income and Wealth, TD No. 520 (October 2004).
D. Jr. MARCHETTI and F. Nucci, Pricing behavior and the response of hours to productivity shocks,
Journal of Money Credit and Banking, TD No. 524 (December 2004).
L. GAMBACORTA, How do banks set interest rates?, European Economic Review, TD No. 542 (February
2005).
P. ANGELINI and A. Generale, On the evolution of firm size distributions, American Economic Review, TD
No. 549 (June 2005).
Page 50
R. FELICI and M. PAGNINI,, Distance, bank heterogeneity and entry in local banking markets, The Journal
of Industrial Economics, TD No. 557 (June 2005).
M. BUGAMELLI and R. TEDESCHI, Le strategie di prezzo delle imprese esportatrici italiane, Politica
Economica, TD No. 563 (November 2005).
S. DI ADDARIO and E. PATACCHINI, Wages and the city. Evidence from Italy, Labour Economics, TD No.
570 (January 2006).
M. BUGAMELLI and A. ROSOLIA, Produttività e concorrenza estera, Rivista di politica economica, TD
No. 578 (February 2006).
PERICOLI M. and M. TABOGA, Canonical term-structure models with observable factors and the dynamics
of bond risk premia, TD No. 580 (February 2006).
E. VIVIANO, Entry regulations and labour market outcomes. Evidence from the Italian retail trade sector,
Labour Economics, TD No. 594 (May 2006).
S. FEDERICO and G. A. MINERVA, Outward FDI and local employment growth in Italy, Review of World
Economics, Journal of Money, Credit and Banking, TD No. 613 (February 2007).
F. BUSETTI and A. HARVEY, Testing for trend, Econometric Theory TD No. 614 (February 2007).
V. CESTARI, P. DEL GIOVANE and C. ROSSI-ARNAUD, Memory for Prices and the Euro Cash Changeover: An
Analysis for Cinema Prices in Italy, In P. Del Giovane e R. Sabbatini (eds.), The Euro Inflation and
Consumers’ Perceptions. Lessons from Italy, Berlin-Heidelberg, Springer, TD No. 619 (February 2007).
B. ROFFIA and A. ZAGHINI, Excess money growth and inflation dynamics, International Finance, TD No.
629 (June 2007).
M. DEL GATTO, GIANMARCO I. P. OTTAVIANO and M. PAGNINI, Openness to trade and industry cost
dispersion: Evidence from a panel of Italian firms, Journal of Regional Science, TD No. 635
(June 2007).
A. CIARLONE, P. PISELLI and G. TREBESCHI, Emerging Markets' Spreads and Global Financial Conditions,
Journal of International Financial Markets, Institutions & Money, TD No. 637 (June 2007).
S. MAGRI, The financing of small innovative firms: The Italian case, Economics of Innovation and New
Technology, TD No. 640 (September 2007).
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