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Temi di discussione(Working papers)
Ap
ril
2008
664
Num
ber
Portfolio selection with monotone mean-variance preferences
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.
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.
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.
1 Introduction
Since the seminal contributions of Markowitz [Ma] and Tobin [To], mean-variance preferences have been extensively used to model the behavior ofeconomic agents choosing among uncertain prospects and have become oneof the workhorses of portfolio selection theory.1 These preferences, denotedby �mv, assign to an uncertain prospect f the following utility score:
U� (f) = EP [f ]� �
2VarP [f ] ;
where P is a given probability measure and � is an index of the agent�saversion 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. Itmay happen that an agent with mean-variance preferences strictly prefersless to more, thus violating one of the most compelling principles of economicrationality. This is especially troublesome in Finance because monotonicityis 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 knownproblem (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 veryrestrictive 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. Supposeshe has to choose between the two following prospects f and g:
States of Nature s1 s2 s3 s4Probabilities 0:25 0:25 0:25 0:25Payo¤ of f 1 2 3 4Payo¤ of g 1 2 3 5
Prospect g yields a higher payo¤ than f in every state. Any rational agentshould prefer g to f . However, it turns out that our mean-variance agentstrictly 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, andShanken [GRS], Kandel and Stambaugh [KS], and MacKinlay and Richardson [MR].
3
The reason why monotonicity fails here is fairly intuitive. By choosing grather than f , the payo¤ in state s4 increases by one unit. This additionalunit increases the mean payo¤, but it also makes the distribution of payo¤smore spread out, thus increasing the variance. The increase in the mean ismore than compensated by the increase in the variance, and this makes ourmean-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 andtractability of mean-variance preferences.Speci�cally, we consider the variational preference �mmv represented by
the choice functional
V� (f) = minQ
�EQ [f ] +
1
2�C (QjjP )
�8f 2 L2 (P ) ;
where Q ranges over all probability measures with square-integrable densitywith respect to P , and C (QjjP ) is the relative Gini concentration index (or�2-distance), a concentration index that enjoys properties similar to those ofthe relative entropy.The preferences �mmv have the following key properties:
� They are monotone and they agree with mean-variance preferenceswhere 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 ofU�: that is, if V 0
� is any other monotone extension of U� outside itsdomain of monotonicity, then
jV� (f)� U� (f)j � jV 0� (f)� U� (f)j
for each prospect f .
Moreover:2This is the set where rU� is positive, called domain of monotonicity of U�.
4
� The functional relation between V� and U� can be explicitly formulated.
� The parameter � retains the usual interpretation in terms of uncertaintyaversion.
� The functional V� preserves second order stochastic dominance.
All these features make the preferences �mmv a natural adjusted version ofmean-variance preferences that satis�es monotonicity. For this reason we callthem monotone mean-variance preferences.
In view of all this, it is natural to wonder what happens in a portfolioproblem à la Markowitz when we use monotone mean-variance preferencesin place of standard mean-variance preferences. This is the main subjectmatter of this paper. Markowitz�s well-known optimal allocation rule undermean-variance preferences is:
��mv =1
�VarP [X]�1 EP
hX �~1R
i;
where ��mv is the optimal portfolio of risky assets, X is the vector of grossreturns 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 preferencesthe optimal allocation rule becomes:
��mmv =1
�P (W � �)VarP [X jW � � ]�1 EP
hX �~1R jW � �
i;
where W is future wealth and � is a constant determined along with ��mmvby solving a suitable system of equations.Except for a scaling factor, the di¤erence between Markowitz�s optimal
portfolio ��mv and the above portfolio ��mmv is that in the latter conditional
moments of asset returns EP [� jW � � ] and VarP [� jW � � ] are used insteadof unconditional moments, so that the allocation ��mmv ignores the part ofthe distribution where wealth is higher than �. As a result, a monotonemean-variance agent does not take into account those high payo¤ states thatcontribute to increase the mean return, but give an even greater contributionto increase the variance. By doing so, this agent does not incur in violationsof monotonicity caused for mean-variance preferences by an exaggerate pe-nalization of �positive deviations from the mean.�This is a key feature ofmonotone mean-variance preferences. We further illustrate this point in Sec-tion 6 by showing how this functional avoids some pathological situations inwhich the more the payo¤ to an asset is increased in some states, the more a
5
mean-variance agent reduces the quantity of it in her portfolio, until in thelimit she ends up holding none.
In the last part of the paper we derive a monotone CAPMmodel based onthe above portfolio analysis with our monotone mean-variance preferences.We �rst show that in our model optimal portfolios satisfy the classic twofund separation principle: (i) portfolios of risky assets optimally held byagents with di¤erent degrees of uncertainty aversion are all proportional toeach other, (ii) at an optimum the only di¤erence between two agents is theamount of wealth invested in the risk-free asset. This separation has signif-icant theoretical implications because it allows to identify the equilibriummarket 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 ofthe 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 monotonicityof the preference functional V�, in the monotone CAPM there are no arbitrageopportunities. This is a key property of the monotone CAPM and is in starkcontrast with what happens with the standard CAPM. In fact, as observed byDybvig 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 equilibriumprices of all assets in a complete-markets economy. This is, instead, possiblein our arbitrage free monotone CAPM, which can thus be integrated in theclassic Arrow-Debreu complete-markets framework.
The paper is organized as follows. Section 2 illustrates in detail monotonemean-variance preferences. Sections 3 and 4 state and solve the portfolio se-lection problem under the proposed speci�cation of preferences. Section 5contains the CAPM analysis. Section 6 presents some examples that illus-trate the di¤erence between the optimal allocation rule proposed here andMarkowitz�s. Section 7 concludes. All proofs are collected in the appendices,along with some general results on monotone approximations of concave func-tionals.
2 Monotone Mean-Variance Preferences
We consider a measurable space (S;�) of states of nature. An uncertainprospect is a �-measurable real valued function f : S ! R, that is, a sto-chastic monetary payo¤.
6
The agent�s preferences are described by a binary relation � on a set ofuncertain prospects. [MMR] provides a set of simple behavioral conditionsthat guarantee the existence of an increasing utility function u : R! R anda convex uncertainty index c : � ! [0;1] on the set � of all probabilitymeasures, such that
f � g , infQ2�
�EQ [u (f)] + c (Q)
� inf
Q2�
�EQ [u (g)] + c (Q)
(1)
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 values0 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 respectto a reference probability measure P .3
Variational preferences satisfy the basic tenets of economic rationality. Inparticular, they are monotone, that is, given any two prospects f and g, wehave f � g whenever f (s) � g (s) for each s 2 S.4
For concreteness, given a probability measure P on (S;�), we considerthe 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 choicefunctional
U� (f) = EP [f ]� �
2VarP [f ] 8f 2 L2 (P ) ;
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 �mv is monotone onthe set G�, which has the following properties.
Lemma 1 The set G� is convex, closed, and
G� =�f 2 L2 (P ) : f � EP [f ] 6 1
�
�. (2)
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 andU� (g) < U� (f).
3The relative entropy of Q given P is EPhdQdP ln
dQdP
iif Q� P and 1 otherwise.
4In the special case in which u is linear, some recent �nance papers (e.g., Filipovicand Kupper [FK] and Kupper and Cheridito [KC]) call monetary utility functions thefunctionals representing variational preferences.
7
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 �mv is economically meaningful. In fact, it says that if we take anyprospect f outside G�, then in every its neighborhood, however small, thereis at least a prospect g that is statewise better than f , but ranked by �mv
below f .The mean-variance preference �mv thus exhibits irrational non-monotone
behavior in every neighborhood, however small, of prospects f outside G�.For this reason �mv has no economic meaning outside G�.
It can be shown that the restriction of �mv to G� is a variational prefer-ence, and
U� (f) = minQ2�2(P )
�EQ [f ] +
1
2�C (QjjP )
�8f 2 G�,
where �2 (P ) is the set of all probability measures with square-integrabledensity with respect to P , and
C (QjjP ) =(EPh�
dQdP
�2i� 1 if Q� P
+1 otherwise
is the relative Gini concentration index (or �2-distance).5
This suggests to call monotone mean-variance preference the relation�mmv on L2 (P ) represented by the choice functional
V� (f) = minQ2�2(P )
�EQ [f ] +
1
2�C (QjjP )
�8f 2 L2 (P ) : (3)
Our �rst result is the following:6
Theorem 2 The functional V� : L2 (P ) ! R de�ned by (3) is the minimalmonotone functional on L2 (P ) such that V� (g) = U� (g) for all g 2 G�; thatis,
V� (f) = sup fU� (g) : g 2 G� and g 6 fg 8f 2 L2 (P ) : (4)
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
i=1Q2i �1, and C (QjjP ) is its continuous
and relative version. We refer to [LV] for a comprehensive study of concentration indices.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 provein Appendix A (Proposition 12).
8
The functional V� is concave, continuous, and in view of this theorem ithas the following fundamental properties:
(i) V� coincides with the mean-variance choice functional U� on its domainof monotonicity G�.
(ii) V� is the minimal monotone extension of U� outside the domain ofmonotonicity G�, and so it is the most cautious monotone adjustmentof the mean-variance choice functional.
(iii) V� is the best possible monotone approximation of U�: if V 0� is any other
monotone extension of U� outside the domain of monotonicity G�, thenV 0� (f) � V� (f) � U� (f) and so
jV� (f)� U� (f)j � jV 0� (f)� U� (f)j 8f 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�;U� (f ^ �f ) else,
where�f = max ft 2 R : f ^ t 2 G�g : (5)
A monotone mean-variance agent can thus be regarded as still using themean-variance functional U� even in evaluating prospects outside the domainof monotonicity G�. In this case, however, the agent no longer considers theoriginal prospects, but rather their truncations at �f , the largest constant tsuch that f ^ t belongs to G�.Besides depending on the given prospect f , the constant �f also depends
on the parameter �. Corollary 16 in Appendix B shows that �f decreases as �increases, and it is the unique solution of the equation EP
�(f � �)�
�= 1=�.
Given two preferences over uncertain prospects, we say that �1 is moreuncertainty averse than �2 if and only if
f �1 c =) f �2 c
for all f 2 L2 (P ) and c 2 R. That is, Agent 1 is more uncertainty aversethan Agent 2 if, whenever Agent 1 prefers the uncertain prospect f to a surepayo¤ 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 orderto keep the paper focused).
9
A mean-variance preference ��mv is more uncertainty averse than another
mean-variance preference� mv if and only if � > . Thus, � can be interpreted
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.
Proposition 4 The preference ��mmv is more uncertainty averse than �
mmv
if and only if � > .
We conclude this section by observing that, unlike U�, the preferencefunctional V� preserves second order stochastic dominance (�SSD).8 This isa further proof of the sounder economic meaning of monotone mean-variancepreferences relative to mean-variance ones.
Theorem 5 Let f; g 2 L2 (P ). If f �SSD g, 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 remarkablefeature of involving, like multiplier preferences ([HS]), a classic concentrationindex. This ensures to V� a good analytical tractability, as the next sectionsshow.
3 The Portfolio Selection Problem
We consider the one-period allocation problem of an agent who has to decidehow 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 returnon the i-th asset after one period is denoted by Xi. The (n � 1) vector ofthe returns on the �rst n assets is denoted by X and the (n � 1) vector ofportfolio weights, indicating the fraction of wealth invested in each of therisky assets, is denoted by �. The return on the (n+ 1)-th asset is risk-freeand 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 perfectlydivisible; there are no transaction costs or taxes; agents are price-takers, in
8Recall that f �SSD g i¤ EPh(f � t)�
i� EP
h(g � t)�
ifor all t 2 R. We refer to
Dana [Da], to which the proof of Theorem 5 is inspired, for references on second orderstochastic dominance.
10
that they believe that their choices do not a¤ect the distribution of assetreturns; there are no institutional restrictions, so that agents are allowed tobuy, sell or short sell any desired amount of any security.9 As a result, � canbe chosen in Rn.We adopt �mmv as a speci�cation of preferences, and so portfolios are
ranked according to the preference functional:
V� (W�) = minQ2�2(P )
�EQ [W�] +
1
2�C (QjjP )
�;
where P is the reference probability measure. Hence, the portfolio problemcan be written as:
max�2Rn
minQ2�2(P )
�EQ [W�] +
1
2�C (QjjP )
�: (6)
Notice that, if the agent�s initial wealth is w > 0, then her end-of-periodwealth is wW�, therefore she solves the problem
max�2Rn
minQ2�2(P )
�EQ [wW�] +
1
2�C (QjjP )
�which �dividing the argument by w �reduces to (6) up to replacement of �with �w.
4 The Optimal Portfolio
In this section we give a solution to the portfolio selection problem outlinedin the previous section. The characterization of the optimal portfolio is givenby the following theorem.10
Theorem 6 The vector �� 2 Rn is a solution of the portfolio selection prob-lem (6) if and only if there exists �� 2 R such that (��; ��) satis�es thesystem of equations:(
�P (W� � �)VarP [X jW� � � ]� = EPhX �~1R jW� � �
i;
EP�(W� � �)�
�= 1=�:
9This assumption can be weakened, by simply requiring that at an optimum institu-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.
11
As observed in Section 2, the second displayed equation guarantees that ��
is the largest constant such thatW��^�� 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 equationsin 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 equationsolver is straightforward: given an initial guess (�; �), one is able to cal-culate the �rst two moments of the conditional distribution of wealth; ifthe moments thus calculated, together with the initial guess (�; �), satisfythe system of equations, then (�; �) = (��; ��) and numerical search stops;otherwise, the search procedure continues with a new initial guess for the pa-rameters.11 In the next section we will solve in this way few simple portfolioproblems in order to illustrate some features of the model.
The optimal allocation rule of Theorem 6 is easily compared to the rulethat would obtain in a classic Markowitz�s setting. In the traditional mean-variance model we would have:
�� =1
�
�VarP [X]
��1EPhX �~1R
i: (7)
The monotone mean-variance model yields:
�� =1
�P (W�� � ��)VarP [X jW�� � �� ]�1 EP
hX �~1R jW�� � ��
i:
Relative to Markowitz�s optimal allocation (7), here the unconditional meanand variance of the vector of returns X are replaced by a conditional meanand a conditional variance, both calculated by conditioning on the eventfW�� � ��g. Furthermore a scaling factor is introduced, which is inverselyproportional 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 isoptimal 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 increasingthe variance more than the mean, thus leading the mean-variance agent toirrationally prefer a strictly smaller prospect. With monotone mean-variancepreferences, this kind of behavior is avoided by arti�cially setting the prob-ability of some high payo¤ states equal to zero. In our portfolio selectionproblem 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.
12
When there is only one risky asset, the optimal quantities ��mmv and ��mv
prescribed, respectively, by our model and by the mean-variance model canbe compared by means of the following result.
Proposition 7 Suppose that S is �nite, with P (s) > 0 for all s 2 S, andthat there is only one risky asset. Then, either
��mmv � ��mv � 0
or��mmv � ��mv � 0:
If, in addition, P�W��mmv > ��
�> 0, then:
��mmv > ��mv if ��mmv > 0;
��mmv < ��mv if ��mmv < 0:
That is, an investor with monotone mean-variance preferences alwaysholds 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, thisis 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 thana mean-variance investor would do. This kind of behavior will be thor-oughly illustrated by the examples in the next section: the intuition behindit is that in some cases a favorable investment opportunity is discarded by amean-variance investor because of non-monotonicity of her preferences, whilea monotone mean-variance investor exploits the opportunity, thus taking amore leveraged position.
5 Monotone CAPM
In this section we show how the standard CAPM analysis can be carried outin 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 agentsdecide the amount of wealth to invest in the risk-free asset; then, they decidehow to allocate the remaining wealth among the risky assets. The outcomeof this second decision is the same for all agents, regardless of their initialwealth or aversion to uncertainty.
Proposition 8 Let �; > 0. If���; ��
�solves the portfolio selection problem
(6) for an investor with uncertainty aversion �, then�� ��; �
�� +
�1� �
�R�
solves it for an investor with uncertainty aversion .
13
Given a � > 0 with �� �~1 > 0, de�ne m = ��� �~1. Hence,
�m =��
�� �~1and Proposition 8 guarantees that m and �m do not depend on the choice of�. The equality �m �~1 = 1 implies that �m is the portfolio held by an investorwho does not invest any of her wealth in the risk-free asset. Following themajority of the literature, we call �m the market portfolio and denote byXm = �m �X its payo¤. In particular, W�m = R + �m �
�X �~1R
�= Xm.
In an economy consisting of monotone-mean variance agents, all investorshold a portfolio of risky assets proportional to the market portfolio. Speci�-cally, an investor with uncertainty aversion � will invest m=� in the marketportfolio and the rest of her wealth in the risk free asset. Like in the standardmean-variance setting, also here the amount of wealth invested in the marketportfolio only depends on the coe¢ cient � of the agent.
All this has strong empirical implications. From market data �moreprecisely, by observing the market values of the assets in the economy �itis 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 �m by solving asystem of equations with observable coe¢ cients.12
Proposition 9 The pair (x�; y�) � (m;�m) solves the following system ofequations�
P (Xm � y)VarP [Xm jXm � y ]x = EP [Xm �R jXm � y ] ;
EP�(Xm � y)�
�= 1=x:
The knowledge ofm and �m makes it possible to determine the equilibriumpricing kernel rVm (Xm), which will become very important momentarilywhen discussing the monotone CAPM. To see why this is the case, we needthe following lemma, which gives some properties of rVm (Xm).
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 theuncertainty aversion coe¢ cient m is a mean of the uncertainty aversion coe¢ cients of the
agents. Speci�cally, m = ��P
j
��j��1��1
where �j is the uncertainty aversion coe¢ cient
of agent j and � is the market value of all assets.
14
(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 easenotation, in what follows we set rV � rVm (Xm).
Lemma 10 also makes it possible to derive our monotone version of theCAPM. 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 Xm be de�ned as above. Then,
EP [Xi]�R = �i�EP [Xm]�R
�; 8i = 1; :::; n; (8)
where
�i =CovP [Xi;rV ]CovP [Xm;rV ]
. (9)
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 Yi the end-of-period payo¤ per share of asset i andpi its current price. Then, Xi = Yi=pi and (8) becomes
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 portfolioconsisting of qi shares of asset i is
nXi=1
qipi =1
REP
" nXi=1
qiYi
!rV
#;
15
which is positive as long asnXi=1
qiYi is positive (rV is positive because Vm is
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 instark contrast with their presence in the standard CAPM model, caused bythe lack of monotonicity of mean-variance preferences. This was observedby Dybvig and Ingersoll [DI], who show that if Xm =2 Gm and the marketis complete, then the pricing rule obtained from standard CAPM is not apositive 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-Debreucomplete-markets economy, with all assets in such an economy priced inequilibrium according to our CAPM.
On the empirical side, our monotone CAPM model can be fully analyzedfrom market data. First observe that the values of the betas (9) can bederived from market data because we just observed that, besides those of Xi,also the values of Xm and 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 betasof the single assets can be estimated by instrumental variables, using rV asan 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 canbe 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 beinferred from market data. The monotone CAPM has thus a sounder theo-retical foundation than the standard CAPM, while retaining its remarkableempirical tractability.
16
6 Some Examples
In this section we present three simple examples to illustrate the optimalportfolio rule we derived above. In every example there are �ve possiblestates 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 notinteresting per se, but it serves as a benchmark and it helps to introduceExample 2, where the two optimal portfolios di¤er. In Example 1 there isonly one risky asset, whose gross return is denoted by X1 and is reported inthe next table, and a risk-free asset, whose gross return R is equal to 1 acrossall states. In this example, the optimal portfolio ��mmv calculated accordingto our rule is equal to the mean-variance optimal portfolio ��mv. Wmv andWmmv represent the overall return to the two optimal portfolios for each stateof the world. The table also displays the value of the constant �� at which itis optimal to truncate the distribution of the return to the portfolio of riskyassets.
P (si) P (si jWmmv � �� ) R X1 Wmv Wmmv
s1 0:1 0:1 1 0:97 0:9375 0:9375s2 0:2 0:2 1 0:99 0:9791 0:9791s3 0:4 0:4 1 1:01 1:0208 1:0208s4 0:2 0:2 1 1:03 1:0620 1:0620s5 0:1 0:1 1 1:05 1:1041 1:1041��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 s5 from 1:05 to 1:10, leaving everything else un-changed. The e¤ect of this change is an increase in both the mean and thevariance of X1, the payo¤ to the risky asset. The optimal behavior accordingto the mean-variance model is to reduce the fraction of wealth invested in therisky asset from 2:083 to 1:3574. According to our model it is also optimalto decrease the position in the risky asset, but less, from 2:083 to 1:8382.
17
P (si) P (si jWmmv � �� ) R X1 Wmv Wmmv
s1 0:1 0:1111 1 0:97 0:9592 0:9448s2 0:2 0:2222 1 0:99 0:9864 0:9816s3 0:4 0:4444 1 1:01 1:0135 1:0183s4 0:2 0:2222 1 1:03 1:0407 1:0551s5 0:1 0 1 1:10 1:1357 1:1838��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 ofthat asset. This behavior can be understood by looking at the distributionsof the overall return in the two tables. By reducing the fraction of wealthinvested in the risky asset, the overall return increases in the states wherethe risky asset pays less than the risk-free asset. On the contrary, the overallreturn 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 compensatedby the fact that we have raised the payo¤ to the risky asset from 1:05 to1:10. Hence, by reducing the amount of wealth invested in the risky asset,the investor gives up some of the extra payo¤ received in state s5 in orderto guarantee himself a higher overall return in the states where the riskyasset has a low payo¤. The problem with this kind of behavior is that it canbecome pathological with mean-variance preferences. The next table showswhat happens if we further increase the payo¤ in state s5.
X1 (s5) 1:05 1:10 1:15 1:20 1:50 2 3��mv 2:0830 1:3574 0:9174 0:6747 0:2465 0:1175 0:0572��mmv 2:0830 1:8382 1:8382 1:8382 1:8382 1:8382 1:8382
The more we increase the payo¤ in state s5, the more the mean-varianceoptimal fraction ��mv of wealth invested in the risky asset decreases, until itgoes to zero when the payo¤ in state s5 becomes very large. In our model thisdoes not happen. At �rst ��mmv decreases, but it then stops decreasing andit remains �xed at the same value, though the payo¤ in state s5 is furtherincreased. The reason why this happens is that, once probabilities have beenoptimally reassigned to states and a zero probability has been assigned tostate s5, any further increases of the payo¤ in s5 are disregarded and haveno in�uence on the formation of the optimal portfolio.
18
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 byX2, is high in the states where X1 is low and low where X1 is high.
P (si) P (si jWmmv � �� ) R X1 X2 Wmv Wmmv
s1 0:1 0:1111 1 0:97 1:05 1:002 1:0231s2 0:2 0:2222 1 0:99 1:00 0:9833 0:9570s3 0:4 0:4444 1 1:01 0:99 1:0061 1:0125s4 0:2 0:2222 1 1:03 0:99 1:0393 1:0985s5 0:1 0 1 1:10 0:99 1:1556 1:3994��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 thetraditional 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 hedgeaway almost completely the risks taken by investing in asset 1. Consider forexample 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 thisportfolio pays o¤ slightly less than the risk-free asset, while in all other statesit pays o¤ more and in some states considerably more. Roughly speaking, ifit was not for the slightly low payo¤ in state s2, there would be an arbitrageopportunity because the portfolio would pay o¤more than the risk-free assetin every state. As a consequence, we would expect an optimal portfoliorule to exploit this favorable con�guration of payo¤s by prescribing to takea highly levered position. As reported in the last table, according to ourmodel it is optimal to take a highly levered position in the risky assets inorder to exploit this opportunity, at the cost of facing a low payo¤ in states2. In contrast, with the mean-variance model the optimal portfolio is muchless aggressive, and the investor is overly concerned with the unique state inwhich 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 themean-variance principle, which avoids the problem of non-monotonicity. In
19
the cases where mean-variance preferences are well-behaved (i.e., monotone)the optimal portfolios suggested by our rule do not di¤er from standardmean-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 empiricaltractability 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 expectthat also our analysis can be extended to an intertemporal framework. Thiswill be the subject of future research, along with an empirical analysis of themonotone CAPM.
8 Acknowledgements
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 tothe participants at the VI Quantitative Finance Workshop (Milan, January2005), 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 delPiemonte Orientale, Università di Salerno, and Università di Udine. Part ofthis 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), andUniversità Bocconi.
A Monotone Fenchel Duality
In this section we recall some important de�nitions of Convex Analysis, andwe prove some properties of monotone extensions of concave functionals thatpave 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 andgenerates E. Denote by E 0 the norm dual of E, and by E 0+ the cone of allpositive, linear, and continuous functionals on E. (See [Ch].)
20
Let : E ! R be a concave and continuous functional. The directionalderivative of at x is
d+ (x) (v) = limt!0+
(x+ tv)� (x)
t8v 2 E:
If the above limit exists for t! 0 and all v 2 E, is Gateaux di¤erentiable atx, 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 E 0 de�ned by
@ (x) = fx0 2 E 0 : (y)� (x) � hy � x; x0i 8y 2 Eg ;
while its Fenchel conjugate � : E 0 ! [�1;1) is given by
� (x0) = infx2E
fhx; x0i � (x)g 8x0 2 E 0:
Finally, the domain of monotonicity of is the set G � E given by
G =�x 2 E : @ (x) \ E 0+ 6= ?
:
Next proposition con�rms the intuition that the set G is where the func-tional is monotone.
Proposition 12 Let E be an ordered Banach space and : E ! R be aconcave and continuous functional with G 6= ?.
(i) The functional ~ : E ! R, given by
~ (x) = minx02E0+
fhx; x0i � � (x0)g 8x 2 E; (10)
is the minimal monotone functional that dominates ; that is,
~ (x) = sup f (y) : y 2 E and y 6 xg 8x 2 E: (11)
(ii) Let x 2 E, x 2 G , d+ (x) (v) � 0 8v 6 0, ~ (x) = (x) :
(iii) Let x0 2 E 0, �~ ��(x0) =
� � (x0) if x0 2 E 0+,�1 otherwise.
(12)
(iv) If � is strictly concave on the intersection of its domain with E 0+, then~ is Gateaux di¤erentiable andr ~ (x) = argminx02E0+ fhx; x
0i � � (x0)gfor all x 2 E.
21
Moreover, if G is convex, for all x 2 E there exists y 2 G such thaty 6 x, and there exists a linear subspace F � G such that jF is linear and � (x0) is attained for all x0 2 E 0+ with x
0jF = jF ; then, ~ is the minimal
monotone functional that extends jG from G to E. In this case,
~ (x) = sup f (y) : y 2 G and y 6 xg = minx02E0+:x0jF= jF
fhx; 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 Eand ~ (x) = (x) for all x 2 G . Moreover, (iii) shows that the Fenchelconjugates of ~ and coincide on the coneE 0+. (iv) is a useful di¤erentiabilityproperty. The �nal part, shows that, under additional assumptions (satis�ed,e.g., by the mean-variance functional), the extension ~ has even strongerminimality properties.13
Proof. By the Fenchel-Moreau Theorem, (x) = infx02E0 fhx; x0i � � (x0)gfor all x 2 E. Set
~ (x) = infx02E0+
fhx; x0i � � (x0)g 8x 2 E: (14)
Let x0 2 G (6= ?) and x00 2 @ (x0) \ E 0+ (6= ?), then
(x) � (x0) + hx; x00i � hx0; x00i 8x 2 E:
Therefore, the functional x00 + ( (x0)� hx0; x00i) is a¢ ne, monotone, and itdominates . Notice that
hx0; x00i � (x0) � hx; x00i � (x) 8x 2 E and � (x00) = hx0; x00i � (x0) :(15)
In particular, x00 2 E 0+ and � (x00) 2 R, hence, for all x 2 E,
�1 < (x) � infx02E0+
fhx; x0i � � (x0)g <1:
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 properconcave, and upper semicontinuous function : E ! [�1;1) on a ordered locallyconvex space. Details are available upon request.
22
(i) Let ' : E ! R be a concave and monotone functional such that' � . Then ' is continuous,14 and ' (x) = infx02E0+ fhx; x
0i � '� (x0)g forall x 2 E.15 In addition, since ' � , then '� � � and
' (x) = infx02E0+
fhx; x0i � '� (x0)g � infx02E0+
fhx; x0i � � (x0)g = ~ (x) 8x 2 E:
This shows that ~ is the minimal concave and monotone functional thatdominates . It is easy to check that the function de�ned by Eq. (11) is theminimal monotone functional that dominates , and it is concave, hence itcoincides 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 x0 2 E,16 (iii) delivers
x00 2 @ ~ (x0) , x00 2 arg minx02E0
nhx0; x0i � ~ � (x0)
o, x00 2 arg min
x02E0+fhx0; x0i � � (x0)g :
This shows that the in�mum in Eq. (14) is attained, and that (iv) holds.(ii) If x 2 G , then there is x00 2 @ (x) \ E 0+, and for all v 6 0
d+ (x) (v) = minx02@ (x)
hv; x0i �Dv; 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)\E 0+ = ?. Since E 0+ is a weak* closed and convexcone and @ (x) is weak* compact, convex, and nonempty, there exists v 2 Esuch that
hv; y0i � 0 < hv; x0i 8y0 2 E 0+; x0 2 @ (x) : (16)
Since E+ is closed, then E+ =�x 2 E : hx; y0i � 0 8y0 2 E 0+
. The left
hand side of Eq. (16) amounts to say that v 6 0, the right hand side thatminx02@ (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 x0 2 E;
@' (x0) = arg minx02E0
fhx0; x0i � '� (x0)g 6= ?:
23
If x 2 G , then there is x00 2 @ (x) \ E 0+ that is
x00 2 arg minx02E0
fhx; x0i � � (x0)g and x00 2 E 0+, then
x00 2 arg minx02E0+
fhx; x0i � � (x0)g :
The �rst line implies (x) = hx; x00i � � (x00), the second that ~ (x) =hx; x00i � � (x00). Conversely, if (x) = minx02E0+ fhx; x
0i � � (x0)g, thenthere exists x00 2 E 0+ such that
hx; x00i � � (x00) = (x) = minx02E0
fhx; x0i � � (x0)g , then
x00 2 arg minx02E0
fhx; x0i � � (x0)g and x00 2 E 0+,
therefore, x00 2 @ (x) \ E 0+ and x 2 G .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 jF is linear and � (x0) is attained for all x0 2 E 0+ with x0jF = jF . Set
(x) = sup f (y) : y 2 G and y 6 xg 8x 2 E:
It is easy to check that : E ! R is the minimal monotone functionalthat extends jG from G to E, and that convexity of G implies that itis concave. Therefore, � ~ , (x) = (x) for all x 2 G , is concave,monotone, and linear on F (where it coincides with ). Hence, for all x 2E,17
(x) = infx02E0
nhx; x0i � � (x0)
o= inf
x02E0+:x0jF= jF
nhx; x0i � � (x0)
o= inf
x02E0+:x0jF= jF
nhx; x0i � � (x0)
o:
For all x0 2 E 0+ such that x0jF = jF ,
� (x0) = infx2E
nhx; x0i � (x)
o� inf
x2G
nhx; x0i � (x)
o= inf
x2G fhx; x0i � (x)g :
(17)Analogously, since is linear on F then, for all x 2 E,
~ (x) = minx02E0+:x0jF= jF
fhx; x0i � � (x0)g . (18)
17If ' : E ! [�1;1] is linear on a subspace F , then '� (x0) = �1 if x0jF 6= 'jF :
24
Since � (x0) is attained for all x0 2 E 0+ such that x0jF = jF , for every x0 withthese properties there exists x0 2 E such that
hx0; x0i � (x0) � hx; x0i � (x) 8x 2 E:
Then x0 2 @ (x0)\E 0+ and x0 2 G , in particular, � (x0) is attained in G .That is, for x0 2 E 0+ such that x0jF = jF ,
� (x0) = infx2G
fhx; x0i � (x)g � � (x0)
where the last inequality descends from Eq. (17). We conclude that
(x) = infx02E0+:x0jF= jF
nhx; x0i � � (x0)
o� inf
x02E0+:x0jF= jFfhx; x0i � � (x0)g = ~ (x)
for all x 2 E, as wanted. �
These results have been recently extended in Filipovic and Kupper [FK-2].
B Proofs
Let f 2 L (P ), we denote by Ff (t) = P (f � t) its cumulative distributionfunction and by gf (t) =
R t�1 Ff (z) dz its integral distribution function.
Next two lemmas regroup some useful properties of integrated distributionfunctions. We report the proofs for the sake of completeness.
Lemma 13 For all z 2 R,
gf (z) =
Z(f � z)� dP
= zP (f � z)�Zf1ff�zgdP
= zP (f < z)�Zf1ff<zgdP
=
Zz � (f ^ z) dP
=
Z z
�1P (f < t) dt:
Proof. Let z 2 R. (f � z)� = (z � f) 1ff�zg, henceZ(f � z)� dP = zP (f � z)�
Zf1ff�zgdP:
25
Moreover,
zP (f � z)�Zf1ff�zgdP = zP (f < z) + zP (f = z)�
Zff<zg
fdP �Zff=zg
fdP
= zP (f < z) + zP (f = z)�Zff<zg
fdP � zP (f = z)
= zP (f < z)�Zf1ff<zgdP:
Since f1ff�zg = (f ^ z)� z1ff>zg, then
zP (f � z)�Zf1ff�zgdP = zP (f � z)�
Z �f ^ z � z1ff>zg
�dP
= zP (f � z)�Z(f ^ z) dP + z
Z1ff>zgdP
= zP (f � z) + zP (f > z)�Z(f ^ z) dP
=
Zz � (f ^ z) dP:
Observe that z � (f ^ z) � 0, and soZz � (f ^ z) dP =
Z 1
0
P (z � (f ^ z) � u) du:
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
f � z � u) f ^ z � (z � u) ^ z ) f ^ z � z � u) z � (f ^ z) � u:
Hence,Zz � (f ^ z) dP =
Z 1
0
P (z � (f ^ z) � u) du
=
Z 1
0
P (f � z � u) du =
Z z
�1P (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
Lemma 14 The function gf : R! [0;1) is continuous and
Ff (z) = lim"!0+
�gf (z + ")� gf (z)
"
�8z 2 R:
That is, Ff is the right derivative of gf , and Ff (z) is the derivative of gf atevery point z at which Ff is continuous. Moreover, setting � = essinf (f), gfis strictly increasing on (�;1), gf � 0 on (�1; �],18 limz!�+ gf (z) = 0+,and limz!1 gf (z) =1.
Proof. The Fundamental Theorem of Calculus guarantees the continuityand derivability properties of gf . Recall that
essinf (f) = sup f� 2 R : P (f < �) = 0g :
If z 2 R and z � essinf (f), for all t < z there exists � > t such thatP (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) =
Z z0
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, gf is strictly increasing on (�;1).Notice that limt!1 Ff (t) = 1. Then, for all n > 1 there exists k � 1
(n; k 2 N) such that Ff (t) > 1� 1nfor all t � k. Therefore,
gf (k + n) =
Z k+n
�1Ff (t) dt �
Z k+n
k
Ff (t) dt
� n
�1� 1
n
�= n� 1:
Since gf is increasing on R, then limz!1 gf (z) =1.If � > �1, gf (�) = 0, continuity and nonnegativity imply limz!�+ gf (z) =
0+. Let � = �1, for all n > 1,
gf (�n) =Z(�n� (f ^ (�n))) dP =
Z(((�f) _ n)� n) dP
=
Z((�f)� ((�f) ^ n)) dP:
18With the convention (�1;�1] = ?.
27
The Monotone Convergence Theorem guarantees that limn!1 gf (�n) = 0:Monotonicity and nonnegativity imply limz!�+ gf (z) = 0
+. �
For the rest of the Appendix we indi¤erently write EP or just E. Denotingby > 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 relationhf; Y i = E [fY ]. Simple computation shows that, for all � > 0,
�f 2 L2 (P ) : rU� (f) > 0
=
�f 2 L2 (P ) : f � EP [f ] 6 1
�
�(19)
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
�: (20)
Proof. Notice that
f ^ t� E [f ^ t] = f1ff�tg + t1ff>tg � tP (f > t)� E�f1ff�tg
�= f1ff�tg + t1ff>tg � t+ tP (f � t)� E
�f1ff�tg
�= (f � t) 1ff�tg + gf (t) :
Since (f � t) 1ff�tg 6 0,
gf (t) �1
�) f ^ t� E [f ^ t] 6 1
�;
i.e., gf (t) � 1�) f ^ t 2 G�.
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. Itfollows that:
f ^ t� E [f ^ t] 6 1
�) essup (f ^ t)� tP (f > t)� E
�f1ff�tg
�� 1
�
) t� tP (f > t)� E�f1ff�tg
�� 1
�
) tP (f � t)� E�f1ff�tg
�� 1
�
) gf (t) �1
�;
i.e., f ^ t 2 G� ) gf (t) � 1�. �
Lemmas 14 and 15 immediately yield the following:
28
Corollary 16 Let f 2 L2 (P )�G�, then g�1f�1�
�= max ft 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 thereexists g 2 L2 (P ) such that jf (s)� g (s)j < " for all s 2 S, g > f , andU� (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 ),
U�� (Y ) =
�� 12�(E [Y 2]� 1) if E [Y ] = 1;
�1 otherwise.(21)
(iv) fY 2 L2 (P ) : E [Y ] = 1g =nY 2 L2 (P ) : h�; Y ijT = U�jT
oand U�� is
attained and strictly concave on this set.
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 gn of fn such thatgn (s)! f (s) for P -almost all s in S. Let A0 = fs 2 S : gn (s)! f (s)g, andfor all n � 1, An = fs 2 S : gn (s)� E [gn] � 1=�g, then P
�Tn�0An
�= 1
and for all s 2Tn�0An,
f (s)� E [f ] = limn!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
limt!0
U� (f + t1A)� U� (f)
t=
Z1ArU� (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
For convenience, (iii) and (iv) are proved together. For all t 2 R, U� (t1S) =t, then for all Y 2 L2 (P )n
Y 2 L2 (P ) : h�; Y ijT = U�jT
o=�Y 2 L2 (P ) : E [Y ] = 1
;
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+�
2hf � E [f ] ; f � E [f ]i
is well de�ned, convex, and Gateaux di¤erentiable. Its Gateaux di¤erentialis
rW� (f) = Y � 1 + � (f � E [f ]) : (22)
Notice that f = ���1Y solves rW�
�f�= 0 (since E [Y ] = 1), and W� at-
tains its minimum onL2 (P ) at f . This implies that U�� (Y ) = inff2L2(P )W� (f)is attained, and
U�� (Y ) = minf2L2(P )
W� (f) =W�
�f�=
��1�Y; Y
����1�Y; 1
�+�
2Var
��1�Y
�= �1
�E�Y 2�+1
�E [Y ] +
�
2
1
�2Var [Y ] = � 1
2�
�E�Y 2�� 1�:
This concludes the proof of Eq. (21). Strict concavity of U�� in its domain isa straightforward application of the the Cauchy-Schwartz inequality. �
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 andcontinuous functional on an ordered Banach space, G� = GU� , Lemma 17and Corollary 16, guarantee that all the hypotheses of Proposition 12 aresatis�ed. For all f 2 L2 (P ),
~U� (f) = minY 2L2+(P ):E[Y ]=1
�E [fY ] +
1
2�
�E�Y 2�� 1��
= minQ2�2(P )
�EQ [f ] +
1
2�C (QjjP )
�= V� (f) ;
and V� has all the desired properties. �
30
Remark 18 Proposition 12.(iv) and Lemma 17.(iv) guarantee that V� isGateaux di¤erentiable, and
rV� (f) = arg minY 2L2+(P ):E[Y ]=1
�E [fY ] +
1
2�
�E�Y 2�� 1��
8f 2 L2 (P ) :
(23)
Theorem 19 Let f 2 L2 (P ). Then
V� (f) =
�E [f ]� �
2Var [f ] if f 2 G�;
E [f ^ �]� �2Var [f ^ �] else,
where � = g�1f�1�
�: 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 )�EQ (f) + 1
2�C (QjjP )
.
That is, V� (f) is the value of the problem:8<: min�E [fY ] + 1
2�E [Y 2]� 1
2�
Y > 0E [Y ] = 1
: (24)
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 isa solution of problem (24) if and only if it is a solution of the constrainedoptimization problem: 8<: min
�E [fY ] + 1
2�E [Y 2]
Y > 0E [Y ] = 1
: (25)
The Lagrangian is
L (Y; �; �) = E (fY ) +1
2�E�Y 2�� E (�Y )� � (E (Y )� 1) ;
with � 2 L2+ (P ), � 2 R. The Kuhn-Tucker optimality conditions are:
f +1
�Y � �� � = 0 (P -a.s.)
E (�Y ) = 0
Y > 0; � > 0E (Y ) = 1
31
Since �; Y > 0, they are equivalent to:
f +1
�Y � �� � = 0 (P -a.s.)
�Y = 0 (P -a.s.)
Y > 0; � > 0E (Y ) = 1
that is,
f +1
�Y � � � 0 P -a.s. (26)�
f +1
�Y � �
�Y = 0 P -a.s. (27)
Y � 0 P -a.s. (28)
E [Y ] = 1 (29)
It is su¢ cient to �nd (Y �; ��) that satisfy (26) - (29) everywhere (notonly P -a.s.).If s 2 fY � > 0g, then by (27) f (s) + 1
�Y � (s)� �� = 0 and
Y � (s) = � (�� � f (s)) : (30)
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 andY � = � (�� � f) 1ff<��g .
By (29),
1 = E [Y �] = E�� (�� � f) 1ff<��g
�= �
���P (f < ��)� E
�f1ff<��g
��;
that is,
gf (��) = ��P (f < ��)� E
�f1ff<��g
�=1
�:
In other words,
�� = g�1f
�1
�
�� �; (31)
and �� is unique. A fortiori, Y � is unique and
Y � = � (�� f) 1ff<�g: (32)
32
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 concludethat 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�f 21ff��g + �21ff��g � 2�f1ff��g
�and
E�(Y �)2
�= �2
�Zf 21ff��gdP + �2P (f � �)� 2�
Zf1ff��gdP
�:
Moreover,
E [fY �] = E�f� (�� f) 1ff��g
�= E
���f1ff��g � �f21ff��g
�= ��
Zf1ff��gdP � �
Zf 21ff��gdP:
Therefore,
V� (f) = E [fY�] +
1
2�E�(Y �)2
�� 1
2�
= ��
Zf1ff��gdP � �
Zf 21ff��gdP+
+�
2
�Zf 21ff��gdP + �2P (f � k)� 2�
Zf1ff��gdP
�� 1
2�
= ��2
Zf 21ff��gdP +
�
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 ^ �] = Eh�f1ff��g + �1ff>�g
�2i� ��� 1�
�2=
Zf 21ff��gdP + �2P (f > k)� �2 � 1
�2+ 2
�
�
=
Zf 21ff��gdP � �2P (f � k)� 1
�2+ 2
�
�:
33
Finally,
E [f ^ �]� �
2Var [f ^ �] = �� 1
�� �
2
�Zf21ff��gdP � �2P (f � k)� 1
�2+ 2
�
�
�= �� 1
����
2
Zf21ff��gdP �
�
2�2P (f � k)� 1
2�+ �
�= ��
2
Zf21ff��gdP +
�
2�2P (f � k)� 1
2�
= V� (f) :
�
Proof of Theorem 3. It is now enough to combine Corollary 16 andTheorem 19. �
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�1f�1�
�we
have
V� (f) = E [f ^ �]��
2Var [f ^ �] ;
rV� (f) = � (�� f) 1ff��g = � (f � �)� ;
rV� (f) = argminQ2�2(P )
�EQ (f) +
1
2�C (QjjP )
�:
� The properties of gf guarantee that � exists, it is unique, and 1�=
�P (f � �)�Rf1ff��gdP ; therefore, P (f � �) > 0.
� Moreover, 1�= �P (f � �)�
Rf1ff��gdP implies 1
�P (f��)+RfdPff��g =
�, and so
rV� (f) = � (�� f) 1ff��g =
�1
P (f � �)+ �
ZfdPff��g � �f
�1ff��g
=
�1
P (f � �)� � (f � E [f jf � �])
�1ff��g:
N
Proof of Theorem 5. By Theorem 13.8 of Chong and Rice [CR], for allf; Y 2 L2 (P ), the set
�RfY 0dP : Y 0 2 L2 (P ) and Y 0 �c Y
coincides with
the interval �Z 1
0
F�1f (1� u)F�1Y (u) du;
Z 1
0
F�1f (u)F�1Y (u) du
�;
34
where �c is the concave order and F�1f is the quantile function of f .19 Con-sider the function � : L2 (P )! [0;1] de�ned by
� (Y ) =
�12�(E [Y 2]� 1) if Y > 0 and E [Y ] = 1;
1 otherwise.
It is easy to check that Y 0 �c Y implies � (Y 0) � � (Y ).20 Let f 2 L2 (P ),
V� (f) = minY 2L2(P )
fE [fY ] + � (Y )g � infY 2L2(P )
�Z 1
0
F�1f (1� u)F�1Y (u) du+ � (Y )
�:
Moreover, for all Y 2 L2 (P ) there exists Y 0 2 L2 (P ) with Y 0 �c Y suchthat
R 10F�1f (1� u)F�1Y (u) du =
RfY 0dP . Since � (Y 0) � � (Y ), thenZ 1
0
F�1f (1� u)F�1Y (u) du+� (Y ) =
ZfY 0dP +� (Y ) �
ZfY 0dP +� (Y 0) ;
hence
V� (f) = infY 2L2(P )
�Z 1
0
F�1f (1� u)F�1Y (u) du+ � (Y )
�:
If f �c g, an inequality of Hardy (see, e.g., [CR, p. 57-58]) deliversZ 1
0
F�1f (1� u)F�1Y (u) du �Z 1
0
F�1g (1� u)F�1Y (u) du 8Y 2 L2 (P ) ;
and V� (f) � V� (g). Let f �SSD g , Theorem 1.1 of [Cn] guarantees thatf � h �c g for some h 2 L2+ (P ). Since V� is monotone, then V� (f) �V� (f � h) � V� (g). �
Proof of Theorem 6. The maximization problem is
sup�2R
V� (W�)
whereW� = R+���X �~1R
�(remember that � 2 Rn, X 2 L2 (P )n, and ~1 is
a vector of 1s). From Theorem 19 we know that V� is Gateaux di¤erentiableand
rV� (W�) =
�1
P (W� � ��)� � (W� � E [W�jW� � ��])
�1fW����g;
19I.e. Y 0 �c Y i¤R� (Y 0) dP �
R� (Y ) dP for all concave � : R! R ([CR]
and [Cn] denote this relation by Y 0 � Y and call it majorization), and F�1f (t) =inf fz 2 R : Ff (z) � tg for all t 2 [0; 1] :20Let Y 0 �c Y . If � (Y ) =1, then � (Y 0) � � (Y ). Else Y > 0 and E [Y ] = 1, then also
Y 0 > 0 and E [Y 0] = 1 (see [CR, p. 62]). The function � (t) = � 12�
�t2 � 1
�is concave,
whence E [� (Y 0)] � E [� (Y )], and � (Y ) = �E [� (Y )] � �E [� (Y 0)] = � (Y 0).
35
where �� solves:
P (W� � ��)��� � EP [W� jW� � �� ]
�=1
�: (33)
Since for all i = 1; : : : ; n
@V� (W�)
@�i= E [rV� (W�) (Xi �R)] ;
the �rst order conditions for an optimum are:
E [XrV� (W�)] = ~1R: (34)
Substituting rV� (W�):
E
�1fW����g
P (W� � ��)X � �
�(� �X) 1fW����gX � E [� �X jW� � �� ] 1fW����gX
��= ~1R
set A = fW� � ��g to obtain
E [X jA ]� �P (A) (E [(� �X)X jA ]� E [X jA ] E [� �X jA ]) = ~1R
the observation that E [(� �X)X jA ] � E [X jA ] E [� �X jA ] = Var [X jA ]�yields:
EhX �~1R jW� � ��
i= �P (W� � ��)Var [X jW� � �� ]�:
These are the �rst n equations. The (n+ 1)-th is the equation which deter-mines ��, that is (33) or (see Lemma 13)
E�(W� � ��)
�� = 1
�:
Concavity of V� guarantees the su¢ ciency of �rst order conditions. �
Proof of Proposition 7. Set �� = ��mmv. The maximization problem itsolves is
max�2R
minY 2Y
�E [(R + � (X �R))Y ] +
1
2�E�Y 2�� 1
2�
�(35)
where Y =�Y 2 RS+ : E [Y ] = 1
. Clearly, Y is convex and compact, and
(��; Y �) is a solution of (35) if and only if it is a solution of
max�2R
minY 2Y
G (�; Y )
36
where G (�; Y ) = E [(R + � (X �R))Y ] + 12�E [Y 2]. Moreover, notice that
G : R�Y! R is continuous, it is a¢ ne in � (for each �xed Y ) and strictlyconvex in Y (for each �xed �). Set v = max�2RminY 2YG (�; Y ), by (aversion of) the Min-Max Theorem (e.g. [?, p. 134]) there exists �Y 2 Y suchthat
v = sup�2R
G��; �Y
�:
Moreover,
G���; �Y
�� min
Y 2YG (��; Y ) = G (��; Y �) = v = sup
�2RG��; �Y
�� G
���; �Y
�;
therefore, G���; �Y
�= minY 2YG (�
�; Y ), strict convexity implies �Y = Y �.In turn, this yields sup�2RG (�; Y
�) = v 6=1 and it cannot be
sup�2R
�R + �E [(X �R)Y �] +
1
2�E�(Y �)2
��=1;
thereforeE [Y � (X �R)] = 0: (36)
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:
R + �� (X �R) +1
�Y � � �� �� = 0; (37)
E [Y �] = 1; (38)
Y � > 0; � > 0; �Y � = 0: (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 [�]) : (41)
Multiply both sides by �, take expectations and use (39) to get:
E [�]� ���Cov [�;X] + �Var [�] = 0
37
and, rearranging terms:
���Cov [�;X] = E [�] + �Var [�] : (42)
Since � > 0, then E [�] � 0 thus:
�� = 0) � = 0) Cov [�;X] = 0; (43)
�� > 0) Cov [�;X] � 0; (44)
�� < 0) Cov [�;X] � 0: (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) �� = ��mv = 0; (47)
�� > 0) �� � ��mv; (48)
�� < 0) �� � ��mv: (49)
From the proof of Theorem 19 �Eq. (31) �we know that �� = g�1W��
�1�
�= ��.
Furthermore, if P (W�� > ��) > 0, since S is �nite, there exists s such that
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 theinequalities in (48) and (49) become strict.
38
Finally, we want to show that ����mv � 0. By contradiction, suppose����mv < 0. Then, either �� > 0 and ��mv < 0 or �� < 0 and ��mv > 0.Suppose �� > 0 and ��mv < 0, since
��mv =1
�
E [X �R]
Var [X];
it must be E [X �R] < 0 and
��E [X �R] < 0: (50)
Clearly, if �� < 0 and ��mv > 0, (50) still holds. Remember that (��; Y �) is
a saddle point for
G (�; Y ) = E [(R + � (X �R))Y ] +1
2�E�Y 2�
and so:G (��; Y �) � G (��; 1S) = R + ��E [X �R] < R
where the last inequality follows from (50). But,
minY 2Y
G (0; Y ) = R +minY 2Y
1
2�E�Y 2�� R
> G (��; Y �) = max�2R
minY 2Y
G (�; Y ) � minY 2Y
G (0; Y ) ;
which is impossible. �
Proof of Proposition 8. Assume8<: �P�W�� � ��
�Var
�X��W�� � ��
��� = E
hX �~1R
��W�� � ��i;
Eh�W�� � ��
��i= 1
�:
(51)
Set
� =�
�� and � =
�
��� �R
�+R:
Then,
W�� = R + �� ��X �~1R
�= R +
�� �X �
�� �~1R
= R�
�R +
�R +
�� �X �
�� �~1R
=
�W� +R
�1�
�
�;
39
and so
W�� =
�W� +R
�1�
�
�and �� =
�� +R
�1�
�
�: (52)
Equation (51) becomes8<: �P (W� � � )Var [X jW� � � ] �� = E
hX �~1R jW� � �
i;
EPh�
�(W� � � )
��i= 1
�:
that is( P (W� � � )Var [X jW� � � ]� = E
hX �~1R jW� � �
i;
E�(W� � � )�
�= 1
:
so that (� ; � ) solves the portfolio selection problem for an investor withuncertainty aversion . �
Proof of Lemma 9. Consider the portfolio selection problem (6) with� = 1. Next we show that
max�2Rn
V1 (W�) = max�2R
V1 (�Xm + (1� �)R)
and m 2 argmax�2R V1 (�Xm + (1� �)R).First observe that, for all � 2 R,
�Xm + (1� �)R = � (�m �X) +�1� ��m �~1
�R = W��m
hence max�2R V1 (�Xm + (1� �)R) � max�2Rn V1 (W�). Conversely, if �1 2argmax�2Rn V1 (W�), then m = �1 � ~1 and �m = �1=
��1 �~1
�or �1 = m�m.
Hence
max�2Rn
V1 (W�) = V1 (W�1) = V1 (Wm�m) = V1
�m�m �X +
�1�m�m �~1
�R�
= V1 (mXm + (1�m)R) � max�2R
V1 (�Xm + (1� �)R)
� max�2Rn
V1 (W�)
as wanted. Now, applying Theorem 6 with Xm instead of X, m must satisfythe following conditions:(
P (mXm + (1�m)R � �)Var [Xm jmXm + (1�m)R � � ]m = E [Xm �R jmXm + (1�m)R � � ] ;
Eh(mXm + (1�m)R� �)�
i= 1:
(53)
40
Moreover, since Xm is the (optimal) �nal wealth of an agent with initialwealth 1 and uncertainty aversion coe¢ cient m, again by Theorem 6 appliedto such an agent it must be the case that
E�(Xm � �m)�
�= 1=m: (54)
But the second equation of previous system is equivalent to
E
"�Xm �
��
m� (1�m)
mR
���#=1
m(55)
and since gXm is strictly increasing (see Lemma 14), then
�
m� (1�m)
mR = �m: (56)
By (56), fmXm + (1�m)R � �g =nXm � �
m� (1�m)
mRo= fXm � �mg
and (55) is equivalent to (54), thus (53) amounts to�P (Xm � �m)Var [Xm jXm � �m ]m = E [Xm �R jXm � �m ] ;
E�(Xm � �m)�
�= 1=m;
as wanted. �
Proof of Lemma 10. (i) First observe that by Remark 20 and Eq. (52) inthe proof of Proposition 8, for all � > 0
rV� (W��) = ���� �W��
�1fW��
���g = ��m��m � m
�W�m
�1fW�m��mg
= 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. �
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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).
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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).
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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).
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L. CASOLARO and L. GAMBACORTA, Redditività bancaria e ciclo economico, Bancaria, Vol. 61, 3, pp. 19-27, TD No. 519 (October 2004).
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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).
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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).
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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).
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S. DI ADDARIO, Job search in thick markets: Evidence from Italy, Oxford Discussion Paper 235, Department of Economics Series, TD No. 605 (December 2006).
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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).
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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).
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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).
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).
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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