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Econometrica, Vol. 81, No. 3 (May, 2013), 1075–1113

ALPHA AS AMBIGUITY: ROBUST MEAN-VARIANCEPORTFOLIO ANALYSIS

FABIO MACCHERONIIGIER, Università Bocconi, 20136, Milano, Italy

MASSIMO MARINACCIIGIER, Università Bocconi, 20136, Milano, Italy

DORIANA RUFFINOUniversity of Minnesota, Minneapolis, MN 55455, U.S.A.

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Econometrica, Vol. 81, No. 3 (May, 2013), 1075–1113

ALPHA AS AMBIGUITY: ROBUST MEAN-VARIANCEPORTFOLIO ANALYSIS

BY FABIO MACCHERONI, MASSIMO MARINACCI, AND DORIANA RUFFINO1

We derive the analogue of the classic Arrow–Pratt approximation of the certaintyequivalent under model uncertainty as described by the smooth model of decision mak-ing under ambiguity of Klibanoff, Marinacci, and Mukerji (2005). We study its scope byderiving a tractable mean-variance model adjusted for ambiguity and solving the cor-responding portfolio allocation problem. In the problem with a risk-free asset, a riskyasset, and an ambiguous asset, we find that portfolio rebalancing in response to higherambiguity aversion only depends on the ambiguous asset’s alpha, setting the perfor-mance of the risky asset as benchmark. In particular, a positive alpha corresponds to along position in the ambiguous asset, a negative alpha corresponds to a short positionin the ambiguous asset, and greater ambiguity aversion reduces optimal exposure toambiguity. The analytical tractability of the enhanced Arrow–Pratt approximation ren-ders our model especially well suited for calibration exercises aimed at exploring theconsequences of model uncertainty on equilibrium asset prices.

KEYWORDS: Ambiguity aversion, mean-variance analysis, robustness, smooth ambi-guity model.

“Crises feed uncertainty. And uncertainty affects behaviour, which feeds the crisis.”

Olivier Blanchard, The Economist, January 29, 2009

1. INTRODUCTION

When a von Neumann–Morgenstern expected utility maximizer with utilityu and wealth w considers an investment h, the Arrow–Pratt approximation ofhis certainty equivalent for the resulting uncertain prospect w+ h is

c(w+ h�P)≈w+EP(h)− 12λu(w)σ

2P(h)�(1)

where P is the probabilistic model that describes the stochastic nature of theproblem.

This classic approximation has two main merits, a theoretical and a practicalone. Its theoretical merit is to show that, for an expected utility agent, thepremium associated with facing risk h is proportional to the variance σ2

P(h) ofh with respect to P . This relation between risk and variance is a central pillar

1We thank Pauline Barrieu, David Bates, Emanuele Borgonovo, Veronica Roberta Cappelli,John Cochrane, Marco Giacoletti, Massimo Guidolin, Lars Hansen, Sujoy Mukerji, Fulvio Ortu,Andrea Resti, Alessandro Sbuelz, Claudio Tebaldi, Piero Veronese, the co-editor, four anony-mous referees, and especially Giulia Brancaccio and Simone Cerreia-Vioglio for helpful com-ments. The financial support of the European Research Council (Advanced Grant, BRSCDP-TEA), the AXA-Bocconi Chair, and the Carlson School of Management at the University ofMinnesota (Dean’s Research Grant) is gratefully acknowledged.

© 2013 The Econometric Society DOI: 10.3982/ECTA9678

1076 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

of risk management. In particular, the coefficient λu(w)= −u′′(w)/u′(w) thatlinks risk premium and volatility is determined by the agent’s risk aversion atw.The practical merit of (1) is in providing the foundation for the mean-variancepreference model, where a prospect f is evaluated through

U(f)=EP(f )− λ

2σ2P(f )�(2)

obtained from (1) by setting w + h = f and λu(w) = λ. This model is theworkhorse of asset management in the finance industry.

The purpose of this paper is to extend the classic Arrow–Pratt analysis to ac-count for model uncertainty: the situation in which the agent is uncertain aboutthe true probabilistic model P that governs the occurrence of different states.If only risk is present, that is, the agent fully relies on a single probabilisticmodel P , then the certainty equivalent c(w+ h�P) of w+ h, the sure amountof money that he considers equivalent to the uncertain prospect w+h, is givenby

c(w+ h�P)= u−1(EP

(u(w+ h)))�(3)

Here u represents the agent’s attitude toward risk. If, in contrast, the agentis not able to identify a single probabilistic model P , but he also considersalternative models Q, then c(w+ h�Q) becomes a variable amount of moneythat depends onQ. Suppose μ is the agent’s prior probability on the space Δ ofpossible models and v is his attitude toward model uncertainty (stricto sensu;see Section 2.2). The rationale used to obtain the certainty equivalent (3) leadsto a (second-order) certainty equivalent

C(w+ h)= v−1(Eμ

(v(c(w+ h))))(4)

= v−1(Eμ

(v(u−1

(E

(u(w+ h))))))�

where c(w + h) is the random variable that associates c(w + h�Q) to eachmodel Q in Δ. This is the smooth ambiguity certainty equivalent of Klibanoff,Marinacci, and Mukerji (2005; henceforth abbreviated KMM).

The case in which the support of the prior μ is a singleton P corresponds tothe absence of model uncertainty. In fact, the agent is fully confident about Pand (4) coincides with (3). Analogously, if v= u, it can be shown that

C(w+ h)= c(w+ h� Q)�where Q is the reduced probability

∫Qdμ(Q) induced by the prior μ. In this

case, the certainty equivalent (4) reduces to (3), where the probabilistic modelP is replaced by Q; in the jargon of decision theory, the agent is ambiguityneutral and the reduced distribution Q represents all the uncertainty he is fac-ing (see Ellsberg (1961, p. 661)). However, if the support of μ is nonsingleton

MEAN-VARIANCE PORTFOLIO ANALYSIS 1077

(there is model uncertainty, an information feature) and v differs from u (thereactions to model uncertainty and to risk differ, a taste feature) the identifi-cation of (3) and (4) no longer holds—model uncertainty cannot be reducedto risk—and the Arrow–Pratt analysis needs to be extended.

The first step in our extension of the Arrow–Pratt analysis is to derive, inSection 3, the analogue of approximation (1) under ambiguity, as captured bythe KMM certainty equivalent (4). Specifically, Proposition 3 shows that

C(w+ h)≈ w+EQ(h)− 12λu(w)σ

2Q(h)(5)

− 12(λv(w)− λu(w)

)σ2μ

(E(h)

)�

where Q = ∫Qdμ(Q) is the reduced probability induced by the prior μ, and

E(h) :Δ→ R is the random variable

Q �→EQ(h)

that associates the expected value EQ(h) to each possible modelQ. Its varianceσ2μ(E(h)), along with the difference λv(w)−λu(w) in uncertainty attitudes, de-

termines an ambiguity premium—the last term in (5)—that is novel relative to(1). In other words, model uncertainty renders volatile the return E(h) of h,thereby affecting the agent’s certainty equivalent. Indeed, (5) shows that λv(w)captures model uncertainty aversion also in “the small,” à la Pratt (1964): ce-teris paribus, the higher λv(w), the greater the ambiguity premium. In turn,this completes the KMM analysis of ambiguity aversion, as discussed at theend of Section 3.

In Section 4, we study those prospects that are unaffected by model uncer-tainty, that is, the special class of prospects for which approximation (5) re-duces to its classic counterpart (1).

The quadratic approximation (5) allows us to extend, in Section 5, the mean-variance model (2). Specifically, by setting w + h = f , λu(w) = λ, λv(w) −λu(w)= θ, and Q= P , we obtain the natural and parsimonious extension

U(f)=EP(f )− λ

2σ2P(f )− θ

2σ2μ

(E(f)

)(6)

of the mean-variance model (2) that is able to deal with ambiguity. This aug-mented mean-variance model is determined by the three parameters λ, θ, andμ, as opposed to the two parameters λ and P of the classic mean-variancemodel. The taste parameters λ and θ represent attitudes toward risk and am-biguity, respectively. Higher values of these parameters correspond to strongernegative attitudes. The information parameter μ determines the variancesσ2P(f ) and σ2

μ(E(f )) that measure the risk and model uncertainty perceived

1078 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

in the evaluation of prospect f . Higher values of these variances correspond topoorer information on prospect’s outcomes and on models.

In Section 6, we study the scope of the augmented mean-variance model(6) via a portfolio allocation exercise. In particular, we study a tripartite port-folio problem with a risk-free asset, a purely risky asset, and an ambiguousone. Relative to more traditional portfolio analyses with a risk-free and a riskyasset only, the addition of an ambiguous asset allows for the study of modeluncertainty. Our portfolio analysis shows that optimal portfolio rebalancing inresponse to higher ambiguity aversion only depends on the ambiguous asset’salpha, setting the performance of the risky asset as benchmark. An asset’s al-pha, it is found, is the component of the expected excess return of the ambigu-ous asset that is ambiguity specific, that is, uncorrelated with pure risk. Moreprecisely, it is the (expected) return of the ambiguous asset in excess of the re-turn on the risk-free asset, less the amount that can be explained as the excessreturn due to the pure risk embedded in the asset.2 When alpha is positive, theasset return offers compensation in excess of its risky component, a compensa-tion that an ambiguity averse agent would need to hold a long position in theasset. Indeed, a positive alpha corresponds to a long position in the ambiguousasset, a negative alpha corresponds to a short position in the ambiguous asset,and greater ambiguity aversion reduces optimal exposure to ambiguity.

Some fundamental asset allocation problems feature a natural tripartitestructure. This is the case for international portfolio allocation problems withdomestic bonds, domestic stocks, and foreign stocks. Our analysis is relevantfor these problems when the information available to investors is such that thetripartite structure may be interpreted as reflecting different types of uncer-tainty (i.e., risk and ambiguity) about the assets. We expect this to be often thecase.3

Related Works

Our work is related to recent papers by Nau (2006), Skiadas (2009), Izhakianand Benninga (2011), and Jewitt and Mukerji (2011), that inter alia also obtainapproximations for the ambiguity premium in the smooth ambiguity model onthe basis of special assumptions. More importantly, these papers do not usethe approximation to extend the mean-variance approach and study portfoliodecisions.

Our findings on the portfolio selection problem share some features withthe ones of Epstein and Miao (2003), Taboga (2005), Boyle, Garlappi, Uppal,and Wang (2012), and Gollier (2011). In particular, Taboga (2005) proposed

2For details, see (29) and its discussion.3See, for example, French and Poterba (1991), Canner, Mankiw, and Weil (1997), and Hu-

berman (2001) for evidence on these and related allocation problems that is inconsistent withexisting static choice models.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1079

a model of portfolio selection based on a two-stage evaluation procedure todisentangle ambiguity and ambiguity aversion. Gollier (2011) investigated thecomparative statics of more ambiguity aversion in a static two-asset portfolioproblem. He showed that ambiguity aversion may not reinforce risk aversionand exhibits sufficient conditions to guarantee that, ceteris paribus, an increasein ambiguity aversion reduces the optimal exposure to ambiguity. Gollier’s in-sight has been confirmed in terms of ambiguity premia by Izhakian and Ben-ninga (2011), who showed, for CRRA and CARA specifications, that such pre-mium may differ qualitatively from the risk premium. Epstein and Miao (2003)used a recursive multiple priors model to study the home bias, while Boyle etal. (2012) employed the concepts of ambiguity and ambiguity aversion in a mul-tiple priors framework to formalize the idea of investor’s “familiarity” towardassets.

In addition, the analytical tractability of the enhanced Arrow–Pratt approx-imation (5) favors empirical tests of our model’s implications to several ob-servationally puzzling (and economically interesting) investment behaviors.These include the home bias puzzle, the equity premium puzzle, as well as theemployer-stock ownership puzzle. For this reason, our paper is also relatedto several papers in the literature that explore the consequences of ambigu-ity aversion on equilibrium prices. Among others, Chen and Epstein (2002)identified separate excess return premia for risk and ambiguity within a repre-sentative agent asset market setting, while Garlappi, Uppal, and Wang (2007)extended a traditional portfolio problem to a multiple priors setting. Caskey(2009) and Illeditsch (2011) studied the effects of “ambiguous” informationon investors’ market trades and valuations. Easley and O’Hara (2010a, 2010b)explained how low trading volumes during part of the recent financial crisismay have resulted from investors’ perceived uncertainty and how designingmarkets to reduce ambiguity may induce participation by both investors andissuers.

By use of recursive versions of the smooth ambiguity model, Ju and Miao(2012) calibrated a representative agent consumption based asset pricingmodel and generated a variety of dynamic asset pricing phenomena that areobserved in the data; Chen, Ju, and Miao (2011) studied an investor’s optimalconsumption and portfolio choice problem4; while Collard, Mukerji, Sheppard,and Tallon (2011) showed the importance of model uncertainty for the analysisof long run risk (LLR) by matching the historical equity premium with a LLRmodel that features endogenously time-varying ambiguity (e.g., increasing dur-ing recessions) based on publicly available data on aggregate consumption anddividend. Hansen and Sargent (2010) considered two risk-sensitivity operators,one of them being a recursive version of the smooth ambiguity model, in a LLRsetup, and showed how sensitive to information beliefs become under modeluncertainty; they showed how the resulting “model uncertainty premia” affect

4The recursive multiple priors case appears in Miao (2009).

1080 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

the price of macroeconomic risk. Finally, Weitzman (2007) showed how modeluncertainty can be important in dynamic asset pricing already under standardexpected utility (and so, without taking into account agents’ specific reactionto model uncertainty) when “persistent” uncertainty may prevent full learn-ing of the true data generating process, so that subjective beliefs on modelskeep being relevant even with large amounts of observations. Though thesedynamic papers are different from our static analysis (where learning issuesare absent), they share with this paper the insight that a proper account ofmodel uncertainty is required to better understand the quantitative puzzles ofasset markets.

2. PRELIMINARIES

2.1. Mathematical Setup

Given a probability space (Ω�F�P), let L2 = L2(Ω�F�P) be the Hilbertspace of square integrable random variables on Ω and let L∞ = L∞(Ω�F�P)be the subset of L2 consisting of its almost surely bounded elements. Given aninterval I ⊆ R, we set

L∞(I)= {f ∈L∞ : essinf f�esssup f ∈ I}�

Throughout the paper, ‖ · ‖ denotes the L2 norm. The space L2 is the naturalsetting for this paper because of our interest in quadratic approximations.

We indicate by EP(X) and σ2P(X) the expectation and variance of a random

variable X ∈ L2, respectively. Moreover, we indicate by σP(X�Y) the covari-ance

σP(X�Y)=EP[(X −EP(X)

)(Y −EP(Y)

)]between two random variables X�Y ∈L2.

The set of probability measures Q on F that have square integrable densityq = dQ/dP with respect to P can be identified, via Radon–Nikodym deriva-tion, with the closed and convex subset of L2 given by

Δ={q ∈L2

+ :∫Ω

q(ω)dP(ω)= 1}�

By Bonnice and Klee (1963, Theorem 4.3), we have the following existenceresult.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1081

LEMMA 1: Given a Borel probability measure μ on Δ with bounded support,5there exists a unique q ∈ Δ such that∫

Ω

X(ω)q(ω)dP(ω)(7)

=∫Δ

(∫Ω

X(ω)q(ω)dP(ω)

)dμ(q)� ∀X ∈L2�

The density q is denoted by∫Δqdμ(q) and is called the barycenter of μ.

Notice that, when restricted to indicator functions 1A of elements of F , (7)delivers

Q(A)=∫Δ

Q(A)dμ(Q)� ∀A ∈ F�(8)

where the identification of each probability measureQ with its density q allowsto write dμ(Q) instead of dμ(q). The probability measure Q is called reductionof μ on Ω. In fact, (8) suggests a natural interpretation of Q in terms of reduc-tion of compound lotteries. For example, if suppμ= {Q1� � � � �Qn} is finite andμ(Qi)= μi for i= 1� � � � � n, then (8) becomes

Q(A)= μ1Q1(A)+ · · · +μnQn(A)� ∀A ∈ F �

Hence, μ can be seen as a lottery whose outcomes are all possible models,which, in turn, can be seen as lotteries that determine the state.

2.2. Decision Theoretic Setup

Given any nonsingleton interval I ⊆ R of monetary outcomes, we considerdecision makers (DMs) who behave according to the smooth model of decisionmaking under ambiguity of KMM,6 that is, DMs who rank prospects throughthe functional V :L∞(I)→ R defined by

V (f )=∫Δ

φ

(∫Ω

u(f (ω)

)q(ω)dP(ω)

)dμ(q)� ∀f ∈L∞(I)�(9)

where μ is a Borel probability measure on Δ with bounded support, and u : I →R and φ :u(I)→ R are continuous and strictly increasing functions.

5A carrier of μ is any Borel subset of Δ having full measure. If the intersection of all closedcarriers is a carrier, it is called support of μ and denoted by suppμ. Since Δ inherits the L2

distance, suppμ is bounded when {‖q‖ :q ∈ suppμ} is a bounded set of real numbers. If F isfinite, the support always exists and is bounded.

6We use the terms decision maker and agent interchangeably in the paper.

1082 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

LEMMA 2: The functional V :L∞(I)→ R is well defined, with V (L∞(I)) =φ(u(I)).

The certainty equivalent function C :L∞(I)→ I induced by V is defined byV (C(f ))= V (f ) for all prospects f , that is,

C(f )= u−1

(φ−1

(∫Δ

φ

(∫Ω

u(f (ω)

)q(ω)dP(ω)

)dμ(q)

))�(10)

∀f ∈L∞(I)�

In the monetary setting of the present paper, where outcomes are amounts ofmoney and prospects are financial assets, it is natural to consider monetary cer-tainty equivalents. To this end, set v=φ◦u : I → R (see KMM (2005, p. 1859)).It is then possible to rewrite (9) as

V (f )=∫Δ

(v ◦ u−1

)(∫Ω

u(f (ω)

)q(ω)dP(ω)

)dμ(q)�(11)

∀f ∈L∞(I)�

and so (10) as

C(f )= v−1

(∫Δ

v

(u−1

(∫Ω

u(f (ω)

)q(ω)dP(ω)

))dμ(q)

)�(12)

∀f ∈L∞(I)�

Here the certainty equivalent C(f ) is viewed as the composition of two mone-tary certainty equivalents,

c(f�q)= u−1

(∫Ω

u(f (ω)

)q(ω)dP(ω)

)and

v−1

(∫Δ

v(c(f�q)

)dμ(q)

)�

This is the approach we sketched in the Introduction, motivated by the papermonetary setting.

In KMM, the function v represents attitudes toward stricto sensu model un-certainty, that is, the uncertainty that agents face when dealing with alternativepossible probabilistic models. The function v is characterized in KMM, alongwith the prior μ, through prospects whose outcomes depend only on modelsand, as such, are only affected by model uncertainty.

Model uncertainty cumulates with the state uncertainty that any nontrivialprobabilistic model features. The combination of these two sources of uncer-tainty determines, in the KMM model, the ambiguity that DMs face in rank-

MEAN-VARIANCE PORTFOLIO ANALYSIS 1083

ing prospects f :Ω→ R. KMM showed that overall attitudes toward ambigu-ity are captured by the function φ. In particular, its concavity characterizesambiguity aversion, which therefore implies positive Arrow–Pratt coefficientsλφ = −φ′′/φ′. Since

λφ(u(w)

) = 1u′(w)

(λv(w)− λu(w)

)�(13)

we conclude that ambiguity aversion amounts to λv − λu ≥ 0, a key conditionfor the paper.7

Ambiguity neutrality is modelled by φ(x)= x, that is, v = u, while absenceof model uncertainty is modelled by a trivial μ with singleton support (i.e., aDirac measure). In both cases, criterion (9) reduces to expected utility, thoughin one case the reduction originates from a taste component (a neutral atti-tude, under which the two sources of uncertainty “linearly” combine via reduc-tion (8)), while in the other case it originates from an information component(absence of a source of uncertainty, i.e., model uncertainty).

3. QUADRATIC APPROXIMATION

Let w ∈ int I be a scalar interpreted as current wealth. To ease notation, wealso denote by w the degenerate random variable w1Ω. Given any prospecth ∈ L∞ such that w+ h ∈ L∞(I), we are interested in the certainty equivalentC(w+ h) of w+ h, that is,

C(w+ h)= v−1

(∫Δ

v

(u−1

(∫Ω

u(w+ h)qdP))

dμ(q)

)�(14)

For all h ∈L∞, the functions

E(h) :q �→∫Ω

hqdP and σ2(h) :q �→∫Ω

h2qdP −(∫

Ω

hqdP

)2

are continuous and bounded on Δ, and so they belong to L∞(Δ�B�μ). Thevariance of E(h) with respect to μ,∫

Δ

(∫Ω

h(ω)q(ω)dP(ω)

)2

dμ(q)

−(∫

Δ

(∫Ω

h(ω)q(ω)dP(ω)

)dμ(q)

)2

�

7See Lemma 12 in the Appendix.

1084 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

is denoted by σ2μ(E(h)). This variance reflects the uncertainty on the expecta-

tion E(h), which, in turn, is implied by model uncertainty. Thus, higher valuesof σ2

μ(E(h)) correspond to a higher incidence of model uncertainty on the ex-pectation of h.

We can now state the second order approximation of the certainty equiva-lent (14).

PROPOSITION 3: Let μ be a Borel probability measure with bounded supporton Δ and let u�v : I → R be twice continuously differentiable with u′� v′ > 0.Then,

C(w+ h)= w+EQ(h)− 12λu(w)σ

2Q(h)(15)

− 12(λv(w)− λu(w)

)σ2μ

(E(h)

) +R2(h)

for all h ∈L∞ such that w+ h ∈L∞(I), where

limt→0

R2(th)

t2= 0�(16)

Moreover, if F is finite, then R2(h)= o(‖h‖2) as h→ 0 in L2.

Notice that the first three components on the right hand side of (15) cor-respond to the Arrow–Pratt approximation of the ambiguity neutral certaintyequivalent u−1(

∫u(w+ h)dQ) of w+ h. Instead, the fourth component rep-

resents the effects of ambiguity: the sign and magnitude of these effects on thecertainty equivalent depend on the difference λv(w) − λu(w). In particular,provided model uncertainty affects the expectation of h, that is, σ2

μ(E(h)) = 0,ambiguity has no effects when the DM is neutral to it at w, that is, λv(w) =λu(w).8 Finally, the fifth component, “the error” tends to zero faster than thesquare of the size of the uncertainty exposure.

The variance σ2Q(h) can be decomposed along the two sources of uncer-

tainty as

σ2Q(h)=Eμ

(σ2(h)

) + σ2μ

(E(h)

)�

State uncertainty, which exists within each model, underlies the average vari-ance Eμ(σ2(h)). Model uncertainty, instead, determines the variance of aver-

8Notice that this may be the case even if v = u, that is, the condition λv(w) = λu(w) is arequirement of ambiguity neutrality in “the small” rather than in “the large.”

MEAN-VARIANCE PORTFOLIO ANALYSIS 1085

ages σ2μ(E(h)). Approximation (15) can thus be rearranged according to the

Arrow–Pratt coefficients of u and v as

C(w+ h)= w+EQ(h)− λu(w)

2Eμ

(σ2(h)

)(17)

− λv(w)

2σ2μ

(E(h)

) +R2(h)�

This formulation shows that, when the indexes u and v are sufficiently smooth,both state and model uncertainty play at most a second order effect in the eval-uation.9 Specifically, risk aversion determines the DM’s reaction to the averagevariance Eμ(σ2(h)) and model uncertainty aversion determines his reaction tothe variance of averages σ2

μ(E(h)).Formulation (17) also suggests the conditions under which the quadratic ap-

proximation is exact. Recall that the classic Arrow–Pratt approximation is ex-act under constant absolute risk aversion and normal distribution of prospecth (see Eeckhoudt, Gollier, and Schlesinger (2005, p. 21)). Specifically, if uis CARA and h has a normal cumulative distribution �(·;m�σ) with meanm and variance σ2, then the Arrow–Pratt approximation takes the exactform

c(w+ h�m)= u−1

(∫u(w+ x)d�(x;m�σ)

)= w+m− 1

2λu(w)σ

2�

This result easily generalizes to our quadratic approximation: exactness holdsunder constant absolute aversion to both risk and model uncertainty, and nor-mal distribution with unknown mean and known variance of prospect h. For-mally, if both u and v are CARA utility functions, and h has a normal cumu-lative distribution �(·;m�σ) with unknown mean m and known variance σ2,then approximation (17) takes the exact form10

C(w+ h)= v−1

(∫v

(u−1

(∫u(w+ x)d�(x;m�σ)

))d�(m; μ�σμ)

)9For standard expected utility, this follows from the Arrow–Pratt approximation. Segal and

Spivak (1990) is a classic study of orders of risk aversion. In Maccheroni, Marinacci, and Ruffino(2011), we studied in detail orders of risk aversion and of model uncertainty aversion in thesmooth ambiguity model.

10Notice that normally distributed random variables belong to L2 and not to L∞. But, thedefinition (4) of the KMM certainty equivalent C(w+ h) does not require the boundedness of h(an assumption maintained in the rest of the paper for mathematical convenience).

1086 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

= v−1

(∫v

(w+m− 1

2λu(w)σ

2

)d�(m; μ�σμ)

)=w+ μ− 1

2λu(w)σ

2 − 12λv(w)σ

2μ�

provided the prior on the unknown means m is given by a normal cumula-tive distribution �(·; μ�σμ) with parameters μ and σμ. Normality of the priorimplies that μ is the mean of the unknown means m and σμ is their vari-ance.

We conclude by observing that formulas (13) and (15) allow to interpret φas capturing ambiguity aversion both in the large and in the small, à la Pratt(1964). In fact, given any u�v1� v2 : I → R twice continuously differentiable withu′� v′

1� v′2 > 0, by (13) it holds that

λφ1

(u(w)

) ≥ λφ2

(u(w)

)⇔ 1

u′(w)

(λv1(w)− λu(w)

) ≥ 1u′(w)

(λv2(w)− λu(w)

)⇔ λv1(w)≥ λv2(w)�

In “the large,” by Corollary 3 of KMM, this implies that agent 1 is more am-biguity averse than agent 2 if and only if he is more model uncertainty aversethan agent 2. In “the small,” by (15), this is equivalent to saying that the ambi-guity premium for agent 1 is greater than the ambiguity premium for agent 2(while risk premia coincide).

In the rest of the paper, we will focus on the case in which the DM is am-biguity averse at w, that is, λv(w)− λu(w) ≥ 0. Similarly, we will assume riskaversion at w, that is, λu(w)≥ 0.

4. APPROXIMATELY UNAMBIGUOUS PROSPECTS

As previously observed, the ambiguity premium

12(λv(w)− λu(w)

)σ2μ

(E(h)

)(18)

may vanish for two reasons: the ambiguity neutrality condition λv(w)= λu(w)or the information condition σ2

μ(E(h)) = 0. In this section, we consider thelatter case since it directly refers to model uncertainty (as captured by the pa-rameter μ) and our main focus is on ambiguity non-neutral behavior.

DEFINITION 4: A prospect h ∈ L2 is approximately unambiguous ifσ2μ(E(h))= 0.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1087

Clearly, if h is approximately unambiguous, then approximation (15) col-lapses to

C(w+ h)=w+EQ(h)− 12λu(w)σ

2Q(h)+R2(h)�

that is, the DM’s evaluation of h is indistinguishable (in the second order ap-proximation) from the certainty equivalent of an (ambiguity neutral) expectedutility maximizer with utility u and beliefs given by the reduced probabilitymeasure Q induced by μ onΩ. The equivalence of (i) and (ii) in the next resultshows that also the converse is true, thus motivating the term “approximatelyunambiguous” for such a prospect.

PROPOSITION 5: For a prospect h ∈ L2, the following properties are equiva-lent:

(i) h is approximately unambiguous;(ii) σ2

μ(E(h))=R2(h);(iii) EQ(h)=EQ′(h) for all Q�Q′ ∈ suppμ.

The equivalence between (i) and (iii) is also noteworthy. It has two differentimplications. First, it says that h is approximately unambiguous if and only ifthe first moment of h is invariant across all models in the support of μ, thatis, according to all models that a DM with prior μ deems plausible.11 Sec-ond, it shows that the set of all approximately unambiguous prospects formsa closed linear subspace of L2, and hence any prospect can be decomposedinto an approximately unambiguous part and a residual ambiguous one (seeAppendix A.3).

Next, we show that, under model uncertainty, there always exist arbitrarilysmall “ambiguous” prospects.

PROPOSITION 6: If λv(w) = λu(w), then the following properties are equiva-lent:

(i) all prospects in L2 are approximately unambiguous;(ii) there is an absorbing12 subset B of L∞ such that w+ h ∈L∞(I) and

C(w+ h)=w+EQ(h)− 12λu(w)σ

2Q(h)+R2(h)� ∀h ∈ B;(19)

(iii) μ is a Dirac measure.

11The interpretation of suppμ as the set of plausible models, à la Ghirardato, Maccheroni, andMarinacci (2004), is formally discussed by Cerreia-Vioglio, Maccheroni, Marinacci, and Montruc-chio (2011, Theorem 21). Also notice that a simple prospect h is unambiguous in the sense ofGhirardato, Maccheroni, and Marinacci (2004) if and only if all of its moments (not only the first)coincide on the support of μ.

12A subset B of a vector space is absorbing if, for any point of the space, there exists a strictlypositive multiple of B that contains the segment joining the point and zero. For example, any ballthat contains the origin is absorbing.

1088 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

This result shows that the only case in which our approximation (15) coin-cides with that of Arrow–Pratt for all “small prospects” (i.e., the prospects inB) is when there is no model uncertainty to start with, that is, the prior μ is de-generate. Otherwise, for each ε > 0, however small, there is a prospect h with‖h‖< ε for which the two approximations differ. In other words, as long as μ isnot trivial and the agent is not ambiguity neutral, ambiguity effects may neverfade away, even approximately for arbitrarily “small prospects” (the counter-parts of “small risks” in risk theory).

This result may seem at odds with Skiadas (2009), who argued that ambiguityeffects may fade in the small for some vanishing nets of prospects. But, thereare at least two nontrivial differences in the analyses; for simplicity, we discussthem in the case Ω= {1�−1}, so that L2 can be identified with R

2:• First, and conceptually more importantly, in our static analysis μ is fixed,

while in the Skiadas (2009) continuous time setup it depends on t; for example,given any p�p′ ∈ (0�1), μt can be the uniform distribution on [p− t�p′ + t] forsome t ∈ (0� ε).13

• Second, condition (ii) of Proposition 6 refers to an absorbing subset, forexample, {(h�k) ∈ R

2 :h2 + k2 < ε}; instead, Skiadas (2009) considered van-ishing nets, which are just paths closing to the origin, for example, {(t�−t) ∈R

2 : t ∈ (0� ε)}.The first point is crucial because it allows model uncertainty to vanish when

μt converges to a Dirac measure.14 A complete analysis can be found in Mac-cheroni, Marinacci, and Ruffino (2011), where, in a nutshell, we showed thatambiguity effects fade away if and only if model uncertainty vanishes (it is a“perturbed” version of Proposition 6 above). On a related note, Hansen andSargent (2011) proposed a continuous time limit of the smooth model in whichmodel uncertainty attitudes survive.

5. ROBUST MEAN-VARIANCE PREFERENCES

Inspired by the quadratic approximation (15), in this section, we generalizestandard mean-variance preferences to account for model uncertainty. Specifi-cally, we consider a DM who ranks prospects f in L2 through the robust mean-variance functional C :L2 → R ∪ {−∞} given by

C(f )=EQ(f )− λ

2σ2Q(f )− θ

2σ2μ

(E(f)

)� ∀f ∈L2�(20)

where λ and θ are (strictly) positive coefficients, and μ is a Borel probabilitymeasure on Δ with bounded support and barycenter Q given by (8).

13Here, we identify Δ with the unit interval [0�1] by assuming that q ∈ [0�1] is the probabilityof state 1.

14In the uniform example, this happens if and only if p= p′, in which case μt weakly convergesto the Dirac measure δp.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1089

As mentioned in the Introduction, this preference functional is fully deter-mined by three parameters: λ, θ, and μ. Its theoretical foundation is given bythe quadratic approximation (15), which shows that (20) can be viewed as a lo-cal approximation of a KMM preference functional (12) at a constant w suchthat λ = λu(w) and θ = λv(w) − λu(w). Thus, the taste parameters λ and θmodel the DM’s negative attitudes toward risk and ambiguity, respectively. Inparticular, higher values of these parameters correspond to stronger negativeattitudes.

In turn, the information parameter μ determines the variances σ2Q(f ) and

σ2μ(E(f )) that measure the risk and model uncertainty that the DM perceives

in the evaluation of prospect f . Higher values of these variances correspond toa DM’s poorer information on prospect’s outcomes and on models.

Since the probability measure Q is the reference model for a DM with priorμ, to facilitate comparison with the classic case we identify the barycenter Q ofΔ with the baseline probability P . That is, in the rest of the paper, we maintainthe following assumption:

ASSUMPTION 1: Q= P .

Under this assumption, C(f ) is always finite and (20) takes the form

C(f )=EP(f )− λ

2σ2P(f )− θ

2σ2μ

(E(f)

)� ∀f ∈L2�(21)

which we will consider hereafter. When the information condition σ2μ(E(f ))=

0 holds, we obtain the standard mean-variance evaluation

C(f )=EP(f )− λ

2σ2P(f )(22)

for prospect f . Approximately unambiguous prospects are thus regarded aspurely risky by robust mean-variance preferences, that is, they form the class ofprospects on which robust and conventional mean-variance preferences agree.

Like standard mean-variance preferences, our robust mean-variance pref-erences (21) separate taste parameters, λ and θ, and uncertainty measures,σ2P(f ) and σ2

μ(E(f )). This sharp separation gives standard mean-variance pref-erences an unsurpassed tractability and is the main reason for their successand widespread use. These key features fully extend to robust mean-variancepreferences, as (21) shows. As a result, they are well suited for finance andmacroeconomics applications and can improve calibration and other quantita-tive exercises. Their scope will be illustrated in detail in the portfolio problemof the next section.

Finally, we expect that a monotonic version of robust mean-variance pref-erences can be derived by suitably generalizing what Maccheroni, Marinacci,Rustichini, and Taboga (2009) established for conventional mean-variancepreferences.

1090 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

6. THE PORTFOLIO ALLOCATION PROBLEM

In this section, we apply the newly obtained robust mean-variance prefer-ences to a portfolio allocation problem. The theoretical difference betweenthese preferences and the smooth ambiguity preferences of KMM is analogousto the one that separates the expected utility approach and the mean-varianceutility approach in the case of portfolio selection under risk only.15

In the final part of the section, we focus on a portfolio of three assets: a risk-free asset, a purely risky asset, and an ambiguous asset. This problem is thenatural extension of the standard portfolio problem (with a risk-free and a riskyasset) to our setting with model uncertainty. As mentioned in the Introduction,international portfolio allocation problems provide a natural application of oursetting with domestic Treasury bonds viewed as risk-free assets, other domesticassets viewed as purely risky assets, and foreign assets viewed as ambiguousassets. This will be our motivating example.

6.1. The General Setting

We consider the one-period optimization problem of an agent who has todecide how to allocate a unit of wealth among n+ 1 assets at time 0. The grossreturn on asset i after one period, i= 1� � � � � n, is denoted by ri ∈L2. Then, the(n× 1) vector of returns on the first n assets is denoted by r and the (n× 1)vector of portfolio weights (indicating the fraction of wealth invested in eachasset) is denoted by w. The return on the (n+ 1)th asset is risk-free and it isequal to a constant rf .

The end-of-period wealth rw induced by a choice w is given by

rw = rf + w · (r − rf1)�where 1 is the n-dimensional unit vector. We assume frictionless financial mar-kets in which assets are traded in the absence of transaction costs and bothborrowing and short-selling are allowed without restrictions. Then, the portfo-lio problem can be written as

maxw∈Rn

C(rw)= maxw∈Rn

(EP(rw)− λ

2σ2P(rw)− θ

2σ2μ

(E(rw)

))�(23)

A simple argument, in the Appendix, delivers the optimality condition[λVarP[r] + θVarμ

[E[r]]]w = EP[r − rf1]�(24)

where:

15An empirical comparison of the two (à la Levy and Markowitz (1979)) goes beyond the scopeof this paper and is left for future research.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1091

• VarP[r] = [σP(ri� rj)]ni�j=1 is the variance–covariance matrix of returns un-der P ,

• Varμ[E[r]] = [σμ(E(ri)�E(rj))]ni�j=1 is the variance–covariance matrix ofexpected returns under μ,

• EP[r − rf1] = [EP(ri − rf )]ni=1 is the vector of expected excess returns un-der P .Thus, the solution to (23) is an ambiguity-adjusted mean-variance portfo-lio whose weights reflect uncertainty about expected returns as captured byVarμ[E[r]]. A key feature of condition (24) is that it allows to make use of thevast body of research on mean-variance preferences developed for problemsinvolving risk to analyze problems involving ambiguity. For example, also herethe optimal investments are smooth functions of the taste parameters λ and θand of the information parameters EP[r − 1rf ], VarP[r], and Varμ[E[r]]. Thisis crucial for the comparative statics analysis. In fact, given the available infor-mation on assets’ returns,16 condition (24) allows to explicitly write the optimalportfolios as functions of the uncertainty attitudes,17 and so to study how dif-ferent attitudes influence optimal holdings. Conversely, given the uncertaintyattitudes of an agent, condition (24) allows to explicitly write the optimal port-folios as functions of the available information on assets’ returns, and so tostudy how information and its quality affect optimal holdings.

Next, we study condition (24) for the case of a single ambiguous asset, andfor the case of one purely risky asset and one ambiguous asset. These two casesare important because they immediately contrast the optimal portfolio solutionwith and without ambiguity.

6.2. One Ambiguous Asset

If n= 1, then there is only one uncertain asset and (24) delivers

w= EP(r)− rfλσ2

P(r)+ θσ2μ(E(r))

�(25)

If r is purely risky, that is, σ2μ(E(r)) = 0, then (25) reduces to the standard

mean-variance Markowitz (1952) solution

w= EP(r)− rfλσ2

P(r)�(26)

Notice that ambiguity does not affect excess returns, so that the difference be-tween (25) and (26) lies in their denominators only. Specifically, an increase

16That is, EP [r − 1rf ], VarP [r], and Varμ[E[r]].17That is, λ and θ.

1092 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

in θσ2μ(E(r))—that is, an increase in either ambiguity aversion θ or ambigu-

ity in expectations σ2μ(E(r))—makes the ambiguous asset less desirable and

increases the DM’s demand for the risk-free asset (a flight-to-quality effect).As shown by Gollier (2011), this intuitive result is not true in general for thesmooth ambiguity preferences of KMM. Gollier (2011) established some con-ditions which ensure that more ambiguity aversion reduces the optimal levelof exposure to uncertainty for KMM preferences; in our simplified quadraticsetting, no additional conditions are needed.

6.3. One Purely Risky Asset and One Ambiguous Asset

We now turn to the case of two uncertain assets with returns rm and re in L2,which we interpret as a domestic and a foreign security index, respectively. Forthis reason, we choose rm to be purely risky for robust mean-variance prefer-ences, that is, σ2

μ(E(rm)) = 0, and re to be ambiguous, that is, σ2μ(E(re)) > 0.

The DM can now invest in a risk-free asset, with rate of return rf , in a purelyrisky asset, with rate of return rm, and in an ambiguous one, with rate of returnre.

Here, condition (24) becomes

λ

[σ2P(rm) σP(rm� re)

σP(rm� re) σ2P(re)

][wmwe

]+ θ

[0 00 σ2

μ

(E(re)

)][wmwe

]=

[EP(rm)− rfEP(re)− rf

]�

that is,

EP(rm)− rf = wmλσ2P(rm)+ weλσP(rm� re)

and

EP(re)− rf = wmλσP(rm� re)+ we[λσ2

P(re)+ θσ2μ

(E(re)

)]�

For convenience, we set

A=EP(re)− rf �B=EP(rm)− rf �C = λσ2

P(rm)�

D= λσ2P(re)+ θσ2

μ

(E(re)

)�

H = λσP(rm� re)�

MEAN-VARIANCE PORTFOLIO ANALYSIS 1093

Then, the optimal portfolio weights associated with the purely risky and theambiguous assets are

wm = BD−HACD−H2

and we = CA−HBCD−H2

�

and CD−H2 > 0.18 We are interested in determining how changes in the pref-erence parameters affect the optimal amounts wm and we, as well as their ratio

wm

we= BD−HACA−HB �

These quantities vary with μ, the prior probability over models, with θ, theparameter of ambiguity aversion, and with λ, the parameter of risk aversion.Before stating the results, we collect all the assumptions we make, most ofwhich are non-triviality assumptions.

OMNIBUS CONDITION: Suppose that A > 0 and B > 0, that is, excess re-turns on uncertain assets are both positive, and CA =HB, that is, the optimalinvestment in the ambiguous asset is nonzero.

The next lemma simplifies our analysis of the effects of varying ambiguity onthe optimal portfolio.

LEMMA 7: Suppose that the Omnibus Condition holds and set σ2μ = σ2

μ(E(re)).Then,

∂(wm/we)

∂σ2μ

= θ

σ2μ

∂(wm/we)

∂θ�(27)

∂wm

∂σ2μ

= θ

σ2μ

∂wm

∂θ� and

∂we

∂σ2μ

= θ

σ2μ

∂we

∂θ�

Notice that, since the optimal portfolio allocation varies with μ (the priorover models) only through σ2

μ (the variance of expected returns), changes in μcan be measured by taking derivatives with respect to σ2

μ. Moreover, variationsin wm, we, and wm/we due to changes in σ2

μ and θ share the same sign. In viewof this result, hereafter we only consider variations in θ and we generally referto them as variations in ambiguity.

18In fact, CD−H2 = λ2[σ2P(rm)σ

2P(re)−σP(rm� re)2] +λθσ2

P(rm)σ2μ(E(re)), with the first sum-

mand nonnegative by the Cauchy–Schwarz inequality and the second one positive since all of itsfactors are positive.

1094 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

6.3.1. Changes in θ

We begin our comparative statics analysis by studying the effects of changesin θ; later, we study the effects of changes in λ.

For the comparative analysis, it is convenient to decompose the excess returnof the ambiguous asset by means of the ordinary least squares coefficients. Tothis end, we project the excess return of the ambiguous asset over the excessreturn of the purely risky one, that is, we consider the solutions of

minα�β∈R

∥∥(re − rf )− (α+β(rm − rf )

)∥∥�As is well known, they are given by

βP(rm� re)= σP(rm� re)

σ2P(rm)

(28)

and

αP(rm� re)=EP(re − rf )−βP(rm� re)EP(rm − rf )�(29)

The beta coefficient βP(rm� re)measures the level of pure risk—as embodied inrm—of asset re; in the finance jargon, βP(rm� re) is a pure risk adjustment.19 Asa result, βP(rm� re)EP(rm−rf ) is what index re is expected to earn/lose, net of rf ,given its level of pure risk sensitivity. The residual component αP(rm� re) of theexpected excess returnEP(re−rf ) is then what re is expected to earn/lose, net ofrf , given its level of uncertainty uncorrelated with pure risk.20 Such uncertaintyis specific to the ambiguous asset.

The next result shows that the sign of αP(rm� re) alone determines the effectof changes in ambiguity on the optimal proportion between risky and ambigu-ous holdings wm and we, as well as on the variation of we. On the other hand,also the sign of βP(rm� re) becomes relevant to describe the variation of wm.

PROPOSITION 8: Suppose the Omnibus Condition holds. Then,

sgn∂

∂θ

(wm

we

)= sgnαP(rm� re)�(30)

19Equation (50) in the Appendix further illustrates this interpretation by showing thatσP(rm� re) = σP(rm� re − rae ). That is, only the purely risky projection re − rae of the ambiguousre enters its covariance with the purely risky rm.

20To see why it is uncorrelated, notice that, by the Hilbert Decomposition theorem,

re − rf =AP(rm�re)+βP(rm�re)(rm − rf )�where AP(rm�re) ∈ L2 is the part of the excess return re − rf that is uncorrelated with rm − rfand αP(rm�re) is the expected value of the uncorrelated part AP(rm�re), that is, αP(rm�re) =EP(AP(rm�re)).

MEAN-VARIANCE PORTFOLIO ANALYSIS 1095

Moreover,

sgn∂we

∂θ= − sgnαP(rm� re) and(31)

sgn∂wm

∂θ= sgnαP(rm� re)βP(rm� re)�

The sign of αP(rm� re) also governs the sign of optimal ambiguous holdingswe.

PROPOSITION 9: Suppose the Omnibus Condition holds. Then

sgn we = sgnαP(rm� re)�(32)

In other words, the agent uses αP(rm� re) as a criterion to decide whether totake a long or short position in the ambiguous asset, that is, to decide in whichside of the market of asset re to be. As anticipated in the Introduction, whenthis term is positive, the ambiguous asset return offers compensation in excessof its risky component, which induces the agent to hold a long position on theasset (a symmetric argument holds in case of negativity).

In the jargon of investment practitioners, our agent “seeks the alpha”(buys/sells the ambiguous fund if αP(rm� re) is positive/negative) and he is awarethat this extra return comes from the ambiguous nature of the investment;“therefore,” he reduces exposure to ambiguity as ambiguity aversion rises.

Thanks to Propositions 8 and 9, we can show how variations in θ affect theoptimal portfolio composition, both in relative and in absolute terms. Two pos-sible cases arise.21

CASE 1: If α is positive, then we > 0 and

�θ > 0 �⇒ �

(wm

we

)> 0 and �we < 0�(33)

where � denotes a small variation.22 Here, an increase in ambiguity aversiondetermines a higher ratio wm/we and a lower optimal we. Since we is positive,a lower optimal we corresponds to a lower exposure to ambiguity. Finally, thesign of the variation in wm coincides with the sign of β, that is, of the covarianceσP(rm� re),

�θ > 0 �⇒ �wm ≷ 0 if and only if βP(rm� re)≷ 0�

21To ease notation, in what follows, we just write α and β omitting P(rm� re).22Not to be confounded with the set of probability measures Δ.

1096 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

When β is positive, higher ambiguity aversion thus results in a higher value ofwm and a lower value of we. Otherwise, the ratio wm/we still increases, but onlybecause the optimal amount wm decreases less than we.

CASE 2: If α is negative, then we < 0 and

�θ > 0 �⇒ �

(wm

we

)< 0� with �we > 0 and �wm < 0�

Here, an increase in ambiguity aversion determines a lower ratio wm/we, ahigher optimal we, and a lower optimal wm (it is easy to check that α< 0 impliesβ > 0). Since we is negative, a higher optimal we corresponds again to lowerexposure to ambiguity.

In sum, depending on the values of the technical risk measures α and β,we have different effects of variations in θ on the composition of the optimalportfolio. But, in any case, our DM:

• goes long on re when α is positive and short otherwise (Proposition 9);• reduces exposure to re as ambiguity increases (Proposition 8).

For example, in an international portfolio interpretation of our tripartite anal-ysis, this means that higher ambiguity results in higher home bias.

6.3.2. Changes in λ

We now study the effects of changes in risk attitudes on the agent’s assetsholdings.

PROPOSITION 10: Suppose the Omnibus Condition holds. Then,

sgn∂

∂λ

(wm

we

)= sgn

∂we

∂λ= − sgnαP(rm� re)�(34)

Variations in the ratio wm/we due to changes in λ thus have opposite signrelative to changes in θ. That is, changes in risk attitudes vary the portfolioproportional composition in the opposite direction to changes in ambiguityattitudes. This confirms the numerical findings of KMM (2005, p. 1878), in thegeneral theoretical setting of this paper.

Moreover, variations in wm/we and we share the same sign, which again isdetermined by αP(rm� re). As to ∂wm/∂λ, we have

∂wm

∂λ= we ∂(wm/we)

∂λ+ wm

we

∂we

∂λ�(35)

If wm and we are both positive, then variations in wm have the same sign asthose in wm/we and we. If wm and we are not both positive, then the relationsamong variations in wm/we and we and variations in wm are more complicated,but can be determined through (35).

MEAN-VARIANCE PORTFOLIO ANALYSIS 1097

7. CONCLUSIONS

In this paper, we study how the classic Arrow–Pratt approximation of the cer-tainty equivalent is altered by model uncertainty. Under the smooth ambiguitymodel of Klibanoff, Marinacci, and Mukerji (2005), we find that the adjustedapproximation contains an additional ambiguity premium that depends bothon the degree of ambiguity aversion displayed by the DM and on the incidenceof model uncertainty on the expectation of the prospect he is evaluating.

Then, we introduce robust mean-variance preferences, which are the coun-terpart for the smooth ambiguity model of standard mean-variance prefer-ences for the expected utility model. We illustrate their scope by studying astatic portfolio problem under model uncertainty. In the special case of a risk-free asset, a purely risky one, and an ambiguous one, the implied comparativestatics of ambiguity aversion carries two noteworthy consequences: (i) portfo-lio rebalancing in response to ambiguity depends solely on the return alphagenerated by the ambiguous asset in excess of the purely risky asset after cor-recting for beta (e.g., the DM takes a long/short position on the ambiguousasset if and only if alpha is positive/negative), and (ii) an increase in ambiguityaversion decreases the optimal exposure in the ambiguous asset.

Finally, we note that the analytical tractability of the enhanced approxima-tion renders our model particularly fit for the study of puzzling investment be-haviors, including the home bias puzzle, the asset allocation puzzle, the equitypremium puzzle, and the employer-stock ownership puzzle. A natural direc-tion of development of this research project is therefore the derivation of arobust CAPM and its calibration.

APPENDIX: PROOFS AND RELATED ANALYSIS

To prove the quadratic approximation (15), we need the following version ofstandard results on differentiation under the integral sign.

LEMMA 11: Let O be an open subset of RN , (Ω�F) be a measurable space,

and g :O×Ω→ R be a function with the following properties:(a) for each x ∈O, ω �→ g(x�ω) is F -measurable;(b) for each ω ∈Ω, x �→ g(x�ω) is twice continuously differentiable on O;(c) the functions g, ∂jg, and ∂jkg are bounded on O × Ω for all j�k ∈

{1�2� � � � �N}.Then,(i) for each probability measure π on F , the function defined on O byG(x)=∫g(x�ω)dπ(ω) is twice continuously differentiable;(ii) the functions ω �→ ∂jg(x�ω) and ω �→ ∂jkg(x�ω) are measurable for all

x ∈O, with

∂jG(x)=∫∂jg(x�ω)dπ(ω)�(36a)

1098 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

∂jkG(x)=∫∂jkg(x�ω)dπ(ω)�(36b)

for all x ∈O and j�k ∈ {1�2� � � � �N}.

A.1. Quadratic Approximation

Here we prove Proposition 3. We assume throughout the section that μ isa Borel probability measure with bounded support on Δ and the functionsu : I → R and v : I → R are twice continuously differentiable, with u′� v′ > 0.We start with a simple lemma.

LEMMA 12: Let φ= v ◦ u−1 :u(I)→ v(I) and ψ= v−1 :v(I)→ I. The func-tions φ and ψ are twice continuously differentiable on u(int I) and v(int I), re-spectively. In particular, there exist ε > 0 such that [w − ε�w + ε] ⊆ int I andM > 1 such that the absolute values of u, v, φ, and ψ—as well as their firstand second derivatives—are bounded by M on [w − ε�w + ε], [w − ε�w + ε],u([w− ε�w+ ε]), and v([w− ε�w+ ε]), respectively. Finally, for all x ∈ int I:

φ′(u(x)) = v′(x)u′(x)

� φ′′(u(x)) = v′′(x)u′(x)2

− v′(x)u′′(x)u′(x)3

�

ψ′(φ(u(x)

)) = 1v′(x)

� ψ′′(φ(u(x)

)) = − v′′(x)v′(x)3

�

If h ∈L∞(Ω�F�P)N and x ∈ RN , set x ·h = ∑N

i=1 xihi ∈L∞. Denote by |·| theEuclidean norm of R

N . The next result yields Proposition 3 as a corollary.

PROPOSITION 13: Let μ be a Borel probability measure with bounded sup-port on Δ and let u�v : I → R be twice continuously differentiable, with u′� v′ > 0.Then, for each h ∈L∞(Ω�F�P)N and all x ∈ R

N such that w+ x · h ∈L∞(I),

C(w+ x · h)= w+EQ(x · h)− 12λu(w)σ

2Q(x · h)(37)

− 12(λv(w)− λu(w)

)σ2μ

(E(x · h)

) + o(|x|2)

as x → 0.

PROOF: Let h = (h1� � � � �hN) and (w.l.o.g.) assume that all the hi’s arebounded.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1099

Clearly, ‖x · h‖sup ≤ ∑N

i=1 |xi|‖hi‖sup; therefore, there exists δ > 0 such that‖x · h‖sup < ε (ε is the one obtained in Lemma 12) for all x ∈ (−δ�δ)N .23 Inparticular, for all x ∈ (−δ�δ)N and all ω ∈Ω,

w− ε <w− ‖x · h‖sup ≤w+ x · h(ω)≤w+ ‖x · h‖sup <w+ ε�that is, w+ x · h(ω) ∈ (w− ε�w+ ε), and so w+ x · h ∈L∞([w− ε�w+ ε])⊆L∞(I). Set O = (−δ�δ)N .

Define

g :O×Ω→ R�

(x�ω) �→ u(w+ x · h(ω)

)�

Next, we show that g satisfies assumptions (a), (b), (c) of Lemma 11:(a) For each x ∈ O, ω �→ g(x�ω) is F -measurable; in fact, ω �→ w + x ·

h(ω) ∈ (w − ε�w + ε) is measurable and u : (w − ε�w + ε)→ R is continu-ous.

(b) For each ω ∈Ω, x �→ g(x�ω) is twice continuously differentiable on O;in fact, given ω ∈Ω, for all x ∈O and all j�k ∈ {1�2� � � � �N},

∂jg(x�ω)= u′(w+ x · h(ω))hj(ω) and

∂jkg(x�ω)= u′′(w+ x · h(ω))hj(ω)hk(ω)�

and the latter equation defines (for fixed ω�j�k) a continuous function on O.(c) The functions g, ∂jg, and ∂jkg are bounded on O × Ω for all j�k ∈

{1�2� � � � �N}; in fact, given j�k ∈ {1�2� � � � �N}, for all (x�ω) ∈O×Ω (choosingM like in Lemma 12),∣∣g(x�ω)∣∣ = ∣∣u(w+ x · h(ω)

)∣∣ ≤M�∣∣∂jg(x�ω)∣∣ = ∣∣u′(w+ x · h(ω))∣∣∣∣hj(ω)∣∣ ≤M‖hj‖sup�∣∣∂jkg(x�ω)∣∣ = ∣∣u′′(w+ x · h(ω))∣∣∣∣hj(ω)∣∣∣∣hk(ω)∣∣ ≤M‖hj‖sup‖hk‖sup�

and indeed a uniform bound K for the supnorms on O ×Ω of all these func-tions can be chosen.

Then (by Lemma 11), for each q ∈ Δ, the function defined on O by

G(x� q)=∫g(x�ω)dQ(ω)

(=

∫Ω

u(w+ x · h)qdP)

(38)

23Take, for example, δ= ε(∑Ni=1 ‖hi‖sup + 1)−1. Then,

‖x · h‖sup ≤N∑i=1

|xi|‖hi‖sup ≤ δN∑i=1

‖hi‖sup = ε(

N∑i=1

‖hi‖sup

)/(N∑i=1

‖hi‖sup + 1

)< ε�

1100 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

is twice continuously differentiable, the functions ω �→ ∂jg(x�ω) and ω �→∂jkg(x�ω) are measurable for all x ∈O, and

∂jG(x� q)=∫∂jg(x�ω)dQ(ω)

(=

∫Ω

u′(w+ x · h)hjqdP)�

∂jkG(x� q)=∫∂jkg(x�ω)dQ(ω)

(=

∫Ω

u′′(w+ x · h)hjhkqdP)�

for all x ∈O and j�k ∈ {1�2� � � � �N}.Notice that, by point (c) above, for all j�k ∈ {1�2� ����N} and all (x� q) ∈

O×Δ, ∣∣G(x� q)∣∣ ≤K� ∣∣∂jG(x� q)∣∣ ≤K� and∣∣∂jkG(x� q)∣∣ ≤K�

and that, by definition, G(x� q) ∈ u([w− ε�w+ ε]), where φ is twice continu-ously differentiable.

Set f =φ ◦G. Next, we show that the function f :O×Δ→ R, with (x� q) �→φ(G(x� q)), satisfies assumptions (a), (b), and (c) of Lemma 11:

(a) For each x ∈ O, q �→ f (x� q) is Borel measurable; in fact, given x ∈ O,the function f (x� ·)=φ(G(x� ·))=φ(〈u(w+ x · h)� ·〉), being a composition ofcontinuous functions, is continuous.24

(b) For each q ∈ Δ, x �→ f (x� q) is twice continuously differentiable on O;this follows from the fact that it is a composition of twice continuously dif-ferentiable functions, specifically, given q ∈ Δ, for all x ∈ O and all j�k ∈{1�2� � � � �N},

∂jf (x� q)=φ′(G(x� q))∂jG(x� q)�∂jkf (x� q)=φ′′(G(x� q))∂kG(x� q)∂jG(x� q)

+φ′(G(x� q))∂jkG(x� q)�and the latter equation defines (for fixed q� j�k) a continuous function on O.

(c) The functions f , ∂jf , and ∂jkf are bounded on O × Δ for all j�k ∈{1�2� � � � �N}; in fact, given j�k ∈ {1�2� � � � �N}, for all (x� q) ∈O×Δ (choosingM like in Lemma 12 and K as above),∣∣f (x� q)∣∣ = ∣∣φ(

G(x� q))∣∣ ≤M�∣∣∂jf (x� q)∣∣ = ∣∣φ′(G(x� q))∣∣∣∣∂jG(x� q)∣∣ ≤MK�∣∣∂jkf (x� q)∣∣ ≤ ∣∣φ′′(G(x� q))∣∣∣∣∂kG(x� q)∣∣∣∣∂jG(x� q)∣∣

+ ∣∣φ′(G(x� q))∣∣∣∣∂jkG(x� q)∣∣ ≤MK2 +MK�

24The duality pairing EP(XY) in L2 is denoted, as usual, by 〈X�Y 〉 for all X�Y ∈L2.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1101

and the latter majorization holds term by term.By Lemma 11, the function defined on O by

F(x)=∫f (x� q)dμ(q)

(=

∫Δ

φ

(∫Ω

u(w+ x · h)qdP)dμ(q)

)(39)

is twice continuously differentiable, the functions q �→ ∂jf (x� q) and q �→∂jkf (x� q) are measurable for all x ∈O, and

∂jF(x)=∫∂jf (x� q)dμ(q) and ∂jkF(x)=

∫∂jkf (x� q)dμ(q)�

for all x ∈O and j�k ∈ {1�2� � � � �N}.Finally, for all x ∈ O and all q ∈ Δ, G(x� q) ∈ u([w − ε�w + ε]) implies

f (x� q) = φ(G(x� q)) ∈ v(u−1(u([w − ε�w + ε]))) = v([w − ε�w + ε]) andF(x) ∈ v([w− ε�w+ ε]). Thus

c(x)=ψ ◦ F(x) ∀x ∈Ois well defined and twice continuously differentiable on O = (−δ�δ)N . Its sec-ond order McLaurin expansion is

c(x)= c(0)+ ∇c(0)x + 12

xᵀ∇2c(0)x + o(|x|2)�(40)

Next, we explicitly compute it using repeatedly the relations obtained aboveas well as those provided by Lemma 12. For all x ∈O,

∂jc(x)=ψ′(F(x))∂jF(x) and

∂jkc(x)=ψ′′(F(x))∂kF(x) ∂jF(x)+ψ′(F(x))∂jkF(x);in particular, for x = 0,

∂jc(0)=ψ′(F(0))∂jF(0) and

∂jkc(0)=ψ′′(F(0))∂kF(0) ∂jF(0)+ψ′(F(0))∂jkF(0)�but F(0)=φ(u(w)) for all j�k ∈ {1�2� � � � �N}:

∂jF(0)=∫Δ

∂jf (0� q)dμ(q)

=∫Δ

φ′(G(0� q))∂jG(0� q)dμ(q)=

∫Δ

φ′(u(w))(∫Ω

u′(w)hjqdP)dμ(q)

1102 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

= φ′(u(w))u′(w)∫Δ

(∫Ω

hjqdP

)dμ(q)

= v′(w)∫Δ

(∫Ω

hjqdP

)dμ(q)= v′(w)EQ(hj)

and

∂jkF(0)=∫Δ

∂jkf (0� q)dμ(q)

=∫Δ

φ′′(G(0� q))∂kG(0� q)∂jG(0� q)+φ′(G(0� q))∂jkG(0� q)dμ(q)

=∫Δ

φ′′(G(0� q))∂kG(0� q)∂jG(0� q)dμ(q)+

∫Δ

φ′(G(0� q))∂jkG(0� q)dμ(q)�where the last equality is justified by the fact that both summands are continu-ous and bounded in q.25 Now∫

Δ

φ′′(G(0� q))∂kG(0� q)∂jG(0� q)dμ(q)=

∫Δ

φ′′(u(w))(∫Ω

u′(w)hkqdP)(∫

Ω

u′(w)hjqdP)dμ(q)

=φ′′(u(w))u′(w)2

∫Δ

〈hk�q〉〈hj�q〉dμ(q)

=(v′′(w)− v′(w)

u′′(w)u′(w)

)Eμ

(〈hk� ·〉〈hj� ·〉)and ∫

Δ

φ′(G(0� q))∂jkG(0� q)dμ(q)=

∫Δ

φ′(u(w))(∫Ω

u′′(w)hjhkqdP)dμ(q)

= v′(w)u′(w)

u′′(w)EQ(hjhk)�

25Boundedness was already observed. Continuity in q descends from G(0� q) = u(w),∂iG(0� q)= 〈u′(w)hi� q〉 for i= j�k, and ∂jkG(0� q)= 〈u′′(w)hjhk�q〉.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1103

Finally,

c(0)=w(41)

for all j ∈ {1�2� � � � �N},∂jc(0)= ψ′(F(0))∂jF(0)

= ψ′(φ(u(w)

))v′(w)EQ(hj)= 1

v′(w)v′(w)EQ(hj)=EQ(hj)�

so that

∇c(0)x =EQ(x · h) ∀x ∈O(42)

and, for all j�k ∈ {1�2� � � � �N},∂jkc(0)= ψ′′(F(0))∂kF(0) ∂jF(0)+ψ′(F(0))∂jkF(0)

= ψ′′(φ(u(w)

))v′(w)2EQ(hk)EQ(hj)

+ψ′(φ(u(w)

))((v′′(w)− v′(w)

u′′(w)u′(w)

)Eμ

(〈hk� ·〉〈hj� ·〉)+ v′(w)u′(w)

u′′(w)EQ(hjhk))

= −v′′(w)v′(w)

EQ(hk)EQ(hj)

+(v′′(w)v′(w)

− u′′(w)u′(w)

)Eμ

(〈hk� ·〉〈hj� ·〉) + u′′(w)u′(w)

EQ(hjhk)

= λv(w)EQ(hj)EQ(hk)

+ (λu(w)− λv(w)

)Eμ

(〈hj� ·〉〈hk� ·〉) − λu(w)EQ(hjhk)= −[

λu(w)σQ(hj�hk)+ (λv(w)− λu(w)

)σμ

(〈hj� ·〉� 〈hk� ·〉)]�Denoting by ΣQ and Σμ the variance–covariance matrixes [σQ(hj�hk)]Nj�k=1 and[σμ(〈hj� ·〉� 〈hk� ·〉)]Nj�k=1,

∇2c(0)= −[λu(w)ΣQ + (

λv(w)− λu(w))Σμ

]and

12

xᵀ∇2c(0)x = −12λu(w)σ

2Q(x · h)(43)

− 12(λv(w)− λu(w)

)σ2μ

(E(x · h)

) ∀x ∈O�

1104 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

This concludes the proof, since replacement of (41), (42), and (43) into (40)delivers

c(x)= w+EQ(x · h)− 12λu(w)σ

2Q(x · h)

− 12(λv(w)− λu(w)

)σ2μ

(E(x · h)

) + o(|x|2)

as x → 0, and c(x)= v−1(F(x))= C(w+ x · h) for all x ∈O. If x ∈ RN \O and

w+ x · h ∈L∞(I), just set

o(|x|2

) = C(w+ x · h)−[w+EQ(x · h)− 1

2λu(w)σ

2Q(x · h)

− 12(λv(w)− λu(w)

)σ2μ

(E(x · h)

)];

the property of vanishing faster than |x|2 as x → 0 has no bite there. Q.E.D.

PROOF OF PROPOSITION 3: We first consider the case in which F is fi-nite. Let A = {A1� � � � �AN} be the family of atoms of F that are assigneda positive probability by P . Then {1A1� � � � �1AN } is a base for L2 and, settingh = (1A1� � � � �1AN ), the map

γ : x �→N∑i=1

xi1Ai = x · h

is a norm isomorphism between RN and L2.26 In particular, choosing δ > 0 as

in the proof of Proposition 13, for all x ∈ γ((−δ�δ)N)= {x · h : x ∈ (−δ�δ)N},

C(w+ x)= w+EQ(x)− 12λu(w)σ

2Q(x)(44)

− 12(λv(w)− λu(w)

)σ2μ

(E(x)

) + o(|x|2)

as x → 0 in RN . Set

R2(x)= C(w+ x)−[w+EQ(x)− 1

2λu(w)σ

2Q(x)

− 12(λv(w)− λu(w)

)σ2μ

(E(x)

)]26Finite dimensionality guarantees that Δ is bounded; therefore—in this case—the assumption

that the support of μ is bounded is automatically satisfied.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1105

for all x ∈ γ((−δ�δ)N), p= mini=1�����N P(Ai), and p= maxi=1�����N P(Ai); then

p

N∑i=1

x2i ≤ ‖x‖2 ≤ p

N∑i=1

x2i �

Now, if xn is a (nonzero) vanishing sequence in γ((−δ�δ)N),

|R2(xn)|

p

N∑i=1

(xn)2i

≤ |R2(xn)|‖xn‖2

≤ |R2(xn)|

p

N∑i=1

(xn)2i

�

and by (44), the three sequences above vanish as n → ∞. That is, R2(x) =o(‖x‖2), since γ((−δ�δ)N) is a neighborhood of 0 in L2.

In the general case, let h ∈ L∞(Ω�F�P). By Proposition 13, for all t ∈ R

such that w+ th ∈L∞(I),

C(w+ th)= w+EQ(th)− 12λu(w)σ

2Q(th)

− 12(λv(w)− λu(w)

)σ2μ

(E(th)

) + o(t2)as t → 0. That is, setting, for all t ∈ R such that w+ th ∈L∞(I),

R2(th)= C(w+ th)−[w+EQ(th)− 1

2λu(w)σ

2Q(th)(45)

− 12(λv(w)− λu(w)

)σ2μ

(E(th)

)]�

it results that limt→0R2(th)/t2 = 0. Moreover, the assumption w+ h ∈ L∞(I)

guarantees that we can consider t = 1 in (45), that is,

C(w+ h)= w+EQ(h)− 12λu(w)σ

2Q(h)

− 12(λv(w)− λu(w)

)σ2μ

(E(h)

) +R2(h)�

as wanted. Q.E.D.

1106 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

A.2. Approximately Unambiguous Prospects

PROOF OF PROPOSITION 5: Property (iii) trivially implies (i), which in turnimplies (ii). To complete the proof, we show that (ii) implies (iii). First noticethat, for all h ∈L2 and all t ∈ R,

σ2μ

(E(th)

) = σ2μ

(tE(h)

) = t2σ2μ

(E(h)

)�(46)

Therefore, σ2μ(E(h))=R2(h) implies

0 = limt→0

σ2μ(E(th))

t2= σ2

μ

(E(h)

)�

It remains to show that σ2μ(E(h))= 0 implies that 〈h� ·〉 is constant on suppμ.

If h ∈L2 and σ2μ(E(h))= 0, then

〈h�q〉 =Eμ(〈h�q〉) =EQ(h)

for μ-almost all q ∈ Δ. If, per contra, there exists q∗ ∈ suppμ such that〈h�q∗〉 = EQ(h), then the continuity of 〈h� ·〉 on Δ implies the existence of anopen subsetG of Δ such that 〈h�q〉 = EQ(h) for all q ∈G. ButG∩ suppμ = ∅,and so μ(G) > 0, a contradiction. We conclude that 〈h�q〉 = EQ(h) for allq ∈ suppμ. Q.E.D.

PROOF OF PROPOSITION 6: Property (i) trivially implies (ii) and (iii) implies(i). Next, we show that (ii) implies (iii). For all h ∈ B, set

F(h)= C(w+ h)−[w+EQ(h)− 1

2λu(w)σ

2Q(h)

]and

G(h)= C(w+ h)−[w+EQ(h)− 1

2λu(w)σ

2Q(h)

− 12(λv(w)− λu(w)

)σ2μ

(E(h)

)];

by (19) and Proposition 3, limt→0 F(th)/t2 = limt→0G(th)/t

2 = 0. Therefore,for all h ∈ B, setting k= 2(λv(w)− λu(w))−1,

σ2μ

(E(h)

) = limt→0

σ2μ(E(th))

t2= lim

t→0kG(th)− F(th)

t2= 0�

MEAN-VARIANCE PORTFOLIO ANALYSIS 1107

Since B is absorbing inL∞, for allA ∈ F , there is ε= εA > 0 such that ε1A ∈ B;thus

σ2μ

(E(1A)

) = σ2μ(E(ε1A))

ε2= 0

and 1A is approximately unambiguous for all A ∈ F . Then, by Proposition 5,for all A ∈ F ,

Q(A)=EQ(1A)=EQ′(1A)=Q′(A) ∀Q�Q′ ∈ suppμ�

that is, Q=Q′. Q.E.D.

A.3. Orthogonal Decomposition

By Proposition 5, the collection M of all approximately unambiguousprospects is easily seen to be a closed linear subspace of L2. Clearly, M con-tains all risk-free (constant) prospects, and its orthogonal complementM⊥ is aclosed subspace of {h ∈ L2 :EP(h) = 0}. The Hilbert Decomposition theoremthen implies the following decomposition of each prospect.

PROPOSITION 14: For each prospect h ∈L2, there exist unique hc ∈ R, hg ∈Mwith EP(hg)= 0, and ha ∈M⊥ such that

h= hc + hg + ha�(47)

Moreover,

σ2P(h)= σ2

P

(hg

) + σ2P

(ha

)(48)

and

σ2μ

(E(h)

) = σ2μ

(E

(ha

))�(49)

In particular, h is approximately unambiguous if and only if ha = 0, and it is risk-free if and only if hg = ha = 0.

PROOF: Let h ∈ L2. By definition of M⊥ and by the Hilbert Decompositiontheorem, decomposition (47) and its uniqueness are easily checked. In particu-lar, EP(h)= hc . Moreover, the maps h �→ hg ∈M and h �→ ha ∈M⊥ are linearand continuous operators.

Since hg and ha are orthogonal and have zero mean, then

σ2P(h)= ∥∥h−EP(h)

∥∥2 = ∥∥EP(h)+ hg + ha −EP(h)∥∥2 = ∥∥hg + ha∥∥2

= ∥∥hg∥∥2 + ∥∥ha∥∥2 = σ2P

(hg

) + σ2P

(ha

)�

1108 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

which proves (48).Finally, observe that 〈h� ·〉 = 〈hc +hg� ·〉 + 〈ha� ·〉 and hc +hg ∈M imply that

〈hc + hg� ·〉 is μ-almost surely constant; thus σ2μ(〈h� ·〉) = σ2

μ(〈ha� ·〉). The restis trivial. Q.E.D.

In view of decomposition (47), the constant hc—which is equal to EP(h)—can be interpreted as the risk-free component of h. Indeed, h= hc if and onlyif σ2

P(h)= 0. The next component, hg, can be viewed as a fair gamble becausehg ∈M and EP(hg) = 0. The sum hc + hg of the first two components is ap-proximately unambiguous. In contrast, (48) and (49) show that the “residual”component ha reflects both risk and ambiguity in pure variability terms (net ofany level effect factored out by the constant hc).

In our portfolio exercise of Section 6.3.1, this implies the orthogonal decom-positions rm = EP(rm) + rgm and re = EP(re) + rge + rae of the purely risky andthe ambiguous assets. In particular, EP(re) + rge = re − rae is the purely riskycomponent of the ambiguous asset. Simple algebra shows that

βP(rm� re)= σP(rm� re − rae )σ2P(rm)

�(50)

which confirms the interpretation of β as a measure of the risk sensitivity of rerelative to rm.

A.4. Portfolio

DERIVATION OF (24): Setting

m = [EP(r1 − rf )� � � � �EP(rn − rf )

]ᵀ� ΣP = [

σP(ri� rj)]ni�j=1

�

Σμ = [σμ

(E(ri)�E(rj)

)]ni�j=1

� Ξ = λΣP + θΣμ�

(23) becomes

maxw∈Rn

{rf + w · m − λ

2wᵀΣPw − θ

2wᵀΣμw

}�

which is equivalent to

maxw∈Rn

(w · m − 1

2wᵀΞw

)�

so that the optimal solution w satisfies Ξw = m. Q.E.D.

MEAN-VARIANCE PORTFOLIO ANALYSIS 1109

PROOF OF LEMMA 7: Since ∂D/∂σ2μ = θ and ∂D/∂θ = σ2

μ, simple algebrashows that

∂(wm/we)

∂σ2μ

= θ B

CA−HB and∂(wm/we)

∂θ= σ2

μ

B

CA−HB�(51)

and this implies the first equality in (27). Analogously,

∂wm

∂σ2μ

= θ(AC −BH)H(CD−H2)2

and∂we

∂σ2μ

= −θ(CA−HB)C(CD−H2)2

�(52)

∂wm

∂θ= σ2

μ

(AC −BH)H(CD−H2)2

and∂we

∂θ= −σ2

μ

(CA−HB)C(CD−H2)2

�(53)

which imply the other equalities in (27). Q.E.D.

PROOF OF PROPOSITION 8: By definition,

βP(rm� re)= H

Cand(54)

αP(rm� re)=EP(re)− rf −βP(rm� re)(EP(rm)− rf

) =A− H

CB�

while, by (51),

∂(wm/we)

∂θ= σ2

μ

B

CA−HB ;

thus

sgn∂

∂θ

(wm

we

)= sgn(CA−HB)= sgn

(CA−HB

C

)= sgnαP(rm� re)�

that is, (30) holds.Moreover, by (53) and (54),

sgn∂we

∂θ= sgn(HB−CA)= − sgnαP(rm� re)�(55)

that is, the first part of (31) holds. Moreover, (53) and (54) again deliver

∂wm

∂θ= −

(−σ2

μ

(CA−HB)(CD−H2)2

C

)H

C= −∂we

∂θβP(rm� re)�

which, together with (55), delivers the second part of (31). Q.E.D.

1110 F. MACCHERONI, M. MARINACCI, AND D. RUFFINO

PROOF OF PROPOSITION 9: By (54),

αP(rm� re)=A− H

CB�

but

we = CA−HBCD−H2

�

which, together with CD−H2 > 0 and C > 0, delivers (32). Q.E.D.

PROOF OF PROPOSITION 10: Direct computation delivers

λ∂(wm/we)

∂λ= −θ∂(wm/we)

∂θ�

which, together with Proposition 8, determines the sign of ∂(wm/we)/∂λ.Moreover,

∂we

∂λ

∂(wm/we)

∂λ= − θσ2

μB

λ2(CD−H2)2

(H2 −C(

D− θσ2μ

));hence

∂we

∂λ

∂(wm/we)

∂λ> 0 ⇔ H2 −C(

D− θσ2μ

)< 0

⇔ σP(rm� re)2 <σ2

P(rm)σ2P(re)�

By the Cauchy–Schwarz inequality, the above relation fails if and only if

re −EP(re)= k(rm −EP(rm)

)for some k ∈ R, but this would imply that re is approximately unambiguous,which is absurd. Q.E.D.

A.5. Proofs of Section 2

PROOF OF LEMMA 1: Consider the duality inclusion ι : suppμ→ (L2)∗ givenby q �→ 〈·� q〉. For all X ∈ L2, the composition X ◦ ι : suppμ → R given byq �→ 〈X�ι(q)〉 = 〈X�q〉 is (norm) continuous and hence Borel measurable onsuppμ, that is, ι is weak* measurable (see, e.g., Aliprantis and Border (2006,Chapter 11.9)). The range of ι is norm bounded since suppμ is norm boundedand ι is an isometry. Therefore, ι is Gelfand integrable over suppμ (Aliprantis

MEAN-VARIANCE PORTFOLIO ANALYSIS 1111

and Border (2006, Corollary 11.53)). In particular, there exists a unique q ∈L2

such that

〈X� q〉 =∫

suppμ

⟨X�ι(q)

⟩dμ(q)� ∀X ∈L2�(56)

By (56), it readily follows that q ∈ Δ. Q.E.D.

PROOF OF LEMMA 2: Let f ∈ L∞(I) and set a= essinf f and b= esssup f .There exists A ∈ F with P(A) = 1 such that f (A) ⊆ [a�b] ⊆ I. Since u is in-creasing and continuous, u(f (ω)) ∈ [u(a)�u(b)] ⊆ u(I) for all ω ∈A. More-over, u ◦ f|A :A → [u(a)�u(b)] is measurable since f|A is measurable and uis continuous. Therefore, u(f ) is defined P-almost surely on Ω, measurable,and u(a)≤ u(f )≤ u(b) P-almost surely. It follows that 〈u(f )� ·〉 :Δ→ R, withq �→ ∫

Ωu(f )qdP , is norm continuous, affine, with range in [u(a)�u(b)] ⊆ u(I).

Therefore, φ ◦ 〈u(f )� ·〉 :Δ → R, with q �→ φ(∫Ωu(f )qdP), is well defined,

norm continuous, with range in [φ(u(a))�φ(u(b))] ⊆ φ(u(I)). Therefore,V (f )= ∫

Δφ ◦ 〈u(f )� ·〉dμ ∈ [φ(u(a))�φ(u(b))] ⊆φ(u(I)) is well defined.

The first part of this proof yields V (L∞(I)) ⊆ φ(u(I)). Conversely, if z =φ(u(x)) for some x ∈ I, then x1Ω ∈ L∞(I) and V (L∞(I)) � V (x1Ω) = ∫

Δφ ◦

〈u(x)� ·〉dμ=φ(u(x))= z. Q.E.D.

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Dept. of Decision Sciences and IGIER, Università Bocconi, Via Sarfatti 25,20136, Milano, Italy; fabio.maccheroni@unibocconi.it,

MEAN-VARIANCE PORTFOLIO ANALYSIS 1113

Dept. of Decision Sciences and IGIER, Università Bocconi, Via Sarfatti 25,20136, Milano, Italy; massimo.marinacci@unibocconi.it,

andDept. of Finance, University of Minnesota, 321 19th Avenue South, Minneapo-

lis, MN 55455, U.S.A.; druffino@umn.edu.

Manuscript received November, 2010; final revision received September, 2012.