This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. and Quig- s), restricting tility paradigm, ected u ear utilitie utility theory Dhaene & Van Wouwe (1999) and hoquet expected ty: the classical exp ure is a mapping from a class of random literature are discussed. sion under uncertainty, axiomatic characterization, rbitrage-free pricing are two sides of the hich most classical risk measures appear to be a contingent claim equals the price of the t risk measures studied recently in the financial number of martingale measures and each of cept (i.e., correspond to lin 10.1524/stnd.2006.24.1.1 tility theory, C DOI utility con e study initiated in Denuit, expected u imary: 91B06, 91B30; Secondary: 62P05 ecision under uncertain rnatives proposed in the ¨ unchen 2006 ture. Mathematically, a risk meas rature disregard the Risk measures have been studied for several decades in the actuarial literature, where This paper complements th R. Oldenbourg Verlag, M c their applicability. Some alte gin’s rank-dependent utility theory. Buildingbroad on classes the of actuarial risk equivalent measures utility are pricing generated, principle, of w particular cases. This approachmathematics lite shows that mos Yaari’s dual approach, maximin variables to thedecision-maker. real line. Economically, a risk measureconsiders should several theories capture for the d preferences of the equivalent utility, risk aversion Key words and phrases: risk measures, theories for deci hedging (in this case:there replicating) may portfolio. When be the noposition. market Moreover, replicating there is portfolio exists incomplete, an and however, them infinite the generates a hedging different strategy priceprices. for could Now, the involve since contingent a claim. the Thishedging risky hedging leads portfolio of depends to explicitly a the on set contingent the of agent’s claim possible attitude involves towards risk, risk. the price of the 1 Introduction and1.1 motivation The needOne for of risk the measures majorIn needs complete for financial risk markets,same measures hedging is problem: and related the a to arbitrage-free pricing price in of incomplete markets. Received: December 15, 2005; Accepted: March 16, 2006 Summary: Risk measurement with equivalent utility principles Michel Denuit, Jan Dhaene, MarcRoger Goovaerts, Laeven Rob Kaas, they appeared under thethat guise risk of measures premiummathematics should calculation satisfy litera principles. have Risk recently measures received and properties considerable attention in the financial Statistics & Decisions 24, 1–25 (2006) / AMS 2000 subject classification: Pr This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ection also amounts to is covered) and 0 (no X ¨ ollmer & Schied (2004). Let us now te if, and only if, their characterizing n the financial mathematics literature of ticular risk measure. In both cases, a set can be given that is regarded as reasonable easure directly by imposing axioms (e.g., ively traded in the (financial) market and l mathematics literature disregard utilities ned in both ways, which allows for a better Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven ial) mathematics are just phrased differently utility theory, Yaari’s dual utility theory or g the risks at hand) to the real numbers. ance. Note that this sel e than 30 years, under the guise of premium s. This will also enlighten the fact that most ¨ uhlmann (1970) and Goovaerts, De Vijlder & Haezen- (corresponding to the case where X ]− X [ Both approaches for measuring risk will be considered in this paper. It will be seen First, a set of axioms for risk measurement Second, a paradigm for decision under uncertainty can be selected, for instance Note that the problem of market incompleteness is particularly relevant in insurance. Recent years have witnessed the emergence i (considering only linear utility functions).insensitivity This to restriction liquidity explains risk, some for instance. problems, like understanding of their intrinsicrisk propertie measures studied recently in the financia opting for a setuncertain of prospects; axioms, but see, thisthen e.g., set obtained Wakker explains by (2004) how an decision-makersthe for choose equivalence cash-flow more between principle: details. thecoverage). The decision-maker is risk indifferent measure between is that many classical risk measures can be obtai briefly explain these two approaches. axioms are. Axiomatizationscriticize can it. then be used to justifythe a Von Neumann–Morgenstern risk expected other measure generalized but utility principles, also for inst to in the situation under consideration. Then,the the agreed form of axioms. the Risk risk measure measures is are deduced from appropria with respect to ordering andon aggregation, random variables, such or as we those canin discussed axiomatize fact a in is functional Section representing also 1.3 preferencesmeasures. below) (which a The axioms risk in (actuarial measure) and financ andfrom the use axioms in economic economics. In indifference (actuarial and argumentson financial) to mathematics random we impose obtain variables axioms (risks) risk whereastwo in approaches economics coincide we is impose explained axioms nicely on acts. by That F the donck (1984). A risk measuredefined is on defined a as measurable a space mapping (representin from a class of random variables of axioms is given, buthow the individuals axioms make either their describe howmeans decisions the that under risk we risk is can or measured either or uncertainty. axiomatize More describe a specifically, risk this m 1.2 How toThere define are a risk basically measure? two ways to define a par This is due to several reasons:the the fact stochastic that nature many ofthe insurance insurance processes fact risks (with that are jumps), securitized notunderlying insurance act traded asset. risks often have an underlying indexa rather sophisticated than theory an of riskstudied measures. in However, the risk actuarial measurescalculation have literature principles; been see, for extensively e.g., mor B 2 This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ] 3 d X Y . for = ]= X ] t ≤ X [ X > [ where Y P [ ] . P ]. Y [ R ⇔ Y . This property [ ∈ X Y ]≤ t y st ≤ ]= , > ] x  X [ X X X [ [ P ⇒ for all Y  ] ,i.e. d y Y = . ≤ st is the distribution function of ] 1 implies X  X are said to be co-monotonic if they X Y eak monotonicity) or with stochastic [ [ F ,wehavethat X Y ]= P a Y rally be divided into three classes: ratio- , Y ] = x ≤ and ] additivity for independent random variables, isk measurement can be entirely based on and where ≤ X X eak monotonicity will not imply strong mono- ] [ X aX ed to measure the riskiness of X X [ such that P [ F , P [ Y . Y
0, ] , which is clearly currency independent (changing Y } > and [ does not depend on the risk itself but only on its under- 0 and ]= min ¨ a uller (2006) for a careful examination of objectivity and X X ξ, X [ { ]≤ ]= y X [ max is objective, then both requirements have the same force. We ≤ For any For any = For any Y , ¨ auerle & M + x ξ ≤ (strong monotonicity). , implies X with st [ R The risk measure ] P  agrees with almost sure comparisons (w ∈ + ) t d contains all the information need all lying distribution, i.e. X − F Two monotonicity requirements can then be envisaged: either we impose that the risk Let us now detail some of the properties that risk measures may or may not satisfy: X 1. In the absence of co-monotonicity, w ( [ Strong monotonicity In the literature itindependency. is This often stated is that however positive a homogeneity is wrong equivalent interpretation. with Take currency as an example E dominance Weak monotonicity denotes the equality in distribution. measure = refer the reader to B monotonicity properties. Positive homogeneity tonicity. However, if Risk measurement 1.3 The axiomaticAxioms approach to to characterize risk a measurement risk measure can gene nality axioms, additivity axioms and technicalaxioms axioms. First such there are as the basicaxioms monotonicity, rationality deal which with sums are of risks.spect satisfied They to describe risk by aggregation. the Let most sensitivity us of mentionthe risk the the additivity for risk measures. co-monotonic measure random with variables, Additivity the re- subadditivityThese or additivity the requirements superadditivity. often are the mostmeasure. Finally, characteristic there for are the technical corresponding requirements, mostly risk continuityusually conditions. They necessary are for obtaining mathematical proofs andor are to typically explain difficult economically. to validate Objectivity distribution functions. This is usually thethat case in actuarial science. This condition ensures The objectivity requirement implies that r Co-monotonicity is apapers useful by notion Dhaene, in Denuit,therein. decision Recall Goovaerts, theory; that Kaas see, the & for random Vyncke variables example, (2002a,b) the and review the references satisfy For two co-monotonic random variables is sometimes called “law invariance”, and phrased as This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder.  is X used to for some ] + ) ] might be equal X ] [ b [ − . X b ( [  + ] X [ ]= is translation equivariant then . X b [ =  ] b ]+ X + [ is changed in that way), but not positively X Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven in both cases. This is of course not the case d mework of Solvency 2 for the calculation [ a ] ] whatever the dependency structure of X , Y [ ]= [ R b ∈ + + + b b . Consequently, if ] aX X [ [ ]= X and 0,  For any ¨ uppelberg (1999), subadditivity is a convenient mathematical + , so that we have translation invariance in this case. It is of ≤ ] > b [ X ] a is often specified: for example, additivity for independent risks [ Y  Y + involved in the definition of positive homogeneity cannot be used to that X , the utility. ]= a b [ ) and b b ( X + u . X from euros to dollars, say. It is rather related to the “volume” of the risk and Y [ which can be defended in the case of insurance pricing (to avoid free lunches  X b and When equality holds, we speak of additivity. In this case, the dependence struc- In Artzner, Delbaen, Eber & Heath (1999), a risk measure that satisfies the properties of Combining positive homogeneity with translation equivariance guarantees that ]= b [ The rationale behind subadditivityextra can risk”. Subadditivity be reflects summarizedAccording the to as Rootzen idea & “a Kl that merger risk does can not be reduced create by diversification. property that does notthe hold aggregation of in risks reality. isimposition The manifested of behavior by penalties. of the award a of risk diversification discounts measure and with the ture respect between to monotonicity, positive homogeneity, translation equivariance and subadditivitya is “coherent” called risk measure. As explained below, coherent risk measures coincide with or additivity forDhaene, co-monotonic Laeven, risks Vanduffel, Darkiewicz (see &on Goovaerts below). the (2004) We desirability for referments. a of detailed the the discussion subadditivity interested axiom reader when to setting solvency capital require- Subadditivity linear, that is, whatever to a function for risk measures used forThese the quantities calculation constitute of amounts solvencyand (regulatory of or premiums. money economic) Such for capital. risk safety, measures in are addition of to the form the provisions course not forbidden toof take risk translation measures equivariance in asof this an situation, solvency. axiom The but for it prescribed the cannot rules construction be in interpreted the in fra the economic concept of solvency capital (theValue-at-Risk at difference level 75 between %) the recognizes this Value-at-Risk reality. at Related of level course 99.5 is % the and requirement the and/or the no-ripoff condition) but in the definition of a risk measure Translation equivariance might becalculate provisions considered or premiums. desirable Indeed it for seemsa to a be certain reasonable amount when risk the risk measure contains appropriate risk measures 4 from dollars to euros only makes sense if also homogeneous. Hence, positive homogeneity is about multiplyingclaims all by euro amounts a of constant the that factor. the Let constant us alsoconvert mention that futurehas currencies a close are relation random with so additivity for co-monotonic risks (seeTranslation below). equivariance satisfies This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 5 ¨ uhlmann ¨ atschmer ] whatever the Y [ tion in the risk-load ) λ sk measures that may − 1 ( ] for independent random ] for co-monotonic random Y + Y [ [ ] X [ + + ] ] λ X X [ [ ≤ ] Y = ) = able leads to other ri f coherent risk measures, it is sometimes ] axioms, and no set is universally accepted. λ ] Y hedge condition, no reduc Y − + he policyholder has no interest in splitting the 1 + ( . X [ X Y + ity. It has been further developed by Kr [ X and λ [ X 1], , ¨ uller (2006) for an extension to unbounded random variables. [0 . . Y Y ∈ λ and and X X ¨ auerle & M For any dependency structure of variables variables ¨ ollmer & Schied (2002), and independently Frittelli & Rosazza Gianin (2002), intro- F duced the concept of convex risktranslation measures, which equivariance satisfy the and properties of convex monotonicity, (2005). The class ofvex coherent risk measures risk that measures satisfyrisk can the be measures positive homogeneity characterized is property. as As ancalled the the extension class the class of of of class convex thework con- of class by o weakly Deprez & coherentmeasures. Gerber risk (1985) measures. already It contains many is nice worthAdditivity results for to co-monotonic on risks mention convex risk that the Co-monotonic risks are bets on thea same hedge event against and the in other. Because this of sense the neither no- of them provides is awarded for a combined policy, resulting in co-monotonic additivity. Additivity for independent risks Additivity for independent risksculation may principle, be since a it reasonable ensures requirement that for t a premium cal- risk in independent components askingfied for by coverage premium to computation several(1985). from insurers. It top The is to also most down;dependent justi- see, general random e.g., representation Borch variables (1962) of isthe or mixed due B risk Esscher to measures principle.principle, Gerber that Despite it & the are does Goovaerts numeroushave additive (1981) not appealing for been features and satisfy of is provided in- the theGoovaerts, known by strong Kaas, Esscher as monotonicity Van Laeven requirement. Heerwaarden, &of Counterexamples risk Kaas Tang measures that & (2004) are providedthat additive Goovaerts for guarantees a (1989). independent monotonicity. random new The variables, This involvingmixed axiomatic obtained an is axiom Esscher risk characterization why measure principle is thatpremiums. a can restricted be version regarded of as the an ordinary mixture of exponential be called “coherent” with just as much reason. Convexity Risk measurement upper expectations introduced by Huber (1981) in robust statistics.satisfying Coherent the risk objectivity measures requirement have been studied by Kusuoka (2001); see(2003) also Inoue and B The terminology “coherent” isany somewhat risk flawed measure that in is thecoherency not is sense “coherent” defined is that with always it inadequate. respect It may to is a suggest worth set that to of mention that Modifying the set of axioms regarded as desir This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. norm. 1 L nce axiom, an in- both hold. X culation, premium calculation  ¨ uhlmann (1970) is obtained by Y , the closure being often meant with X and Y sures should satisfy and then determining nk-dependent expected utility (RDEU, in to be separable across mutually exclusive .  Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven l representation, one could determine the co-monotonic independe Y hich refer to directly meaningful empirical X the concept of rationality is embodied in the  X ,thatis, then Y is closed for every Y } st X and nce-reinsurance, premium cal  X  X have the same distribution function then the decision-maker is Y | If Y Y { and : is complete, transitive and reflexive. X   If The set respect to the topology of weak convergence, associated with the indifferent between Utility theories are typically presented in an axiomatic form. The axioms are intended Instead of stating a set of axioms that risk mea Fishburn (1982) and Yaari (1987) considered the following four axioms on the pref- Strong monotonicity Weak order Continuity Then, a fifth axiom isences. added Whereas that there specifies is the a particularlast rather form one large of consensus continues the about to indicator thependence be of four axiom, prefer- subject axioms which listed to requires above, debate. the preferences events. The Alternative EU theories axiomatization proposed hinges by(1989) on Quiggin replaced the (1982), this Yaari inde- condition (1987)tuitive by and and a Schmeidler appealing type condition of requiringwhen that hedging effects the are usual absent. independence axiom only holds 6 1.4 Risk measuresGoovaerts, and Kaas, economic Dhaene decision under &properties uncertainty that Tang risk (2004) measures should gaveapplications have several are (namely, insura examples determined by where the evidently realities of the the actuarial considering an insurer whoseshort). preferences The distortion are premium principle characterized proposed by by Denneberg (1990)can expected and Wang be utility (1996) (EU, obtained in viaThe indifference distortion-exponential premium arguments principle based& has on Desli been distortion (2003) similarly utility and derivedshort) (DU, Heilpern by theory. in (2003) See Tsanakas using short). also ra LuanWouwe (2001). (1999) The where present paper onlyRDEU). follows EU Denuit, Dhaene and & DU Van were consideredas (with reasonable a consistency brief requirements: excursion to axioms, with the possiblethat addition here, axioms of impose the rationalcase postulate behavior for in of the decision sets risk under ofuse aversion. uncertainty. axioms As The intuitive defining it axioms difference risk was measures, is such the the as preference transitivity, foundations typically w from top to down,Dhaene, capital Vanduffel, Tang, allocation, Goovaerts, Kaas solvency & margin Vyncke and (2004). setting provisions). Seemathematically also the corresponding functiona functional form of the riskthe measure equivalent via economic utility indifference arguments. premium For principle instance, defined by B properties of preferences, andstructural technical richness axioms of the such model as and continuity, serve which to simplify describe the the mathematicalerence analysis. relation Objectivity This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . 7 ] is B ] [ 1 , C 0 , denoted ]≤ C →[ A [ B C : f events are given ⇒ C is the set of events, B such that  ] ⊆ 1 is called coherent if there , . 0 A C . Capacities are real-valued B . , B B ∈ B →[ ∈ A ∈ B A : B , P A for all with many possible probability measures. Quiggin (1982). The application of the ] for all case the probabilities o ogy, for example, a coherent risk measure A ] ” in order to generate a safety loading. We , denoted by ler (1989) and the rank-dependent expected A \  certainty: the Choquet expected utility (CEU, [ 1989), the maximin expected utility (MEU, in P  [ M C ∈ sup P − 1andforany 1 ]= on of these sets. The capacity A ]= ]= [ A  [ C [ of probability measures C C that generalize the notion of probability distributions. Formally, we can associate a dual capacity. The dual capacity of M 0, B C exceed 1. Or the sum of the capacities of two subsets may be strictly [∅] = C . This set may be finite or infinite. Associated with  , is defined by Uncertainty is often represented with the help of capacities. Specifically, we assume In this paper, we will consider in addition to the classical EU and DU theories several In the financial mathematics literature, and the actuarial papers in the realm of financial To each capacity C Otherwise, the capacity has littledisjoint structure. subsets In may particular, the sum ofless the than capacities the of capacity two of the uni as a capacity if utility (RDEU, inactuarial short) equivalent utility theory pricing principle proposed in theseof by paradigms risk will measures generate considered most so classes far in the financial mathematics literature. Decision-making under risk andaccording decision-making to under the condition uncertainty that are in differentiated the former short) theory proposed by Gilboa & Schmeid 1.5 Risk and uncertainty will also see below thatis in not economics a terminol measure of risk but a measure of uncertainty. that the uncertainty a decision makerdenoted faces can as be described by a non-empty set of states, Risk measurement alternative paradigms for decision under un in short) theory proposed by Schmeidler ( while in the latter casemodels they have been are introduced not. which With associate regard some to kind decision-making of under probabilities uncertainty, tomathematics, events. the probability distribution of aa random risk variable does defined not on need a tounspecified). measurable space, be The and fixed majority the (a of probability risk measure actuarial isand can papers just statistics, be in left uses which the a framework randomprobability of space, variable not is probability on just defined theory a on measurable space aThe probability particular measure and may be fixed unknown, underlying but itthat is many fixed weight and functions unique. operating This on does events not can prevent instance, be the used to actuarial calculate practice the in premiums. life For insuranceremaining consists lifetime in replacing by the a distribution “more of the dangerous one taken to be a sigma-algebra of subsets of exists a non-empty set functions defined on As we will see, coherent capacities are closely related to coherent risk measures. a capacity is a normalized monotonic set function. More precisely, This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , ] A [ (1.1) C ]= . A A , and defined [ , provided the ) and a concept cv C R X  is always preferred → ] X R agree on [ . : E  φ M ] A ∈ [ ). It is interesting to note that P P Y M is given by the interval is preferred over ∈ ] sup P B X , [ ] ∈ E A A [ ¨ uhlmann (1970) introduced the equivalent P Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven sion-maker), these paradigms are replaced M ∈ inf P is preferred over  eauneuf, Cohen & Meilijson (2004). X for all concave functions = ]  ) ] ⇒ Y A [ Y and the Choquet expected utility (CEU, in short), respec- φ( sk analysis, some valuable lessons may be drawn from C sk measurement, most risk measures can be obtained in [ , cv E ]  A itself. An alternative definition of risk aversion requires a risk [ X X ]≤ C )  X φ( [ E ⇔ does not involve any vagueness and all priors ¨ ollmer & Schied (2004), the close connections between risk measurement , a natural and widely used measure of the degree of uncertainty (or imprecision, Y A C cv  Decisions under risk are usually performed in the EU or RDEU paradigms. In sit- We thus have a concept of weak risk aversion ( X averse individual to behave according to the concave order, denoted as of strong risk aversion ( strong risk aversion is definedthe with dispersive the order help have of been aChateauneuf used stochastic & to Cohen order define (2000) relation. alternative and Relations attitudes Chat like towards risk; see, e.g., uations of uncertainty (i.e.,probability in distribution situations available to whereby the there the deci does subjective utility not theory existtively. a given objective to the random prospect the event expectations exist. This order extends the concept of mean preserving increase in risk. by 1.7 Outline ofThis the paper paper aims tomental demonstrate subject that, of as financial pointed ri out by Geman (1999), in theutility principle funda- in theand context appealing of approach setting to insurance ri premiums. Considering this simple insurance. Actuaries have used indifferencedecision under arguments uncertainty based for on decades, after economic B theories for a way that enlightensincluding their intrinsic F properties. Unfortunately, apartand from economic decision some theory authors haveindependently been developed disregarded their by own financial axiomatictwo mathematicians, theories. approaches. who This paper tries to reconcile the 1.6 Aversion to riskThere and are uncertainty many notions ofkinds risk of risk aversion. Chateauneuf aversion: weak, & monotonic,this Cohen left paper, (2000) monotonic, we distinguish right restrict five monotonic ourselves andThe to first strong. one the In defines following an individual two to be standard risk averse concepts if of the sure risk gain aversion. vagueness, ambiguity) associated with an event 8 The dual capacity is useful whencapacity losses are considered instead of gains. Given a coherent The interval (1.1) captures uncertainty in the sense of Knight (1921). If This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ] 9 u | Y (2.1) − . Facing ) w · calculated [ ( . Therefore ] u X EU X [ U , has to choose ) · and acts in order ( 0 and normalizing ) u · with ( = u ] u w is covered, the expected | (w). X X ] u ) − X ( w ]= [ u u . Putting [ | . A classical reference for EU is 0 (2.2) E X ] EU w [ u as the solution of the equation U = | = ]  Y EU is then evaluated by [ and with a utility function X P u and utility function [

U d X w ) EU X w = is obtained when X ; the left-hand side of (2.1) represents the U ubjective utility theory of Savage (1954) for ] (  X u ]− u X

[ X ]≥ X  [ u | r assuming the random financial loss  is fair in terms of utility: the right-hand side of (2.1) X ]− ]= [ ] EU u X then he compares | X [ U , EU [ X Y [ U + if EU w and U  . Y is equal to the utility without X P EU X U , we get the so-called “equivalent utility principle”: ): risk aversion and diminishing marginal utility of wealth are syn- u u , he sets its price for coverage X is preferred over X Note that in the EU framework, the agent’s attitude towards risk and the agent’s at- If the decision-maker, with initial wealth according to this principle is the root of the equation between random losses the utility function a random loss Von Neumann & Morgenstern (1947). The s 2.3 Equivalent EUConsider risk a measures decision-maker with initial wealth onymous. Nevertheless, risk aversion expresses anmarginal attitude utility towards risk expresses while an decreasing attituderisk towards wealth. aversion Thus, being in encapsulated EU inwealth theory, the the rather utility concept than function of of is attitudes a towards property risk. of attitudes towards to maximize the expected utility. The prospect decision under uncertainty differs fromprobability EU distribution only by its interpretation of the underlying and chooses the loss which gives rise to the highest expected utility. 2.2 Risk aversion The two definitions ofcoincide risk with aversion the are concavity of equivalentaversion the implies under automatically utility EU strong function: risk theory, for aversion. where an EU they decision-maker, both weak risk titude towards wealth are forevercharacteristics bonded together of (since they are both derived from the represents the utility of not not covering The classical expected utilityaxiom. Every (EU, EU in decision-maker short) possesses some theory utility is function identified by the independence Risk measurement 2 Equivalent utility2.1 risk measures in EU EU theory theory Condition (2.1) expresses that expected utility of the decision-make (2.1) means that, provided an amount of utility of wealth with and This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. ¨ ell (3.2) (2.4) (2.3) (3.1) 0and can be = ] X ) . is valued at [ 0 x ( d f X

) x ( X . F ]

f f | exists and that the utility X . 0 1], as we immediately find [  +∞ , f = X the decision-maker uses the , x

DU [0 0 ]  ), U X . ∈ ≥ x + [ ] ( ) p has to be convex. E such that a prospect x x X  , d f f F cX p

( , DU

) ) ≤ U x 1] is non-decreasing with exp cx ) ( [ , fied independence axiom, Yaari (1987) has Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven p X − E ( [0 ]= ( F f c ln

X [ f exp c → 1 E defined as − − 1] ] 1 . Yaari’s model has been further developed by Ro 1 , f ]= ]≤

| )) f X [0 x = | [ X : ( [ ) X X [ x f , then (2.2) admits an explicit solution and 0 =−∞ ( F DU c x ( u DU f  U U ]=− f | X [ DU and the utility of not accepting the risk. U 1. Instead of using the tail probabilities X = ]− If we assume that the moment generating function of ) X 1 [ ( f its “distorted expectation” For the strong risk aversion, the distortion function Note that the exponentialterms principle of ruin (2.4) theory; also see, e.g., possessesTsanakas Kaas, & an Goovaerts, Desli Dhaene appealing (2003). & interpretation Denuit (2001, in Section 5.2) or that 3.2 Risk aversion Contrary to the EU case, theFor two the concepts of weak risk risk aversion aversion, are it no more suffices equivalent that in DU. Under the same set of axioms but with a modi function of the insurance company is of the form 3 Equivalent utility risk3.1 measures in DU DU theory theory shown that there must exist a “distortion function” distorted tail probabilities for some positive constant expressed as Here the distortion function (1987), and an alternative axiomatization isCarlier proposed & in Dana Guriev (2003) (2001). for We interesting also representations of refer to 10 which can be interpreted as an equality between the expected utility of the income This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , 11 X (3.6) (3.3) (3.7) (3.5) (3.4) (3.5) . and that ] f f | : corresponds X f ≡ is defined by [ ) f p , is defined by X . ( ] DU 1 x with values in the X . U d − X [ x

) p ) 1 x , . ( 1 0 . X ( ]=− 1 f . F ≤ ∈ |

≤ . q f 1] p X  x , f ≤ ∞

[0 [− ≤ 0 0 X  ∈ 0 DU , x − for + q U ] d x , the p-quantile risk measure (or Value- X ) d ) [ , q

1 , for each distortion function ( . It can be shown that F

,  1 ) p )  ), f 0 1 ure” in the sense of a Choquet-integral, see 1 x − , X x ( ( , , − F DU X 0 p − 1 ( 1 U F p 1 x ures have been introduced by Denneberg (1990)

− ∈ (  > f is a non-increasing function of f 1 p ]= ,p )) f  − − } otherwise if q x ] 1 | − p ( 1 1 f 0 [ 1 X

| 0 1 Forany p in min ≥ = F X  ( [ 0 DU = ) ) = f −∞ x x ] U = ) ( (  The Tail-VaR at level p, denoted by TVaR DU X x f ) X [ ( U q F f p ( | f =− R ]= X TVaR ∈ [ concave. Furthermore, we have that x { f 1]. The risk measure (3.5) can be interpreted as a “distorted expectation” of inf , ⇔ = with distortion function ) p ( Many risk measures can be cast into the form (3.5). Let us give some prominent The equivalent DU risks measures are obtained as the solution of the indifference 1 (3.5) convex − X f F The dual distortion function is again a distortion function. It is clear that Example 3.2 (Tail-VaR) It is the arithmeticwith average distortion of function the quantiles of X, from p on, and corresponds to interval [0 to at-Risk) for a random variable X with given distribution function F Risk measurement 3.3 Equivalent DULet risk us introduce measures the dual distortion function evaluated with a “distorted probability meas Denneberg (1994). Risk measures (3.5)actuarial are literature, often distortion called risk distortion meas riskand measures. Wang In (1996). the Axiomatic characterizationspremiums have of been proposed insurance in Goovaerts prices & Dhaene leading(1997). (1998) to and Wang, Young& distorted Panjer examples. Example 3.1 (Value-at-Risk) One immediately finds that Solving (3.4) gives equation This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. (3.9) (3.10) together B is then called on C P is a strict subclass f .  . . (3.8) +  ) ) ) q ] ( . Such a capacity U X f ( [ P  d f E ) ◦ ) q f − U − = X 1 − bias of the VaR when used for a portfolio ( Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven (  2004) for portfolio allocation with mixtures 1 C 1 ( concave are subadditive. The VaR therefore E − X 1 f F is differentiable, (3.8) can be rewritten as − X + 1 F f is an example of a risk measure that is coherent, ] 0 in (3.5) and to revert the order of the integrations.  For any random variable X, consider the risk mea-  X is in general not additive for co-monotonic risks. ) E [ q = E ( = f ] (3.10) ] = d X X ] ) [ (3.10) [ x X ( , and then define [ X f F 0  by )) x ( X F ( f Note that the class of distortion risk measures (3.5) with concave The Dutch risk measure VaR’s are the building blocks of the distortion measures (3.5). To see this, it is enough Distortion risk measures (3.5) with Hence, the Dutch risk measure although it is notcoherent risk a measures distortion are not risk necessarily measure. additive for The co-monotonic example risks. also illustrates the fact that of the class of coherent riskDhaene, measures, Vanduffel, Tang, as Goovaerts, is Kaas shown & by Vyncke the (2004). following example, taken from Example 3.3 (Dutch risk measure) sure with a distortion function with fat-tailed distributions. fails to be subadditive. Danielsson, Jorgensen,proved Samorodnitsky, Sarma & that de for Vries (2005) demonstrated most that practical VaR is applications subadditiveprovided VaR for the is the tails are relevant subadditive. not tails super Moreis of fat not (i.e., precisely, all defined, distributions fat with they such so as tailed fat theKitano distributions, tails Cauchy (2005) that law, the demonstrated with first tail the moment index considerable strictly less than 1). Inui, Kijima & This is a particularrandom case variables by of Kaas, Van the Heerwaarden & “Dutch Goovaerts premium (1994). principle”, defined for non-negative 4.1 CEU theory A possible device to construct a capacity is to take a probability measure 4 Equivalent utility risk measures in CEU theory to replace 12 Formulas (3.8)–(3.9) show that the spectralcases risk measures of of Acerbi distortion (2002) risk aremeasures particular measures. can It be can represented as be(2005); mixtures see further of also shown Tail-VaR’s, as Bassett, Koenker that pointed & out coherentof Kordas ( and by Tail-VaR’s, distortion Pflug Inui risk (2002). & Kijima One then finds that any distortion risk measure can be written as Note that when the distortion function This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . , . ] is C 13 Y C 1 , C (4.1) 0 and ∈[ X t α d ] t is concave and defined directly , is convex then C )> C B , one may wonder X for some distortion for any C ( ∈ C u P Y [ B ◦ , C f A ecial case that characterizes +∞ = 0 is finite, there exist capacities  with respect to the capacity C for all  ) + ] t . Moreover, with . The decision weights used in the X t representing the preference relation B ( d B [

u 1 C ∈ l is set in the framework of uncertainty is preferred over CEU A e decision-maker better off. Specifically, such that ]− U ]+ Y ed utility (CEU, in short) theory. In CEU t ndence axiom into co-monotonic indepen- A P α) [ is a capacity and integration is in the sense of C )> − is concave. Furthermore, if C X for any 1 ( C ( ] C ]≥ u [ d A B l will overweight large outcomes if ) + [ C is convex. Additivity is a sp X

∩ C X ( C A α u 0 [ −∞ ]≤ C   A is a monotonic function in [ then is concave. See also Leitner (2005) for related results. ] = t ]+ C u ]= Y is evaluated by is convex then B C X ∪ )> , C u X A | ( [ X u [ C [ C CEU . The answer is in general negative: even if f U is a utility of wealth function, u is preferred over If the reverse inequality holds in (4.1) then the capacity is called concave. Note that Due to the absence of an objective probability distribution, the definition of uncertainty Schmeidler (1989) modified the indepe X convex, that is, The corresponding preferences areestablished that thus the decision-maker convex. is Chateauneuf, uncertainty-averse if, Dana and only & if, the Tallon capacity (2000) on events, there is no necessity for any intermediate evaluation of probabilities. if the dual capacity is concave and satisfies where Choquet; see Choquet (1953). The functional a decision-maker is uncertainty-averse if,if for every pair of random variables as defined by Anscombe & Aumann (1963). aversion has to be based on thethe random variables idea themselves. Schmeidler (1989) that suggests the a outcomes convex and combination makes of an non-co-monotonic uncertainty-avers random variables smooths Risk refers to situations where theresented perceived by likelihoods probability of events distributions, of whereasinformation interest available uncertainty can to the refers be decision-maker rep- to is too situationsability imprecise where to measure. be Schmeidler’s the summarized (1989) by CEU a mode prob- in the CEU model is thus the Choquet integral of 4.2 Uncertainty aversion Risk measurement a distorted probability measure. Conversely, for a given capacity whether there exists a probability measure that cannot be obtained ase.g., non-decreasing Laeven transformations (2005) of for probability a measures. simple See, counterexample. dence. This gave risetheory, to the prospect the Choquet expect Note that the integralsdefined on since the right-hand side are Riemann integrals and they are well function and the utility function will overweight low outcomes if computation of the Choquet integra This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , n B ]= ∈ . C t A , d u , ,...,α ] | t ) 0 1 ] P A . X > ≥ d [ [  1 X X P B . [ α ( f ∈ C C  CEU ) d U A C X ( ]= +∞ − 0 A for any core  sup [ ∈ 1  C P for all + X . t ] d P is concave. = A

[ d 1 ) C C ]=− C d X X ( X [ is taken to be the identity function ]− u t ]≥ − u A  > [ ⇒ )  rest concerning the uncertainty aversion e preference for diversification in CEU. P C  X (

− [ C . Equality holds for all random variables Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven P C , is preferred over C inf core

i d ∈ P X is concave then X 0 i and −∞ ,thatis, f are sometimes called lower and upper Choquet α −  = C 1 identity  C

C = = P d n i d X d ) X C  X d X − ≤− is defined as ( X ]− u   X C C ) [  d C −  ( X  is then deduced from the indifference equation probability measures inf  core ]= in CEU), with respect to and
∈ is convex, and if 1. This concept is obviously related to the convexity for risk mea- ] C CEU P X , = C C C U = , d is additive. This inequality can be used to explain the bid-ask spread ) = i u X = C | α C ( C n 0 1  d X = that are all considered as equivalent by the agent (i.e., n i identity [ X | core n X   X [ CEU is assumed to be convex. We then get for the preference indicator simply the U CEU C is convex then The core of the capacity The risk measure There are thus two properties of special inte ,..., U f 0 with 1 X if This integral is positively homogeneous, monotonic and translation equivariant. sures. They established thatagent preference having for a convex portfolio capacity diversification and is a concave equivalent utility to function. the 4.3 Equivalent CEUMost risk often measures in the financial mathematics literature, ≥ Since Schmeidler (1986), it isnon-empty and well-known the that Choquet when integral the is given capacity by is convex, its core is and Choquet integral of 14 uncertainty neutrality. Probability distributions areboth special concave cases and of convex. In capacities the which case are of a distorted probability This relates the CEU model to the MEU one, presented inin the CEU next section. theory, namelyity. the Chateauneuf non-emptiness & of Tallon the (2002)The considered core agent th and is the said convexity of to the exhibit preference capac- for···= diversification if, for any random variables Note that, in general, integrals, respectively. More precisely, if, and only if, observed in some financialThe markets, according integrals to Chateauneuf, Kast & Lapied (1996). This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , ] 15 ⇔ B and [ Y C  icv  ]= . A X C [ correspond to , provided the } C R 1 → /α, ]⇒ p R B rding to a probability {  [ : (defined by µ M φ min the Borel subsets of ∈ icv = B ]=  P ) ,

A ] p [ is intimately connected with the P 1 ( has been considered in insurance, f µ d ) , ) 0 C C X d d ( X =[ X u aximin expected utility (MEU, in short) −    tions; see Kadane & Wasserman (1996) for   such that hat in the additive case, all events are unam- − he safety loading acco C , inf C d X ]=  M is concave; it is superadditive for convex ( , C u | ted as mixtures of Tail-VaR. X [ MEU U for all non-decreasing concave functions ] ) Y . φ( ) [ C E ( the Lebesgue measure). They characterized coherent symmetric capacities (as ]≤ ) µ X Note however that Marinacci (1999) proved that symmetric and coherent Choquet After Denneberg (1990), the risk measure Castagnoli, Maccheroni & Marinacci (2004) noticed that the presence on the market Kadane & Wasserman (1996) considered φ( [ the extreme points of the setrisk of measures all can distortion functions. be This represen directly implies that spectral capacities turn out toexistence be of additive a under single non-trivial areduces unambiguous to fairly event a single mild (in point) the condition. issince sense enough More to excluding that specifically, make the the them the existence additive. interval This ofgeneral (1.1) is a even a very a very stringent strong single property assumption non-trivial (recall unambiguous t event seems in biguous). This is obviously related(2004) to mentioned the above. result of Castagnoli, Maccheroni & Marinacci a definition). If the capacity isWasserman further (1996) proves assumed that to the be distortion functions concave then Section 4 in Kadane & those with doubly star-shaped distortion func with symmetric capacities (that is, capacities 5 Equivalent utility5.1 risk measures in MEU MEU theory theory Gilboa & Schmeidler (1989) suggested the M model, also called multiple priorresentation model. by In the this functional context, preference relations have a rep- E Risk measurement so that the length of the interval size of core e.g., by Wang (1996), Wang, Young & Panjer(2003) (1997) under and the De name Waegenaere, of Kast Choquet & pricing.type Lapied An of overview risk paper measures on for the solvency applicability purposes& is of Vyncke Dhaene, this (2004). Vanduffel, Tang, Chateauneuf, Goovaerts, Kaas Kast && Lapied (1996), Sherris Wang (2000, (2003) 2002) used andPelsser Hamada the (2004). This same risk mechanism measureIt for is is continuous, pricing subadditive monotonic if, in and and financial only co-monotonic if, additive. markets; see also of assets without bid-ask spreadsprecisely, turns Choquet if prices an into standard insurancedistortion expectations. in More company agreement sets with t the increasing concave order expectations exist) then itor assigns zero either safety strictly loading positive tonot safety prevent all the loading contracts. use Despite to of Choquet its allpositive risk theoretical safety contracts measures loading, interest, (since to this actuarial avoid risk result almost theory should certain imposes strictly ruin). This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. (5.1) ,that ,that  C is defined by M is not necessarily P d C

scenarios in Artzner, , } X is the core of M ]− M ∈ X [ P does not have to be its core. ]|

A M [ .  P P { M d ∈ X inf P inf  = Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven are called generalized city, Gilboa & Schmeidler (1989) postulated ]=  ed by Castagnoli, Maccheroni & Marinacci M . Moreover, when an arbitrary (closed and ] ∈ A sup ) [ P M M aces (see Delbaen (2002)), though in that case , C C ( u

is a closed and convex set of additive probability ]= X X core [ M , u ]− happens to be convex, | X X [ C [ is a moment space (or more generally, when  M MEU MEU U U = ]= gives the class of coherent risk measures in the sense of Artzner, Delbaen, 0 C is given and one defines x , u = | M ) X x [ is a utility function and ( u u CEU Risk measures of the form (5.1) have been considered for a long time in actuarial Risk measures of this type have been us In addition to the usual assumptions on the preference relation such as transitivity, U is, a set of integral constraints); see, e.g., Section 5.3(1984) of for Goovaerts, De a Vijlder & detailed Haezendonck account. measures. Thus, CEU coincides with MEU when the set “the important conceptual aspects area buried under measure a theoretic and mass topological of nature,” technical as complications is of stated by Huberscience, (1981). especially when Delbaen, Eber & Heath (1999).a Huber risk (1981) measure is proved coherent for if,result the and remains case only if, of valid it a for has finite more an set upper general expectation sp representation. This Eber & Heath (1999). More precisely, the equivalent utility principle then yields 6 Equivalent utility6.1 risk measures in RDEU RDEU theory theory In this section, we presenta a certain theory extent. which The combines the rank-dependentproposed EU by expected and Quiggin utility (1982). the (RDEU, It DU is inthe assumptions, a short) properties to generalization of theory of continuity, was EU transitivity and first the and DU fact weak theories that monotonicity. that The preserves cumulative nameto probabilities comes a are from particular transformed, outcome so depends that on the its new rank weight among assigned possible outcomes. We do not give convex) set 5.2 Equivalent MEUTaking risk measures (2002) to compute insurance(5.1) premiums. in robust Huber statistics. (1981) The already elements of considered functionals 16 where whence it follows that uncertainty aversion and certainty-independenceindependence to axiom derive MEU is theory. weakerdegenerate The distribution than functions certainty- corresponding standard to certainty independence are considered). (since only mixings with completeness, continuity and weak monotoni convex. Furthermore, even if This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. = 17 ]≤ ) t f p d , ( ) α u ] | f t X [ . For descrip- )> f X ( RDEU u [ U P ( f objective probabilities . p 0 we get the DU paradigm, , we get the EU model, that +∞ d = p

t x )  p = = ( + ) 1 ) t if, and only if, p − x X d ( (

F f X from EU with a distortion function u 1

) u · − ( α u ) s suggested to consider convex distortion 0 ]  t erent concepts. First, there is outcome risk comes and attitudes towards probabilities, hem when choosing among risky prospects. 1 α . ssibly non-additive) subjective probabilities, dler (1989) do not take ] f )> | X ]= , whereas if ( X is normally taken to be concave or linear. There is to the fortune ] α [ u f u u [ Y | , P DU X u ( [ | U f . In this case, we have X ]

[ EU 1 . It is easy to see that if , U ]= Y f 0 0 , =−∞ RDEU t ∈[ ]= (that plays the role of a probability perception function). Such (that plays the role of utility on certainty) in conjunction with  U α f u ]= identity f | identity where , X , ] [ u u | f Following the usage of Knight (1921) in distinguishing uncertainty (i.e., am- | for some X (this property is akin to risk aversion, as shown above). , X } [ u [ f 1 | RDEU Y [ U /α, RDEU p RDEU from DU. For interesting applications to insurance, see, e.g., Chateauneuf, Cohen & { Simple distortion functions are those given by the one-parameter family Just as in EU, the utility function Under the rank-dependent expected utility model, a decision-maker is characterized U U ) · RDEU ( f that is, is, tive purposes, Quiggin (1982) originallyextreme proposed low an probability outcomes. S-shaped Other function, author overweighting min This family has been considered, e.g., in Bassett, Koenker & Kordas (2004). Remark 6.1 biguity) and risk, Gilboa (1987) and Schmei as given but consider subjective probabilities. Capacitiescompared are then on used the and prospects basis are weak of monotonicity is Choquet fulfilled, integrals. this Wakker approach is (1990) equivalentbetween to established the RDEU. that, The two subtle provided approaches difference canknow be the summarized objective as probabilities follows: but in distort RDEU, t decision-makers As RDEU distinguishes attitudes towardsrisk out aversion in RDEU mustaversion, combine associated two with diff the idea that the marginal utility of wealth is declining. This is 6.2 Risk aversion less consensus concerning the appropriate shape for the distortion function functions In CEU, decision-makers use theirderived own from (po some capacity. with a similar expression for U Risk measurement the set of axioms connectedaxiom with appearing RDEU in since Quiggin the (1982) modified doesother version not sets of of admit the axioms, a see independence clear e.g. behavioralRDEU Chateauneuf interpretation. & approach For Wakker combines (1999) and the Abdellaoui utility (2002). The function Meilijson (1997). by a utility function a distortion function a decision-maker prefers the fortune This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. , f x and = (6.2) (6.1) X .This ) is non- p implies x ( u and does f u . It can be icx icv .If  c is concave and for all  u . The equivalent p f ,

≤ ] . ) X p [  ( t . Note that the restriction E f

d

icv u ] . t   is linear or exponential and and pessimism are reasonable = f if, and only if, u u  , f )> u icv is non-decreasing and concave,

, for all real constants cX X u 

( c u ] X ]− [ exp ive, more restrictive, characterization of [ ]= to probability preferences. Risk aversion X E mism: the decision-maker adopts a set of Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven c [ P  [

if, and only if, for any random variable  f p RDEU +∞ U RDEU 0  whatever the distortion function U for all  ) ]≤ ] p = f is not equivalent to preference for certainty over risk. ln f . Therefore, concavity of ) , | ≤ is a convex distortion function, Heilpern (2003, Theorem 1) X 0 c 1 u icv ( X | ) f [ u al cases (broadly speaking, if p  may be regarded as pessimistic if X ( [ ]= f f DU X U [ ( u RDEU U u ]≤ f is preferred over , ] u | X [ X [ E The equivalent utility risk measures solving (6.1) are great variety. Provided The transformation Let us now turn to the definition of risk aversion as preservation of is convex (see, e.g., Quiggin (1993)). Convexity of the distortion function RDEU f it is moreover positively homogeneous, co-monotonic additive andin subadditive. Note this that case, wevariables only are in very back speci to the DU paradigm. It is additive for independent random concave utility that is, proved that the resulting risk measure is translation equivariant, preserves decreasing and concave and RDEU risk measure solves is the identity). not entail unjustified safety loading, that is 6.3 Equivalent RDEUHeilpern risk (2003) and measures Tsanakas &can Desli be (2003) introduce considered a as classin the of solutions risk Heilpern of measures which theU (2003) indifference equations establishes for RDEU. that Lemma 1 provided pessimism leads to a definition of risk aversion in terms of in probability weighting correspondsdecision to weights pessi that yieldsthan an the expected mathematical value expectation. for An a alternat transformed riskyto prospect concave utility lower functions does not prohibitpossible risk to seeking model behaviors in risk RDEU. seeking It with is indeed diminishing marginal utility ofcomes wealth. from the fact that shown that a decision-maker will be consistent with a strong form of pessimism that can beis roughly overweighted summarized most, as follows: and the worst the outcome bestEU outcome theory, underweighted preservation most. of Unlike the situation in 18 the standard notion of risk aversionfunction. from the Second, EU theory there defined by are concavity attitudes of the specific utility In general, there is no explicit solutionfunction to of this the equation. form A (2.3) notable for exception which is with utility conditions for risk aversion. This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 19 risk: for example, according neral behavior is nevertheless difficult to functions. Disregarding utility of wealth onal change in capital requirements. This ifferent theories for decision under risk and ng sensitivity to portfolio size and to risk ures insensitive to liquidity (that is, in the EU case). Moreover, the risk measure (6.2) is strongly p = ) p ( f It is now also clear that risk measures encountered in the financial mathematics In particular, the risk measure (6.2) is additive for independent random variables In particular, the distortion exponential risk measure (6.2) proposed by Tsanakas & The RDEU framework simultaneously accounts for utility functions and distorted does not alwaysutility seem function reasonable makes risk from meas an economic viewpoint. The linearity of the uncertainty. As such it makes clear the economic rationale behind theseliterature risk correspond measures. to linear utility of wealth 7 Discussion In this paper, wethe have actuarial derived equivalent utility the principle great in majority d of the classical risk measures using Risk measurement only if monotonic, translation equivariant and convex. Tsanakas & Deslithe (2003) studied properties in detail ofaggregation. this They risk conclude that measure thisrisk regardi risk measure measure for behaves smaller approximately portfolios asinduced of by a risks, the coherent utility while function for becomesaggregation larger prevalent, issues and portfolios gradually the the increases. sensitivity risk to aversion liquidity and risk to positive homogeneity, changesis in unchanged, the should size only of affect a a portfolio, given proporti that its composition determine since it is difficultwill to know have at the which point mostRDEU the influence. utility risk As or measures the pointed are distortiongeneral. out function convex by under Tsanakas mild & conditions. Desli Note (2003), that equivalent CEUDesli is (2003) even combines more theis properties of obtained the as DU autility equivalent special function and case (2.4) (2.3) of risk isrisk equivalent measures. used. RDEU measures. It It risk Whether inherits measures thecloser properties when properties to from those an of of both exponential the (2.4) (2.4)on or distortion the and those exponential underlying of DU risk risks equivalent equivalent DU that measuresexponential risk are risk are measures measure examined. is depends For approximately to equivalent relatively to some smalland the extent inherits portfolios, equivalent their DU the properties. risk For distortion measures larger portfolios,become for an which issue, liquidity the and effect of risk the aggregation utilityexponential function risk (2.3)becomes measure prevalent and inherits the its distortion properties from (2.4). The definition of what a large disregards that very large portfolios might produce very largeit losses that, difficult in turn, for can the make holderThis of has the been portfolio pointed out, to e.g., raise by sufficient Dhaene, cash Goovaerts to & meet Kaasprobabilities: (2003). his the agent obligations. is equipped with both aRisk utility measures function and emerging a probability from distortion. case RDEU of inherit a linear properties utility from functionwhile, both equivalent in DU EU the risk case measures and of are DU: obtained aare as linear in found. special probability the In cases distortion general, function, equivalentin the EU between properties risk of measures of these equivalent two RDEU extreme cases. risk Their measures ge lie somewhere This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 15, 3, 424–444. 2, 477–492. 31, 267–284. . Springer Verlag, Astin Bulletin 70, 717–736. 0 that are often used (1999). Coherent risk = stortion of a probability. 38, 132–148. γ (2002). Insurance premia Econometrica generalization of the von when Econometrica x ankelijkheden in verzekerings- en 26, 1505–1518. (2004). Pessimistic portfolio alloca- nderzoeksfonds K.U. Leuven (GOA/02: 0andln Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven =  (1963). A definition of subjective probability. γ ´ ees” ARC 04/09-320 is gratefully acknowledged Journal of Financial Econometrics 34, 199–205. 9, 203–228. for (2006). Stochastic orders and risk measures: Consis- γ 113, 199–222. γ (2003). The core of convex di x Insurance: Mathematics and Economics = Mathematical Methods in Risk Theory ) x ( u ¨ uller, A. (2002). A genuine rank-dependent Insurance: Mathematics & Economics (1985). Premium calculation from top down. (1970). The financial support of the Belgian Government under the contract Journal of Banking and Finance ¨ ele en statistische aspecten van afh (1962). Equilibrium in a reinsurance market. (2002). Spectral measures of risk: A coherent representation of subjective Mathematical Finance ¨ ele, financi ¨ ¨ ¨ uhlmann, H. uhlmann, H. auerle, N., & M ¨ ele portefeuilles). Roger Laeven acknowledges the support of the Dutch Organi- 89–101. Carlier, G., & Dana, R. A. B New York. B Journal of Economic Theory Castagnoli, E., Maccheroni, F.,consistent with & the Marinacci, market. M. Bassett, G. W., Koenker, R., &tion Kordas, and G. Choquet expected utility. B tency and bounds. Borch, K. Annals of Mathematical Statistics Acerbi, C. risk aversion. Anscombe, F. J., & Aumann, R. J. Artzner, Ph., Delbaen, F.,measures. Eber, J.-M., & Heath, D. Abdellaoui, M. Neumann–Morgenstern expected utility theorem. Alternatives to (6.2) can be obtained by substituting other classes of utility functions to [8] [9] [5] [6] [7] [2] [3] [4] [1] [10] [11] in economic applications. Acknowledgements. “Projet d’Actions de Recherche Concert zation for Scientific Research (No. NWO 42511013). by Michel Denuit, who also thanks the Banque Nationaleunder de Belgique grant for financial “Risk support Measuresacknowledge and Economic the Capital”. financial Jan support Dhaene and of Marc the Goovaerts financi O Actuari References the exponential one (2.3). One could thinkutility of functions CARA given (for constant by absolute risk aversion) 20 portfolio is depends oncontrolled the by modifying specific the situationnumerical risk illustrations and in aversion preferences Tsanakas parameter & of Desli of its (2003) the support holder. this exponential evidence. It function. can The be This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 21 19, , Kluwer Methods 18, 137– 5 (Greno- . Kluwer Academic utility model: A new Economic Theory 40, 547–571. 18, 25–46. (2004). Choquet insurance (2000). Optimal risk sharing Journal of Mathematical Eco- (2004). Four notions of mean- Finance 32, 359–370. (1999). The economics of insur- (1997). New tools to better model (2003). Choquet pricing and equi- Bulletin of the Swiss Association of 6, 323–330. (1996). Choquet pricing for financial Fuzzy Measures and Integrals Annales de l’Institut Fourier 14, 481–485. . Springer Verlag, Berlin. Journal of Risk and Uncertainty (1999). An axiomatization of cumulative (2002). Diversification, convex preferences 4, 179–189. (2000). Choquet expected .A.,&Tallon,J.-M. (1985). On convex principles of premium calculation. hoquet expected utility model. Non-Additive Measure and Integral Choquet-expected-utility. 63, 3–5. Journal of Mathematical Economics Mathematical Finance & Tallon, J.-M. Mathematical Finance (1994). (1990). Distorted probabilities and insurance premiums. (2002). Coherent risk measures on general probability spaces. In: (1953). Theory of capacities. 1999(2), 137–175. Insurance: Mathematics and Economics 34, 191–214. (2005). Subadditivity re-examined: The case for Working Value-at-Risk. Paper, Chateauneuf, A., & Cohen, M. Academic Publishers, pp. 289–313. Chateauneuf, A., Cohen, M., & Meilijson,behavior I. under risk and uncertainty: An overview. approach to individual behavior underbish, M., uncertainty Mirofushi, T., and Sugeno, M., to editors, social welfare. In: Gra- Chateauneuf, A., Cohen, M., & Meilijson, I. Publishers, Boston. Denuit, M., Dhaene, J.,ance: & A Van review and Wouwe, some M. recentActuaries developments. preserving increases in risk,expected risk utility attitudes model. and applications to the rank-dependent rules and equilibria with of Operations Research Deprez, O., & Gerber, H. U. Insurance: Mathematics & Economics De Waegenaere, A., Kast, R., & Lapied, A. Chateauneuf, A., Dana, R librium. Castagnoli, E., Maccheroni, F., & Marinacci, M. pricing: A caveat. Chateauneuf, A., and non-empty core in the C markets with frictions. Denneberg, D. nomics Chateauneuf, A., Kast, R., & Lapied, A. Denneberg, D. 509–523. Chateauneuf, A., &prospect Wakker, theory P. for P. decision under145. risk. Choquet, G. ble), 131–295. Danielsson, J., Jorgensen, B. N., Samorodnitsky,C. G. G., Sarma, M., &EURANDOM. de Vries, Delbaen, F. Essays in Honour of Dieter Sondermann [13] [14] [15] [25] [26] [27] [16] Risk measurement [12] [18] [24] [17] [23] [19] [20] [21] [22] This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. (2002a). European Insurance: Insurance: (2002b). The 34, 505–516. , 221–227. Insurance Premi- Belgian Actuarial 7, 44–59. Research Report OR . Reidel Publishing Co, (1984). (2004). Some new classes (2nd Edition). De Gruyter, ective non-additiveprobabilities. (2003). Economic capital allocation Expected Utility 18, 141–153. (2002). Putting order in risk measures. Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven (1981). On the representation of additive , M. J., Kaas, R., & Vyncke, D. ffel, S., Darkiewicz, G., & Goovaerts, , Goovaerts, M. J., Kaas, R., & Vyncke, D. , M. J., Kaas, R., & Vyncke, D. 16, 65–88. Stochastic Finance Scandinavian Actuarial Journal (1998). On the characterization of Wang’s class of F., & Haezendonck, J. , M. J. (1989). Maxmin expected utility with a non-unique 26, 1473–1486. 31, 3–33. 31, 133–161. (2004). North American Actuarial Journal Insurance: Mathematics and Economics (2002). Convex measures of risk and trading constraints. 6, 429–447. The Foundationsof oovaerts Transactions of the 26th International Congress of Actuaries osazza Gianin, E. 2, 113–124. (1982). (1999). Learning about risk: Some lessons from insurance. (1987). Expected utility with purely subj 4, 53–61. Journal of Mathematical Economics (2004). Can a coherent risk measure be too subadditive? , Department of Applied Economics, K. U. Leuven. . North-Holland, Amsterdam. ¨ ¨ ollmer, H., & Schied, A. ollmer, H., & Schied, A. ums Goovaerts, M. J., & Dhaene, J. premium principles. 4, 121–134. Goovaerts, M. J., Kaas, R., Dhaene, J., & Tang, Q. of consistent risk measures. prior. principles of premium calculation. Gilboa, I. Gilboa, I., & Schmeidler, D. Journal of Mathematical Economics Goovaerts, M. J., De Vijlder, Finance Review Gerber, H. U., & G Journal of Banking & Finance Geman, H. Berlin. Frittelli, M., & R Dordrecht. F F Finance and Stochastics Dhaene, J., Denuit, M., Goovaerts The concept of comonotonicity in actuarial scienceMathematics and and finance: Economics Theory. derived from risk measures. Dhaene, J., Denuit, M., Goovaerts Dhaene, J., Goovaerts, M. J., & Kaas, R. concept of comonotonicity in actuarial science and finance: Applications. Mathematics and Economics Bulletin Fishburn, P. C. Dhaene, J., Laeven, R. J.M. A., J. Vandu 0431 Dhaene, J., Vanduffel, S., Tang, Q. (2004). Capital requirements, risk measures and comonotonicity. [42] [43] [39] [40] [41] [38] [37] [36] [34] [35] 22 [28] [29] [30] [33] [31] [32] This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. . 23 61, Ordering of The Geneva Modern Actuarial Advances in Mathe- (1994). Journal of Mathemat- (2004). A comonotonic Sankhya Ð Series A ricing using probability Journal of Mathematical (2001). Insurance: Mathematics and . Houghton Mifflin Company, zation of zero-utility principle. 10, 19–47. 29, 853–864. tions with anticipated utility theory. 9, 597–608. 33, 67–73. 72, 299–311. 26, 117–137. . Wiley, New York. (2005). VaR is subject to a significant positive (1996). Symmetric, coherent, Choquet capacities. (2003). Contingent claim p 286, 237–247. Essays on Risk Measures and Stochastic Dependence Risk, Uncertainty, and Profit (2005). On the significance of expected shortfall as a coherent Robust Statistics 24, 1250–1264. 3, 83–95. (2005). (2005). Robust representation of convex risk measures by prob- Sherris, M. Finance and Stochastics (1999). Upper probability and additivity. Applied Mathematical Finance 41, 994–1006. 31, 23–35. . CAIRE Educations Series, Brussels. (1921). (2003). A rank-dependent generali Journal of Banking and Finance (2001). On law invariant coherent risk measures. (1981). (2005). Dilation monotonous Choquet integrals. . Kluwer Academic Publishers, Dordrecht. (2001). On microfoundation of the dual theory of choice. 35, 581–594. (2003). On the worst conditional expectation. (2001). Insurance premium calcula Statistics and Probability Letters ¨ atschmer, V. 358–361. ical Economics Luan, C. ASTIN Bulletin Marinacci, M. Tinbergen Institute Research Series 360, Thela Thesis, Amsterdam. Leitner, J. matical Economics Laeven, R. J. A. Actuarial Risks Boston. Kr ability measures. Kadane, J., & Wasserman, L. A. Annals of Statistics Knight, F. H. Kusuoka, S. Risk Theory Kaas, R., Van Heerwaarden, A. E., & Goovaerts, M. J. risk measure. Inui, K., Kijima, M., & Kitano, A. bias. Kaas, R., Goovaerts, M. J., Dhaene, J., & Denuit, M. Inoue, A. Analysis and Applications Inui, K., & Kijima, M. Hamada, M., & distortion operators: Methods from insurancefinancial risk theory. pricing and their relationship to Heilpern, S. Insurance: Mathematics and Economics Papers on Risk and Insurance Theory Huber, P. J. Goovaerts, M. J., Kaas, R., Laeven, R. J. A., & Tang, Q. Guriev, S. image of independence forEconomics additive risk measures. [60] [61] [59] [58] [56] [54] [55] [57] [53] [51] [52] [49] [50] [46] [47] [48] Risk measurement [44] [45] This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. British Proceedings of 8, 550–555. (1989). Properties of 21, 173–183. 29, 453–463. 31, 73–75. Journal of Economic Behavior Choquet-expected utility theory Theory of Games and Economic . Wiley, New York. (1997). Axiomatic characterization Insurance: Mathematics & Economics , Routledge, pp. 20–35. (1947). ity Press, Princeton. 67, 15–36. Denuit -- Dhaene -- Goovaerts -- Kaas -- Laeven 97, 143–159. 97, 253–261. (1999). A single number cannot hedge against Theory and Decision (2003). Risk measures and theories of choice. Ambio-Royal Swedish Academy of Science Austrian Journal of Statistics . Princeton Univers Uncertainty in Economic Theory: A Collection of Essays in Generalized Expected Utility Theory: The Rank Dependent 57, 571–587. ¨ ing Paper, Erasmus University Rotterdam. uppelberg, C. 26, 71–92. The Foundations of Statistics Insurance: Mathematics & Economics 9, 959–981. Economic Journal 3, 323–343. (2004). On the applicability of the Wang transform for pricing (1989). Subjective probability and expected utility without addi- (1986). Integral representation without additivity. 32, 213–234. (2004). Preference axiomatizations for decision under uncertainty. (1990). Under stochastic dominance (2002). Coherent risk measures and convex combinations of the con- (1993). (2000). A class of distortion operators for pricing financial and insur- (1996). Premium calculation by transforming the layer premium den- (1982). A theory of anticipated utility. (2002). A universal framework for pricing financial and insurance risks. (1954). Journal of Risk and Insurance (1987). Risk aversion in Quiggin and Yaari’s rank-order model of choice . Kluwer Academic Publisher, Boston. Econometrica ASTIN Bulletin ¨ ell, A. sity. Wang, S. S. ance risks. and anticipated utility are identical. Wakker, P. P. In: Gilboa, I., editor, Wang, S. S. Wang, S. S. ASTIN Bulletin Wang, S. S., Young, V. R., & Panjer, H. H. Wakker, P. P. Honor of David Schmeidler’s 65th Birthday of insurance prices. the American Mathematical Society Actuarial Journal Van Heerwaarden, A. E., Kaas,the R., Esscher & premium Goovaerts, calculation M. principle. 8, J. 261–267. Von Neumann, J., & Morgenstern, O. Behavior (2nd edition) Schmeidler, D. tivity. Tsanakas, A., & Desli, E. Rootzen, H., & Kl economic catastrophes. Savage, L. Schmeidler, D. financial risks. Work Pflug, G. C. ditional Value at Risk. Model Ro under uncertainty. and Organisation Quiggin, J. Pelsser, A. A. J. Quiggin, J. [77] [75] [76] [78] [79] [74] [72] [73] [70] [71] [67] [68] [69] [63] [66] [65] [64] 24 [62] This article is protected by German copyright law. You may copy and distribute this article for your personal use only. Other use is only allowed with written permission by the copyright holder. 25 55, 95– Econometrica Jan Dhaene Catholic University of Leuven Center for Risk and InsuranceB-3000 Studies Leuven Belgium [email protected] and University of Amsterdam Dept. of Quantitative Economics 1018 WB Amsterdam The Netherlands Rob Kaas University of Amsterdam Dept. of Quantitative Economics 1018 WB Amsterdam The Netherlands [email protected] (1987). The dual theory of choice under risk. ´ e Catholique de Louvain 115. Yaari, M. E. Michel Denuit Institut des Sciences Actuarielles & Institut de Statistique Universit 1348 Louvain-la-Neuve Belgium [email protected] Marc Goovaerts Catholic University of Leuven Center for Risk and Insurance3000 Studies Leuven Belgium [email protected] and University of Amsterdam Dept. of Quantitative Economics 1018 WB Amsterdam The Netherlands Roger Laeven University of Amsterdam Dept. of Quantitative Economics 1018 WB Amsterdam The Netherlands [email protected] Risk measurement [80]