Strongly Consistent Multivariate Conditional Risk Measures

Strongly Consistent Multivariate Conditional Risk Measures annes offmann Thilo Meyer-Brandis regor Svindland January 11, 2016 Abstract We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced offmann et al. 2015. Further, in analogy to the univariate case in Föllmer 2014, we prove that under law-invariance strong consistency implies that multivariate conditional risk measures are necessarily multivariate conditional certainty equivalents. Keywords: multivariate risk measures, systemic risk measures, strong consistency, systemic risk, law-invariance, conditional certainty equivalents. MSC 2010 classifications: 91B30, 9199 1 Introduction Over the recent years the study of multivariate risk measures ρ : L d F R, 1.1 that associate a risk level ρx to a d-dimensional vector X = X 1,..., X d of random risk factors at a given future time horizon T, has increasingly gained importance. ere, L d denotes the space of d-dimensional bounded and - measurable random vectors on a probability space Ω, F, P for some σ-algebra F, i.e. in this article we restrict the analysis to bounded risk factors X for technical simplicity. In particular since the financial crisis, it has become Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, ermany. Emails: hannes.hoffmann@math.lmu.de, meyer-brandis@math.lmu.de and gregor.svindland@math.lmu.de. 1

clear that the traditional approach to individually manage the stand-alone risks X i of single institutions in a financial network by a univariate risk measure i.e. d = 1 in 1.1 fails to encompass the perilous systemic risk which arises due to the interconnectedness and corresponding contagion effects between the system entities. This has triggered a rapidly growing literature on how to define more appropriate multivariate risk measures, also referred to as systemic risk measures, that are capable of capturing systemic risk by assessing the risk of a financial system as a whole, and on how to identify those financial institutions of the system which shall be deemed as systemic, i.e. those components of the system which might prevent the system from working properly. See e.g. Bisias et al. 2012 and the references therein for an excellent survey on systemic risk. Most of the systemic risk measures proposed in the literature are of the form ρx = η ΛX, 1.2 where one first aggregates the d-dimensional risk factors X into a univariate risk by some aggregation function Λ : R d R, and then applies a univariate risk measure η : L F R. Some prominent examples are the Contagion Index of Cont et al. 2013 or the SystRisk of Brunnermeier and Cheridito 2014. Chen et al. 2013 were the first to describe axiomatically this intuitive type of multivariate risk measures ρ which allow for a decomposition 1.2 on a finite state space. In Kromer et al. 2014b this has been extended to general L p -spaces. An interesting extension of the static viewpoint of deterministic risk measurement in 1.1 is to consider conditional risk measures which allow for systemic risk measurement under varying information. A conditional multivariate risk measure is a map ρ : L d F L, 1.3 that associates to a d-dimensional risk factor a -measurable bounded random variable, where F is a sub-σ-algebra. We interpret ρ X as the risk of X given the information. In the present literature, conditional risk measures have mostly been studied within the framework of univariate dynamic risk measures, where one adjusts the risk measurement in response to the flow of information that is revealed when time elapses. For a good overview on univariate dynamic risk measures we refer the reader to Acciaio and Penner 2011 or Tutsch 2007. One possible motivation to study conditional multivariate risk measures is thus the extension from univariate to multivariate dynamic risk measures, and to study the question of what happens to the risk of a system as new information arises in the course of time. In the context of multivariate risk measures, however, also a second interesting and important dimension of conditioning arises, besides dynamic conditioning: Risk measurement conditional on information in space in order to identify systemic relevant structures. In that case represents for example information on the state of a subsystem, and one is interested in questions of the type: ow is the overall risk of the system affected, given that a subsystem is in distress? Or how is the risk of a single institution affected, given the entire 2

system is in distress? The systemic risk measures CoVaR of Adrian and Brunnermeier 2011 or the systemic expected shortfall of Acharya et al. 2010 can be considered conditional systemic risk measures of this type. Also, in Föllmer 2014 and Föllmer and Klüppelberg 2014 the authors analyze so-called spatial risk measures, which are univariate conditional risk measures where again the conditioning is with respect to information on the state of subsystems. In analogy to 1.2, we observe that the conditional systemic risk measures proposed in the literature can be written in the form ρ = η Λ X, 1.4 where now η : L F L is a univariate conditional risk measure, and Λ is a conditional aggregation function. In offmann et al. 2015 we extend the axiomatic approach of Chen et al. 2013 and Kromer et al. 2014b to the conditional setting 1.3 and give a characterization of multivariate risk measures which allow for a decomposition 1.4. When dealing with families of univariate conditional risk measures, a frequently imposed requirement is that the conditional risk measurement behaves consistent in one way or another with respect to the flow of information. In particular, in the literature on univariate dynamic risk measures most often the so-called strong consistency is studied; c.f. Detlefsen and Scandolo 2005; Cheridito et al. 2006; Cheridito and Kupper 2011; Kupper and Schachermayer 2009; Penner 2007. Two univariate conditional risk measures ρ and ρ with corresponding σ-algebras F are called strongly consistent if for all X, Y L F ρ X ρ Y = ρ X ρ Y, 1.5 i.e. strong consistency states that if Y is riskier than X given the information, then this risk preference also holds under less information. The purpose of this paper is to extend the concept and characterization of strong consistency to multivariate conditional risk measures. To this end, we define strong consistency of two multivariate conditional risk measures ρ and ρ with F as in 1.5 for any d-dimensional risk vectors X and Y, i.e. the interpretation of strong consistency for multivariate risk measures is in exact analogy to the univariate case. As a first main result we prove that the members of any family of strongly consistent multivariate conditional risk measures are necessarily of type 1.4. A requirement here, which is automatically satisfied in the univariate case, is that the family contains a terminal risk measure ρ F : L d F L F under full information F. Such a terminal risk measure is nothing but a statewise aggregation rule for the components of a risk X L d F. In the univariate case, if X L F, there is of course no aggregation necessary. Indeed letting the terminal risk measure correspond to the identity mapping, i.e. ρ F = id, we have that any univariate risk measure ρ with F is strongly consistent with ρ F by monotonicity, so the existence of such a terminal risk measure which is strongly consistent with the other risk measures of the 3

family is no further restriction. In the truly multivariate case, however, it is very natural that also under full information, there is a rule for aggregating risk over the dimensions, and the risk measures in the family should be consistent with this terminal aggregation rule. If this is the case, we show, as already mentioned, that the members of the family are necessarily of type 1.4. Indeed we show that by strong consistency the risk measures inherit a property called risk-antitonicity in offmann et al. 2015 from the terminal risk measure. This property is the essential axiom behind allowing for a decomposition of type 1.4; see Theorem 3.12. Along the path to this result we characterize strong consistency in terms of a tower property. It is well-known, see e.g. Tutsch 2007, that for univariate conditional risk measures which are normalized on constants η a = a for all a L, strong consistency 1.5 is equivalent to the following tower property: ρ X = ρ ρ X for all X L F. 1.6 The recursive formulation 1.6 is often more useful than 1.5 when analyzing strong consistency. The formulation 1.6, however, cannot be extended in a straight forward manner to the multivariate case. Firstly, note that 1.6 is not even well-defined in the multivariate case since ρ X is not a d-dimensional random vector but a random number. Secondly, also in the univariate case the equivalence 1.5 1.6 only holds for risk measures that are normalized on constants, which in the monetary univariate case is implied up to a normalization by requiring that this class of risk measures satisfy cash-additivity η X + a = η X a. For multivariate risk measures there is neither a canonical extension of the concept of cash-additivity nor is it clear that such a property is desirable at all. In a first step we therefore derive a generalization of the recursive formulation 1.6 of strong consistency for not necessarily cash-additive multivariate risk measures. Indeed, under some typical regularity assumptions, one of our first results is that two multivariate conditional risk measures ρ and ρ with F are strongly consistent if and only if for all X L d F ρ X = ρ f ρ ρ X1 d, 1.7 where 1 d is a d-dimensional vector with all entries equal to 1, and fρ is the well-defined inverse function of the function f ρ associated to ρ given by f ρ : L L ; α ρ α1 d. 1.8 The map f ρ describes the risk of a system where each component is equipped with the same amount of -constant cash α. Note that if ρ is a univariate risk measure that is normalized on constants then f ρ = id is minus the identity map and 1.7 reduces to 1.6. In this sense, for a multivariate risk measure ρ the generalization of the normalization on constants property that is suited for our purposes is the requirement f ρ = id. Further, we remark that one can always normalize a given conditional risk measure ρ by putting ρ := f ρ ρ X. 1.9 4

Then ρ is a multivariate conditional risk measure with f ρ = id. After studying strong consistency for general families of multivariate conditional risk measures, we move on to give a characterization of strongly consistent multivariate conditional risk measures which are also conditionally law-invariant. In contrast to before we do not require consistency with respect to a risk measure under full information, but with respect to the initial risk measure given the trivial information {, Ω}. These studies were triggered by the results obtained in Föllmer 2014 for univariate risk measures, where it is shown that the only family of univariate, strongly consistent, conditional, cash-additive, convex risk measures is the family of conditional entropic risk measures, i.e. the conditional risk measures are conditional certainty equivalents of the form ρ X = u E P [ux ], X L F, with deterministic utility function ux = a + be βx or ux = a + bx, where a R and b, β > 0 are constants. We also remark that Kupper and Schachermayer 2009 showed this characterization for the case of dynamic risk measures by an alternative proof. In the multivariate case we will see that every strongly consistent family of multivariate conditionally law-invariant conditional risk measures consists of risk measures of type ρ X = f ρ f u EP [ux ], X L d F, 1.10 where u : R d R is a multivariate utility function and f u x := ux1 d, x R. In other words they can be decomposed into the function f ρ in 1.8 applied to a multivariate conditional certainty equivalent fu EP [ux ]. Moreover, we will derive the decomposition 1.4 from 1.10, i.e. in terms of u and f u. Structure of the paper In Section 2 we introduce our notation and multivariate conditional risk measures. Moreover, we give the definition and some auxiliary results for the function f ρ which has been motivated above. In Sections 3 and 4 we prove the results mentioned above for two strongly consistent conditional risk measures, where the law-invariant case is studied in Section 4. Throughout Section 5 we extend these results to families of multivariate conditional risk measures. 2 Definitions and basic results Throughout this paper Ω, F, P is a probability space. For d N we denote by L d F := {X = X 1,..., X d : X i L Ω, F, P i} the space of equivalence classes of F-measurable, P-almost surely a.s. bounded random vectors. It is a Banach space when equipped with the norm X d, := max i=1,...,d X i where F := esssup F is the supremum norm for F L Ω, F, P. We will use the usual componentwise orderings on R d and L d F, i.e. x = x 1,..., x d y = 5

y 1,..., y d for x, y R d if and only if x i y i for all i = 1,..., d, and similarly X Y if and only if X i Y i a.s. for all i = 1,..., d. Furthermore 1 d and 0 d denote the d-dimensional vectors whose entries are all equal to 1 or all equal to 0, respectively. Definition 2.1. Let F. A conditional risk measure CRM is a function possessing the following properties: ρ : L d F L, i There exists a position with zero risk, i.e. 0 Im ρ. ii Strict Antitonicity: X Y and PX > Y > 0 implies ρ X ρ Y and P ρ X < ρ Y > 0. iii -Locality: For all A we have ρ X1 A + Y 1 A C = ρ X1 A + ρ Y 1 A C. iv Lebesgue property: If X n n N L d F is a d, -bounded sequence such that X n X P-a.s., then ρ X = lim ρ X n P-a.s. We remark that the properties in Definition 2.1 are standard in the literature on conditional risk measures. Note that strict antitonicity is sometimes also referred to as strong sensitivity in the literature. In order to stress the dimension we often use the term univariate conditional risk measure for a conditional risk measure as defined in Definition 2.1 with d = 1 and we typically denote it by η. For d > 1 the risk measure ρ of Definition 2.1 is called multivariate conditional risk measure. A standard assumption on univariate CRMs is cash-additivity, i.e. η X + α = η X α for all α L, which in particular implies that we postulate a certain behavior of the risk measure η on -constants α L which turns out to be helpful in the study of consistency. Since we do not require this property - given that a multivariate analogue is tricky to define and probably not reasonable to ask for - we will have to extract the behavior of a CRM on constants in the following way. Definition 2.2. For every CRM ρ we introduce the function and the corresponding inverse function f ρ : L L ; α ρ α1 d f ρ : Im f ρ L ; β α such that f ρ α = β. 6

Remark 2.3. Note that the strict antitonicity of ρ implies that the inverse function fρ in Definition 2.2 is well-defined. Indeed let β Im f ρ and α 1, α 2 L such that f ρ α 1 = β = f ρ α 2. Suppose that PA > 0 where A := {α 1 > α 2 }. Then by strict antitonicity and -locality we obtain that β1 A + ρ 0 d 1 A C = ρ α 1 1 d 1 A + ρ 0 d 1 A C = ρ α 1 1 d 1 A ρ α 2 1 d 1 A = ρ α 2 1 d 1 A + ρ 0 d 1 A C = β1 A + ρ 0 d 1 A C, and the inequality is strict with positive probability which is a contradiction. Thus we have that Pα 1 > α 2 = 0. The same argument for {α 1 < α 2 } yields α 1 = α 2 P-a.s. Next we will show that properties of ρ transfer to f ρ and fρ. Since the domain of fρ might be only a subset of L, we need to adapt the definition of the Lebesgue property for fρ in the following way: If β n n N Im f ρ is a sequence which is lower- and upper-bounded by some β, β Im f ρ, i.e. β β n β for all n N, and such that β n β P-a.s., then fρ β n fρ β P-a.s. Note that this alternative definition of the Lebesgue property is equivalent to Definition 2.1 iv if the domain is L. The properties strict antitonicity and locality of f ρ or fρ are defined analogous to Definition 2.1 ii and iii. Lemma 2.4. Let f ρ and fρ be as in Definition 2.2. Then f ρ and fρ are strictly antitone, -local and fulfill the Lebesgue property. Proof. For f ρ the statement follows immediately from the definition and the corresponding properties of ρ. Concerning the properties of fρ, we start by proving strict antitonicity. Let β 1, β 2 Im f ρ such that β 1 β 2 and Pβ 1 > β 2 > 0. Suppose that PA > 0 where A := { fρ β 1 > fρ β 2 }. Then β 1 1 A + f ρ 01 A C = f ρ f ρ β 1 1 A + f ρ 01 A C f ρ f ρ β 2 1 A = β2 1 A + f ρ 01 A C, = f ρ f ρ β 1 1 A and the inequality is strict on a set with positive probability since f ρ is strictly antitone. This of course contradicts β 1 β 2. ence fρ β 1 fρ β 2. Moreover, as Pβ 1 > β 2 = P f ρ f ρ β 1 > f ρ f ρ β 2 > 0 we must have f ρ β 1 f ρ β 2 with positive probability, i.e. P f ρ β 1 < f ρ β 2 > 0. Now we show that fρ is -local. Let β 1, β 2 Im f ρ as well as A be arbitrary. Further let α i = fρ β i, i = 1, 2, i.e. f ρ α i = β i. Then we have that f ρ α 1 1 A + α 2 1 A C = f ρ α 1 1 A + f ρ α 2 1 A C = β 1 1 A + β 2 1 A C. 7

Thus f ρ β 1 1 A + β 2 1 A C = α 1 1 A + α 2 1 A C. Finally for the Lebesgue property let β, β Im f ρ and let β n n N Im f ρ be a sequence with β β n β for all n N and β n β P-a.s. Consider the bounded sequences βn u := sup k n β k and βn d := inf k n β k, n N which converge monotonically almost surely to β, i.e. βn u β βn u β for all n N which by antitonicity of fρ f ρ β, we observe that the sequence f ρ β u n n N β P-a.s. and βn d β P-a.s. Since yields fρ β fρ βn u is uniformly bounded in L. Note that by the same argumentation also the sequences fρ βn d n N and fρ β n are uniformly bounded in n N L. Next we will show that β u n Im f ρ for all n N. Fix n N and set recursively A n n := {β u n = β} and A n k := {β u n = β k }\ k i=n A n i, k n, then it follows from induction that A n k, k n 1. Since sup {β, β k : k n} = max {β, β k : k n}, we have that C k n k An is a P-nullset. It follows from -locality and the Lebesgue property of f ρ that f ρ fρ β1 A n n + fρ β k 1 A n k k n = β1 A n n + f ρ = β1 A n n + lim m lim m m k=n f ρ β k 1 A n k 1 k n An k m β k 1 A n k + f ρ 0 1 k m An k k=n = β1 A n n + k n β k 1 A n k = β u n, which implies βn u Im f ρ. By a similar argumentation we obtain βn d Im f ρ. Recall that βn u β P-a.s. which by antitonicity of fρ implies that the sequence f ρ βn u is isotone and thus α = lim n N fρ βn u exists in L. It follows from antitonicity and the Lebesgue property of f ρ that β = lim β u n = lim f ρ f ρ β u n = f ρ α, and hence that indeed α = fρ β. Analogously, we obtain that f ρ ˆα = β for ˆα = lim fρ βn, d and thus ˆα = α = fρ β. ence, by antitonicity of fρ fρ β = lim fρ βn u lim inf f ρ β n lim sup fρ β n lim fρ βn d = fρ β, so lim fρ β n = fρ β, i.e. fρ has the Lebesgue property. 8

An important observation that will be needed later on is that the domain of f ρ is equal to the image of ρ, i.e. f ρ ρ X is well-defined for all X L d F. Lemma 2.5. For a CRM ρ : L d F L it holds that ρ L d F = f ρ L. Proof. Clearly, ρ L d F f ρ L. For the reverse inclusion let X L d F. Our aim is to show that there exists an α L such that ρ X = f ρ α. 2.1 Define P := { α L : f ρ α ρ X }. As X d, 1 d X X d, 1 d we have that X d, P, so P. Moreover, P is bounded from above by X d, since if A := {α > X d, } for α L has positive probability, then by -locality and strict antitonicity f ρ α1 A = f ρ α1 A 1 A f ρ X d, 1 A 1 A = f ρ X d, 1 A ρ X1 A where the first inequality is strict with positive probability, so α P. By - locality it also follows that P is upwards directed. ence, for α := esssup P there is a uniformly bounded sequence α n n N P such that α = lim α n P-a.s.; see Föllmer and Schied 2011 Theorem A.33. Thus it follows that α L and f ρ α = lim f ρ α n ρ X, i.e. α P. Let B := {f ρ α > ρ X} and note that by the Lebesgue property B = n N{f ρ α + 1/n > ρ X} P-a.s. ence, if PB > 0 it follows that PB n > 0 for some B n := {f ρ α + 1/n > ρ X}. Note that B n and that α 1 B C n + α + 1/n1 Bn P by -locality of f ρ. But this contradicts the definition of α. ence, PB = 0. Sometimes it will be useful to normalize the CRM in the following sense: 9

Definition 2.6. We call a CRM ρ : L d F L normalized on constants if f ρ α = α for all α L. Indeed let ρ : L d F L be a CRM and define ρ := f ρ ρ. Then ρ is a CRM Lemma 2.4 and Lemma 2.5 which is normalized in the sense of being normalized on constants as defined above. We call ρ the normalized CRM of ρ. 3 Strong consistency In this section we study consistency of CRMs. We consider the most frequently used consistency condition for univariate risk measures in the literature which is known as strong consistency and extend it to the multivariate case. We refer to Detlefsen and Scandolo 2005, Cheridito et al. 2006, Cheridito and Kupper 2011, Kupper and Schachermayer 2009, and Penner 2007 for more information on strong consistency of univariate risk measures, and to Feinstein and Rudloff 2013, 2015 for an extension to set-valued risk measures. Kromer et al. 2014a also study a kind of consistency for multivariate risk measures, however, as we will point out in Remark 4.11 below, their definition of consistency differs from our approach. For the remainder of this section we let and be two sub-σ-algebras of F such that, and let ρ : L d F L and ρ : L d F L be the corresponding CRMs. Definition 3.1 Strong consistency. The pair {ρ, ρ } is called strongly consistent if ρ X ρ Y ρ X ρ Y X, Y L d F. 3.1 Strong consistency states that if one risk is preferred to another risk in almost surely all states under more information, then this preference already holds under less information. Our first result shows that strong consistency can be equivalently defined by a recursive relation. Lemma 3.2. Equivalent are: i {ρ, ρ } is strongly consistent; ii For all X L d F it holds that ρ X = ρ f ρ ρ X 1 d, where f ρ was defined in Definition 2.2. Proof. i ii: As for all X L d F ρ X = ρ f ρ ρ X 1 d, 10

it follows from strong consistency that ρ X = ρ f ρ ρ X 1 d. ii i: Let X, Y L d F be such that ρ X ρ Y. Then by antitonicity of fρ and ρ it follows that ρ X = ρ fρ ρ X 1 d ρ fρ ρ Y 1 d = ρ Y. Remark 3.3. Let η and η be two univariate CRMs, where η is normalized on constants, i.e. η α = α for all α L. Then f η α = fη α = α and thus strong consistency is equivalent to η F = η η F, F L F. Remark 3.4. If {ρ, ρ } is strongly consistent so is the pair of normalized CRMs { ρ, ρ } as defined in Definition 2.6 and vice versa. Since f ρ = f ρ = id strong consistency of the normalized CRMs is equivalent to ρ F = ρ ρ F 1 d, F L F, in analogy to Remark 3.3. In the following lemma we will show that strong consistency of {ρ, ρ } uniquely determines the normalized CRM ρ. Lemma 3.5. If {ρ, ρ } is strongly consistent, then ρ uniquely determines the normalized CRM ρ = f ρ ρ. Proof. Suppose that there are two CRMs ρ 1 and ρ2 which are strongly consistent with respect to ρ, i.e. ρ f ρ ρ 1 1 X 1 d = ρ X = ρ f ρ ρ 2 2 X 1 d, X L d F. We will show that f ρ 1 ρ 1 X = f ρ 2 ρ 2 X. Suppose that there exists an { } X L d F such that A := f ρ 1 ρ 1 X > f ρ 2 ρ 2 X has positive probability. Then, by the -locality of ρ 1 and ρ2, we obtain ρ X1 A = ρ f ρ ρ 1 1 X1 A 1 d = ρ f ρ ρ 1 1 X 1 A 1 d ρ ρ 2 X 1 A 1 d = ρ f ρ 2 X1 A 1 d f ρ 2 = ρ X1 A. 3.2 where the inequality 3.2 is strict with positive probability as ρ is strictly antitone, and hence we have a contradiction. Reverting the role of ρ 1 and ρ2 in the definition of A proves the lemma. 11 ρ 2

In offmann et al. 2015 we studied under which conditions a multivariate conditional risk measure can be decomposed as in 1.4, i.e. into a conditional aggregation function and a univariate conditional risk measure. We will pursue showing that strong consistency of {ρ, ρ F } is already sufficient to guarantee a decomposition 1.4 for both ρ and ρ F. To this end we need to clarify what we mean by a conditional aggregation function: Definition 3.6. We call a function Λ : L d F L F a conditional aggregation function if it fulfills the following properties: Strict isotonicity: X Y and PX > Y > 0 implies ΛX ΛY and P ΛX > ΛY > 0. F-Locality: ΛX1 A + Y 1 A C = ΛX1 A + ΛY 1 A C for all A F; Lebesgue property: For any uniformly bounded sequence X n n N in L d F such that X n X P-a.s., we have that ΛX = lim ΛX n P-a.s. Moreover for F, we call Λ a -conditional aggregation function if in addition ΛL d J L J for all J F. Remark 3.7. The name -conditional aggregation function refers to the fact that Λx L for all x R d. Thus every conditional aggregation function is at least a F-conditional aggregation function. As for conditional risk measures we define: Definition 3.8. For a conditional aggregation function Λ : L d F L F let and f Λ : L F L F; F ΛF 1 d f Λ : Im f Λ L F; F such that f Λ F =. Lemma 3.9. Let Λ : L d F L F be a conditional aggregation function. Then f Λ and f Λ are strictly isotone, F-local, and fulfill the Lebesgue property. Moreover, ΛL d F = f ΛL F and ΛX = Λ f Λ ΛX1 d for all X L d F. The well-definedness of f Λ follows as in Remark 2.3. Further the proof of Lemma 3.9 is analogous to the proofs of Lemma 2.4 and Lemma 2.5 and therefore omitted here. In order to state the decomposition result for strongly consistent CRMs, we first recall the main result from offmann et al. 2015 adapted to the framework of this paper in Proposition 3.11 for which we need the following definition. 12

Definition 3.10. We say that a function ρ : L d F L has a continuous realization ρ,, if for all X L d F there exists a representative ρ X, of the equivalence class ρ X such that ρ : R d Ω R; x, ω ρ x, ω is continuous in its first argument P-a.s. Proposition 3.11. Let ρ : L d F L be a CRM and suppose that there exists a continuous realization ρ, which satisfies risk-antitonicity: ρ Xω, ω ρ Y ω, ω P-a.s., implies ρ X ρ Y. Then there exists a -conditional aggregation function Λ : L d F L F and a univariate CRM η : Im Λ L such that ρ X = η Λ X for all X L d F and This decomposition is unique. η Λ X = Λ X for all X L d. 3.3 Proof. Since ρ is antitone, R d x ρ x is antitone. It has been shown in offmann et al. 2015 Theorem 2.10 that this property in conjunction with the fact that ρ has a continuous realization which fulfills risk-antitonicity is sufficient for the existence and uniqueness of a function Λ : L d F L F which is isotone, F-local and fulfills the Lebesgue property and a function η : Im Λ L which is antitone such that ρ = η Λ and η Λ x = Λ x for all x R d. 3.4 Note that in the proof of Theorem 2.10 in offmann et al. 2015 Λ is basically constructed by setting Λ Xω = ρ Xω, ω, which implies that Λ is necessarily F-local even though this is not directly mentioned in the paper. Indeed in offmann et al. 2015 we do not require or mention locality at all. It remains to be shown that Λ is a -conditional aggregation function, η is a univariate CRM on Im Λ, and that 3.3 holds. First of all, we show that F- locality and 3.4 imply 3.3. To this end denote by S the set of F-measurable simple random vectors, i.e. X S if X is of the form X = k i=1 x i1 Ai, where k N, x i R d and A i F, i = 1,..., k, are disjoint sets such that PA i > 0 and P k i=1 A i = 1. Now let X L d. Pick a uniformly bounded sequence kn X n n N = S such that A n i for all i = 1,..., k n, n N, i=1 xn i 1 A n i n N and X n X P-a.s. Then by 3.4, F-locality and the Lebesgue property of Λ and ρ we infer that Λ X = lim Λ X n = = lim k n i=1 lim k n i=1 Λ x n i 1 A n i ρ x n i 1 A n i = lim ρ X n = ρ X, 13

which proves 3.3. Next we show that Λ is a -conditional aggregation function. The yet missing properties which need to be verified are strict antitonicity and that Λ is -conditional. The latter follows from offmann et al. 2015 Lemma 3.1. As for strict antitonicity let X, Y L d F with X Y such that PX > Y > 0. Then by isotonicity of Λ we have that Λ X Λ Y. Suppose that Λ X = Λ Y P-a.s., then ρ X = η Λ X = η Λ Y = ρ Y which contradicts strict antitonicity of ρ. Thus Λ fulfills all properties of a -conditional aggregation function. As for η, note that by Lemma 3.9 for all F Im Λ we have that where f Λ η F = η Λ f Λ F 1 d = ρ f Λ F 1 d, 3.5 are strictly monotone, -local, and fulfill the Lebesgue property, so does η, i.e. η is a univariate CRM on Im Λ. was defined in Definition 3.8. Since ρ and f Λ Theorem 3.12. Let ρ : L d F L and ρ F : L d F L F be CRMs such that {ρ, ρ F } is strongly consistent. Moreover, suppose that f ρ F ρ F x R for all x R d. 3.6 If ρ has a continuous realization ρ,, then there exists a -conditional aggregation function Λ : L d F L F and a univariate CRM η : Im Λ L such that and ρ X = η Λ X for all X L d F 3.7 η Λ X = Λ X for all X L d. Let Λ F := ρ F and η F := id so that ρ F = η F Λ F for the F-conditional aggregation function Λ F and the univariate CRM η F. Then Λ F X Λ F Y = Λ X Λ Y X, Y L d F, 3.8 i.e. Λ and Λ F are strongly consistent. Conversely, suppose that the CRM ρ : L d F L satisfies 3.7, then {ρ, ρ F } is strongly consistent where ρ F := Λ is a CRM. We remark that in Theorem 3.12 we require consistency of the pair {ρ, ρ F } where ρ F is a CRM given the full information F. Note that ρ F is apart from the sign simply a conditional aggregation function as defined in Definition 3.6, so ρ is required to be consistent with some aggregation function under full information. This also explains Λ F. For d = 1 this consistency is automatically satisfied by monotonicity and the aggregation is simply the identity function, 14

and clearly the assertion is trivial anyway. For higher dimensions, Theorem 3.12 states that if there exists an aggregation function which is consistent with ρ, then ρ is automatically of type 3.7. Clearly, if we already know that 3.7 holds true, then ρ is consistent with ρ F = Λ. Consistency with an aggregation under full information is a very natural requirement, because even under full information, so without risk, typically the losses still need to be aggregated in some way, and therefore any CRM under less information should respect this aggregation. Note also that the condition 3.6 is a slight strengthening of being normalized on constants, the latter being automatically satisfied by the very definition of the normalization fρ F ρ F ; see above. The following proof of Theorem 3.12 is based on two observations: ρ F is necessarily risk-antitone as defined in Proposition 3.11. Strong consistency in turn implies that risk-antitonicity of ρ F is passed on backwards to ρ, and hence Proposition 3.11 applies. Proof of Theorem 3.12: In case we already know that 3.7 holds, then by antitonicity of η it follows that {ρ, Λ } is strongly consistent, and clearly Λ : L d F L F is also a CRM. Thus the last assertion of Theorem 3.12 is proved. In order to show the first part of Theorem 3.12, we recall that the only property which remains to be shown in order to apply Proposition 3.11 is riskantitonicity of ρ : For this purpose we first consider simple random vectors X, Y S where S was defined in the proof of Proposition 3.11. Note that there is no loss of generality by assuming that X = n i=1 x i1 Ai S and Y = n i=1 y i1 Ai S, i.e. the partition A i i=1,...,n of Ω is the same for X and Y. Suppose that ρ Xω, ω ρ Y ω, ω P-a.s. It follows that ρ x i, ω ρ y i, ω for all ω A i \N, i = 1,..., n, where N is a P-nullset. We claim that this implies f ρ ρ x i f ρ ρ y i for all i = 1,..., n. 3.9 In order to verify this, we first notice that as ρ and ρ F are strongly consistent and by 3.6 we have for all x R d that f ρ ρ x = f ρ ρ f ρ F ρf x 1 d = f ρ F ρf x R. 3.10 ere we also used that the normalization fρ ρ is normalized on constants. In other words fρ ρ x i and fρ ρ y i are real numbers. Next we define B i := {ω Ω ρ x i, ω ρ y i, ω}. Then A i \ N B i and hence PB i > 0 for all i = 1,..., n. Using antitonicity and -locality of fρ we obtain f ρ ρ x i 1 Bi = f ρ ρ x i 1 Bi 1Bi f ρ ρ y i 1 Bi 1Bi = f ρ ρ y i 1 Bi. As f ρ ρ y i are indeed real numbers, 3.9 follows. 15

Now by strong consistency of {ρ, ρ F }, F-locality of ρ F 3.10 as well as antitonicity of ρ we obtain ρ X = ρ f ρ F ρf X 1 d n = ρ i=1 = ρ Y, f ρ ρ x i 1 Ai 1 d = ρ n i=1 f ρ F ρf x i 1 Ai 1 d ρ n i=1 and f ρ F, and by f ρ ρ y i 1 Ai 1 d which proves risk-antitonicity for simple random vectors X, Y S. For general X, Y L d F with ρ Xω, ω ρ Y ω, ω for P-a.e. ω Ω we can find uniformly bounded sequences X n n N, Y n n N S such that X n X and Y n Y P-a.s. for n. Then by antitonicity ρ X n ω, ω ρ Xω, ω ρ Y ω, ω ρ Y n ω, ω for P-a.s. Therefore, ρ X n ρ Y n and the Lebegue property of ρ yield ρ X = lim ρ X n lim ρ Y n = ρ Y. Thus ρ is risk-antitone and we apply Proposition 3.11. ence, there is a - conditional aggregation function Λ : L d F L F and a univariate CRM η : Im Λ L such that ρ = η Λ and η Λ X = Λ X for all X L d. Let X, Y L d F such that Λ F X = ρ F X ρ F Y = Λ F Y and let X n n N S and Y n n N S be uniformly bounded sequences such that X n X and Y n Y P-a.s. for n. Again there is no loss in assuming that both X n and Y n for given n N are defined over the same partition, i.e. X n = k n i=1 xn i 1 A n and Y i n = k n i=1 yn i 1 A n. By the F-locality and antitonicity of i ρ F it follows that for all n N k n i=1 ρ F x n i 1 A n i ρ F X ρ F Y k n i=1 ρ F y n i 1 A n i. As fρ F ρ F x n i and fρ F ρ F yi n are real numbers according to assumption 3.6 and as the above computation shows that fρ F ρ F x n i fρ F ρ F yi n on A n i, we obtain, as above that indeed fρ F ρ F x n i fρ F ρ F yi n, i = 1,..., k n. Now strong consistency and 3.3 imply that Λ x n i = ρ x n i = ρ f ρ F ρ F x n i ρ f ρ F ρ F y n i = ρ y n i = Λ y n i 16

and hence by -locality of Λ Λ X n = k n i=1 Λ x n i 1 A n i k n i=1 Λ y n i 1 A n i = Λ Y n. Finally we conclude with the Lebesgue property that Λ X = lim Λ X n lim Λ Y n = Λ Y. Remark 3.13. We know from Lemma 3.9 that the inverse function f Λ of f Λ is isotone and that Λ X = Λ f Λ Λ X1 d for all X L d F. Therefore it can be shown as in Lemma 3.2, that 3.8 is equivalent to f Λ Λ X = f Λ F ΛF X, for all X L d F. Note that we cannot write the recursive form of the strong consistency of two CRMs ρ and ρ F as above, since f ρ is only defined on L and not on L F in constrast to f Λ. In the following Theorem we summarize our findings from Proposition 3.11 and Theorem 3.12 on CRMs which extend the results in offmann et al. 2015 for strong consistency: Theorem 3.14. If ρ : L d F L is a CRM with a continuous realization ρ, and satisfies fρ ρ x R for all x R d, then the following three statements are equivalent i ρ, is risk-antitone; ii ρ is decomposable as in 3.7; iii ρ is strongly consistent with some aggregation function Λ : L d F L F, i.e. {ρ, Λ} is strongly consistent. Proof. The equivalence of ii and iii has been shown in Theorem 3.12 and that i implies ii follows from Proposition 3.11. Finally, the proof of Theorem 3.12 shows that iii implies i. 4 Conditional law-invariance and strong consistency As in the previous section, if not otherwise stated, throughout this section we let and be two sub-σ-algebras of F such that, and let ρ : L d F L and ρ : L d F L be the corresponding CRMs. 17

Definition 4.1. A CRM ρ is conditional law-invariant if ρ X = ρ Y whenever the -conditional distributions µ X and µ Y of X, Y L d F are equal, i.e. if PX A = PY A for all Borel sets A BR d. In case = {, Ω} is trivial, conditional law-invariance of ρ is also referred to as law-invariance. In the law-invariant case we will often have to require a little more regularity of the underlying probability space Ω, F, P: Definition 4.2. We say that Ω, F, P is atomless, if Ω, F, P supports a random variable with continuous distribution; conditionally atomless given F, if Ω, F, P supports a random variable with continuous distribution which is independent of. The next lemma shows that conditional law-invariance is passed from ρ forward to ρ by strong consistency. The proof is based on Föllmer 2014. is conditionally law- Lemma 4.3. If {ρ, ρ } is strongly consistent and ρ invariant, then ρ is also conditionally law-invariant. Proof. Let X, Y L F such that µ X = µ Y and let A := {ρ X > ρ Y }. Then the random variables X1 A and Y 1 A have the same conditional distribution given. As ρ is conditionally law-invariant and strongly consistent with ρ we obtain ρ f ρ ρ X1 A + ρ 0 d 1 A C 1d = ρ X1 A = ρ Y 1 A = ρ fρ ρ Y 1 A + ρ 0 d 1 A C On the other hand, by strict antitonicity of ρ and f ρ ρ fρ ρ X1 A + ρ 0 d 1 A C 1d ρ f ρ ρ Y 1 A + ρ 0 d 1 A C 1d. 1d, and the inequality is strict with positive probability if PA > 0. Thus A must be a P-nullset and interchanging X and Y in the definition of A shows that indeed ρ X = ρ Y. While in Theorem 3.12 we had to require that the strongly consistent pair {ρ, ρ } satisfies = F, in this section we in some sense require the opposite extreme, namely that = {, Ω} is trivial while F. Assumption 1. For the rest of the section we assume that = {, Ω}. For simplicity we will write ρ := ρ = ρ {,Ω}. Lemma 4.4. Let {ρ, ρ } be strongly consistent and suppose that ρ is law-invariant and thus ρ is conditionally law-invariant by Lemma 4.3. If Ω,, P is an atomless probability space and X L d F is independent of, then fρ ρ X = fρ ρx. 18

The proof of Lemma 4.4 is adapted from Kupper and Schachermayer 2009. Proof. We distinguish three cases: Suppose that fρ ρ X fρ ρx and strictly smaller with positive probability. Then by strong consistency f ρ ρx = f ρ ρ f ρ ρ X 1 d < fρ ρ f ρ ρx 1d = f ρ ρx, by strict antitonicity of ρ which is a contradiction. Analogously it follows that it is not possible that fρ ρ X fρ ρx and Pfρ ρ X > fρ ρx > 0. There exist A, B such that PA = PB > 0 and fρ ρ X > fρ ρx on A and f ρ ρ X < fρ ρx on B. Then we have for an arbitrary m = a1 d where a R that ρx1 A + m1 A C = ρ fρ ρ X1 A + m1 A C 1 d = ρ fρ ρ X 1 A 1 d + m1 A C < ρ fρ ρx 1A 1 d + m1 A C 4.1 and similarly ρx1 B + m1 B C > ρ f ρ ρx 1B 1 d + m1 B C. 4.2 owever, as X is independent of the random vector X1 A +m1 A C has the same distribution under P as X1 B +m1 B C. Note that also fρ ρx 1A + a1 A C and fρ ρx 1B +a1 B C share the same distribution under P. ence, as ρ is law-invariant, 4.1 and 4.2 yield a contradiction. Now we are able to extend the representation result of Föllmer 2014 to multivariate CRMs. Theorem 4.5. Let Ω,, P be atomless and let Ω, F, P be conditionally atomless given. Suppose that ρ is law-invariant. Then, {ρ, ρ } is strongly consistent if and only if ρ and ρ are of the form ρx = g f u EP [ux] for all X L d F 4.3 and ρ X = g f u EP [ux ] for all X L d F 4.4 19

where u : R d R is strictly increasing and continuous, fu : Im f u R is the inverse function of f u : R R; x ux1 d and g : R R and g : L L are strictly antitone, fulfill the Lebesgue property, 0 Im g Im g, and g is -local. In particular, for any CRM of type 4.3 or 4.4 we have that g = f ρ g = f ρ, where f ρ and f ρ are defined in Definition 2.2. The common function u : R d R appearing in 4.3 and 4.4 can be seen as a multivariate utility where u being strictly increasing means that x, y R d with x y and x y implies ux > uy. So fu EP [u ] and fu EP [u ] are conditional certainty equivalents in the univariate case d = 1 we clearly have fu = u. Thus if ρ and/or ρ in Theorem 4.5 are normalized on constants and hence f ρ id or f ρ id, then ρ and/or ρ equal minus certainty equivalents. But 4.3 and 4.4 also comprise other prominent classes of risk measures. For instance if f ρ = f u or f ρ = f u, then ρ X = E P [ux] is an multivariate expected utility whereas ρ X = E P [ux ] is a multivariate conditional expected utility. Proof. For the last assertion of the theorem note that since u is a deterministic function, we have for α L that f ρ α = ρ α1 d = g f u EP [uα1 d ] = g f u fu α = g α and analogously we obtain f ρ g. Next we prove sufficiency in the first statement of the theorem: Let ρ and ρ be as in 4.4 and 4.3. It is easily verified that ρ and ρ are conditionally lawinvariant CRMs. Furthermore, since fu is strictly increasing and g is strictly antitone and -local, we have for each X, Y L d F with ρ X ρ Y that E P [ux ] E P [uy ]. But this implies that also E P [ux] E P [uy ] and thus that ρx ρy, i.e. {ρ, ρ } is strongly consistent. Now we prove necessity in the first statement of the theorem: We assume in the following that ρ and ρ are normalized on constants and follow the approach of Föllmer 2014 Theorem 3.4. The idea is to introduce a preference order on multivariate distributions µ, ν on R d, BR d with bounded support given by µ ν ρx > ρy, with X µ and Y ν. ere BR d denotes the Borel-σ-algebra on R d and X µ means that the distribution of X L d F under P is µ. It is well-known that if this preference 20

order fulfills a set of conditions, then there exists a von Neumann-Morgenstern representation, that is µ ν ux µdx < ux νdx, 4.5 where u : R d R is a continuous function. Sufficient conditions to guarantee 4.5 are that is continuous and fulfills the independence axiom; cf. Föllmer and Schied 2011 Corollary 2.28. We refer to Föllmer and Schied 2011 for a definition and comprehensive discussion of preference orders and the mentioned properties. Suppose for the moment that we have already proved 4.5. Note that strict antitonicity of ρ implies that δ x δ y whenever x, y R d satisfy x y and x y. ence ux = us δ x ds > us δ y ds = uy, and we conclude that u is necessarily strictly increasing as claimed. Now we prove 4.5: The proof of continuity of is completely analogous to the corresponding proof in Föllmer 2014 Theorem 3.4, so we omit it here. The crucial property is the independence axiom, which states that for any three distributions µ, ν, ϑ such that µ ν and for all λ 0, 1], we have λµ + 1 λϑ λν + 1 λϑ. Since Ω, F, P is conditionally atomless given, we can find X, Y, Z L d F which are independent of such that X µ, Y ν and Z ϑ. Furthermore, since Ω,, P is atomless, we can find an A with PA = λ. It can be easily seen that X1 A +Z1 A C λµ+1 λϑ and Y 1 A +Z1 A C λν+1 λϑ. Moreover, since µ ν, we have that ρx ρy. As {ρ, ρ } is strongly consistent and as ρ is law-invariant, we know from Lemma 4.3 that ρ is conditionally lawinvariant. This ensures that we can apply Lemma 4.4 to the random vectors X and Y which are independent of. Therefore, by -locality of ρ and recalling Remark 3.4 ρ X1 A + Z1 A C = ρ ρ X1 A + Z1 A C 1 d = ρ ρ X1 A 1 d ρ Z1 A C1 d = ρ ρx1 A 1 d ρ Z1 A C1 d ρ ρy 1 A 1 d ρ Z1 A C1 d = ρ Y 1 A + Z1 A C, which is equivalent to λµ + 1 λϑ λν + 1 λϑ. Thus there exists a von Neumann-Morgenstern representation 4.5 with a continuous and strictly increasing utility function u : R d R. In the next step we define f u : R R; x ux1 d. Then f u is strictly increasing and continuous and thus fu exists. Let µ be an arbitrary distribution on R d, BR d with bounded support and X µ. Then ρ X d, 1 d ρx ρ X d, 1 d 21

and hence f u X d, = ux δ X d, 1 d dx ux µdx ux δ X d, 1 d dx = f u X d,. The intermediate value theorem now implies the existence of a constant cµ R such that f u cµ = ux µdx cµ = fu ux µdx. Finally, since δ cµ1d µ, we have ρx = ρ cµ1 d = cµ = f u ux µdx = fu EP [ux]. ence, we have proved 4.3 with g id. Define ψ X := fu EP [ux ], X L d F, then we have seen in the first part of the proof that ψ is a CRM which is strongly consistent with ρ. Moreover, ψ is normalized on constants. Thus it follows by Lemma 3.5 that ρ = ψ. If ρ and/or ρ are not normalized on constants, then considering the normalized CRMs fρ ρ and fρ ρ as introduced after Definition 2.6, the result follows from ρ = f ρ fρ ρ and ρ = f ρ fρ ρ, i.e. g = f ρ and g = f ρ. Recall Theorem 3.12 where we proved that if a multivariate CRM ρ is strongly consistent in a forward looking way with an aggregation ρ F under full information F and ρ F fulfills 3.6, then the multivariate CRM can be decomposed as in 3.7. The following Theorem 4.6 shows that we also obtain such a decomposition 3.7 under law-invariance by requiring strong consistency of ρ in a backward looking way with ρ given trivial information {, Ω}. When stating Theorem 4.6 we will need an extension of f ρ to L F: Suppose that the process R a f ρ a allows for a continuous realization. Due to the fact that ρ is strictly antitone and -local, we can find a possibly different realization f ρ, such that f ρ : R Ω R : x f ρ x, ω is continuous and strictly decreasing in the first argument for all ω Ω. Note that there exists a well-defined inverse f ρ, ω of f ρ, ω for all ω Ω. Now define the functions f ρ : L F L F; F f ρ F ω, ω 4.6 and f ρ : Im f ρ L F; F f ρ F ω, ω, 22

where we with the standard abuse of notation identify the random variable f ρ F ω, ω or f ρ F ω, ω with the equivalence classes they generate in L F. By construction of f ρ we have that f ρ L J L J for all σ-algebras J such that σ f ρ a,, a R J F, c.f. offmann et al. 2015 Lemma 3.1. By definition f ρ is also F-local and has the Lebesgue property due to continuity of R a f ρ a, ω. Moreover, -locality and continuity also imply that indeed f ρ X = f ρ X for all X approximation by simple random variables, so f ρ is indeed an extension of f ρ to L F. Theorem 4.6. Under the same conditions as in Theorem 4.5 let {ρ, ρ } be strongly consistent. Then ρ can be decomposed as where Λ : L d F L F; ρ = η Λ, is a {, Ω}-conditional aggregation function, X f ρ f u ux η : Im Λ R; F U E P [UF ] is a law-invariant univariate certainty equivalent given by the deterministic utility U : Im ρ R; a f u f ρ a which is strictly increasing and continuous. ere u : R d R is the multivariate utility function from Theorem 4.5. If the function R a f ρ a has a continuous realization, then ρ can be decomposed as ρ = η Λ, with where η Λ X = Λ X, for all X L d, Λ : L d F L F; X f ρ fu ux is a σ f ρ a, : a R- conditional aggregation function f ρ a, denotes a continuous realization with strictly increasing paths; η : Im Λ L ; F U E P [U F ] is a univariate conditional certainty equivalent; the stochastic utility U : Im Λ L F; F f u f ρ F is strictly isotone, F-local, fulfills the Lebesgue property and U Im U L L ; 23

f ρ is given in 4.6. Moreover, it holds that U Λ = u = U Λ 4.7 are deterministic and independent of the chosen information or {Ω, }. Finally we also have that f Λ Λ = fu u = f Λ Λ, i.e. {Λ, Λ } is strongly consistent as defined in 3.8. Proof. By Theorem 4.5 we have that ρ X = f ρ f u E P [ux ] = f ρ f u EP [ fu f ρ fρ f u ux ], where u and f u are given in Theorem 4.5. ence, recalling the definitions of U, η, and Λ, we have ρ = η Λ. It can be readily seen that U as well as U, and thus also Λ, are F-local, strictly isotone, and fulfill the Lebesgue property. As f ρ L J L J for all σ-algebras J such that σ f ρ a, ω : a R J F, the same also applies to Λ = f ρ fu u and we conclude that Λ is a σ f ρ a, ω : a R-conditional aggregation function. Moreover, for X L d η Λ X = f ρ f u ux = U ux = Λ X. The result for ρ follows similarly to the proof above without requiring a continuous realization and by using the canonical extension of f ρ from R to L d F, i.e. f ρ F ω = f ρ F ω for all ω Ω and F L F. We remark that 4.7 is the crucial fact which ensures that ρ and ρ are strongly consistent and conditionally law-invariant. In Theorem 4.6 we have seen that basically every CRM which is strongly consistent with a law-invariant CRM under trivial information can be decomposed into a conditional aggregation function and a univariate conditional certainty equivalent. For the rest of this section we study the effect of additional properties of the CRMs on this decomposition. For instance, we want to identify conditions under which the univariate conditional certainty equivalent is generated by a deterministic instead of a stochastic utility function; see Corollary 4.7. Also we study what happens if the univariate CRMs η and η from Theorem 4.6 are required to be strongly consistent; see Corollary 4.9. Corollary 4.7. In the situation of Theorem 4.6, if ρ is normalized on constants, then ΛX = fu ux, X L d F, and ηf = ρf 1 d = f u E P [f u F ], F L F. If ρ is normalized on constants, then similarly Λ X = f u ux, X L d F, 24

and η F = ρ F 1 d = f u E P [f u F ], F L F. In particular the univariate conditional certainty equivalent η is now given by the deterministic univariate utility function f u, and thus η is conditionally lawinvariant. If both ρ and ρ are normalized on constants, then Λ = Λ. Remark 4.8. Suppose that ρ and ρ from Theorem 4.6 are normalized on constants and that for all F, L F, m, λ R with λ 0, 1 as well as ρf 1 d + m1 d = ρf 1 d m 4.8 ρ λf 1 d + 1 λ1 d λρf 1d + 1 λρ1 d. 4.9 Recalling Corollary 4.7 it follows that ηf = ρf 1 d is cash-additive 4.8 and convex 4.9. Since f u is a deterministic function it can be easily checked that η and η are strongly consistent conditionally law-invariant univariate CRMs. Therefore we are in the framework of Föllmer 2014. There it is shown that the univariate CRMs must be either linear or of entropic type, i.e. f u x = ax + b or f u x = ae βx + b, x R, for constants a, b, β R with a, β > 0, which implies that η F = E P [ F ] or η F = 1 β log E P [ e βf ] and similarly for η. Clearly, this also has consequences for the aggregation function Λ = Λ = fu u since x ux1 d = f u x is either of linear or exponential form. For instance, a possible aggregation would be given by ux 1,..., x d = a d i=1 w ix i + b, where w i 0, 1 for i = 1,..., d such that d i=1 w i = 1, because f u x = ax + b. In this case the aggregation function is simply Λx = d i=1 w ix i. Corollary 4.9. In the situation of Theorem 4.6, suppose that η and η defined on all of L F. Then {η, η } are strongly consistent if and only if are η = ũ E P [ũf ] and η = ũ E P [ũf ] for a continuous and strictly increasing utility function ũ : R R. Moreover, the corresponding conditional aggregation functions are given by Λ = f ρ f u u and Λ = f ρ f u a u, where a F = αf + β, F L F, is a positive affine transformation given by α, β L with Pα > 0 = 1. 25