On Kusuoka Representation of Law Invariant Risk Measures

MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of Law Invariant Risk Measures Alexander Shapiro School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 3332, ashapiro@isye.gatech.edu In this paper we discuss representations of law invariant coherent risk measures in a form of integrals of the average value-at-risk measures. We show that such an integral representation exists iff the dual set of the considered risk measure is generated by one of its elements, and this representation is uniquely defined. On the other hand, representation of risk measures as a maximum of such integral forms is not unique. The suggested approach gives a constructive way for writing such representations. Key words: coherent risk measures; law invariance; average value-at-risk; Fenchel-Moreau theorem; comonotonic risk measures MSC2 subject classification: Primary: 9B5 OR/MS subject classification: Primary: decision analysis; secondary: risk History: Received September 25, 211; revised May 14, 212. Published online in Articles in Advance November 14, 212. 1. Introduction. In this paper we discuss Kusuoka [8] representations of law invariant coherent risk measures. We give a somewhat different view of such representations from the standard literature. In particular, we show that for comonotonic risk measures the corresponding representation is unique, while such uniqueness does not hold for general law invariant coherent risk measures. Let us recall some basic definitions and results. Consider a probability space F P and the space = L p F P, p 1, of measurable functions Z (random variables) having finite pth order moment; for p = the space = L F P is formed by essentially bounded measurable functions. Unless stated otherwise all probabilistic statements, in particular expectations, will be with respect to the reference probability distribution P. A function is said to be a coherent risk measure if it satisfies the following axioms (Artzner et al. [2]): (A1) Monotonicity: If Z, Z and Z Z, then Z Z. (A2) Convexity: tz + 1 tz tz + 1 tz for all Z, Z and all t 1. (A3) Translation equivariance: If a and Z, then Z + a = Z + a. (A4) Positive homogeneity: If t and Z, then tz = tz. The notation Z Z means that Z Z for a.e.. For a thorough discussion of coherent risk measures we refer to Föllmer and Schield [6]. Risk measures satisfying axioms (A1) (A3) are called convex. A risk measure is said to be law invariant if Z depends only on the distribution of Z; i.e., if Z and Z are two distributionally equivalent random variables, then Z = Z. For p 1 the space = L p F P is paired with the space = L q F P, where q 1 is such that 1/p + 1/q = 1, with the corresponding scalar product Z = Z dp Z (1) In that case coincides with the linear space of continuous linear functionals on ; i.e., is the dual of the Banach space. For = L F P the situation is more involved; the dual of the Banach space L F P is quite complicated. The standard practice, which we follow, is to pair = L F P with the space = L 1 F P by using the respective scalar product of the form (1). It is known (Ruszczyński and Shapiro [12]) that (real-valued) convex risk measures are continuous in the norm topology of the space L p F P. Thus, for p 1 it follows by the Fenchel-Moreau Theorem that Z = sup Z Z (2) where = sup Z Z Z is the corresponding conjugate functional. For p = we pair L F P with the space L 1 F P, rather than with its dual space of continuous linear functionals. In that case we assume that is lower semicontinuous in the respective paired topology, which is the weak 142

Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS 143 topology of L F P. Then the Fenchel-Moreau Theorem can be invoked to establish the corresponding dual representation (2). If the risk measure is convex and positively homogeneous (i.e., is coherent), then is an indicator function of a convex and closed, in the respective paired topology, set, and hence the dual representation (2) takes the form Z = sup Z Z (3) The set is uniquely defined and is given by = { Z Z Z } (4) Because of the axioms (A1) and (A3), the set consists of density functions ; i.e., if, then and dp = 1 (cf., Ruszczyński and Shapiro [12]). We refer to the set as the dual set of the corresponding coherent risk measure. For p 1 the dual set is weakly compact, and hence the maximum in (3) is attained for any Z. On the other hand, for p = the space L F P is paired with L 1 F P. In that case the dual set L 1 F P is not necessarily compact in the respective paired topology (which is the weak topology of the Banach space L 1 F P), and the corresponding maximum may not be attained (see Example 1 in the next section). An important example of law invariant coherent risk measures is the average value-at-risk measure (also called the conditional value-at-risk or expected shortfall measure), which can be defined as { } AV@R Z = inf t + 1 1 ƐZ t + (5) t or equivalently as AV@R Z = 1 1 V@R Z d (6) where 1, F Z is the cumulative distribution function of Z and V@R Z = inft F Z t (7) is the corresponding left side quantile. It is natural to use here the space = L 1 F P. The dual set of AV@R is { } = 1 1 dp = 1 L F P (8) For a given Z, AV@R Z is monotonically nondecreasing in, and it follows from (6) that AV@R Z is a continuous function of 1. It is possible to show that AV@R = Ɛ, and AV@R Z tends to ess supz as 1. Therefore we set AV@R 1 Z = ess supz. In that case, for AV@R 1 Z to be real valued, we will have to use the space = L F P. The dual set of AV@R 1, for L F P paired with L 1 F P, consists of density functions { } = dp = 1 L 1 F P (9) It is interesting to note that, as was pointed out earlier, for = AV@R with 1 the dual set is weakly compact, and hence the maximum in the dual representation (3) is always attained. On the other hand, for = AV@R 1 with = L F P, the maximum in the respective dual representation may be not attained. Unless stated otherwise we assume in the remainder of this paper that the probability space F P is nonatomic. It was shown by Kusuoka [8] that any law invariant coherent risk measure can be represented in the following form: Z = sup AV@R Z d (1) where is a set of probability measures on the interval 1. Actually the original proof in Kusuoka [8] was for the L F P space; for a discussion in L p F P spaces, see, e.g., Pflug and Römisch [11]. It could be noted that if a measure has a positive mass at = 1, then the right-hand side of (1) is + for any unbounded from above function Z. Therefore for L p F P with p 1, all measures should be on the interval 1. Also, if (1) holds for some set, then it holds for the convex hull

144 Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS of. Moreover, as the following proposition shows, we can take the topological closure of this convex hull with respect to the weak topology of probability measures (see, e.g., Billingsley [3] for a discussion of the weak topology). Therefore it can be assumed that the set is convex and closed, and hence by Prohorov s theorem, is compact, in the weak topology of probability measures. Proposition 1. Let L p F P be a coherent risk measure representable in the form (1) for some set of probability measures on the interval 1. Then Z = sup where is the closure of in the weak topology of probability measures. AV@R Z d Z L p F P (11) Proof. For a given Z L p F P we have that = AV@R Z is a real valued continuous and monotonically nondecreasing function on the interval 1. If Z is essentially bounded, and hence 1 is bounded, then the assertion follows in a straightforward way from the definition of weak convergence (see Billingsley [3, p. 7]). In general, let k, k = 1 be a sequence of probability measures converging (in the weak topology) to a probability measure. Recall that the weak topology of probability measures on the interval 1 is metrizable and hence can be described in terms of convergent sequences. We need to show that d Z (12) Indeed, for any 1 we have lim k 1 d k = 1 d (13) where 1 A denotes the indicator function of set A. By adding a constant to the function Z if necessary, we can assume without loss of generality that AV@R Z >, and hence > for all 1, and thus It follows from (13) and (14) that 1 d k Moreover, by the monotone convergence theorem: lim 1 and hence (12) follows. This completes the proof. d k Z k = 1 (14) 1 d Z (15) 1 d = d (16) 2. A preliminary discussion. Let T be one-to-one onto mapping, i.e., T = T iff = and T =. It is said that T is a measure-preserving transformation if image T A = T A of any measurable set A F is also measurable and PA = PT A (see, e.g., Billingsley [3, p. 311]). Let us denote by = the set of one-to-one onto measure-preserving transformations T We have that if T, then T 1 ; and if T 1, T 2, then their composition 1 T 1 T 2. That is, forms a group of transformations. The measure-preserving transformation group was used in Jouini et al. [7] to study coherent risk measures. Let us observe that since the probability space F P is nonatomic, we have that two random variables Z, Z have the same distribution iff there exists a measure-preserving transformation T such that Z = Z T (see, e.g., Jouini et al. [7, Lemma A.4]). This implies that a risk measure is law invariant iff it is invariant with respect to measure-preserving transformations; i.e., iff for any T it follows that Z T = Z for all Z. For a measurable function denote by = T T (17) the corresponding orbit of. By the above discussion, forms a class of distributionally equivalent functions. 1 Composition T = T 1 T 2 of two mappings is the mapping T = T 1 T 2.

Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS 145 Proposition 2. A convex risk measure is law invariant iff its conjugate function is invariant with respect to measure-preserving transformations; i.e., iff for any T it follows that T = for all. Proof. For any and T we have { T = sup Z { = sup Z { = sup Z } T Z dp Z } ZT 1 dp Z } Z dp Z T where we made a change of variables: Z = Z T 1. If is law invariant and hence is invariant with respect to measure-preserving transformations, then Z T = Z, and thus it follows that T =. Conversely, recall that we have that =. Therefore, if is invariant with respect to measure-preserving transformations, then by the same argument it follows that is invariant with respect to measure-preserving transformations. Corollary 1. A coherent risk measure is law invariant iff its dual set is invariant with respect to measure-preserving transformations; i.e., iff for any it follows that. Definition 1. We say that the dual set, of a law invariant coherent risk measure, is generated by a set if Z = sup Z Z (18) where =. In particular, if = is a singleton, we say that is generated by. Note that since is law invariant, we have by Proposition 2 that the set forms a subset of the dual set. It follows from (18) that the topological closure (in the respective paired topology) of the convex hull of set coincides with the dual set. Recall that an element is said to be an extreme point of if 1, 2, t 1, and = t 1 +1 t 2 implies that 1 = 2 =. For p 1 the dual set is convex and weakly compact, and hence by the Krein-Milman Theorem coincides with the topological closure (in the weak topology) of the convex hull of the set of its extreme points. Example 1. Consider = AV@R with 1. The corresponding dual set is described in Equation (8). Let us show that the dual set is generated by its element 2 = 1 1 1 A, where A F is a measurable set such that PA = 1. The corresponding set can be written as Let Z and = 1 1 1 B B F PB = 1 t = inft P Z t 1 } = V@R 1 Z Suppose for the moment that P Z = t = ; i.e., the cumulative distribution function (cdf) F t of Z is continuous at t = t. Then for B = Z t we have that PB = 1. It follows that } sup Z = sup {1 1 Z dp PB = 1 = 1 1 Z dp B F B B Moreover, Z dp = + t df t, and hence B t sup Z = AV@R Z (19) Since the set of random variables Z having continuous cdf forms a dense subset of, it follows that formula (19) holds for all Z. We obtain that the set is generated by. We also have that coincides with the set of extreme points of the set. Indeed, if, then for a.e. A and for a.e. \A. It follows that if 1, 2, t 1, and 2 By 1 A we denote the indicator function of the set A; i.e., the 1 A = 1 if A, and 1 A = if A.

146 Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS = t 1 + 1 t 2, then 1 = 2 = ; i.e., is an extreme point of. The same arguments can be applied to every element of to show that all points of are extreme points of. Now let be such that. Then there exists > and a set B F of positive measure such that for a.e. B it holds that and + 1 1. Let us partition B into two disjoint sets B 1 and B 2 such that PB 1 = PB 2 = PB/2. Then 1 = + 1 B1 1 B2, 2 = + 1 B2 1 B1, and = 1 + 2 /2. That is, is not an extreme point of. On the other hand, consider = AV@R 1. Its dual set is the set of density functions (see (9)). This dual set does not have extreme points and is not generated by one of its elements. Since F P is nonatomic, we can assume without loss of generality that is the interval 1 equipped with its Borel sigma algebra and uniform reference distribution P. Lemma 1. Let be the interval 1 equipped with its Borel sigma algebra and uniform distribution P, and let be the corresponding group of measure-preserving transformations. Then for any measurable function 1 there exists a monotonically nondecreasing right-hand side continuous function 1 and T such that = T a.e. Proof. We can view as a random variable defined on the probability space F P. Consider its cumulative distribution function F t = P t and the right side quantile F 1 = supt F t 1 Note that the function = F 1 is monotonically nondecreasing right-hand side continuous and, since the distribution P is uniform on 1, the random variables and have the same distribution. Therefore, as it was pointed out above, there exists T such that = T a.e. If is generated by an element, then is also generated by any. It could be remarked that an element, viewed as a function of 1, is defined up to a set of measure zero. Therefore by making the transformation T for an appropriate T, we can assume that the generating element of is a monotonically nondecreasing right-hand side continuous function on the interval 1. More accurately, it should be said that is a member of the class of functions corresponding to the element. By assuming that is right-hand side continuous, we uniquely specify such a monotonically nondecreasing member function. Definition 2. Let, 1 be two integrable functions. We say that is majorized by, denoted, if d d 1 and d = d (2) The relation defines a partial order; i.e., for,, it holds that: (i), (ii) if and, then =, (iii) if and, then. Remark 1. For monotonically nondecreasing functions, 1 the above concept of majorization is closely related to the concept of dominance in the convex order. That is, if and are monotonically nondecreasing right-hand side continuous functions, then they can be viewed as right side quantile functions = FX 1 and = FY 1 of some respective random variables X and Y. It is said that X dominates Y in the convex order if ƐuX ƐuY for all convex functions u such that the expectations exist. Equivalently this can be written as (see, e.g., Müller and Stoyan [1]) F 1 X d F 1 Y d 1 and ƐX = ƐY Note that ƐX = F 1 X d and ƐY = F 1 Y d. The dominance in the convex (concave) order was used in studying risk measures in Föllmer and Schield [6] and Dana [4], for example. This is also related to the concept of majorization used in the theory of Shur convexity. That is, consider a finite set n = 1 n with the sigma algebra of all its subsets and assigned equal probabilities 1/n. This probability space can be viewed as a discretization of the probability space = 1 equipped with uniform distribution. Random variables x n on that space can be identified with vectors x = x 1 x n n, and the set of measure-preserving transformations with the group of permutations of the set 1 n. It is said that vector y n is majorized by vector x n if n n n n x i y i k = 1 n 1 and x i = y i (21) i=k i=k where x 1 x n are components of vector x arranged in increasing order (see, e.g., Marshall and Olkin [9]). i=1 i=1

Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS 147 It is not difficult to show that the following properties hold. (P1) If, then for a monotonically nondecreasing on 1 function Z it holds that Z d Z d (22) (P2) If Z and are monotonically nondecreasing on the interval 1 functions, then sup T ZT d = Z d (23) Theorem 1. Let be a law invariant coherent risk measure. Suppose that the dual set is generated by (monotonically nondecreasing). Then: (i) for every Z maximum in the dual representation (3) is attained at some element of the set, (ii) if, then, (iii) if is monotonically nondecreasing and, then, (iv) is generated by an element iff, (v) the set coincides with the set of extreme points of. Proof. Note that if a function Z is monotonically nondecreasing, then by (P2) Z = sup Z = sup T ZT d = Z d = Z (24) Now consider an element Z. We can choose T such that Z T is monotonically nondecreasing on the interval 1. By (24) we have then that T 1 Z = Z T = Z T = Z (25) where the last equality follows since is law invariant. That is, T 1 is a maximizer in the dual representation (3). This proves the assertion (i). Consider. It follows that is a density function, i.e., and d = 1. By (4) and taking Z = 1 1 we obtain that d = Z Z = Z = d Since d = d = 1, it follows that. This proves (ii). Conversely, suppose that is monotonically nondecreasing and. Then for a Z choose T such that Z = Z T is monotonically nondecreasing on the interval 1. Since is law invariant we have that Z = Z. By (22) and (24) it follows that Z = Z d Since is monotonically nondecreasing we also have that Z d Z T 1 d = Z d Z d It follows that Z Z, and hence by (4) that. This proves (iii). Now suppose that there exists generating which does not belong to. We can assume that is monotonically nondecreasing. Consider = sup 1 (26) Since, the set 1 is nonempty, and >. Since we have that there is such that < for all. For Z = 1 1 it follows that Z = d > d = Z (27) Since is monotonically nondecreasing, we also have that Z T Z for any T. It follows that and hence cannot be a generating element of. This proves (iv). sup T Z < Z (28) T

148 Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS Let us prove the assertion (v). We have that if, 1, 2 and = t 1 + 1 t 2, then T, 1 T, 2 T, and T = t 1 T + 1 t 2 T for any T. Therefore, if one of the points of is extreme, then all point of are extreme. Suppose that = t 1 + 1 t 2 for some 1, 2 and t 1, and that 1. Then there exists a subinterval I = 1 of 1 such that either for i = 1 or i = 2, we have that i for all I and i > on some subset of I of positive measure. For Z = 1 1 we have i Z = Z i d = i d > d = Z d (29) and since Z i Z, it follows that Z > Z dp. This, however, contradicts (24). This shows that is an extreme point of, and hence all points of are extreme points. Finally, we need to show that if is an extreme point of, then. We argue by a contradiction. Suppose that is an extreme point of and. Consider = 2 T for such T that is monotonically nondecreasing. By (ii) we have that, and hence d = 2 d d for any 1. Since d = d = 1, we also have that d = 1. It follows that, and hence by (iii) we have that and thus T 1. It remains to note that T 1 +/2 =, and hence since is an extreme point of it follows that =, a contradiction. It is also possible to give the following characterization of the generation of the dual set by its element. It is said that an element of is the greatest element of if for any it follows that. Clearly if the greatest element exists, it is unique. Proposition 3. Let be the dual set of a law invariant coherent risk measure. If possesses a greatest element, then is generated by. Conversely if is generated by and is monotonically nondecreasing, then is the greatest element of. Proof. Suppose that possesses a greatest element. We have that if and = T is monotonically nondecreasing on 1 for some T, then. Therefore is monotonically nondecreasing on 1. We also have by (P1) that if Z is monotonically nondecreasing and, then Z Z. It follows that the maximum of Z over is attained at, and hence is generated by. The converse assertion follows from Theorem 1(ii). 3. AV@R representations. Recall that random variables X and Y are said to be comonotonic if X Y is distributionally equivalent to FX 1 1 U FY U, where U is a random variable uniformly distributed on the interval 1 (see, e.g., Dhaene et al. [5] for a discussion of comonotonic random variables). A risk measure is said to be comonotonic if for any two comonotonic random variables X, Y it follows that X + Y = X + Y. Theorem 2. Let = L p F P, p 1, and be a law invariant coherent risk measure with the respective dual set. Then: (i) the set is generated by an element iff there exists a probability measure on the interval 1 such that Z = AV@R Z d Z (3) (ii) the probability measure in representation (3) is defined uniquely, (iii) the risk measure is comonotonic iff the set is generated by a single element, (iv) the risk measure is comonotonic iff the set of extreme points of consists of a single distributionally equivalent class. Note that the measure in the above theorem is defined on the interval 1; i.e., it has mass zero at the point = 1. To prove this theorem we need the following lemma. Lemma 2. Let 1 + be a monotonically nondecreasing, right-hand side continuous function such that d = 1. Then there exists a uniquely defined probability measure on 1 such that = 1 1 d 1 (31)

Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS 149 Proof. We can view in (31) as a right-hand side continuous, monotonically nondecreasing on the interval 1 function and the corresponding integral as the Lebesgue-Stieltjes integral. Suppose for the moment that = and define Then = 1 + d = d + 1 d + d = 1 d d (32) and hence (31) follows. If >, then we can add the mass to the measure at =. Note that if (31) holds for some measure, then 1 = d = 1 1 d d = 1 1 d d = d and hence is a probability measure. To show uniqueness of the measure we proceed as follows. Using integration by parts we have 1 1 d = 1 d1 1 = 1 1 2 d Suppose that there are two measures 1 and 2 which give the same function. Since measure 1 (and measure 2 ) is defined by the corresponding function up to a constant, we can set 1 = 2. Then for the function = 1 2 /1 we have the following equation: d = 1 (33) 1 with =. It follows that is continuous and differentiable and satisfies the equation d d = (34) 1 The last equation has solutions of the form = c1 1 for some constant c. Since =, it follows that c =, and thus =, and hence 1 = 2. Existence of probability measure satisfying (31) is shown in Föllmer and Schield [6, Lemma 4.63]. In the above lemma we also prove its uniqueness. If is a step function, i.e., = n i=1 a i1 ti 1, where a i, i = 1 n, are positive numbers and t 1 < t 2 < < t n < 1, then we can take = n a i 1 t i 1 ti 1 1 (35) i=1 Proof of Theorem 2. As was pointed out earlier, since F P is nonatomic, we can assume without loss of generality that is the interval 1 equipped with its Borel sigma algebra and uniform reference distribution. Suppose that is generated by some element. Then is also generated by = T for any T. Therefore by making an appropriate transformation we can assume that is monotonically nondecreasing on the interval 1 and right-hand side continuous. Consider Z. We have that Z = Z T for any T. We can choose T such that Z T is monotonically nondecreasing on the interval 1. Then Z = sup T ZT d = sup T ZT d = Z T d (36) Now let be a probability measure on 1 such that Equation (31) holds. It follows by (36) that Z = Z T 1 1 d d = Z T 1 1 d d (37) Since the dual set of AV@R is generated by the function 1 1 1 1 and Z T is monotonically nondecreasing, we have that Z T 1 1 d = AV@R Z T (38)

15 Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS and since measure AV@R is law invariant we also have that AV@R Z T = AV@R Z. Together with (37) this implies (3). Conversely, suppose that the representation (3) holds for some probability measure. Consider Z and let T be such that Z T is monotonically nondecreasing on the interval 1. Then Equation (38) holds, and hence (37) follows. Consequently for defined in (31) we obtain that (36) holds. This completes the proof of the assertion (i). As far as uniqueness of is concerned, lets make the following observations. By Proposition 3 the nondecreasing right-hand side continuous function generating the set is defined uniquely. Moreover, the representation (3) holds for some probability measure iff Equation (31) holds. Hence the uniqueness of follows by Lemma 2. This proves the assertion (ii). Now suppose that is generated by some element. Let X, Y be comonotonic variables; i.e., X Y is distributionally equivalent to FX 1 1 U FY U, where U is a random variable uniformly distributed on the interval 1. This means in the considered setting that both functions X, Y 1 are monotonically nondecreasing, and hence X + Y is also monotonically nondecreasing. Let T be such that = T is monotonically nondecreasing on the interval 1. Then the maximum in (3) is attained at for both X and Y and for X + Y ; i.e., X = X, Y = Y, and X + Y = X + Y. It follows that X + Y = X + Y, and hence is comonotonic. Conversely, suppose that is comonotonic. Let us argue by a contradiction; i.e., assume that is not generated by one of its elements. Then there are two monotonically nondecreasing variables X, Y such that the maximum in the right-hand side of (3), for Z = X and Z = Y, is not attained at the same. We have that there is such that X + Y = X + Y = X + Y (39) Moreover, it follows that X X and Y Y (4) with at least one of the inequalities in (4) being strict. Thus X + Y < X + Y, and hence is not comonotonic. This completes the proof of (iii). Denote by the set of extreme points of. Note that since the set is convex and weakly compact, we have that for any Z the maximum of Z over is attained at an extreme point of (this follows from the Krein-Milman Theorem), and hence Z = sup Z (41) Suppose that is comonotonic. Then by (iii) we have that is generated by an element. Thus by Theorem 1(v) we obtain that =. Conversely, suppose that =. Then by (41) we obtain that Z = sup Z; i.e., is generated by. It follows by (iii) that is comonotonic. This completes the proof of (iv). Remark 2. If the representation (3) holds with the measure having positive mass at = 1, then we can write the corresponding risk measure as = 1 + AV@R 1, where Z = AV@R Z d with being the part of the measure restricted to the interval 1 and normalized by factor 1 1. In that case we will have to use the space = L F P for to be real valued. The corresponding dual set = 1 1 + 2, where 1 is the dual set of and 2 is the dual set of AV@R 1. The set 1 is generated by one of its elements, while the set 2 is not (see the discussion of Example 1). Therefore it is essential in the assertions (i), (iii), and (iv) of Theorem 2 that p <. By combining the assertions (i) and (iii) of the above theorem we obtain (for p < ) that a law invariant coherent risk measure is comonotonic iff it can be represented in the integral form (3). For L F P spaces this was first shown by Kusuoka [8]. Uniqueness of the measure in representation (3) also holds for L F P spaces. Remark 3. By (6) we have that the representation (3) can be written as Z = 1 1 V@R Z d d = V@R Z d (42) where is given in (31); i.e., is monotonically nondecreasing on a 1 function generating the dual set of. Risk measures of the form (42) were called spectral risk measures in Acerbi [1].

Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS 151 Unless stated otherwise, we assume in the remainder of this section that = L p F P, with p 1. Let be the dual set of a law invariant coherent risk measure and let be the set of extreme points of. As was pointed out earlier, the set is invariant with respect to measure preserving transformations T, and hence is given by the union of distributionally equivalent classes. In each such class we can choose a monotonically nondecreasing right-hand side continuous function. Let be the collection of all such functions, and thus =. By (41) it follows that Z = sup sup Z (43) i.e., is generated by the set. For we have by Theorem 2 that the risk measure = sup Z is comonotonic and Z = AV@R Z d Z (44) with probability measure given by (32). Combining (43) and (44) gives a constructive way for writing the Kusuoka representation (1) with =. Furthermore, we can take the topological closure (in the weak topology) of the convex hull of (see Proposition 1). Example 2. Consider the absolute semideviation risk measure Z = ƐZ + c ƐZ ƐZ + (45) with constant c 1 and = L 1 F P. If the probability of the event Z = ƐZ is zero, then the maximizer of Z over is given by { 1 c if Z < ƐZ = (46) 1 + c1 if Z > ƐZ where = PrZ > ƐZ (e.g., Shapiro et al. [13, p. 278]), and this is an extreme point of the dual set. For Z monotonically nondecreasing on 1 the corresponding maximizer (which is monotonically nondecreasing) is { 1 c if < 1 = (47) 1 + c1 if 1 1 Since variables Z such that Z ƐZ w.p.1 form a dense set in the space, it is sufficient to consider maximizers (extreme points) of the form (47). Clearly the set of extreme points here is not generated by one element of, and hence this risk measure is not comonotonic. For an appropriate choice of Z the parameter can be any number of the interval 1. Therefore we can take here the generating set to be the set of functions of the form (47) for 1. By applying transformation (35) to an of the form (47) we obtain the corresponding probability measure with mass 1 c at = and mass c at = 1 (note that here the function is piecewise constant, and recall that mass is assigned to at = ). Consequently the risk measure of the form (45) has the following Kusuoka representation: { Z = sup 1 cav@r Z + cav@r 1 Z } (48) 1 That is, the corresponding set can be taken as = { } 1 c + c1 1 where denotes the measure of mass one at 1 (as was discussed above, one can also take the topological closure of the convex hull of that set). Recalling that AV@R Z = Ɛ and using definition (5), we can write representation (48) in the form Z = sup inf Ɛ{ } Z + ct Z + cz t + t 1 = inf sup t 1 Ɛ { Z + ct Z + cz t + } (49) The above representation (49) can be also derived directly (e.g., Shapiro et al. [13, p. 32]).

152 Mathematics of Operations Research 38(1), pp. 142 152, 213 INFORMS While the integral representation (3) is unique, the max-representation (1) is not necessarily unique. Consider, for example, a risk measure representable in the integral form (3). Then we can take =. On the other hand, Ɛ = AV@R and for any law invariant coherent risk measure it holds that Ɛ. Therefore, we always can add a measure of mass one at the point, of the interval 1, to. This will not change the maximum in the max-representation (1). Therefore, it makes sense to talk about minimal representations of the form (1). Let be a law invariant coherent risk measure and let 1 2, be a sequence of convex compact (in the weak topology) sets such that the representation (1) holds for every i. Consider = i=1 i. Since is the intersection of a sequence of nonempty nested convex compact sets, it is nonempty convex, and compact. Recall that the weak topology on the space of probability measures on 1 can be described in terms of a Prohorov metric (and the obtained metric space is separable), e.g., Billingsley [3, pp. 72 73]. Therefore, the weak topology can be described in terms of convergent sequences. It follows that the representation (1) also holds for the set. Indeed, consider an element Z and let i i, i = 1 be a measure at which the maximum in the right-hand side of (1) is attained. By passing to a subsequence if necessary, we can assume that i converge to some. It follows by Proposition 1 that Z = AV@R Z d, and since 1 this implies that the representation (1) holds for as well. This implies that there exists a set for which the representation (1) holds, while it does not hold for any convex compact strict subset of ; i.e., there exists a minimal representation of the form (1). If the dual set is generated by one of its elements, then = is a singleton and the corresponding representation (1) is minimal. By Theorem 2(ii) such a representation with single measure is uniquely defined. It is an open question whether minimal representations are unique in general. Acknowledgments. N1481114. This research was partly supported by the NSF award DMS-914785 and ONR award References [1] Acerbi C (22) Spectral measures of risk: A coherent representation of subjective risk aversion. J. Banking Finance 26:155 1518. [2] Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math. Finance 9:23 228. [3] Billingsley P (1999) Convergence of Probability Measures, Second ed. (John Wiley & Sons, New York). [4] Dana RA (25) A representation result for concave Schur concave functions. Math. Finance 15:613 634. [5] Dhaene J, Denuit M, Goovaerts MJ, Kaas R, Vyncke D (22) The concept of comonotonicity in actuarial science and finance: Theory. Insurance: Math. Econom. 31:3 33. [6] Föllmer H, Schield A (24) Stochastic Finance: An Introduction in Discrete Time, 2nd ed. de Gruyter Studies in Mathematics (De Gruyter, Berlin). [7] Jouini E, Schachermayer W, Touzi N (26) Law invariant risk measures have the Fatou property. Adv. Math. Econom. 9:49 72. [8] Kusuoka S (21) On law-invariant coherent risk measures. Kusuoka S, Maruyama T, eds. Advances in Mathematical Economics, Vol. 3 (Springer, Tokyo), 83 95. [9] Marshall AW, Olkin I (1979) Inequalities: Theory of Majorization and Its Applications, Vol. 143. Mathematics in Science and Engineering Series (Academic Press, New York). [1] Müller A, Stoyan D (22) Comparisons Methods for Stochastic Models and Risks (John Wiley & Sons, New York). [11] Pflug GCh, Römisch W (27) Modeling, Measuring and Managing Risk (World Scientific Publishing Co., London). [12] Ruszczyński A, Shapiro A (26) Optimization of convex risk functions. Math. Oper. Res. 31:433 452. [13] Shapiro A, Dentcheva D, Ruszczyński A (29) Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia).