Worst VaR Scenarios. Paul Embrechts a Andrea Höing a, Giovanni Puccetti b. 1 Introduction

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1 Worst VaR Scenarios Paul Embrechts a Anrea Höing a, Giovanni Puccetti b a Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerlan b Department of Mathematics for Decisions, University of Firenze, Firenze, Italy Abstract The worst-possible Value-at-Risk for a non-ecreasing function ψ of n epenent risks is known when n = 2 or the copula of the portfolio is boune from below. In this paper we analyze the properties of the epenence structures leaing to this solution, in particular their form an the implie functional epenence between the marginals. Furthermore we criticise the assumption of the worst-possible scenario for VaR-base risk management an we provie an alternative approach supporting comonotonicity. Key wors: Value-at-Risk, epenent risks, copulas, comonotonic risks J.E.L. Subject Classification: G10 Subj, Class.: IM01, IM12, IM52 1 Introuction Consier an insurer holing a portfolio consisting of n policies with iniviual risks X 1,...,X n over a fixe time perio. Given some measurable function ψ : R n R, a relevant task in insurance mathematics is the investigation of the risk position associate with ψ(x 1,...,X n ), when the marginal istributions of the single risks are known. Actuarial examples of the function ψ inclue n i=1 x i, simply characterizing the aggregate claim amount eriving from the policies or n i=1 h i (x i ) an h( n i=1 x i ), proviing the risk positions for a reinsurance treaty with retention functions h i, i = 1,...,n an a global reinsurance treaty with global retention function h, respectively. Corresponing author. Tel.: ; fax: aress: ahoeing@math.ethz.ch (Anrea Höing). 1 The authors woul like to thank Alessanro Juri (ETH Zürich/UBS) an Pablo Parrilo (ETH Zürich) for helpful iscussions. Preprint submitte to Elsevier Science

2 The problem of fining the best-possible lower boun on the istribution function (f) of ψ(x 1,...,X n ) has receive a consierable interest in insurance mathematics. From a financial risk management point of view, the problem is equivalent to fining the worst-possible Value-at-Risk (VaR) for the corresponing aggregate position. Here we refer to the Introuction in Embrechts an Puccetti (2004) for etails. Moelling the interepenence arising in a ranom portfolio calls for the use of copulas. If a lower boun on the copula of the vector (X 1,...,X n ) is given, the above problem is fully solve an the bouns provie in Embrechts et al. (2003) are sharp. In the no-information case the latter o hol only if n = 2. Rather than treating the technical proof of such results, for which we refer to the above cite references, in this paper we analyse in more etails the properties of their solutions. We concentrate mainly on the no-information case, when a lower boun on the copula of the portfolio is not available an a solution is known only for twoimensional portfolios. Without loss of generality, we stuy the optimizing copula for the sum of two epenent risks, which is well-known to iffer from comonotonicity. In particular we iscuss its shape, its implications in terms of epenence an we criticise it as not being a rational scenario for an insurance company. Finally, we provie an alternative optimization approach leaing to a suitable measure of risk, which supports the assumption of comonotonicity for a pruent evaluation of the VaR for the aggregate position. 2 Preliminaries an funamental results In this section we present some well-known concepts about copulas an briefly recall the funamental results about the problem of bouning the VaR for functions of epenent risks. For more etails about copulas, we refer to Nelsen (1999). 2.1 Value-at-Risk an epenence structures On some probability space (Ω, A, È), let the ranom vector X := (X 1,...,X n ) represent a portfolio of risks. Given a measurable function ψ : R n R we face the problem of fining the supremum of the VaR for the aggregate position ψ(x) over the class of possible fs for X having fixe marginals F 1,...,F n. Definition 1 Let ϕ : R R be a non-ecreasing function. Its generalize left continuous inverse is the function ϕ 1 : R R efine by ϕ 1 (y) := inf{x R ϕ(x) y}. For 0 α 1 the Value-at-Risk at probability level α for a ranom variable Y with istribution function G is its α-quantile, i.e. VaR α (Y ) := G 1 (α). 2

3 Of course, quantiles of the f of ψ(x) can be compute if the joint istribution function F(x 1,...,x n ) = È[X 1 x 1,..., X n x n ] is known. At this point, the notion of copula becomes useful. Definition 2 An n -imensional copula is an n-imensional istribution function restricte to [0, 1] n having stanar uniform marginals. We enote with C n the family of n-imensional copulas. Given a copula C C n an a set of univariate marginals F 1,...,F n, we can always efine a f F on R n having these marginals by F(x 1,...,x n ) := C(F 1 (x 1 ),...,F n (x n )). (1) Hence, given n fs F 1,..., F n, we let X C = (X 1,..., X n ) be the ranom vector on R n having a copula C satisfying (1). Observe that this copula is unique for continuous marginal fs. Conversely, Sklar s Theorem (Sklar (1973), Theorem 1) states that there always exists C C n coupling the marginals of a fixe f F trough (1). Comparing copulas pointwise an efining the riskiness of a epenence structure through this comparison, we recall that any copula C lies between the lower an upper Fréchet bouns W(u 1,...,u n ) := ( n i=1 u i n + 1) + an M(u 1,...,u n ) := min 1 i n u i, namely W C M. (2) Observe that, contrary to M, the lower Fréchet boun W is not a istribution function for n > 2. Ranom variables couple through C = M (C = W, respectively) are calle comonotonic (countermonotonic). The inepenence copula is enote by Π(u 1,..., u n ) := n i=1 u i. Remark 3 Comonotonicity characterizes the risks of the portfolio as being increasing functions of a common ranom variable. It is therefore a strong epenence an measure of epenence such as Kenall s τ or Spearman s ρ will escribe M as a perfect structure, i.e. τ(m) = ρ(m) = 1. It is precisely this representation which motivates the use of the concept of comonotonicity in financial applications. Moreover, assuming comonotonicity leas to almost all the computational benefits of inepenence, yieling, in aition, a pruent scenario in many contexts as we will emphasize in Section 4. For an in epth iscussion of comonotonicity, see Dhaene et al. (2001). 2.2 Bouns on value at risk for functions of epenent risks We now recall the two funamental results being the object of our analysis. For a proof of both theorems an further iscussions, we refer to Embrechts an Puccetti 3

4 (2004) an the references therein. For a copula C an marginals F 1,...,F n, efine σ C,ψ (F 1,...,F n )(s) := C(F 1 (x 1 ),...,F n (x n )), {ψ<s} τ C,ψ (F 1,...,F n )(s) := sup C(F 1 (x 1 ),..., F n 1 (x n 1 ), Fn (ψ x n (s))), x 1,...,x n 1 R where ψ x n (s) := sup{x n R ψ(x n, x n ) < s} for x n := (x 1,...,x n 1 ) R n 1. In the following, we refer to non-ecreasing functions ψ : R n R as being non-ecreasing in each component. Remark 4 Observe that σ C,ψ (F 1,...,F n )(s) = È[ψ(X C ) < s] for X C having marginals F 1,...,F n. In the Appenix, Proposition 23, we show that the operator τ C,ψ in (3) is actually the left-continuous version of a f, i.e. there exists a ranom variable K with È[K < s] = τ CL,ψ(F 1,...,F n )(s). This result extens a claim of Denuit et al. (1999), p. 37. As first note in Schweizer an Sklar (1974) for the sum of two risks, if C L M there oes not exist a measurable real function g such that K = g(x), with X having marginals F 1,...,F n. Theorem 5 Let X C = (X 1,..., X n ) be a ranom vector on R n (n > 1) having marginal istribution functions F 1,...,F n. Assume that there exists a copula C L such that C C L. If ψ : R n R is non-ecreasing, then for every real s we have σ C,ψ (F 1,...,F n )(s) τ CL,ψ(F 1,...,F n )(s). (3) Translate in the language of VaR, the above statement becomes for every α in the unit interval. VaR α (ψ(x 1,...,X n )) τ CL,ψ(F 1,...,F n ) 1 (α) The bouns state in Theorem 5 are pointwise best-possible an cannot be tightene if n = 2 or a lower boun C L > W on the copula of the portfolio X C is assume. Theorem 6 Further to the hypotheses of Theorem 5, we assume that ψ is also right-continuous in its last argument. Define the copula C α : [0, 1] n [0, 1] as max{α, C L (u)}, if u = (u 1,..., u n ) [α, 1] n, C α (u) := min{u 1,...,u n }, otherwise, where α = τ CL,ψ(F 1,...,F n )(s). Then this copula attains boun (3), i.e. σ Cα,ψ(F 1,..., F n )(s) = α. (4) The latter theorem motivates the investigation of the epenence structures leaing to the worst-case VaR scenario when n = 2 or C L > W. For C L = W an n > 2 the boun state in (3) is still vali but no more sharp. 4

5 3 Analysis of the worst-case portfolios The aim of the present paper is to give more insight into the shape of the copula yieling the worst-possible VaR for ψ(x C ) = ψ(x 1,...,X n ) an to unerstan the implie epenence between the marginals. Uner all possible epenence structures, the worst-case scenario for the VaR at level α is given by the copula minimizing È[ψ(X C ) < s] over s-regions epening on α. Inee, accoring to Definition 1 with m ψ (s) := inf C C n{è[ψ(xc ) < s]}, s R, (5) we have that VaR α (ψ(x C )) m 1 ψ (α), α [0, 1]. The problem at han becomes also the characterization of the copula minimizing m ψ, or equivalently maximizing m ψ (s) = 1 m ψ (s) = sup C C n {È[ψ(X C ) s]}, s R. (6) Such a copula will be referre to as a worst-case scenario for the aggregate position ψ(x C ). We use the term scenario to inicate a (possibly egenerate) set of probability measures in line with Artzner et al. (1999). Analogous to the above efinitions, in the presence of partial information, we write m CL,ψ (respectively m CL,ψ) an the infimum (supremum) is taken over all C C n satisfying the bounary conition C C L. In the next subsections, we concentrate on the sum of risks (generalizations to nonecreasing continuous functions ψ being straightforwar) an we choose C L = W. See however Section 5 for some comments on the latter choice of no epenence information. 3.1 Two-imensional portfolios If we take two risks, the boun given in Theorem 5 cannot be tightene an there always exists a two-imensional copula meeting that boun at a given point s. We restate Theorem 6 in this particular case. Theorem 7 Let X C = (X 1, X 2 ) be a ranom vector on R 2 having marginal istribution functions F 1, F 2. Define the copula C α : [0, 1] 2 [0, 1], max{α, W(u)}, if u = (u 1, u 2 ) [α, 1] 2, C α (u) := min{u 1, u 2 }, otherwise, where α = τ W,+ (F 1, F 2 )(s). Then this copula attains boun (3), i.e. σ Cα,+(F 1, F 2 )(s) = α. (7) 5

6 γ B s 0.9 γ B s A s 0.7 A s Fig. 1. Support of the copula C α, sets A s,b s an curve γ for: N(0,1)-N(1,2)-normal marginals an s = 1 (which gives α = ) (left); LN(0.4,1)-LN(0.4,1)-marginals an s = 4 (α = ) (right). Proofs of Theorem 7 can be foun in Frank et al. (1987) an Rüschenorf (1982). Our aim here is to restate the problem of maximizing (6) from a geometric point of view an illustrate the properties of the optimizing copulas leaing to the worstcase scenario for VaR. Without loss of generality, in what follows, we take continuous, increasing marginals. Let moreover G s := {(x 1, x 2 ) R 2 x 1 + x 2 s} an h : R 2 [0, 1] 2, h(x 1, x 2 ) := (F 1 (x 1 ), F 2 (x 2 )). The basic iea is to use the function h to transport the optimization problem on the unit square [0, 1] 2. In fact, U C := h(x C ) is a ranom vector, with istribution function C on [0, 1] 2. The function h is invertible, hence we have that È[X C G s ] = È[h(X C ) h(g s )] = µ C (A s ), where µ C is the measure corresponing to C on [0, 1] 2 an A s := h(g s ) = {(u 1, u 2 ) [0, 1] 2 F 1 1 (u 1 ) + F 1 2 (u 2 ) s}. The maximization function (6) can now be rewritten as m + (s) = sup C C 2 {µ C (A s )}. (8) For α = 1, (3) leas to σ C,+ (F 1, F 2 )(s) = 1 for every copula C, hence take α [0, 1). The bounary of A s is the image of the curve γ : R [0, 1] 2, γ(t) := (F 1 (t), F 2 (s t)). In Figure 1 the curve γ elimiting the set A s is rawn, with the support of the copula C α, in case of normal (N) an log-normal (LN) marginals. 6

7 The copula C α is uniformly istribute on its support, hence, efining B s := {(u 1, u 2 ) [0, 1] 2 u 1 + u 2 = 1 + α} we have µ Cα (B s ) = 1 α. As note in Nelsen (1999), p. 187, this is the crucial property leaing to the statement of Theorem 7. In fact, when 0 < α < 1, the continuity of the F i s implies that α = τ W,+ (F 1, F 2 )(s) = F 1 (x 1 ) + F 2(s x 1 ) 1 (9) for some x 1. Hence the curve γ meets the segment B s at least in one point. The technical (an for general n an C L > W rather laborious) part of the proof consists in showing that γ always lies below the segment B s, hence A s B s an µ Cα (A s ) µ Cα (B s ) = 1 α. Noting that µ Cα (A s ) 1 α, from Theorem 5 we obtain (7). For α = 0, instea, the existence of a tangent point between γ an B s is not necessary, since the copula W yiels the theorem. Analogous geometric consierations can be given for the case C L > W an for non-ecreasing continuous ψ. Remark 8 Observe that the geometric properties of the support of C α, illustrate in Figure 1, can be extene to a whole family of copulas, which implies that the epenence structure leaing to the worst-case VaR is not unique. Let Ĉ2 α an C2 α enote the family of copulas leaing to the worst possible VaR an the family of copulas sharing their support on [α, 1] 2 with C α, respectively. Formally: Ĉ 2 α : = {C C2 σ C,+ (F 1, F 2 )(s) = α}, C 2 α : = {C C2 C(u 1, u 2 ) = C α (u 1, u 2 ) for α u 1, u 2 1}. Observe that we can write Ĉ2 α = {C C2 µ C (A s ) = µ Cα (A s )}. In particular, it trivially follows that, every copula in C 2 α attains boun (3), since C 2 α Ĉ2 α. We now focus on the epenence implie by the copulas in C 2 α. The support R α := {(u 1, u 2 ) [0, α) 2 u 1 = u 2 } {(u 1, u 2 ) [α, 1] 2 u 1 + u 2 = 1 + α} of the copula C α implicitly efines the epenence of the couple ranom variables by the substitution u i = F i (x i ), i = 1, 2. In fact, if the copula C α couples X 1 an X 2 into the ranom vector X Cα an if we assume F 1, F 2 to be increasing on their omain, then we have X 2 = g(x 1 ), where the function g : R R is efine as F 2 1 (F 1 (x)), if x < F1 1 (α), g(x) := (10) F 2 1 (1 + α F 1 (x)), otherwise. 7

8 Analogously, every other copula in C 2 α efines a functional epenence ientical to that of g for x F1 1 (α). For example, the copula Cα 1 given by Cα 1 (u max{c L (u 1, u 2 ), α}, when (u 1, u 2 ) [α, 1] 2, 1, u 2 ) := otherwise, u 1 u 2 α, couples two marginals, which are inepenent if the first lies below the threshol F1 1 (α) an behaves like C α otherwise. Figure 2 compares R α with the support of Cα Fig. 2. Supports of the copulas C 0.5 (left) an C (right). Merging the two marginals by C α is therefore equivalent to letting X 2 = g(x 1 ). Hence, the two risks are mutually completely epenent. Moreover, the copula C α is a so-calle shuffle-of-m an hence implies that X 1 an X 2 are strongly piecewise strictly monotone functions of each other, in the sense efine in Mikusiński et al. (1991). Nevertheless, measures of epenence such as Kenall s τ or Spearman s ρ escribe C α as a non-perfect structure when 0 α < 1, i.e. τ(c α ), ρ(c α ) < 1. This is ue to the fact that this copula only represents piecewise comonotonicity. Mathematically, the epenence structure inuce by C α is, however, as strong a the one inuce by M, since the two variables couple by C α are in a oneto-one corresponence. Finally, note that every f on R 2 efine by applying a Ĉ 2 α-copula to the given set of marginals has a singular component, i.e. is mixe with a continuous istribution having zero erivative except for a set of Lebesgue measure zero. For instance, C α is singular on its whole omain, whereas Cα 1 only on [α, 1] 2. For more etails about singular istribution functions see Billingsley (1995), Section 31 an Nelsen (1999), p. 23. At this point, it is relevant to note that, in general, M / Ĉ2 α when 0 α < 1, the case α = 1 being the trivial one in which Ĉ2 α = C2. This provies a further geometric proof that comonotonicity oes not lea to the worst possible scenario for VaR an emphasizes the non-coherence of VaR as state in Artzner et al. (1999). Suppose that X 1 an X 2 are ientically istribute with unboune, absolutely continuous f having positive ensity f. If f is eventually ecreasing, it is easy to 8

9 show that for s large enough we have that α = 2F(s/2) 1, while σ M,+ (F, F) = F(s/2) > α. (11) A necessary conition for M to be in Ĉ2 α is that the point (α, α) lies in [0, 1]2 \ A s. Equation (11) implies that this conition is not satisfie for s large enough. Finally, M Ĉ2 0 if an only if A s = [0, 1] 2, i.e. the sum X 1 + X 2 is È-a.s. boune from below by the threshol s. In this case the problem of bouning the VaR for the sum oes not arise. We conclue that, apart from pathological cases of no actuarial importance, we have that σ M,+ (F 1, F 2 )(s) > m + (s) when 0 α < 1. This equation shows that the assumption of comonotonicity among the risks of the portfolio may lea to a angerous uner-valuation of the VaR for the aggregate position. At first, the worst epenence scenario coul seem to be the one implie by M, since uner comonotonicity it is inee known that every ranom variable is a non-ecreasing function of the other, so that high values for the first imply high values for the secon. Theorem 6 provies a eeper view on this issue, stating, instea, that for every threshol s such that α < 1, there exists a copula C α yieling a value for the VaR which is higher than that of comonotonicity. The following example further stresses the fact that M oes not belong, in general, to Ĉ2 α. Example 9 Let X 1 be normally istribute N(0, 1) with f Φ an put X 2 = X 1 to obtain È[X 1 + X 2 = 0] = 1. The copula escribing this epenence is the countermonotonic copula W, uner which X 2 is a non-increasing function of X 1. Accoring to Theorem 6, m + (0) = 0. In this set-up, the maximizing solution of (8) is then the structure of epenence which is opposite to comonotonicity (note that happens whenever α = 0), for which we have instea: σ M,+ (Φ, Φ)(0) = 1/2. Figure 3 (left) illustrates how, in the case of stanar normal marginals, for every positive s R, there exists a copula C Ĉ2 α such that σ C,+ (Φ, Φ)(s) < σ M,+ (Φ, Φ)(s). In the same figure (right) we also provie the shape of the bivariate istribution obtaine by applying C α to stanar normal marginals for s = (α = ). The reaer shoul compare this figure to Figure 2 (right). 9

10 inepenence comonotonicity lower boun = worst case Fig. 3. Range for È[X 1 + X 2 < s] for a N(0,1)-N(0,1)-portfolio. Together with the inepenence an comonotonic value we plot the lower boun m + (s) (left); the ensity of the istribution of (X 1,X 2 ) obtaine by applying the copula C to a N(0,1)-N(0,1)-portfolio is given in the figure on the right. 3.2 Two-imensional uniform portfolios We now state some simple results for uniform marginals that will turn out to be useful in unerstaning the n-imensional case. Proposition 10 Let the hypotheses of Theorem 7 be satisfie with F 1, F 2 uniformly istribute on the unit interval. Then Ĉ 2 α = C2 α. (12) Proof If α = 1, Ĉ2 1 = C 2 1 = C 2. Let 0 α = s 1 < 1 an C Ĉ2 α. 1 1 H20 H 10 A s D 1 H21 H 00 E 2 u 2 u 2 H22 H 11 D 2 H23 α α E 1 0 α 1 0 u 1 α u 1 1 Fig. 4. Sets efine in Proposition 10. Observe that, for uniform marginals, the bounary of A s coincies with B s. For the region unerlying such bounary we efine E i := {(u 1, u 2 ) [0, 1] 2 u 1 + u 2 < s, u i α}, i = 1, 2 10

11 as illustrate in Figure 4 (left). By the efinition of copula an C Ĉ2 α we have that µ C (E i A s ) = µ C (E i ) + µ C (A s ) = 1 α, i = 1, 2, µ C (A s ) = 1 α, which implies µ C (E i ) = 0, i = 1, 2 an µ C ([0, α) 2 ) = α. For the upper region we introuce the following partition: i(1 α) H ni := (α α (i + 1)(1 α), α + ] 2 n 2n+1 2 n (α + (2n i)(1 α) 1 α 2 n 2, α + (2n i)(1 α) ] n+1 2 n for n 0 an i = 0,...,2 n 1. See Figure 4 (right). In particular, consier H 00 = ( 1+α 2, 1]2 an let C i : = {(u 1, u 2 ) [0, 1] 2 u 1 + u 2 > 1 + α, u i 1 + α }, i = 1, 2, 2 D 1 : = {(u 1, u 2 ) [0, 1] 2 u 1 + u 2 = 1 + α, α u α }, 2 D 2 : = {(u 1, u 2 ) [0, 1] 2 u 1 + u 2 = 1 + α, 1 + α 2 < u 1 1}. Using the properties of a copula an consiering that E 1 an E 2 have zero µ C - measure, we have that µ C (H 00 ) + µ C (C 1 ) + µ C (D 1 ) = α 2 µ C (C 1 ) + µ C (D 1 ) = 1 + α 2 = 1 α, 2 α = 1 α 2 an hence µ C (H 00 ) = 0. Analogously, applying the same arguments to the upperright triangles of the squares [α, 1 + α 2 ] [ 1 + α 2, 1] an [ 1 + α 2, 1] [α, 1 + α ], 2 respectively, we obtain that µ C (H 10 ) = µ C (H 11 ) = 0. By iteration we have that µ C (H ni ) = 0 for all n 0, i = 0,...,2 n 1 an we trivially obtain µ C ( n=0 2 n 1 i=0 H ni ) = 0. Hence the only possibility for C is to assign probability mass (1 α) to the set D 1 D 2 = B s, which implies that C C 2 α. Remark 11 With respect to (10), for 1 s 2 an X C = (X 1, X 2 ) having stanar uniform marginals an copula C = C α, X 2 = g(x 1 ), where g : [0, 1] 11

12 [0, 1] is the linear function x, if x < s 1, g(x) = s x, otherwise. The above remark, together with Lemma 10, imply that the copula C of a uniform portfolio X C = (X 1, X 2 ) belongs to Ĉ2 α if an only if È[X 1 + X 2 = s X 1 + X 2 s] = 1. (13) 3.3 Multiimensional portfolios Though the boun (3) hols in arbitrary imensions, Theorem 7 fails to be vali if we take n > 2. Proposition 12 below shows in a simple way that, if we choose uniformly istribute marginals, it is not always possible to choose a copula C so as to obtain m ψ (s) = τ C,+ (F 1,...,F n )(s) =: α. Analogously to Ĉ2 α in the previous section, we efine Ĉ n α = {C Cn σ C,+ (F 1,..., F n )(s) = α}. Proposition 12 Let X C = (X 1,...,X n ) be a ranom vector having marginal fs uniformly istribute on [0, 1]. Take n > 2 an n 1 < s < n. Then Ĉn α =. Proof Let S n := n i=1 X i an note that, for uniform marginals, we have α = s n+1. If there exists k {1,...,n 2} such that È[S n k < s k] = 1 we have È[S n s] = 0 an the statement trivially hols. Suppose then È[S n k s k] > 0 for all k {1,...,n 2}. In this case we have È[S n s] = È[S n s, S n 1 s 1] + È[S n s, S n 1 < s 1] = È[S n s S n 1 s 1] È[S n 1 s 1], since X n is uniformly istribute on [0, 1]. Proceeing by iteration we obtain È[S n s] = È[S n s S n 1 s 1]... È[S 3 s n + 3 S 2 s n + 2] È[S 2 s n + 2 X 1 s n + 1](n s). (14) Assume now that Ĉn α, i.e. there exists XC = (X 1,...,X n ) with copula C Ĉ n α. It immeiately follows that È[S n s] = È[X X n s] = n s an hence all factors in (14), apart from the last one, must be equal to one. In particular, 12

13 inepenence 0.1 comonotonicity worst case lower boun s Fig. 5. Range for È[X 1 + X 2 + X 3 < s] for a stanar uniform portfolio. Together with the inepenence an comonotonic scenario, we plot the worst-case value m + (s) which iffers from the lower boun τ CL,+(s) given by (3). this yiels that È[X 1 + X 2 s n + 2 X 1 s n + 1] = 1, (15) È[S 3 s n + 3 S 2 s n + 2] = 1. (16) Accoring to (13), (15) implies that which, together with (16), leas to È[S 2 = s n + 2 S 2 s n + 2] = 1, 1 = È[S 3 s n + 3 S 2 s n + 2] = È[X 3 1 S 2 s n + 2] = È[X 3 1, S 2 s n + 2]). È[S 2 s n + 2] The latter equation is clearly a contraiction to the fact that X 3 is uniformly istribute on [0, 1]. Remark 13 The boun given in (3) fails to be sharp when n > 2 an C L = W. This erives from the fact that W is not a copula for n > 2, i.e. for more than two ranom variables it is impossible for each of them to be almost surely a nonincreasing function of each of the remaining ones. In Rüschenorf (1982), the worst-case VaR for uniform an binomial marginals is provie. Till now, this is the only known analytical result. In fact, the optimum epenence for uniform marginals oes not solve the general problem, showing that, contrary to the twoimensional case for n > 2, the epenence structure maximizing (6) may epen upon the choice of the marginals. In Embrechts an Puccetti (2004), however, an improve boun for the VaR is provie. Figure 5 illustrates the optimum values for uniform portfolios. 13

14 4 Evaluating risk through comonotonicity In the following, we show that the assumption of comonotonicity among the X i s may lea to a pruent evaluation of the risk associate with the aggregate position ψ(x). To this purpose, we first illustrate that such kin of epenence leas to the more angerous scenario with respect to both stop-loss an supermoular orer. Then, changing the optimization approach iscusse in the previous sections, we show that comonotonicity also arises as a suitable epenence assumption for our original VaR problem. 4.1 Stochastic orers an comonotonicity In this section we provie some motivation for the assumption of comonotonicity among risks base on stochastic orers. In this framework we illustrate an important application in actuarial mathematics. We first recall some concepts about stochastic orers. Definition 14 Let X an Y be two real ranom variables. We say that X is smaller than Y in stop-loss orer an we write X sl Y if for all non-ecreasing convex functions g : R R we have provie the expectations E[g(X)], E[g(Y )] are finite. E[g(X)] E[g(Y )], (17) The stop loss-orer compares one-imensional ranom variables. A multiimensional stochastic orer implying stop-loss orer is the so-calle supermoular orer, i.e. the orer base on a comparison of integrals of supermoular functions f : R n R satisfying f(x y) + f(x y) f(x) + f(y), for all x, y R n, where x y (x y) is the componentwise maximum (minimum) of x, y. Definition 15 Let X an Y be two n-imensional ranom vectors. We say that X is smaller than Y in supermoular orer an write X sm Y if for all supermoular functions g : R n R, E[g(X)] E[g(Y )], (18) provie the expectations E[g(X)], E[g(Y )] are finite. The next theorem recalls two important results about supermoular an stop-loss orers. In particular it states that comonotonicity represents the worst possible epenence scenario with respect to both such orers. 14

15 Theorem 16 Let X C = (X 1,...,X n ) be a n-imensional ranom vector having marginal istributions F 1,...,F n an copula C. Let ψ : R n R be a nonecreasing supermoular function. Then (a) X C sm X M, (b) ψ(x C ) sl ψ(x M ). Proof As note in Müller (1997), part (a) follows from Theorem 5 in Tchen (1980). Since ψ(x C ) sl ψ(x M ) hols if an only if (18) hols for all non-ecreasing convex functions g : R R for which expectations exists, to prove part (b) it is sufficient to show that for such a function g the function g ψ is supermoular. This follows from Lemma 2.2(b) in Bäuerle (1997). Remark 17 Note that Theorem 16 (b) applies to a large class of interesting functionals, incluing ψ(x) = n i=1 h i (x i ), where the h i s are non-ecreasing (see also Müller (1997)) an ψ(x) = h( n i=1 x i ) for h non-ecreasing an convex. For more examples of supermoular functions or some interesting methos of constructing them, Marshall an Olkin (1979), pp is the stanar reference. Here we want to point out that Theorem 16 (b) oes not apply to (6) because the inicator function of the set {ψ(x) s} is not supermoular. Consier again a portfolio of risks X C = (X 1,...,X n ). In insurance mathematics if ψ(x C ) is to be insure with a retention level, the net premium E[ψ(X C ) ] + is calle the stop-loss premium. A stop-loss premium is etermine once the retention an the multivariate f of X C are given. Hence we set π C,ψ (F 1,...,F n )() := E[ψ(X C ) ] +, P ψ () := sup C C n {π C,ψ (F 1,...,F n )()}. (19) Next proposition states that the univariate stop-loss orer base on the comparison of integrals of non-ecreasing convex functions is equivalent to the stochastic orer base on the comparison of stop-loss premiums. It also explains the name stop-loss for the stochastic orer sl. Proposition 18 Let X an Y be two real ranom variables. Then the following statements are equivalent: (a) X sl Y, (b) For all real E[X ] + E[Y ] +. (20) A simple proof can be foun in Müller an Stoyan (2002), Theorem

16 Accoring to Proposition 18, part (b) of Theorem 16 is equivalent to P ψ () = π M,ψ (F 1,...,F n )() (21) for all non-ecreasing supermoular functions ψ, real retention an arbitrary imension n. Hence π C,ψ (F 1,...,F n )() is maximize over C n when the fixe marginals of the portfolio have a comonotonic joint istribution, provie ψ is a non-ecreasing supermoular function. Observe that this is a much stronger result if confronte with Theorem 6. It is remarkable to note that this solution is not unique pointwise. In Figure 6 (right) we plot the ensity of a f on R 2 which, though iffering from comonotonicity (left), maximizes π C,+ (Φ, Φ)(0) over C n. However, M is the only epenence structures that attains (19) for all real retentions. Fig. 6. Densities of two-imensional istributions obtaine by comonotonic epenence (left) an by maximizing π C,+ (Φ,Φ)(0) over C 2 (right). 4.2 Changing the optimization approach in the VaR problem Theorem 5 provies a lower boun for the probability σ C,ψ (F 1,...,F n )(s), respectively, an upper boun for the VaR α (ψ(x 1,...,X n )). Hence an insurance company holing the risky position ψ(x C ) knows that VaR α (ψ(x C )) τ W,ψ (F 1,...,F n ) 1 (α), C C n. This equation represents the worst-case VaR scenario an can be very expensive for the insurer to hanle. Recalling the efinition of the family of copulas reaching the above level α, i.e. Ĉ n α = {C C n σ C,ψ (F 1,...,F n )(s) = α}, two quite natural conitions the insurer may ask to be satisfie are (a) Ĉn α, (b) Ĉn α oes not epen on s. 16

17 1 0.9 r = r = r = r = r = lower boun comonotonicity C r scenarios Fig. 7. Range for È[X 1 + X 2 < s] uner ifferent epenence scenarios for a stanar uniform portfolio. Inee (a) requires the boun to be sharp, i.e. that there exists a structure of epenence which etermines the worst-case VaR scenario. Conition (b) requires the worst scenario for the insurance company to be inepenent of the threshol s or, equivalently, inepenent of the parameter α that is chosen by the company or by the regulator to evaluate the aggregate risk. In the previous sections we showe that (a) is violate when n > 2 but it hols in the two-imensional case, while (b) is violate even when n = 2. Our aim here is to change the optimization approach so that the solutions satisfy conitions (a) an (b). In orer to o this, we efine the worst-case VaR scenario over a suitable range for the threshol s, rather than on a single value. Figure 7 explains how that can be one. In this graph we plot σ Cr,+(F 1, F 2 )(s) for ifferent values of r (0, 1) in case of two uniform marginals together with the best-possible lower boun m + (s) an the comonotonic curve σ M,+ (F 1, F 2 )(s). As a consequence of Theorem 7, every copula C r gives a lower boun that meets the curve m + (s) at the corresponing threshol an then becomes one. The intuition behin this plot is that the comonotonic copula, though never meeting the boun m + (s), is closer to it than any other copula on average. This iea can now be formalize by introucing a loss function Λ to measure the error committe by evaluating the risky position using a fixe copula C C n rather than the appropriate worst-possible structure of epenence. We then integrate the loss function over a suitable set B an we search for the infimum over the class of all n-copulas. Defining { } r ψ := inf Λ[σ C,ψ (F 1,...,F n )(s) m ψ (s)]s C C n B (22) for some non-ecreasing functional Λ : [0, 1] R + 0, we therefore introuce a measure for the istance between the worst an the comonotonic scenario in Figure 7, where Λ can be viewe as a weighting function. Let Ĉn (B) enote the set of 17

18 copulas leaing to (22). To focus our attention, we choose B = [, ). Theorem 19 Let Λ = I an B = [, ). Then, for every real threshol an non-ecreasing supermoular function ψ satisfying E[ψ(X M )] <, we have that M Ĉn ([, )). Proof Let m ψ(s)s <. Note that m ψ (s) epens only on the fixe marginals, so we obtain { } r ψ = inf [σ C,ψ (F 1,...,F n )(s) m ψ (s)] s C C n { } = sup [È[ψ(X C ) s] m ψ (s)] s C C n { } = m ψ (s) s sup È[ψ(X C ) s] s = m ψ (s) s sup C C n C C n { } È[ψ(X C ) > s] s, where the last step is obtaine since È[ψ(X C ) = s] can be positive at most for countably many values of s, so that the last two integrals containe in the brackets are the same. Finally, recalling that È[ψ(X C ) > s] s = E[ψ(X C ) ] +, it follows that r ψ = = = { } m ψ (s)s sup È[ψ(X C ) > s] s C C n m ψ (s)s sup C C n {E[ψ(X C ) ] + } m ψ (s)s P ψ (). Since E[ψ(X M )] is finite, (21) finally implies that M Ĉn ([, )). If the integral m ψ(s)s =, trivially Ĉn ([, )) = C n. Remark 20 Theorem 19 is vali for right-open intervals but it oes not hol in full generality. For instance, if we fix a trivial interval consisting of a single point, we go back to the original VaR problem. In such case, M oes not lea to the worst-possible scenario. Remark 21 For the above theorem, the only relevant portfolios (X 1,...,X n ) are those for which m ψ(s)s is finite. This technical conition is satisfie for all marginal istributions of interest. As escribe in the previous sections an illustrate in Figure 8 for the ensity of the N(0, 1)-N(0, 1)-portfolio of Example 9, 18

19 uner the optimizing copula, the mass m + (s) is concentrate on the upper triangle. On the remaining strips, where only one marginal takes high values, the mass is zero. Observe that, by the efinition of copula, m + (s) can not excee the sum of the marginal tails relative to these strips, i.e. 2Φ(s/2). In arbitrary imensions, with the same argument, we have that n m ψ (s) s F i (s/n) s < (23) i=1 for marginal fs F 1,...,F n satisfying /n F i(s)s <. The latter trivially hols if the marginal rvs are integrable. In particular, if X 1,..., X n enote losses an assume only positive values, conition (23) is satisfie provie that their expectations are finite. The finiteness of m ψ(s)s is equivalent to the finiteness of the expectation of the rv K in Proposition 23. We prove this result in full generality for non-ecreasing, continuous ψ an increasing marginals in Proposition 24 in the Appenix Fig. 8. Density an contour plot of the joint istribution of (X 1,X 2 ) maximizing È[X 1 + X 2 s] in Example 9. The main issue unerlying Theorem 19 is that, even if the comonotonic epenence structure oes not lea to the worst-case scenario for the original VaR problem, if an insurance company wants to boun well σ C,ψ (F 1,...,F n )(s) for all threshols in [, ) in the sense efine by (22), comonotonicity provies a pruent evaluation for the aggregate risk. The next theorem elivers an insight into an extension of Theorem 19 for general loss functions Λ, i.e. every increasing, convex function Λ : [0, 1] R + 0 satisfying Λ(0) = 0. For a copula C, let e C,ψ (s) := σ C,ψ (F 1,...,F n )(s) m ψ (s) enote the error function in (22). 19

20 Theorem 22 Consier X C = (X 1,..., X n ) with marginal istributionsf 1,...,F n an let ψ be as in Theorem 19 with 0 m ψ (s)s <. Then there exists C [ C, C ] such that for all loss functions Λ: Λ[σ C,ψ (F 1,...,F n )(s) m ψ (s)]s for every C, where Λ[σ M,ψ (F 1,...,F n )(s) m ψ (s)]s C := sup{ R e C,ψ (s) e M,ψ () s an e M,ψ (s) > e C,ψ (s) on some interval ( C, )}, C := inf{ R e C,ψ (s) e M,ψ (s) s }. (24) Proof Theorem 19 yiels e C,ψ(s)s e M,ψ(s)s for all R. The latter integrals are finite since 0 m ψ (s)s <. Denote with B(, ) an m the Borel sets on (, ) an the Lebesgue measure, respectively. Applying Chong (1974), Theorem 1.6, Theorem 2.1 an Corollary 1.2 to e C,ψ an e M,ψ, with Φ = Λ an (X, Λ, µ) = (X, Λ, µ ) = ((, ), B(, ), m), (24) is equivalent to [e C,ψ (s) u] + s [e M,ψ (s) u] + s (25) for all u R. By efinition, C C an e C,ψ e M,ψ on [ C, ) implying C C. Assume now < C < C an let ( C, C]. Choosing u = e M,ψ ( C ) in (25) we have that [e C,ψ (s) u] + C s = [e C,ψ (s) e M,ψ ( C )] + s + = C C [e C,ψ (s) e M,ψ ( C )] + s< [e M,ψ (s) u] + s, C [e C,ψ (s) e M,ψ ( C )] + s [e M,ψ (s) e M,ψ ( C )] + s which conclues the proof. With respect to a copula C an any loss function Λ, comonotonicity is hence a suitable epenence scenario on [ C, ) an it provies a pruent VaR on the corresponing α-set. Note that, for a copula C, the set in the efinition of C may be empty an hence C arbitrarily small. Contrarily, since ψ(x C ) sl ψ(x M ), we have that C < if the two fs cross finitely many times. Unfortunately, in general, C an C may become arbitrary large. For instance, if the function ψ is unboune, Rüschenorf (1981), Theorem 5 yiels the existence of a copula Ĉ epening on ψ, s an the marginals such that m ψ (s) = σĉ,ψ (F 1,...,F n )(s) σ M,ψ (F 1,...,F n )(s), 20

21 where, in general, Ĉ M implying sup C C n C =. Analogously there exist epenence structures leaing to sup C C n C =. We therefore conclue that an extension of Theorem 19 to loss functionals can only exist for suitable subclasses of C n. 5 The Presence of Information Throughout this paper we mainly assume a no-information scenario an put the lower Fréchet boun W as lower boun for the unknown copula C in Theorem 5. From a mathematical point of view this is an unsatisfactory initial assumption, since, for n > 2, W is not a copula anymore an the boun provie in Theorem 6 fails to be sharp. However, we want to warn the reaer from choosing some a priori assumption such as C Π an we emphasize that similar restrictions may lea to a critical unervaluation of the portfolio risk. The assumption C Π, for instance, is justifie by the iea that the worst-var scenarios is implie by the so-calle positive lower orthant epenent risks with È[X 1 x 1,...,X n x n ] È[X 1 x 1 ]... È[X n x n ]. Unfortunately, this is not true an restricting the optimization to the class {C Π} substantially changes the initial problem, since it oes not allow to focus on riskier portfolios, as long as m + > m Π,+. This is a consequence of the fact that the componentwise orering in the class C n is not complete an, putting a lower boun on a copula, exclues all copulas not comparable to such boun. To point out this aspect, we observe that countermonotonicity plays a central role in the efinition of the family Ĉn α arising from (6), whereas every copula shuffle with it is not comparable with the inepenence scenario. 6 Conclusions In this paper we focus on the copulas leaing to the worst-possible VaR for a function of epenent risks an we emphasize that comonotonicity oes not lie in this family. Such worst-case scenarios epen upon the level α where the VaR is evaluate an therefore are not reasonable from a practical point of view. Moreover, these solutions are known only for two-imensional portfolios or in presence of partial information. The investigation of optimal bouns in arbitrary imensions with no prior information remains open. Therefore, we provie an alternative approach supporting the assumption of comonotonicity in a pruent evaluation of the quantiles of the aggregate position. 21

22 Appenix The operator τ C,ψ In this section we exten a claim of Denuit et al. (1999), p. 37 by showing that the operator τ C,ψ (F 1,...,F n )(s) = sup C(F 1 (x 1 ),..., F n 1 (x n 1 ), Fn (ψ x n (s))), x 1,...,x n 1 R with ψ x n (s) = sup{x n R ψ(x n, x n ) < s}, for fixe x n R n 1, is actually the left-continuous version of a f. Proposition 23 For ψ : R n R non-ecreasing, there exists a ranom variable K such that τ C,ψ (F 1,...,F n )(s) = È[K < s]. Proof Since ψ is non-ecreasing in each component, τ C,ψ (s) := τ C,ψ (F 1,..., F n )(s) is a non-ecreasing function. Hence, we have to show that it is also left-continuous an that its right an left limits converge to one an zero, respectively. To prove that lim s τ C,ψ (s) = 1, we fix ε > 0 an efine u ε = (u ε 1,...,uε n ) as a vector satisfying F i (u ε i ) 1 ε n, i = 1,...,n. The existence of such a vector is straightforwar, since F 1,...,F n are non-efective fs. By efinition, the function ψ û is non-ecreasing an its right limit is either ε n finite or infinite. Suppose it is finite. For every real s, it follows that which implies sup{x n R ψ(u ε n, x n ) < s} lim ψ s û (s) =: R <, ε n ψ(u ε n, x n) s for all x n R. Therefore ψ(u ε n, R) =, which contraicts ψ having R as its range. Hence R = an it is always possible to select a real s ε, epening only on ε, such that ψ û ε n (s ε) > u ε n implying For τ C,ψ (s ε ) we also obtain that τ C,ψ (s ε ) = F n (ψ û ε n (s ε)) F n (u ε n ) 1 ε n. sup C(F 1 (x 1 ),...,F n 1 (x n 1 ), Fn (ψ x n (s ε ))) x 1,...,x n 1 R C(F 1 (u ε 1),...,F n 1 (u ε n 1), Fn (ψ (s û ε))) ε n W(F 1 (u ε 1),..., F n 1 (u ε n 1), F n (ψ û ε n (s ε))) (1 ε )n n + 1 = 1 ε, n 22

23 an, since τ C,ψ is non-ecreasing, τ C,ψ (s) τ C,ψ (s ε ) 1 ε for every s s ε. Hence the right limit converges to one. Similarly, for the left limit, we fix ε > 0 an choose u ε satisfying F i (u ε i) < ε, i = 1,..., n for which L := lim s ψ û ε n (s) =. It is always possible to select a real s ε, epening only on ε, such that ψ û ε n (s ε) < u ε n an F n (ψûε n (s ε)) < F n (u ε n ) < ε. Let now A u ε := {x n R n 1 x i u ε i for some i {1,...,n 1}} an A u ε := {x n R n 1 x i > u ε i for all i = 1,..., n 1}. Then sup C(F 1 (x 1 ),...,F n 1 (x n 1 ), Fn (ψ x n (s ε ))) x n A u ε sup M(F 1 (x 1 ),..., F n 1 (x n 1 ), Fn (ψ x n (s ε ))) x n A u ε F i (u ε i ) < ε. (26) If x A u ε, then ψ x n (s ε ) ψ û ε n (s ε) an hence sup C(F 1 (x 1 ),...,F n 1 (x n 1 ), Fn (ψ x n (s ε ))) x n A u ε sup C(F 1 (x 1 ),...,F n 1 (x n 1 ), Fn (s ε))) (ψûε n x n A u ε Fn (ψ (s û ε)) < ε. ε n (27) From (26) an (27) we have that τ C,ψ (s ε ) < ε. Since τ C,ψ is non-ecreasing, τ C,ψ (s) < τ C,ψ (s ε ) < ε for all s < s ε an the left limit goes to zero. It remains to show that τ C,ψ is left-continuous. For non-ecreasing functions f : R R left-continuity is equivalent to lower-semicontinuity. By Ruin (1974), p. 39, the supremum of any collection of lower-semicontinuous function is lowersemicontinuous. It is sufficient then to show that C(F 1 (x 1 ),...,F n 1 (x n 1 ), F n (ψ x n (s))) is left-continuous in s for every x n R n 1. By uniform continuity of C, leftcontinuity an non-ecreasingness of F n, an non-ecreasingness of ψ x n (s), the problem is reuce to show that ψ x n (s) is left-continuous. By efinition, it is nonecreasing an hence, for every real s, there exists l(s) := lim x s ψ x n (x). Assume now that ψ x n (s) is not left-continuous. Then there exists s 1 with l(s 1 ) < ψ x n (s 1 ). Let l(s 1 ) be finite (otherwise there is nothing to prove). Then, for arbitrary positive ε, we have sup{x n R ψ(x n, x n ) < s 1 ε} l(s 1 ) < 23

24 an hence, whenever x n l(s 1 ), it follows that ψ(x n, x n ) s 1 ε. Since ε is arbitrary, it follows that ψ(x n, l(s 1 )) s 1, contraicting the fact that ψ(x n, x n ) < s for every x n < ψ x n (s 1 ), which conclues the proof. The following proposition states that, uner suitable assumptions, the rv K has finite expectation. Proposition 24 Let X C have increasing, continuous marginals F 1,...,F n, ψ : R n R be non-ecreasing continuous an increasing in the last argument. Then, for K as in Proposition 23, E[K] const + m ψ(s)s < for all R if an only if E[ψ(X M )] <. Proof A ranom variable Y with f F has finite expectation if an only if 0 F(x)x < an 0 F(x)x <, which is equivalent to E[Y ] + < for all R. We therefore have to show that m ψ (s) s < E[ψ(X M ) ] + < (28) for all R. By the efinition of m ψ (s), we have that for R, E[ψ(X M ) ] + = an hence immeiately follows. P[ψ(X M ) s] s m ψ (s) s Assume now that the rhs of (28) hols an let U be uniformly istribute on [0, 1]. By Dhaene et al. (2001), Theorem 2 we have that E[φ(U) ] + = E[ψ(X M ) ] + <, where φ : [0, 1] R, φ(u) := ψ(f 1 1 (u),...,f 1 n (u)). Uner the assumptions of the theorem, φ is continuous an φ(φ 1 (s)) = s. We can write m ψ (s) 1 τ W,ψ (F 1,...,F n )(s) = 1 sup x n R n 1 {(F 1 (x 1 ) + + F n 1 (x n 1 ) + F n (ψ x n (s)) n + 1) + } inf x n R n 1{F 1(x 1 ) + + F n 1 (x n 1 ) + È[X n ψ x n (s)]}. (29) 24

25 Choosing x n = (F1 1 (φ 1 (s)),...,fn 1(φ 1 (s))) in (29) an since ψ is increasing in the last argument, (s) = F 1 ψ x n n (φ 1 (s)). Integrating, we finally obtain: m ψ (s)s = = = n n i=1 n i=1 n i=1 È[X i Fi 1 (φ 1 (s))]s È[F 1 i (U) F 1 (φ 1 (s))]s i n È[U φ 1 (s)]s È[φ(U) φ(φ 1 (s))]s i=1 È[φ(U) s]s = ne[φ(x M ) ] + <. References Artzner, P., Delbaen, F., Eber, J. M. an Heath, D., Coherent measures of risk. Mathematical Finance 9, Bäuerle, N., Monotonicity results for MR/GI/1 queues. Journal of Applie Probability 34, Billingsley, P., Probability an Measure. Thir E. John Wiley & Sons, Chichester. Chong, K.-M., Some extensions of a theorem of Hary, Littlewoo an Pólya an their applications. Canaian Journal of Mathematics 26 (6), Denuit, M., Genest, C. an Marceau, É., Stochastic bouns on sums of epenent risks. Insurance: Mathematics an Economics 25, Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. an Vyncke, D., The concept of comonotonicity in actuarial science an finance: theory. Insurance: Mathematics an Economics 31, Embrechts, P., Höing, A. an Juri, A., Using copulae to boun the Valueat-Risk for functions of epenent risks. Finance & Stochastics 7, Embrechts, P. an Puccetti, G., Bouns on Value-at-Risk. Preprint. Frank, M.J., Nelsen, R.B. an Schweizer, B., Best-possible bouns on the istribution of a sum a problem of Kolmogorov. Probability Theory an Relate Fiels 74, Marshall, A. W. an Olkin, I., Inequalities: Theory of Majorization an its Applications. Acaemic press, New York. Mikusiński, P., Sherwoo, F. an Taylor, M. D., Probabilistic interpretations of copulas an their convex sums. Avances in Probability Distributions with Given Marginals (Giorgio Dall Aglio, S. Kotz an G. Salinetti Es.), Kluwer Acaemic Publisher, Amsteram, pp Müller, A., Stop-loss orer for portfolios of epenent risks. Insurance: 25

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