Collective Risk Models with Dependence Uncertainty

Transcription

1 Collectie Risk Models with Dependence Uncertainty Haiyan Liu and Ruodu Wang ebruary 27, 2017 Abstract We bring the recently deeloped framework of dependence uncertainty into collectie risk models, one of the most classic models in actuarial science. We study the worst-case alues of the Value-at-Risk (VaR and the Expected Shortfall (ES of the aggregate loss in collectie risk models, under two settings of dependence uncertainty: (i the counting random ariable (claim frequency and the indiidual losses (claim sizes are independent, and the dependence of the indiidual losses is unknown; (ii the dependence of the counting random ariable and the indiidual losses is unknown. Analytical results for the worst-case alues of ES are obtained. or the loss from a large portfolio of insurance policies, an asymptotic equialence of VaR and ES is established. Our results can be used to proide approximations for VaR and ES in collectie risk models with unknown dependence. Approximation errors are obtained in both cases. Key-words: collectie risk model; Value-at-Risk; Expected Shortfall; dependence uncertainty; asymptotic equialence AMS Classification: 91B30; 91G10; 62P05. Department of Statistics and Actuarial Science, Uniersity of Waterloo, Waterloo, ON N2L3G1, Canada ( h262liu@uwaterloo.ca. Department of Statistics and Actuarial Science, Uniersity of Waterloo, Waterloo, ON N2L3G1, Canada ( wang@uwaterloo.ca. 1

2 1 Introduction The question we address in this paper comes from a practical challenge of measuring large insurance portfolios using a risk measure under model uncertainty at the leel of the dependence among indiidual claims and the number of claims. The aggregate loss of an insurance company (the total amount paid on all claims occurring oer a fixed period is often modelled by a sum of random ariables, S N = Y Y N, (1.1 where Y 1, Y 2,... are non-negatie random ariables and N (random or deterministic takes alues in non-negatie integers. Nowadays the simple model (1.1 is taught in practically eery undergraduate actuarial science course on loss models; see, for instance, standard textbooks Kaas et al. (2008 and Klugman et al. (2012. When N is a non-random positie integer, (1.1 is called an indiidual risk model, in which Y 1, Y 2,... represent losses from each indiidual policy and N is the number of policies. When N itself is random, (1.1 is called a collectie risk model. or portfolio analysis, indiidual risk models are a priori the most natural, whereas for ruin theoretic problems, collectie risk models are more natural. In the classic treatment of collectie risk models, Y 1, Y 2,... are iid random ariables representing indiidual claim sizes, and the counting random ariable N is assumed to be independent of (Y 1, Y 2,.... This classic assumption on the independence of N, Y 1, Y 2,... proides great mathematical conenience and elegance, as well as nice interpretations. In some practical situations, the claims or losses Y 1, Y 2,..., in indiidual risk models or collectie risk models are dependent, and they may also be dependent on the number of claims N. Think about, for instance, the losses from wind and flood damage in a certain region; see Kousky and Cooke (2009 for related real-life examples. In the context of collectie risk models or the closely related setting of compound Poisson processes, certain types of dependence among N, Y 1, Y 2,... are studied. or instance, see Cheung et al. (2010, Albrecher et al. (2014 and Landriault et al. (2014 for recent deelopment on dependent Sparre Anderson risk models, and see Denuit et al. (2005 for a comprehensie treatment of dependent losses in actuarial science. Due to the high dimensionality of the joint model and sometimes ited data, it is often difficult to accurately model or justify a dependence structure. In such situations, the dependence between Y 1, Y 2,..., is completely or partially unknown, and this setting is nowadays referred to as dependence 2

3 uncertainty and extensiely deeloped in the past few years. See Bernard et al. (2014 and Embrechts et al. (2014 for a general discussion on dependence uncertainty. In this paper, we bring in the framework of dependence uncertainty into collectie risk models. We assume that Y 1, Y 2,... are identically distributed as in classic collectie risk models, but we do not assume a particular model for the dependence structure among random ariables in (1.1. Two different practical settings will be considered: (i N is independent of Y 1, Y 2,... and the dependence structure of Y 1, Y 2,... is unknown. (ii The dependence structure of N, Y 1, Y 2,... is unknown. rom the perspectie of risk management, we are particularly interested in quantifying S N by certain risk measures under dependence uncertainty, a crucial concern for risk management in the presence of model uncertainty. The two most popular risk measures in banking and insurance are the Value-at-Risk (VaR and the Expected Shortfall (ES, also called TVaR in actuarial science. The VaR of a risk X at the confidence leel α (0, 1 is defined as VaR α (X = inf{x R : (x α}, X L 0, (1.2 and the ES of a risk X at the confidence leel α (0, 1 is defined as ES α (X = 1 1 α 1 where is the distribution function of the random ariable X. α VaR γ (Xdγ, X L 1, (1.3 Risk measures for indiidual and collectie risk models are well studied; see for instance Cai and Tan (2007 for optimal stop-loss reinsurance for these models under VaR and ES, and Hürann (2003 for ES bound for compound Poisson risks. It is well-known that an analytical calculation of the distribution of S N, as well as VaR α (S N and ES α (S N, is often unaailable (see Klugman et al. (2012. Approximation, simulation or numerical calculation is often needed. We study the worst-case alues of VaR α (S N and ES α (S N, under the two settings (i and (ii aboe. The recent literature on dependence uncertainty has focused on the indiidual risk model, in which N = n in (1.1 is non-random. Analytical calculation of the worst-case VaR α (S n is generally unaailable; some analytical results can be found in Wang et al. (2013 and Jakobsons et al. (2016, and an efficient numerical algorithm is established in Puccetti and Rüschendorf (2012 and Embrechts et al. (2013. In the meanwhile, as a well-known result, the worst-case alue of ES α (S N for a nonrandom N = n is simply equal to the sum of the indiidual ES α alues, and this worst-case alue is attained by comonotonic Y 1,..., Y n. 3

4 Due to a natural connection between collectie risk models and indiidual risk models, using a collectie model as in setting (i can be also seen as one of the seeral ways to introduce partial dependence information into risk aggregation; see the Appendix for details and a comparison. or more studies of risk aggregation with partial dependence information, see Bernard et al. (2017a,b,c, Bernard and Vanduffel (2015, Bignozzi et al. (2015 and Puccetti et al. (2016, The main contributions of this paper are summarized as follows. Based on the classic theory of stochastic orders, we first derie some conex ordering inequalities for collectie risk models and thereby obtain analytical formulas for the worst-case alues of ES. Using the results on ES for collectie risk models, we are able to study the worst-case alues of VaR α (S N and ES α (S N as E[N] increases to infinity, that is, a ery large insurance portfolio. or simplicity the reader may think of the case where N is Poisson-distributed with parameter E[N], the most classic choice for the counting random ariable N. In both settings (i and (ii, under some moment and conergence conditions, we show that the worst-case alues of VaR α (S N and ES α (S N enjoy ery nice asymptotic properties. In particular, one can approximate them using the asymptotic equialent E[N]ES α (Y 1 and the conergence rates are obtained in both settings. The results can be used to approximate VaR and ES of a large insurance portfolio since it is straightforward to calculate E[N]ES α (Y 1. Mathematically, our results generalize the asymptotic equialence results for homogeneous indiidual risk models (i.e. N in (1.1 is non-random in Wang and Wang (2015. The rest of this paper is organized as follows. In Section 2, we present basic notation and definitions, stochastic orders, properties of ES, and some preinary results on VaR-ES risk aggregation with dependence uncertainty. In Section 3, we study collectie risk models with dependence uncertainty and obtain formulas for the worst-case ES. In Section 4, we establish asymptotic equialence results under setting (i and gie the conergence rate under this setting. In Section 5, asymptotic equialence results under setting (ii are gien, albeit stronger regularity conditions are needed compared to the case of setting (i. In Section 6, a brief conclusion is drawn. 2 Preinaries 2.1 Some notation Let (Ω,, P be an atomless probability space in which all random ariables are defined. Assume that (Ω,, P is rich enough such that for any random ariable X that appears in the paper, there exists 4

5 a random ariable independent of X. Let L p, p R +, be the set of random ariables with finite p-th moment, and L be the set of bounded random ariables; in this paper R + = [0,. or two random ariables X and Y, we write X = d Y if they hae the same distribution. or a distribution, let X be the set of random ariables with distribution, and for N L 0, let X N be the set of random ariables in X independent of N. Let X 0 be the set of counting random ariables (i.e. taking alue in {0, 1,..., }. or a sequence Y = (Y i, i N, we write (with a slight abuse of notation Y X if Y i X, i N, and similarly for Y X N. Denote by YN the set of random sequences with marginal distribution and independent of N, that is, Y N = {(Y 1, Y 2,... X : (Y 1, Y 2,... is independent of N}. Note that for a sequence Y = (Y i, i N, there is a subtle difference between Y X N and Y YN : the latter requires independence between the sequence Y and N, whereas the former only requires pair-wise independence between N and Y i for i N. Throughout, for N X 0 and Y = (Y i, i N L 0, write N S N = Y i, where by conention 0 Y i = 0. In the following, wheneer S N or S n appears, it implicitly depends on Y = (Y i, i N which should be clear from the context. In collectie risk models, Y i, i N are always assumed to be identically distributed, since Y i represents the claim size of the i-th claim from a pool of policies, not the loss from a specific policy. We also assume Y i, i N to be integrable; otherwise ES α (Y 1 is infinite for α (0, 1. In the case when the claim size Y 1 is not integrable, ES is not a proper risk measure to use in insurance practice; see, for instance, the general discussion on applicability of risk measures in McNeil et al. (2015. or p (0, 1 and any non-decreasing function, we write 1 (p = inf{x R : (x p}. It is well known that for any random ariable X with distribution, 1 (U = d X, where U is any U[0, 1]-distributed random ariable. 2.2 Stochastic orders Definition 2.1. or X, Y L 1, X is said to be smaller than Y in conex order (resp. increasing conex order, denoted by X cx Y (resp. X icx Y, if E[ f (X] E[ f (Y] for all conex functions (resp. in- 5

6 creasing conex functions f : R R, proided that the aboe expectations exist (can be infinity. or a general introduction to conex order and increasing conex order, see Müller and Stoyan (2002 and Shaked and Shanthikumar (2007. Conex order is closely associated with the concept of comonotonicity. Definition 2.2. Two random ariables X and Y are comonotonic if (X(ω X(ω (Y(ω Y(ω 0 for (ω, ω Ω Ω (P P-a.s. Comonotonicity of X and Y is equialent to the existence of a random ariable Z L 0 and two non-decreasing functions f and g, such that X = f (Z and Y = g(z almost surely. We say that seeral random ariables X 1,..., X n are comonotonic if X i and X j are comonotonic for each pair of i, j = 1,..., n 1. See Dhaene et al. (2002 and Rüschendorf (2013 for an oeriew on comonotonicity. Gien random ariables X 1, X 2,..., X n, the following lemma presents an upper bound for sums S n = X 1 +X 2 + +X n in the sense of conex order; see Theorem 7 of Dhaene et al. (2002 and Theorem 3.5 of Rüschendorf (2013. In particular, Rüschendorf (2013, Chapter 3 contains two different proofs and a brief history of this celebrated result. Lemma 2.1. or any random ector (X 1,..., X n (L 1 n we hae where X c i X X n cx X c Xc n, d = X i, i = 1,..., n, and X c 1,..., Xc n L 1 are comonotonic. Another property about increasing conex order and comonotonicity is gien in the following lemma, which is Corollary 3.28 (c of Rüschendorf (2013. Lemma 2.2. or X, Y, X c, Y c L 1 such that X c, Y c are comonotonic, X d = X c, Y d = Y c and X c Y c L 1, we hae XY icx X c Y c. The stochastic inequality in the aboe lemma holds for eery monotonic supermodular function of X and Y; see Theorem 2 of Tchen (1980 and Theorem 2.1 of Puccetti and Wang ( Generally, this definition is stronger than assuming that X 1 and X j are comonotonic for j = 2,..., n; see for instance, Example 4 of Cheung et al. (

7 2.3 Some properties of ES In this paper, we will frequently use some well-known properties of the Expected Shortfall defined in (1.3; see e.g. McNeil et al. (2015 for details on properties of ES. Lemma 2.3. or α (0, 1, ES α : L 1 R satisfies: for any X, Y L 1, (i Monotonicity: ES α (X ES α (Y if X Y P-a.s; (ii Cash-inariance: ES α (X m = ES α (X m for any m R; (iii Subadditiity: ES α (X + Y ES α (X + ES α (Y; (i Positie homogeneity: ES α (λx = λes α (X for any λ > 0; ( Law-inariance: ES α (X = ES α (Y if X = d Y; (i Comonotonic additiity: ES α (X + Y = ES α (X + ES α (Y if X and Y are comonotonic. The reader is referred to öllmer and Schied (2011, Chapter 4 and Delbaen (2012 for interpretations of these standard properties in the literature of risk measures. The following lemma is also well known in the literature of conex order (see Theorem 4.A.3 of Shaked and Shanthikumar (2007. Lemma 2.4. or X, Y L 1, X icx Y if and only if ES α (X ES α (Y for all α (0, 1. As a consequence of Lemma 2.4, for α (0, 1, ES α preseres increasing conex order (and hence conex order. Another property that will be used later is the L 1 -continuity of ES below; for a proof of this property, see, for instance, Sindland (2008. Lemma 2.5. or α (0, 1, ES α : L 1 R is continuous with respect to the L 1 -norm. Recalling the definition of the L 1 -continuity, the aboe lemma means that for a sequence of random ariables X 1, X 2,... and X L 1, as n, E[ X n X ] 0 implies that ES α (X n ES α (X. 2.4 VaR-ES asymptotic equialence in risk aggregation We gie some preinary results on the VaR-ES asymptotic equialence in risk aggregation, which will be useful to the main results in this paper. 7

8 Lemma 2.6 (Corollary 3.7 of Wang and Wang (2015. or any distribution and Y X, The result in (2.1 can be rewritten as sup Y X VaR α (S n = ES α (Y, α (0, 1. (2.1 n n sup Y X VaR α (S n = 1, α (0, 1, (2.2 n sup Y X ES α (S n proided that 0 < ES α (Y <, Y X. Result of type (2.2 is called an asymptotic equialence between VaR and ES. The equialence (2.2 was shown in Puccetti and Rüschendorf (2014 and Puccetti et al. (2013 under different conditions, and with generality in Wang and Wang (2015. or equialence of type (2.2 under the setting of inhomogeneous marginal distributions and for risk measures other than VaR and ES, see Embrechts et al. (2015, Wang et al. (2015 and Cai et al. (2017. The conergence rate of (2.2 is gien in the following lemma. Lemma 2.7 (Corollary 3.8 of Wang and Wang (2015. Suppose that the distribution has finite p-th moment, p 1, and ES at leel α (0, 1 is non-zero. Then as n, sup Y X VaR α (S n sup Y X ES α (S n = 1 o ( n 1/p 1. 3 Collectie risk models with dependence uncertainty 3.1 Setup and a motiating example In this section, we study the worst-case alues of VaR and ES for collectie risk models. As mentioned in the introduction, we consider two different settings of dependence uncertainty: (i the number of claims N is independent of the claim sizes Y 1, Y 2,... and the dependence structure of Y 1, Y 2,... is unknown; (ii the dependence structure of N, Y 1, Y 2,... is unknown. We refer to the setting (i as the classic collectie risk model with dependence uncertainty and to the setting (ii as the generalized collectie risk model with dependence uncertainty. Using the notation introduced in Section 2, for some distribution on R + (i.e. non-negatie claim sizes, setting (i reads 8

9 as Y Y N and setting (ii reads as Y X. The quantities of interest in setting (i are and the quantities of interest in setting (ii are sup VaR α (S N and sup ES α (S N, (3.1 Y Y N Y Y N sup VaR α (S N and sup ES α (S N. (3.2 Y X Y X It turns out that under both settings (i and (ii, the worst-case alue of ES is straightforward to calculate, whereas an analytical formula for the worst-case alue of VaR is not aailable. This is similar to the well-studied case of indiidual risk models; see Embrechts et al. (2014 for a reiew on worst-case VaR aggregation when N is non-random. Before we carry out a theoretical treatment, we illustrate with a simple example in the theory of loss models by comparing an indiidual risk model and a corresponding collectie risk model formulation. Assume both models admit dependence uncertainty, and we ealuate worst-case ES for both models as in (3.2. We shall see that the ES bound is largely reduced by knowing the distribution of the claim frequency, as opposed to an uncertain distribution of the claim frequency implied by the indiidual risk model with dependence uncertainty. Example 3.1. Let n = Consider an indiidual risk model S = n X i, where for i = 1,..., n, X i follows a distribution such that P(X i > x = e x, x 0. If we assume that X 1,..., X n are independent, then the collectie reformulation of S is gien by S N = N Y i, where N follows the Poisson distribution with parameter λ = 40 (denoted by Pois(40, Y i follows an Exponential distribution with mean 1 (denoted by Expo(1, i N, and N, Y 1, Y 2,... are independent. Below we assume that only N and (Y i, i N are independent, but the dependence among X 1,..., X n and the dependence among Y 1, Y 2,..., are uncertain. Take α = To ealuate the corresponding 9

10 worst-case ES α alues, we hae 2 sup Y Y N sup X i X,i n N ES α Y i = , ES α n X i = 800. As we can see from the numerical results, the knowledge of N Pois(40 greatly reduces the worstcase ES alue, as compared to the indiidual risk model. In the sequel, we shall inestigate the VaR and ES bounds for collectie risk models under dependence uncertainty. 3.2 VaR and ES bounds for collectie risk models In this section we establish some explicit formulas for VaR and ES bounds in (3.1 and (3.2. We first proide a simple result on conex order for collectie risk models with unknown dependence. Lemma 3.1. Suppose that (Y i, N L 1 X 0, i N, hae identical joint distributions and NY 1 L 1. We hae N Y i cx NY 1. (3.3 Proof. irst, one can easily erify E[ N Y i ] = E[NY 1 ] and hence both sides of (3.3 are in L 1. Let D = {n {0, 1,... } : P(N = n > 0} be the range of N. Denote by n the conditional distribution of Y 1 gien N = n for n D. Let f be a conex function such that both E[ f ( N Y i ] and E[ f (NY 1 ] are properly defined. or n D, there exist some U[0, 1]-distributed random ariables U n 1,..., Un n such that It follows from Lemma 2.1 that Summing up oer n D yields E[ f (Y Y n N = n] = E[ f ( 1 n (U n n (U n n]. E[ f (Y Y n N = n] E[ f (n 1 n (U n 1 ] = E[ f (ny 1 N = n]. E[ f (Y Y N ] E[ f (NY 1 ], and hence by definition, (3.3 holds. 2 the first alue is calculated ia Theorem 3.3 (see below and the aerage of 100 repetitions of simulation with a sample of size 100,000, and the second alue is calculated analytically. 10

11 As a special case of Lemma 3.1, if N is in L 1 and independent of the identically distributed random ariables Y 1, Y 2,... L 1, then (3.3 holds. This particular result will be used later. To deal with setting (ii in which the dependence structure between N and Y 1, Y 2,... is unspecified, we gie a result in the following lemma on increasing conex order instead of conex order. Note that for X, Y L 1, X cx Y implies that E[X] = E[Y]. Since E[S N ] depends on the dependence structure between N and Y 1, Y 2,..., conex order between collectie risk models under different dependence structures cannot be expected. Lemma 3.2. Suppose that the distribution on R + has finite second moment, and N X 0 L 2. or Y 1, Y 2,... X, we hae where Y X and N, Y are comonotonic. N Y i icx NY, Proof. Note that NY L 1 by Hölder s inequality. Define X n = n Y i I {N i}, n N and X = Y i I {N i}. Note that P(X > X n 0 as n, and hence P(X < = 1. Thus X is a finite random ariable. Then we hae X n X almost surely and hence X n X in distribution. Since 1 (γ is weakly continuous at each 0 for which s 0 1 (s is continuous at s = γ (see e.g. Cont et al. (2010, we hae VaR γ (X n VaR γ (X almost eerywhere in γ [0, 1]. (3.4 or any Y X and any α (0, 1, we hae N ES α Y i = ES α Y i I {N i} = 1 1 α 1 (by (3.4 = α (atou s Lemma (subadditiity of ES (by Lemmas 2.2 and 2.4 (comonotonic additiity of ES α α VaR γ (X dγ VaR γ(x n dγ n inf ES α (X n n n inf ES α Yi I {N i} n inf n = inf n n ES α YI{N i} ES α (L 1 -continuity of ES = ES α (NY. n YI {N i} Since ES α ( N Y i ESα (NY for all α (0, 1, by Lemma 2.4, we hae N Y i icx NY. 11

12 Remark 3.1. Using the same proof, the stochastic inequality in Lemma 3.2 can be generalized to the random sum of non-identically distributed random ariables as follows. Suppose that N X 0, Y i 0, i N, and N Y c i L 1, where Y c i d = Y i, i N, and Y1 c, Yc 2,... and N are comonotonic. Then we hae N Y i icx N Yi c. With the help of Lemmas 3.1 and 3.2, we arrie at the worst-case alues of ES for collectie risk models under dependence uncertainty. Theorem 3.3. Suppose that is a distribution on R +, N X 0 and Y, Y X such that N, Y are independent and N, Y are comonotonic. (i If Y, N L 1, then (ii If Y, N L 2, then sup ES α (S N = ES α (NY, α (0, 1. (3.5 Y Y N sup ES α (S N = ES α (NY, α (0, 1. (3.6 Y X Proof. Note that NY L 1 since N, Y are independent. Since Y X N for any Y YN, we hae sup ES α (S N sup ES α (S N. Y Y N Y X N By Lemma 2.4, ES preseres increasing conex order. urther, by Lemmas 3.1 and 3.2, we hae sup ES α (S N sup ES α (S N ES α (NY and sup ES α (S N ES α (NY. Y Y N Y X N Y X It suffices to take Y 1, Y 2,... to be identical to Y X N to show that N ES α(s N ES α (NY in (i and to take Y 1, Y 2,... to be identical to Y X to show sup Y X ES α (S N ES α (NY in (ii. The results in Theorem 3.3 are consistent with simple intuition. Assume that the riskiness of an insurance portfolio is measured by an ES. If the number of claims and the claim sizes are independent, then, in the worst-case dependence scenario, all claims are comonotonic. If the number of claims and the claim sizes are also dependent, then in the worst-case dependence scenario, all claims are comonotonic and they are further comonotonic with the number of claims. This could for instance be close to reality in the case of insurance losses from flood damage in an area, where the claim sizes and the number of claims are largely determined by the magnitude of the flood, and hence they are all 12

13 positiely correlated. Thus, the portfolio of insurance policies with heay positie dependence has the most dangerous dependence structure, if an ES is the risk measure in use. Note that such an intuition is not alid for the risk measure VaR. The alues of ES α (NY and ES α (NY in (3.5 and (3.6 are straightforward to calculate. or (3.5, one needs to calculate the distribution of NY, which is the product of two independent random ariables. This inoles a one-step conolution after a logarithm transformation. or (3.6, note that NY d = G 1 (U 1 (U, where U is U[0, 1]-distributed and G is the distribution of N. In that case, its ES is simply ES α (NY = 1 1 α 1 α G 1 (u 1 (udu, which is as simple as calculating the ES of any known distribution. The following corollary gies an ES ordering for an indiidual risk model with dependence uncertainty, a collectie risk model under setting (i, and a collectie risk model under setting (ii. Corollary 3.4. Suppose that is a distribution on R + with finite first moment, N X 0 and E[N] N. We hae the following orders sup ES α S E[N] sup ES α (S N sup ES α (S N, α (0, 1. (3.7 Y X Y Y N Y X Proof. Since Y Y N implies Y X, the second inequality follows immediately. To show the first inequality, take Y X N. Note that from the properties of ES, and from Theorem 3.3, sup ES α S E[N] = E[N]ESα (Y = ES α (E[N]Y, Y X sup ES α (S N = ES α (NY. Y Y N By Theorem 3.A.33 of Shaked and Shanthikumar (2007, E[N]Y cx NY. The rest of the proof follows since ES preseres conex order as in Lemma 2.4. In the case of N, Y L 2, the order in (3.7 can be formulated as follows. or N X 0, Y d = Y such that N, Y are independent and N, Y are comonotonic, we hae E[N]ES α (Y ES α (NY ES α (NY, α (0, 1. (3.8 As for the problem of the worst-case alue of VaR for collectie risk models, there is no simple analytical formula, as expected from classic results on dependence uncertainty. Note that VaR α is 13

14 dominated by ES α, for α (0, 1; thus VaR α (S N ES α (S N for all model settings. rom Theorem 3.3, we hae sup VaR α (S N ES α (NY and sup VaR α (S N ES α (NY, (3.9 Y Y N Y X where, N, Y and Y are as in Theorem 3.3. In the next two sections, we will see that sup VaR α (S N ES α (NY and sup VaR α (S N ES α (NY, Y Y N Y X if N is large (in some sense. That is, the inequalities in (3.9 are almost sharp and can be used to approximate VaR. Remark 3.2. In Lemma 3.2 and Theorem 3.3 (ii, we require Y, N L 2 so that NY L 1 ; recall that L 1 is the domain of ES α. One may also use the slightly more general assumption that N L p and Y L q for some p, q > 1 such that 1/p + 1/q = 1. 4 Asymptotic results for classic collectie risk models 4.1 Setup and objecties The rest of the paper is dedicated to the study of an analog of the asymptotic equialence in (2.2 for collectie risk models. Recall that throughout we write N( S N( = Y i, Y = (Y 1, Y 2,.... or some distribution on R +, and a counting random ariable N( with parameter, the analog of (2.2 in setting (i is and the analog of (2.2 in setting (ii is N( VaR α S N( N( ES α ( S N( = 1, (4.1 sup Y X VaR α S N( = 1. (4.2 sup Y X ES α S N( Here, indicates that the expected number of claims goes to infinity. The parameter is interpreted as the olume of the insurance portfolio, and it can be chosen as, for instance, E[N(]. One of the key assumptions we propose is N(/ 1 in L 1. This assumption naturally holds if N( is a Poisson random ariable with parameter > 0, or N( is the partial sum of a short-range 14

15 dependent stationary sequence (so that a law of large numbers holds. Indeed, the problem we study in this paper first appeared as a question of measuring large insurance portfolios under dependence uncertainty, where N( is a Poisson random ariable with a large parameter. Moreoer, an insurance company can analyze effects from potential extension of business by measuring the insurance portfolio as increases. Results under setting (i are presented in this section and results under setting (ii are gien in Section 5 below. Since ES α (Y is straightforward to calculate and thus seres as a basis for approximation of the two worst-case alues of interest, we present our results in terms of the two ratios N( VaR α S N( sup Y Y N( ES α S N( and. ES α (Y ES α (Y We also establish conergence rates in both cases. 4.2 VaR-ES asymptotic equialence Theorem 4.1. Suppose that the distribution on R + has finite first moment, Y X, and {N(, 0} X 0 such that N(/ 1 in L 1 as. Then for α (0, 1, N( VaR α S N( sup Y Y N( ES α S N( = = ES α (Y. (4.3 Proof. By the independence of N( and Y, and N( Hence, N(Y E N(Y L 1 1, we hae Y E N( 1 E [Y] 0, as. L 1 Y. Continuity of ES with respect to the L 1 -norm implies N(Y ES α = ES α (Y. (4.4 rom Theorem 3.3 (i, we hae By (4.4 and the positie homogeneity of ES, we hae Thus, we obtain the second equality in (4.3. sup ES α (S N( = ES α (N(Y. (4.5 Y Y N( N( ES α (S N( = ES α (Y. 15

16 Since L 1 -conergence implies conergence in probability, N( δ > 0, there exists an M 1 > 0 such that for all M 1, ( N ( P 1 > δ < ε. Write S N( = Y Y N(. Define L 1 1 yields that for any ε > 0 and S N( = S N( if N(/ 1 δ, 0 if N(/ < 1 δ. Since S N( S N(, we hae VaR α+ε ( S N( VaRα+ε ( S N( = inf { t R : P ( S N( t α + ε } = inf { t R : P ( S N( t, N(/ 1 δ + P (0 t, N(/ < 1 δ α + ε } inf { t R : P ( S N( t, N(/ 1 δ α } inf { t R : P ( S (1 δ t α } = VaR α ( S (1 δ. (4.6 By Lemma 2.6, for any ε 2 > 0, there exists an M 2 > 1/ε such that for all > M 2, sup Y X VaR α ε S (1 δ > ES α ε (Y ε 2. (1 δ Thus, for the aboe ε > 0 and > max{m 1, M 2 }, VaR α S N( N( N( VaR α ε ( S (1 δ = sup Y X VaR α ε S (1 δ (1 δ > [ES α ε (Y ε 2 ] (1 δ 1 > [ES α ε (Y ε 2 ] (1 δ ε, (1 δ which implies On the other hand, Therefore, sup N( inf N( VaR α ( S N( Thus, we obtain the first equality in (4.3. VaR α ( S N( ES α (Y. N( ES α (S N( = ES α (Y. N( VaR α S N( = ES α (Y. 16

17 Theorem 4.1, together with Lemma 2.6, suggests that for α (0, 1 and Y X, the following fie quantities are all asymptotically equialent as : (i N( (ii N( VaR α ( S N( ; ES α ( S N( ; (iii sup Y X VaR α ( S ; (i sup Y X ES α ( S ; ( ES α (Y. Hence, one may use ( aboe (straightforward to calculate to approximate the other four quantities. The approximation error, that is, the conergence rate in Theorem 4.1, is studied in the following section. Remark 4.1. Since VaR α ES α, the quantity in (i is smaller than or equal to the quantity in (ii, and similarly for (iii and (i. Another obseration is that sup Y X ES α ( S = ESα (Y ES α (Y. rom Corollary 3.4, the quantity in (i is smaller than or equal to the quantity in (ii, proided that E[N(] =. Howeer, there is no general order between (i and ( (or (i; when we approximate N( VaR α ( S N( with ESα (Y, it is not clear which one is larger. See Theorem 4.2 below for more detailed analysis on their relationship. 4.3 Rate of conergence Theorem 4.2. Suppose that the distribution on R + has finite p-th moment, p 1, Y X, E[Y] > 0, and sup q E N( 1 c for some q > 0, c > 0. Then for α (0, 1, 2C 1/2 q/2 + o ( 1/p 1 + o ( q/2 N( VaR α ( S N( ES α (Y 1 C q + o ( q, (4.7 and N( ES α S N( 1 ES α (Y C q + o ( q, (4.8 where C = c 1 α. Proof. Let δ = C q/2, η = o( q/2. Note that q (c + η E N( ( q E 1 c N( 1 + N(/ 1 >δ, and ε = c+η δ q. Clearly ε = c(1 α q/2 + N( ( 1 dp > δ P N ( 1 > δ. Hence, ( N ( P 1 > δ < c + η δ q = ε. 17

18 This implies VaR α S N( VaRα ε S (1 δ as shown in (4.6. By Lemma 2.7, we hae N( VaR α S N( sup Y X VaR α ε S (1 δ (1 δ ES α ε (Y ES α (Y sup Y X ES α ε S (1 δ ES α (Y [ 1 o ( 1/p 1] (1 δ 1 ES α ε (Y ES α (Y. (4.9 Note that 1 ES α ε (Y ES α (Y = ( 1 1 α α+ε α VaR γ (Y dγ 1 α 1 α+ε α ε VaR γ (Y dγ ES α (Y ε 1 α. Therefore, ES α ε (Y ES α (Y Plugging the aboe inequality into (4.9, one has VaR α S N( N( ES α (Y Thus, we obtain the first inequality in ( ε 1 α. In the next step we show (4.8. rom Theorem 3.3 (i, ES α S N( By the subadditiity of ES, we hae Similarly, ES α ( N( ES α (Y = ES α ( N( N( [ 1 o ( 1/p 1] (1 δ 1 ( 1 ε 1 α = 1 2 C q/2 o ( 1/p 1 o ( q/2. ES α (Y Y + Y N( Y = ES α (N(Y. ES α (Y ( N( ES α Y + ES α Y ( ES α (Y + ES N( α Y Y. It follows that ES α ( N( ES α (Y ES α Y Y N( Y ES α ( Y N( Y N( Y Y,. and Therefore, N( ES α Y N( ES α (Y ES α Y Y ES α ( Y N( N( ( ES α Y ES α (Y ES α Y N( Y 1 1 α E N( 1 E [Y], Y. (

19 which implies Thus we obtain (4.8 as ( ES N( α Y 1 ES α (Y C q + o( q. sup Y X N( ES α S N( 1 ES α (Y C q + o( q, and the second inequality in (4.7 is automatically implied since VaR α is dominated by ES α. In Example 4.1 of the next section, we will see that q = 1/2 for Poisson(-distributed N(. In this case, assuming p 4/3 (typically true, the conergence rate in the left-hand side of (4.7 is led by O( 1/4 and the one in (4.8 is led by O( 1/2. Admittedly, the conergence rate O( 1/4 is not ery fast in general, and its applicability for approximation depends on the models and the magnitude of. Howeer, for risk management purpose, one should be on the conseratie side; as such, the faster rate O( q in the right-hand side of (4.7 and in (4.8 is more important in practice. In Example 4.4 below, we will see that the term O( q for the upper bounds in Theorem 4.2 is sharp. 4.4 Some examples Example 4.1 (Poisson number of claims. As the primary example, suppose that N( follows a Poisson distribution with parameter. We check the conditions and parameters in Theorems 4.1 and 4.2. Clearly, N(/ 1 in L 1 as by the L 1 -Law of Large Numbers. Indeed, note that E N( = 2e +1!, and further by Stirling s formula and some elementary analysis, one has +1 2e! 1/2 2 π, which means Therefore in Theorem 4.2, c = 1/2 E 2 π and q = 1/2. N( 1 2 = π. Example 4.2 (Non-random number of claims. Suppose that N( equals. Then q = in the conditions of Theorem 4.2, and the lower bound on VaR conergence rate gien in (4.7 is equialent to Lemma 2.7. Example 4.3 (Non-random claim sizes. Suppose that Y is not random and sup q E 1 c for some q > 0, c > 0. In this case, we hae a conergence rate that is slightly stronger than the one 19 N(

20 gien in (4.7, 1 2C 1/2 q/2 o ( q/2 N( N( VaR α ( S N( ES α (Y ES α S N( ES α (Y 1 + C q + o( q, (4.11 where C = c 1 α. Compared with Theorem 4.2, the term o(1/p 1 in the lower bound for VaR conergence disappears. This is quite natural since the term o( 1/p 1 is due to the randomness of Y as suggested by Lemma 2.7. To see the first inequality in (4.11, let ε = α and δ = C q/2. or large enough, which imply Therefore, N( ( N( P VaR α ( S N( ES α (Y ( 1 N( > δ < ε and P N( VaR α 1 δ. = VaR α (N( The rest of (4.11 comes from Theorem 4.2. < 1 δ < α, 1 δ 1 2C 1/2 q/2 o ( q/2. Example 4.4 (Sharpness of the rate in the right-hand side of (4.7 and in (4.8. or some q > 0, take N( = + 1 q and let be a degenerate distribution of a constant, say 1. In this case, S N( = N( is not random, and obiously VaR α (S N( = ES α(s N( = O( q. This shows that the leading term q in the right-hand side of (4.7 and in (4.8 is sharp up to a constant scale, een in the case when Y 1, Y 2,... and N( are deterministic. 5 Asymptotic results for generalized collectie risk models In this section, we study the more complicated setting (ii in which N and Y 1, Y 2,... are not necessarily independent, and their joint distribution is also uncertain. We hae similar results as in Theorem 4.1 and Theorem 4.2 under stronger regularity conditions. 20

21 5.1 VaR-ES asymptotic equialence Theorem 5.1. Suppose that the distribution on R + has finite second moment, Y X, and {N(, 0} X 0 such that N(/ 1 in L 2 as. Then for α (0, 1, sup Y X VaR α S N( sup Y X ES α S N( = = ES α (Y. (5.1 Proof. rom Theorem 3.3, for fixed > 0, we hae sup Y X ES α ( S N( = ESα ( N(Y, (5.2 where Y X is comonotonic with N(. Hölder s inequality implies E N(Y Y E N( 2 1 E [ (Y 2] 0, as. Hence, N(Y L 1 Y. As a consequence, continuity of ES with respect to the L 1 -norm implies Therefore, ES ( α N(Y / ( = ES α Y = ES α (Y. sup Y X ES α (S N( ES α (N(Y = = ES α (Y. Thus we obtain the second equality in (5.1. one has or the first equality in (5.1, N( L 2 1 implies that for any ε > 0 and δ > 0, for large enough, ( N ( P Similarly to the proof of Theorem 4.2, we hae 1 > δ < ε. ES α ε (Y ES α (Y 1 ε 1 α, and sup Y X ES α (S N( ES α (Y sup Y X VaR α ( S N( ES α (Y [ 1 o ( 1/p 1] (1 δ 1 ES α ε (Y ES α (Y. Thus, and we obtain the first equality in (5.1. sup Y X VaR α S N( = ES α (Y, 21

22 5.2 Rate of conergence In this section we proide the conergence rate in generalized collectie risk models. Similarly to Theorem 5.1, stronger regularity conditions are required as compared to results in Section 4. Theorem 5.2. Suppose that the distribution on R + has finite p-th moment, p 2, Y X, E[Y] > 0, and sup r E N( 1 2 c for some r > 0 and c > 0. Then we hae ( c 2 1 α Proof. Let δ = ( c 1 α r 1/3, η = ( r E 1/3 r/3 + o ( 1/p 1 + o ( r/3 sup Y X VaR α ( S N( sup Y X ES α S N( 1 ES α (Y N( o( r/3. Similar to the proof of Theorem 4.2, we hae ( N ( P 1 > δ < ε and + ES α (Y E[Y 2 ] ES α (Y 1 (5.3 c 1 α r/2 + o ( r/2. ( c, and ε = c+η r. Clearly ε = ( c(1 α 2 r 1/3 δ + 2 ES α ε (Y ES α (Y 1 ε 1 α. Moreoer, sup Y X VaR α ( S N( ES α (Y sup Y X VaR α ε S (1 δ (1 δ ES α ε (Y sup Y X ES α ε S (1 δ ES α (Y [ 1 o ( 1/p 1] (1 δ 1 ( 1 ε ( 1 2 c 1 α 1 α 1/3 r/3 o ( 1/p 1 o ( r/3. Thus we obtain (5.3. The first inequality in (5.4 comes from the fact that ES α dominates VaR α. By (4.10 and Hölder s inequality, we hae N( ( ES α Y ES α (Y ES α Y N( Y 1 1 α E N( 1 2 E [ Y 2]. As a consequence, sup Y X ES α S N( 1 ES α (Y E[Y 2 ] ES α (Y c 1 α r/2 + o ( r/2. Thus we obtain the second inequality in (5.4. Example 5.1 (Poisson number of claims, reisited. Suppose that N( follows a Poisson distribution with parameter. We can check the parameters in Theorem 5.2. Since E[ N(/ 1 2 ] = Var(N(/ 2 = 1/, we hae r = 1 and c = 1. Therefore, the leading term in the left-hand side 22

23 of (5.3 is O( 1/3, which conerges to zero faster than O( 1/4 as in Example 4.1 under setting (i. This is intuitie as sup Y X VaR α S N( supy X N( remains the same order O( 1/2. VaR α ( S N(. The right-hand side of ( A remark on the dependence between the claim frequency and the claim sizes In Sections 4 and 5, we studied the asymptotic equialence of VaR and ES in two settings. A natural question that follows would be whether an asymptotic equialence holds also for specified dependence structures between N( and Y 1, Y 2,... other than independence. That is, whether the following it holds, where ˆX N( sup Y ˆX N( VaR α S N( = 1 (5.5 sup Y ˆX N( ES α S N( X is the set of random ariables with distribution and a pre-specified dependence structure (copula with N(. Note that from Lemma 3.1, the worst-case ES can be calculated as ES α (N(Y, where Y ˆX N(. In general, the knowledge on the dependence structure of (N(, Y i, i = 1, 2,..., would put some restrictions on the dependence structure of (Y 1, Y 2,... ; the latter was assumed to be arbitrary in our settings (i and (ii, as well as in the classic setup of dependence uncertainty. With the effect of dependence uncertainty demolished, (5.5 may no longer hold true. This is eidenced by the following (rather extreme example where N(, Y i are comonotonic for i = 1, 2,... (note that this does not necessarily imply that Y 1, Y 2,... are comonotonic since N( is discrete. or other pre-specified dependence structures between (N(, Y i, i = 1, 2,..., the question of (5.5 requires a case-by-case study. Assume that the distribution has finite second moment, {N(, 0} X 0 such that N(/ 1 in L 2 as, and Y X c,. Denote by Xc, X the set of random ariables with distribution and comonotonic with N(. In this case one still has the ES conergence as in Theorem 4.1, sup Y X c, ES α S N( = ES α (Y, α (0, 1, (5.6 whereas the VaR conergence sup Y X c, VaR α S N( = ES α (Y, α (0, 1, (5.7 may fail to hold. 23

24 To see (5.6, by Hölder s inequality, we hae E N(Y Y E N( 1 Hence, N(Y 2 E [ Y 2] 0, as. L 1 Y. rom Lemma 3.1, we hae sup Y X c, ES α S N( = ESα (N(Y, and (5.6 follows from the continuity of ES with respect to the L 1 -norm. To see that (5.7 may not hold true, we simply gie a counter-example. Take any α (0, 1. Let be a Bernoulli distribution with parameter (1 α/2, and assume that for each > 0, there exists a positie integer f such that P(N( > f = (1 α/2. or fixed and any Y 1, Y 2, X c,, we hae {Y i = 1} = {N( > f } almost surely for each i = 1, 2,..., and hence Y 1, Y 2,... are almost surely equal. As a consequence, there is indeed no dependence uncertainty: S N( = N(Y 1 almost surely. Since P(N(Y 1 > 0 P(Y 1 > 0 = (1 α/2, we hae sup Y X c, VaR α (S N( = VaR α (N(Y 1 = 0. Therefore, sup Y X c, VaR α S N( = 0. Thus (5.7 does not hold noting that ES α (Y > 0. 6 Conclusion In this paper, we study the worst-case alues of VaR and ES of the aggregate loss in collectie risk models under two settings of dependence uncertainty. Analytical formulas for the worst-case alues of ES are obtained. or both settings, an asymptotic equialence of the VaR and ES for a random sum of risks is established under some general moment and regularity conditions. The conditions in our main results are easily satisfied by common models, including the classic compound Poisson collectie risk models. Our main results suggest that under dependence uncertainty, we can use ES α (Y to approximate the worst-case risk aggregation when the risk measure is VaR α or ES α and is large enough; the approximation error is also obtained in terms of some moment and conergence rate of the claim sizes and the claim frequency. There are arious methods deeloped to incorporate dependence information into risk aggregation with model uncertainty. Practically, one should choose a formulation depending on information 24

25 aailable to the user; collectie risk models become releant should the claim frequency distribution be aailable. or indiidual risk models, arious types of dependence information were considered, such as moment information and ariance information by Bernard et al. (2017a,b, positie dependence or independence information by Bignozzi et al. (2015 and Puccetti et al. (2017, and factor relation by Bernard et al. (2017c. Information similar to the aboe types can be naturally incorporated into the collectie risk model. This leads to promising future research directions; we anticipate great mathematical and practically releant implications. Acknowledgements The authors thank an Editor and two referees for arious constructie comments which hae greatly improed the paper. The authors are also grateful to Mario Wüthrich for helpful discussions which motiated this study of dependence uncertainty in collectie risk models. Haiyan Liu acknowledges financial support from the Uniersity of Waterloo and the China Scholarship Council. Ruodu Wang would like to thank IM at ETH Zurich for supporting his isit in 2015, and he acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada (RGPIN A Appendix Below we discuss the difference between a collectie risk model and a corresponding indiidual risk model. Let N be the counting random ariable which is bounded by some n N, i.e. N n, and Y i, i N (in fact, only Y 1,..., Y n are used. A random sum S N may be written in two ways: a collectie risk model S N = N Y i and an indiidual risk model n n S N = Y i I {N i} = Z i, (A.1 (A.2 where Z i = Y i I {N i}, i = 1,..., n. Note that this setup is different from the collectie reformulation in Example 3.1, where one starts with a homogeneous indiidual risk model with small probability of loss from each indiidual risk, and arries at a Poisson collectie risk model. In the recent literature of dependence uncertainty for an indiidual risk model, Z 1,..., Z n in (A.2 are assumed to hae an arbitrary dependence. In our collectie risk model, although S N may be written 25

26 as in (A.2, the dependence among Z 1,..., Z n is not arbitrary anymore, as it is drien by a common random ariable N. There are further essential differences, if we look at the two formulations more closely under the two settings of dependence uncertainty studied in this paper. (i N and the sequence Y 1, Y 2,... are independent. In this case, the distribution of Z i can be determined by that of Y i and N. Denote this distribution by i. We can consider the worst-case risk measure (take an ES for instance in our model sup Y Y N N ES α Y i (A.3 and in the classic model sup Z i X i,i n ES α n Z i. Clearly, through (A.1 and (A.2, the collectie risk model formulation Y Y N (A.4 in (A.3 is a submodel of the indiidual risk model formulation Z i X i, i n in (A.4, and hence the worstcase alue in (A.3 should be smaller than or equal to the one in (A.4. We shall illustrate this difference with a numerical example where one has See Example A.1 below. sup Y Y N N ES α Y i < sup Z i X i,i n ES α n Z i. (ii The dependence between N and the sequence Y 1, Y 2,... is also unknown. In this case, the distribution of Z i, and the conditional distribution of Z i gien N are both unknown. Hence, no existing result in the literature of dependence uncertainty that we are aware of can be applied to this setting. Example A.1. Let n = 10. Suppose that for i = 1,..., n, Y i follows Expo(1, and N follows the binomial distribution with parameters n and 1/3 (denoted by Bin(n, 1/3, independent of {Y i, i N}. or i = 1,..., n, denote the distribution of Y i I {N i} by i. Take α = By Theorem 3.3, we can calculate sup Y Y N sup Z i X i,i n N ES α Y i = ES α(ny 1 = , ES α n Z i = n ES α (Z i = , 26

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