Institut de Science Financière et d Assurances

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1 Institut de Siene Finanière et d Assuranes Les Cahiers de Reherhe de l ISFA ROBUSTNESS ANALYSIS AND CONVERGENCE OF EMPIRICAL FINITE-TIME RUIN PROBABILITIES AND ESTIMATION RISK SOLVENCY MARGIN Stéphane Loisel Christian Mazza Didier Rullière Cahier de Reherhe WP ) Université Claude Bernard Lyon 1 Retrouvez la liste omplète des Cahiers de reherhe de l ISFA à l adresse :

2 Robustness analysis and onvergene of empirial finite-time ruin probabilities and estimation risk solveny margin Stéphane Loisel, Université Claude Bernard Lyon 1 - Eole ISFA - 50, avenue Tony Garnier - F Lyon Cedex 07 Christian Mazza, Department of Mathematis, University of Fribourg, Pérolles, Chemin du Musée 23, CH-1700 Fribourg, Switzerland Didier Rullière. Université Claude Bernard Lyon 1 - Eole ISFA - 50, avenue Tony Garnier - F Lyon Cedex 07 Abstrat We onsider the lassial risk model and arry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to ompute the related influene funtions. We also prove the weak onvergene of a sequene of empirial finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the onepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To ontrol this robust risk measure, an additional initial reserve is needed and alled Estimation Risk Solveny Margin ERSM). We apply our results to show how portfolio experiene ould be rewarded by ut-offs in solveny apital requirements. An appliation to atastrophe ontamination and numerial examples are also developed. Key words: Finite-time ruin probability, robustness, Solveny II, reliable ruin probability, asymptoti Normality, influene funtion, Estimation Risk Solveny Margin ERSM). JEL Classifiation odes: G22, C60. Corresponding author. Tel , fax addresses: Stephane.Loisel@univ-lyon1.fr Stéphane Loisel), hristian.mazza@unifr.h Christian Mazza), Didier.Rulliere@univ-lyon1.fr Didier Rullière). Cahiers de Reherhe de l ISFA, WP )

3 1 Introdution No matter whether it is for risk apital alloation, for solveny requirements, or just for risk measurement, most atuaries traditionally start by fitting the orresponding data with some distribution using log-likelihood maximization, moment-based methods, or other statistial proedures, and then ompute the probability of ruin, the Value-at-Risk, or some relevant risk-related quantity based on probabilisti models involving the fitted distribution. Robust statistis is a huge field, extensively studied in the seventies and in the eighties, in partiular by Hampel 1974) and Huber 1981), and provides powerful onepts for sensitivity studies. Reently, Mareau and Rioux 2001) pointed out the importane of robust statistial methods in risk theory, and provided sensitivity results for infinite-time ruin probabilities. Atuaries are nowadays more interested in finite-time ruin probabilities, within a time-horizon between 1 and 10 years. Robust estimation of finitetime ruin probabilities is really in the spirit of pillar I of Solveny II. Robust CVaRs were used in different papers for portfolio seletion for example. Finite-time ruin probabilities were studied in several papers, in partiular Piard and Lefèvre 1997), Rullière and Loisel 2004), De Vylder 1999), and Ignatov, Kaishev and Krahunov 2001). But surprisingly, no robustness analysis of the finite-time ruin probability has appeared in the literature yet to our knowledge. Similarly, asymptoti Normality of estimators of infinite-time ruin probabilities has been studied by Croux and Veraverbeke 1990) and more reently by Bening and Korolev 2000). Consisteny of bootstrap estimators of finite and infinite-time ruin probabilities had also been studied by Frees 1986) and Hipp 1989). Estimation risk has been designated as one of the risks that should be taken into aount in the Solveny II projet. Despite this motivation, as far as we know, asymptoti Normality of estimators of finite-time ruin probabilities had neither been proved nor used to take estimation risk into aount. In this paper, we first ontinue on the trak of Mareau and Rioux 2001) and we takle the robustness analysis of finite-time ruin probabilities in the lassial risk model. We then prove the onvergene of the resaled error on the finite-time ruin probability toward a Gaussian random variable if omputations are arried out with the empirial laim amount distribution. We ompute expliitly the variane of this distribution and an thus define and quantify the reliable finite-time ruin probability. This Value-at-Risk of the estimator of the finite-time ruin probability has to be ontrolled to over estimation risk, whih requires an additional solveny apital ompare to the ase where one only ontrols the empirial finite-time ruin probability : we define this apital as the Estimation Risk Solveny Margin ERSM). Our paper is organized as follows: in setion 2, we reall the lassial risk model and the literature about omputation of lassial finite-time ruin probabilities. We derive some sensitivity results that are going to be useful in the sequel. We also introdue the onept of influene funtion and briefly reall its main properties. In setion 3 we ompute influene funtions of finite-time ruin probabilities and of some related quantities, using some formulas of Piard and Lefèvre 1997) and Rullière and Loisel 2004) as starting points. In setion 4 we study some properties of the influene funtion, in partiular large laim ontamination in atastrophe risk. We use a result from Hoeffding 1948) to show the weak onvergene of a sequene of empirial ruin probabilities to a Gaussian proess in setion 5. In setion 6, we explain how to use the influene funtion and our results to 2

4 get a more robust determination of solveny apital requirements with reliable finite-time ruin probabilities : the required apital is the sum of the apital that is required to have a probability of ruin based on empirial laim size distribution less than ɛ and of the Estimation Risk Solveny Margin ERSM). The goal of ERSM is to take estimation risk into aount in the spirit of Solveny II. The value of this margin may be easily obtained thanks to the Gaussian approximation derived in setion 5. The impat of exluding some types of atastrophe risks in insurane or reinsurane treaties is also obtained with a very simple formula. Numerial examples illustrate the developed methods in setion 7. In partiular, we show that the better the experiene of the ompany about laim sizes is, the lower the estimation risk solveny margin ERSM) is. The experiene of the ompany about laim sizes is quantified by the number of observed laim amounts in the database. The higher this number, the smaller the ERSM. 2 The lassial risk model : sensitivity analysis and influene funtion We will onsider a lassial risk proess R t ) t 0 defined as follows : for t 0, R t = u + t S t, where u is the non-negative amount of initial reserves, > 0 is the premium inome rate. The umulated laim amount up to time t is desribed by the ompound Poisson proess N t S t = W i, where amounts of laims W i, i = 1, 2,... are non-negative independent, identially-distributed random variables, distributed as W. As usual S t = 0 if N t = 0. The number of laims N t until t 0 is modeled by an homogeneous Poisson proess N t ) t 0 of intensity λ. Claim amounts and arrival times are assumed to be independent. We are interested in the robust estimation of finite-time ruin probabilities. Let us denote by ψu, t) the probability of ruin before time t with initial reserve u : ψu, t) = P [ s [0, t], R s < 0 R 0 = u], u 0, t > 0, and let ϕu, t) = 1 ψu, t) be the probability of non-ruin within time t with initial reserve u. As we onsider finitetime ruin probabilities, no profit ondition has to be satisfied from a theoretial point of view. 2.1 Sensitivity analysis We show here that derivatives of the finite-time non ruin probability ϕu, t) with respet to, λ or u may be easily obtained as funtions of derivatives of the density f St x) of S t 3

5 with respet to x or λ. Some details or parts of proof are given in Appendix Continuous laim amount distribution Consider first the ase of ontinuous laim amount distributions. Note that derivatives of the density of the umulated laim amount S t up to time t an easily be obtained by differentiation of the ontinuous version of Panjer s formula : x y f St x) = λ 0 x f W y)f St x y)dy, x > 0, 1) where f St and f W respetively are the p.d.f. of S t and W. Proposition 1 Let k N. Then for u,, t > 0 suh that f W differentiable on [0, u + t], is k-times ontinuously k k t ϕu, t) = uk u f k k S t u + t) 0 u f k S x u + x)ϕ0, t x)dx. For u,, t > 0 suh that f W is ontinuously differentiable on [0, u + t], t ϕu, t) = tf S t u + t) xf S t u + x)ϕ0, t x) + f St u + x) ϕ0, t x)dx, 0 This provides a self-iterative proess to determine ϕ0, x) for u = 0). In the disrete ase, expressing partial derivatives of finite-time ruin probabilities in terms of derivatives of some f St provides natural reursive omputation shemes Disrete laim amount distribution In the ase of integer-valued laim amounts, we an either use finite-differene alulus instead of differentiation, or study the partiular behavior of ϕu, t) as u varies for example. As explained in Rullière and Loisel 2004), ruin and ruin at inventory are exatly the same, provided that the set of inventory dates Ω is hosen as Ω = {τ ]0, t], u + τ N \ {0}}. This set of inventory dates Ω depends on u, t and but not on λ. Set x + = maxx, 0). Proposition 2 The partial derivatives of order k w.r.t. λ of finite-time non-ruin probabilities starting from zero an be written as follows : 4

6 0, λ ϕ n ) = n [ E 1 W + S ) n/ n + k 0, λ ϕ n ) ) [ n k k = C i k k 1) k i E ] n ϕ 0, n ), k 1, 2) 1 W i + S n/ n ) + ], k 0. [ 1 ) ] Proof: ϕ0, t) = E S t, and A.3) give the result. t + Proposition 3 For k 1, partial derivatives of order k w.r.t. λ of finite-time non-ruin probabilities an be written as follows : k k ϕu, t) = λk λ P [R k t 0] s Ω k C i i k λ P [R i s = 0] k i ϕ0, t s). 3) λk i Proof: This follows from Proposition 2 in Appendix and results of Rullière and Loisel 2004). Remark 1 Some results onerning derivatives of ruin probabilities involve the distribution of W + S t. This will also be the ase for some results about influene funtions in the next setions, in partiular for propositions 15, and 16 in the ase of large laim ontamination. If we add a laim at time zero, we an link the involved ruin probability with a ruin probability in the so-alled dual risk model, in whih the risk proess dereases at a deterministi rate and has upward jumps. We an also link the involved ruin probability with the probability that a lassial proess reahes an upper barrier. For more details, see Mazza and Rullière 2004). 2.2 Influene funtions It is unlikely that the real laim amount proess is exatly the one whih has been hosen for statistial inferene. At best, it might orrespond to a model that is lose to the starting model, for example a small ontamination of it. Therefore, one needs estimators that are effiient and that do not hange muh if a small hange ours in the inputs of the model. Estimators of this kind are alled robust. The influene funtion, whih was introdued by Hampel 1974) to study the infinitesimal behavior of real-valued funtionals, is one of the main tools in robustness theory to measure the impat of a small perturbation of the model on the outputs. Definition 1 Influene Funtion IF)) Assume that T is a funtional of a distribution F. The influene funtion at point x R is defined as the limit when it exists) IF x [T] = lim s 0 TF s,x) ) TF ) s 5

7 where F s,x) is defined for x R and 0 < s < 1 by for u R, F s,x) u) = s1 x u + 1 s)f u). In the sequel, for eah quantity related to the ontaminated distribution F s,x), we use the exponent s,x). Given a random sample X 1,, X n distributed aording to some distribution funtion F, let F n denote the assoiated empirial distribution. The influene funtion has two main uses: it allows the study of the influene of perturbations of the data on the values taken by the funtional T, and it permits, under some regularity assumptions to ath the asymptoti variane when the resaled proess weakly onverges toward a Gaussian random variable as n, where Var n TF n ) TF ) ) AF, T), AF, T) = see Huber 1981), Hampel 1974) or Hampel et al. 1986)). R IF x [TF )]) 2 F dx) 4) 3 Computation of the influene funtion We assume here that W is integer-valued, with P [W = 0] = 0 whih is not restritive, see for example De Vylder 1999) or Rullière and Loisel 2004)). In this setion, we provide algorithms to ompute influene funtions of finite-time non-ruin probabilities and of some related quantities. Set Π i = P [W = i], i N. We assume that the distribution F of a single laim amount is ontaminated, in the sense that we add some probability mass at point x N. x an in general be any real number, but we present here the simpler ase where x N for the sake of larity. This is also onsistent with the fat that laim amounts are integer-valued in reality. As for u, t > 0 and x N, IF x [ψ u, t)] = IF x [ϕ u, t)], we an treat symmetrially the probability of ruin or of non-ruin before t. Given j N and τ R, onsider the funtions h 0 τ) = e λτ and h j τ) = λτ j j i Pr [W = i] h j i τ). For τ > 0, we have h j τ) = P [S τ = j] for j N. Set Π i = P [W = i] for i N, with Π 0 = 0. Similarly, for j N and τ R \ {0}, one sets Then j H j τ) = h i τ) and H j j τ) = h i τ) 1 i ). τ H j 0) = h j 0) and H j 0) = 1 λ E [W 1 W j], 6

8 h j τ) = Pr [ S τ/ = j ], H j τ) = Pr [ S τ/ j ] and H j τ) = 1 τ E [ τ S τ ) 1S τ j ]. Proposition 4 IF for single laim probabilities) IF x [Π i ] = 1 x=i Π i, i N. Proof: We see that Π s,x) = s1 x=i + 1 s)π i, and the result is straightforward. In order to determine the influene funtion of the probability of ruin, we need to give the influene funtion for quantities P [S t = j], j N, t R. Notie that for τ < 0 omputations are formal and do not have any probabilisti meaning. Nevertheless, these formal omputations will be useful for the final results, as in Rullière and Loisel 2004). The influene funtion of these probabilities will be written as follows: IF x, j τ) = IF x [ P [ Sτ/ = j ]] j N, τ R +. Proposition 5 IF for aggregated laim amount probabilities) IF x, y τ) = λτ x y 1 x yp [ S τ = y x] P [ S τ = y] + Proof: Using Panjer s reursion, one obtains P [ S τ Thus, we an obtain both P [ S τ s,x) = y ] and P [ S τ y λτ i y Π iif x, y i τ). 5) = y] = y λτ i Π y ip [ S τ = y i]. = y] reursively as y varies. We an then either onsider the differene between P [ S τ s,x) = y ] and P [ S τ = y] and alulate the limit when s tends to zero, or diretly differentiate P [ S τ s,x) = y ] with respet to s, and then take s = 0. Aording to Panjer s formula, the seond term on the right-hand side of the equality is redued to P [ S τ = y]. As for j N we an define h j τ) for τ R even if it loses its probabilisti interpretation, see Rullière and Loisel 2004) for example), the definition of IF x, j τ) may be extended to the general ase τ R for j, x N simply as follows : IF x, j τ) = s h j s,x) τ). 6) s=0 Proposition 6 Algorithm for IF related to aggregate laim amounts) The following iterative sheme provides both aggregated laim amount distributions and the orresponding influene funtions, for τ R and x, y N : IF x, y τ) = λτ x y 1 x yh y x τ) h y τ) + h y τ) = y λτ y i y Π ih y i τ), λτ i y Π iif x, y i τ), 7

9 where h 0 τ) = e λτ. and IF x, 0 τ) = λτ λτ e 1x=0 Proof: The first equation is given by Proposition 6, the seond is an expression of Panjer s formula. For initial values, one an hek that for x 0, P [S t = 0] = P [ S t s,x) = 0 ] = P [N t = 0], so that IF x, 0 t) = 0, t R. For x = 0, P [ S t s,x) = 0 ] = e λt1 s). Takás s result see Takás 1962a) and Seal 1969)) implies that ϕ 0, n ) = n j=0 n j n P [ S n = j] = n j=0 n j n h jn), n N. 7) As a diret onsequene, we get the influene funtion of the finite-time non-ruin probability starting from zero. Proposition 7 IF for ϕ0, t)) IF x [ϕ 0, n )] = n j=0 n j n IF x, jn). Proof: Differentiate ϕ ) 0, n for the ontaminated single amount distribution, take s = 0 and apply then equation 6). Computations for a time n / N and an initial reserve u / N an be done by appliation of formulas 2.10) and 2.11) in Rullière and Loisel 2004).For the sake of larity, we onsider u, n N in the sequel. Several ways have been proposed to ompute finite time ruin probabilities with initial reserves u N see Rullière and Loisel 2004), Piard and Lefèvre 1997)). By onditioning by the last time the proess R s ) s 0 reahes zero before time t, we get: ϕ u, n ) = H u+n n) n h u+k k) H n k n k). 8) k=1 As a diret onsequene, we get the following reursive sheme for the influene funtion of the finite-time non-ruin probabilities ϕu, t), u N. Set IF H x, j τ) = IF x [ Hj τ) ]. Then IF H x, j τ) = j IF x, i τ) 1 i ), τ [ )] with IF H x, 0 0) = 0. In partiular, IF H x, n n) = IF x ϕ 0, n, n N. 8

10 Proposition 8 IF for ϕu, t) - first method) IF x [ϕ u, n )] u+n = IF x, i n) n k=1 n IF x, u+k k) H n k n k) k=1 h u+k k)if H x, n k n k). Proof: Taking the derivative of 8) for the ontaminated single amount distribution, setting s = 0 and applying then 6) gives the required result. One might use also alternative formulas of Piard and Lefèvre 1997) or diret reursive formulas. These formulas and the orresponding shemes are given in Appendix. During the implementation of the algorithms, one may take are to ompute eah quantity only one. In partiular, sine omputations of h j τ) and IF x, j τ) involve alulation of h i τ) and IF x, i τ), i j, these quantities should be stored and summed at the right time. Notie also that some fators do not depend on perturbation point x, whih enables us to ompute influene funtions for a set of values of x in a shorter time. Some of the above sums may be interpreted as influene funtions of quantities like H..), H..) or ϕ0,.). The disussion on the omparison between omputation times for these three methods an be diretly adapted from Rullière and Loisel 2004). Remark 2 Previous omputations of influene funtions in propositions 8 and 20 for times n/, n / N and initial reserves u / N an be done by adaptation of formulas 2.7) and 2.11) in Rullière and Loisel 2004). It will sometimes be neessary to find the initial reserve u R + respeting some onstraints for ruin probabilities and influene funtions, so that an adaptation of previous formulas given in Appendix) may be useful. 4 Properties of influene funtions assoiated to ruin probabilities In this setion, we first show that the influene funtion of the finite-time ruin probability is non-dereasing, bounded and onstant after a ertain threshold. This leads us to study the partiular properties of the influene funtion for large ontamination points x. The situation is quite simple in this ase sine eah laim amount replaed by x will ause ruin. Nevertheless, the event one laim is replaed by x is strongly dependent on the number of laims on the onsidered period. A first approah an onsist in studying the risk proess given the number of laims, but we onsider here the umulated laim amount proess, whih is suffiient to determine the probability of ruin. This analysis is partiularly relevant for lines of business that may be exposed to atastrophe risk. Let us start with an intuitive, and simple result: Proposition 9 Monotoniity of IF) For all u 0 and t > 0, is non-dereasing in x. IF x [ψ u, t)] = IF x [ϕ u, t)] 9

11 Proof: For eah random path of R t s,x), for any x > x, if R t s,x) reahes the lower barrier 0, then a fortiori R t s,x ) also reahes 0. It follows that ψ s,x ) u, t) ψ s,x) u, t), and the result holds. The following results are true for τ R, but when τ < 0, usual probabilities have to understood formally. In what follows, we assume without loss of generality that τ > 0. Proposition 10 IF for x = 0) IF 0,y τ) = P [ S τ = y] + y λτ i y Π iif 0,y i τ). 9) Proposition 11 Reall that h j s,x) τ) = P [ S τ/ s,x) = j ]. For x > j, we have h s,x) j τ) = exp λτ ) s h j τ1 s)), x, j N, x > j. Proof: Sine x > j, we have P [ S τ/ s,x) = j ] = + n=0 P [ N τ/ s,x) = n ] P [ W s,x) n = j ]. P [ W s,x) n = j ] = 1 s) n P [W n = j]. Proposition 12 Aggregate laim amount IF for large x) For x > j, the influene funtion IF x, j τ) does not depend on x and is given by IF x, j τ) = λτ h jτ) λ λ h jτ), x, j N, x > j. 10) Proof: From proposition 11), taking derivatives at s = 0, and using 6), with eventually τ h τ jτ) = λ h λ jτ). We also remark that for large x, 5) beomes IF x, y τ) = h y τ) + y λτ i y Π iif x, y i τ). 11) We then hek that equation 10) satisfies this last equality using A.2). It is rather diret to get the ruin probability influene funtion [ )] from the aggregate laim amount influene funtion, sine we an simplify IF x ϕ u, n in proposition 8). An interesting link ours then between sensitivity with respet to parameter λ and influene funtion, as shown in the next propositions. 10

12 Proposition 13 Let ϕ s,x) u, n, λ) be the probability of ruin for ontaminated laim amounts when we assume a Poisson intensity λ. For x > u + n, we have ϕ s,x) u, n ), λ = ϕ u, n ), λ1 s) exp λ n ) s. 12) Proof: Let N be the number of laims replaed by x before time n/. We see that ϕ s,x) u, n ) [, λ = P [N = 0]P T s,x) u > n ] N = 0, when x > u + n. Then ϕ s,x) u, n, λ) = P [ N = 0 N n/ = k ] [ P T s,x) u > n ] k N N = 0 N n/ = k. Denote by N s a Poisson proess of intensity λ1 s). Then ϕ s,x) u, n, λ) = P [ Nn/ s = k ] [ P T u > n ] k N N n/ = Nn/ s Nn/ s = k exp λ n ) s, and the result follows. Proposition 14 ruin probability IF for large x) For x > u + n, we have IF x [ϕ u, n )] = λ u, λ ϕ n ) λ n ϕ u, n ) Proof: Plug 10) in 8), and use 8). Another way to get this formula is to differentiate 12) with respet to s, and then to onsider this derivative at s = 0. Derivatives with respet to λ may be simplified by using relations A.1), A.3) and 3). As an example, we get: Proposition 15 For x > j, the influene funtion IF x, j τ) does not depend on x and is given by 13) IF x, j τ) = λτ P [ W + S τ/ = j ], x, j N, x > j. 14) Proof: This is a onsequene of A.3) and 10). Another possibility is to hek that this equation satisfies 11). Use Panjer s formula for h y τ), and develop P [ W + S τ/ = y ] aording to its natural onvolution, one may hek that ) y Π i P [ S τ/ = y i ] 1 i y = λτ y i y Π ip [ W + S τ/ = y i ]. 11

13 Expressing P [ W + S τ/ = y i ] as a natural onvolution sum, the result is obtained with a mere sum inversion, heking then that one sum an be suppressed thanks to Panjer s formula. Proposition 16 For x > n, the influene funtion of the non-ruin probability without initial reserve does not depend on x and is given by IF x [ϕ 0, n )] = λn [ E 1 W + S ) ] n/ n + 15) = λn P [ W + S n/ n ] + λ n jp [ W + S n/ = j ]. j=0 16) Proof: Plug 14) in the expression of ϕ 0, n ) as given in 7). Remark 3 For x > u+n, the influene funtion of the non-ruin probability without initial reserve does not depend on x and is given by non-reursive sums involving only distributions of S t and of W. Sine these expressions are quite long, they are not given here. They may be obtained by diret insertion of equation 14) into, for example, Proposition 8. 5 Weak onvergene of finite-ruin probabilities based on empirial distribution In this setion we show that the resaled empirial finite-time non-ruin probability starting from zero onverges in distribution to a Gaussian distribution. To this end, let us onsider the empirial finite-time non-ruin probability with zero initial reserve and within time horizon t > 0 ϕ N N t 0, t) = P s [0, t], t Yj N < 0, 17) where the ) Yj N are i.i.d. random variables drawn from the empirial distribution of a j 1 random sample {Y 1,..., Y N } of size N 1 from the distribution of W. Given that and ϕ0, t) = 1 t E N t t ϕ N 0, t) = 1 t E N t t j=1 W j j=1 + j=1 Y N j + 18), 19) we may rewrite the differene between the finite-time non-ruin probability and its estimate based on the empirial distribution of W as ϕ0, t) ϕ N 0, t) = P [N t = k] E 1 k t W j 1 Φ k 1 t N k k Y i1,..., Y ik ), j=1 + 1 i 1,...,i k N 12

14 where, for k 1, and for y 1,..., y k R, we set Φ k y 1,..., y n ) = E 1 t t k y j. j=1 + One reognizes a typial von Mises funtional, losely related to U-statistis, for whih many asymptoti results are known see for example Hoeffding 1948), Von Mises 1947) and Gotze 1984)). [ 1 ) ] Let ϕ k 0, t) = E t t kj=1 W j, k 1, with ϕ 0 0, t) = 1, and onsider the proess + ξ N k = P [N t = k] ϕ k 0, t) 1 N k 1 i 1,...,i k N Φ k Y i1,..., Y ik ), k, N 1. We get thus a sequene ζ N ) N 1 taking values in the Banah spae l 2 R + ), where for all N 1, ζ N is defined as ζ N = ) ξk N, k 1 whih indues a measure on the spae R +. From Theorem 7.4 of Hoeffding 1948), the finite-dimensional projetions weakly onverge to a Gaussian distribution: ) K 0 1, N ξ N 1,..., ξk N 0 ZK0 in distribution as N +, where for K 0 1, Z K0 and the K 0 K 0 ovariane matrix Γ K0 follows a Gaussian distribution with mean vetor 0 K0 = 0,..., 0) defined for 1 i, j K 0 by Γ K0 ) ij = ijp [N t = i]p [N t = j]e [ ϕ Y ) i 1 ϕ i ) ϕ Y ) j 1 ϕ j )], with ϕ x) k 0, t) = E [ 1 t t x S t) + N t = k ], and ϕ x) 0, t) = E [ 1 t t x S t) + ] we omit the argument 0, t) for more larity). Notie that for laim amounts taking values in δn = {0, δ, 2δ,... }, where δ > 0 and t > 0 are fixed, we an assume without restrition that laim amounts take values in {δ, 2δ,... } just hange the intensity λ into λ1 P W = 0)) and P W = kδ) into P W = kδ) 1 P W = 0) for k 1). Then, if N t > t δ, 13

15 ruin is ertain sine we have at least t δ + 1 jumps of size greater or equal to δ. This is true both for empirial and true distributions of W. Therefore, for all N 1, ξ N k = 0 for all k K + 1, where K = t δ. Theorem 1 If laim amounts take values in δn \ {0}, N ϕ0, t) ϕ N 0, t) ) Z in distribution as N +, where Z N 0, V 0 ), with variane V 0 = V Y [ λtϕ Y ) 0, t) ] = V Y [IF Y [ϕ0, t)]], 20) with ϕ x) 0, t) = E [ 1 t t x S t) + ], x N, and where Y is a r.v. distributed as W. Remark 4 Notie that the identity between varianes given in Theorem 1 orresponds to the general relation between asymptoti varianes and influene funtions given in 4). Remark 5 This theorem is only valid for u = 0. We leave the theoretial proof of the general ase u > 0 for future researh, but provide a kind of omputer-aided proof in the numerial analysis setion to show that the methods we propose to ompute the Estimation Risk Solveny Margin to be defined in setion 6.1) are implementable. The ase u > 0 is important for appliations to Estimation Risk Solveny Margin see setion 6.1). Proof: Theorem 7.4 of Hoeffding 1948) yields that the limiting variane is given by K K V 0 = Γ K0 ) ij. j=1 Hene V 0 = v 1 v 2, with K K v 1 = E Y ip [N t = i]ϕ Y ) i 1 jp [N t = j]ϕ Y ) j 1 j=1 and K K v 2 = ijp [N t = i]p [N t = j]ϕ i 0, t)ϕ j 0, t). j=1 Using the identities ip [N t = i] = λtp [N t = i 1] and ϕ Y ) k = 0, k > K, 14

16 one obtains that v 1 = λt) 2 E Y P [N t = i]ϕ Y ) i j=0 jp [N t = j]ϕ Y ) j, so that v 1 = λt) 2 E Y [ ϕ Y ) 0, t) ) 2 ]. We an show that [ Y ) ϕ i = E Y ϕ i 1]. Using the above arguments, one gets that v 2 = λt) 2 P [N t = i]e Y [ ϕ Y ) i 0, t) ] j=0 P [N t = j]e Y [ ϕ Y ) j 0, t) ], and therefore v 2 = λt) 2 E Y [ ϕ Y ) 0, t) ]) 2. We next onsider the last identity 20) V Y [ λtϕ Y ) 0, t) ] = V Y [IF Y [ϕ0, t)]]. Given 0 < s < 1, let ε be a generi Bernoulli random variable with P ε = 1) = s = 1 P ε = 0). Then the random variable εy + 1 ε)w has F s,y) u) as a distribution funtion. One must thus onsider the following limit, lim s 0 [ 1 Eε,W ) t s ] ] k k ). ε i y + 1 ε i )W i ) + E W [t W i ) + Using independene and Fubini s Theorem, one is led to onsider first the integral over ε given W ] k k k I w := E ε [t y ε i W i + ε i W i ) +. The olletion of i.i.d. random variables ε 1, ε 2,, ε k ) an be seen as orresponding to random subsets J of {1, 2,, k}, of law P J) = s J 1 s) k J, where J denotes the size of J with J = k ε i. Then we an write k I w = s J 1 s) k J k t yn W i + ε i W i ) +. n=0 J =n i J We shall see that only the first two terms orresponding to n = 0 and n = 1 ontribute to the limit: one first hek the behavior of [ ] E W Iw lim Ĩw, s 0 s where we set Ĩw = t k W i ) +. Let I w = I 0 w + I 1 w, where k k Iw 0 = 1 s) k t W i ) + and Iw 1 = s1 s) k 1 y j=1t W i ) +. i j 15

17 Using the fat that 1 s) k 1)/s) k as s 0, one gets the equivalent expression ] ] k k 1 ke W [t W i ) + + k1 s) k 1 E W [t y W i ) +. The next step onsists in taking the variane of the above random variable when Y is distributed like W, and is independent of the W i. The first term is onstant, and therefore the variane is given by, olleting the terms related to k, V Y k 0 λt) k e λt ke W [t Y k! k 1 = V Y λt λt) k e λt E W [t Y k 0 k! = V Y [ λtϕ Y ) 0, t) ], W i ) + ] k W i ) + ] whih orresponds to the required identity. It remains to hek that the terms related to n 2 do not ontribute to the limit s 0. This follows from bounded onvergene. In the ase where laim amounts follow a ontinuous distribution F W, we an approximate F W with a sequene F Wp )p 1 of disretized versions of F W suh that W p takes values in 1N for p 1, in the sense that for all x R, p F Wp x) F W x) as p tends to infinity. Denote respetively by ϕ p 0, t) and ϕ N p 0, t) the finite-time ruin probability with laim amount distribution F Wp and the related empirial version. Clearly, for a fixed N 1, n [ ϕp 0, t) ϕ n p0, t) ]) 1 n N onverges in distribution to n [ϕ0, t) ϕ n 0, t)] ) 1 n N as p +. As the weak onvergene of a family of measures on the Banah spae l 2 R + ) is ensured by the weak onvergene of the finite-dimensional projetions see for example Billingsley 1999), hapter 1.5), we get that N [ ϕp 0, t) ϕ N p 0, t) ]) N 1 onverges weakly toward N [ ϕ0, t) ϕ N 0, t) ]) N 1, as p +. For given p 1, Theorem 1 yields that N ϕp 0, t) ϕ N p 0, t) ) 16

18 onverges in distribution to where Z p N 0, σ 2 p), σ 2 p = V Wp [ λtϕ W p) p 0, t) ], and [ 1 ϕ x) p 0, t) = E t x Sp t ) t + ]. Here, S p t orresponds to the umulated laim amount up to time t for individual laim amounts distributed as W p. As σ 2 p = V Wp [ λtϕ W p) p 0, t) ] V W [ λtϕ W ) 0, t) ], p, Z p onverges in distribution to with Hene σ 2 = Z N 0, σ 2 ), [ lim p + σ2 p = V W λtϕ W ) 0, t) ]. N ϕ0, t) ϕ N 0, t) ) Z in distribution as N +, with the following ommutative diagram : N ϕp 0, t) ϕ N p 0, t) ) d p + N ϕ0, t) ϕ N 0, t) ) d N + N + Z p N 0, σ 2 p) d p + d Z N 0, σ 2 ). Remark 6 Note that as the infinite-time ruin probability starting from zero only depends on the laim size distribution through its expeted value, the asymptoti variane of tends to the asymptoti variane of N ϕ0, t) ϕ N 0, t) ) N E[W1 ] µ N W ) multiplied by λ 2 / 2 as t goes to infinity, where µ N W is the random) empirial average of W 1 obtained from an N-sample of the laim size distribution. From the entral limit theorem and from the same way of reasoning as above, the asymptoti variane of N ϕ0, t) ϕ N 0, t) ) onverges to as t tends to +. λ 2 2 V ar W 1) 17

19 6 Some appliations of influene funtions 6.1 Reliable finite-time ruin probabilities Reall that ψu, t) denotes the finite-time ruin probability, whih is seen as a funtional of the laim amount distribution F. Similarly, ψ N u, t) denotes the random finite-time ruin probability, with laim amounts drawn from the empirial distribution F N assoiated with an i.i.d. sample of distribution F. Definition 2 The reliable finite-time ruin probability ψ N,reliable 1 ε u, t) is the 1 ε)-quantile of the random) bootstrapped finite-time ruin probability ψ N u, t): { [ ψ N,reliable 1 ε u, t) = inf P ψ N u, t) s ] ε }. s 0 We heked in the previous setions that ψ N u, t) an be approximated for large laim size databases see setion 7.2 to know what large N means in pratie) by a Gaussian random variable of mean ψu, t) when u = 0. Numerial simulations seem to onfirm the asymptoti Normality of ψ N u, t), for arbitrary u see Setion 7.1). When ψ N u, t) is approximately Gaussian of mean ψu, t) and variane V u /N, one an onsider the approximation ψ N,reliable 1 ɛ u, t) = ψu, t) + Vu N Φ 1 1 ɛ), where Φ denotes the distribution funtion of a standard Normal r.v., and where V u is the asymptoti variane of N ϕu, t) ϕ N u, t) ) : V u = V Y [IF Y [ϕu, t)]], u 0, 21) whih an be obtained from Setions 3 and 5. Setion 7 gives examples where the omputation time required to estimate the variane is reasonable, but the omputation time an heavily inrease when disretization step δ beomes smaller. One judiious hoie is ε = 2.5%, as the 97.5 perentile of a Gaussian µ, σ 2 ) random variable an be approximated by µ + 2σ. In this ase, one gets the pragmati approximation Φ 1 1 ɛ) 2. If u η and u η,ε are respetively defined as the initial apital required to ensure that ψu η, t) η and ψ N,reliable 1 ε u η,ε, t) η, the Estimation Risk Solveny Capital ERSM η,1 ε an be defined as the additional apital needed to take estimation risk into aount : ERSM η,1 ε = u η,1 ε u η, whih an be obtained from the results of Setion 7. It might be thus interesting to determine solveny requirements from ψ N,reliable 97.5% η with 1 η < 99.5%, rather than from 18

20 ψ %. This might lead to a gain in robustness, as 99.5% safety levels are almost impossible to handle in pratie. We give examples of values of η that lead to values of u η,97.5% of the same magnitude as u % in setion 7.2. In pratial ases, one may ignore the exat distribution F. If the laim size database ontains N 1 observed laim amounts O N = {w 1,..., w N }, then estimators of ψu, t), ψ N,reliable N,reliable 1 ε u, t) and ψ 1 ɛ u, t) may be obtained, for example, by respetive plug-in estimators, ψ O N u, t), ψ O N,reliable 1 ε u, t) and ψ O N,reliable 1 ɛ u, t), when F is replaed by the empirial distribution funtion F ON from O N. These estimators may also suffer from estimation risk. From Propositions 5, 6, 7 and 8, and with the same kind of reasoning as in Setions 3 and 4, it an be shown that the influene funtion of the influene funtion of the ruinprobability is bounded. From 4) and from Proposition 15, we obtain that estimators ψ O N u, t), ψ O N,reliable 1 ε u, t) and ψ O N,reliable 1 ɛ u, t) are robust aording to Hampel s definition, as their influene funtions are bounded. 6.2 Catastrophe laim ontamination For infinite-time ruin probabilities, in the ase of heavy-tailed laim amount distributions, it would be better to use some methods from the theory of extremes. However, in the ase of finite-time ruin probabilities, we want to point out here a very simple relation that gives the impat of the ontamination of data by large laim amounts. Assume that the solveny apital requirements of an insurane ompany are determined in suh a way that the finite-time ruin probability is less than ε. For some lines of business exposed to atastrophe risk, the following question arises: if the risk orresponding to laims that are larger than a given deterministi amount M > 0 are transferred using reinsurane or seuritization, what is the effet of this transfer on the ruin probability? Is the derease of the required apital level enough to finane this risk transfer in order to maintain the same premium inome rate? Is it possible to determine easily the given amount M neessary to get a given level of ruin probability? Consider the trunated random variable W suh that P [ W = x ] = P [W = x], 0 < x < M and P [ W = 0 ] = P [W M]. Reall that P [W = 0] = 0. Let Ñ t ) t 0 be the Poisson proess with intensity λp [W > M]) defined for t 0 by Ñ t = k 1 1 Ti t1 Wi >M, where T i ) i 1 is the sequene of jump instants of N t ) t 0. Ñ t represents of ourse the number of laims of size larger than M up to time t 0. Denote by ϕu, t) the finite-time non-ruin probability in the modified model. From the total probability formula, the lassial finite-time non-ruin probability satisfies the following equation : for all u, M and t 0, 19

21 ϕu, t) = P [ s t, u + t S t 0 Ñ t = 0 ].P [ Ñ t = 0 ] + P [ s t, u + t S t 0 Ñ t > 0 ].P [ Ñ t > 0 ] 22) As Ñt) t 0 and N t Ñt) t 0 are two independent Poisson proesses, P [ s t, u + t S t 0 Ñ t = 0 ] is exatly ϕu, t). If besides M > t, then any laim of size larger than M auses ruin, and onsequently P [ s t, u + t S t 0 Ñ t > 0 ] = 0. As P [ Ñ t = 0 ] = e λp[w >M]t, equation 22) simplifies for M > t into ϕu, t) = ϕu, t)e λp[w >M]t. 23) If follows that when it makes sense, determining the minimal value M 0 of M > t suh that ϕu, t) 1 ɛ is straightforward sine P [W > M] = 1 [ ] ϕu, t) λt ln. ϕu, t) This leads to the following ondition : P [W > M] 1 [ ] 1 ɛ λt ln, ϕu, t) and so M has to be greater than M 0 = V ar α W ), where the Value-at-Risk level α is given by α = 1 [ ] 1 ɛ λt ln. ϕu, t) Equation 23) may also be used to evaluate the influene of large laims and the impat of an underestimation of atastrophe risk. 7 Numerial examples In this setion, we first show how the influene funtion of finite-time ruin probabilities may look like. Then we analyze the impat of the size of the laim size database on the asymptoti variane of the estimator of finite-time ruin probabilities and thus on the estimation-risk solveny margin ERSM), obtained from differene between the reserves that are needed to ontrol the reliable finite-time ruin probability whose definition and main properties were given in subsetion 6.1) and the ones needed to ontrol the empirial finite-time ruin probability. 20

22 Figure 1. Aspet of the influene funtion IF x [P [S t = u]] = IF x, u t) as a funtion of x, for λ 0 = 1, = 1.1, t = 10, for disrete exponentially distributed laim amounts with δ = Numerial analysis of influene funtions The results presented hereafter have been obtained for parameters λ 0 = 1, = 1.1, and t = 10. We first onsider the ase where W 0 is exponentially distributed with parameter 1. We then define the distribution funtion F δ of a disrete laim amount W δ with F δ iδ) defined on eah interval [iδ, iδ + δ[, suh that F δ iδ) = 1 δ [iδ,iδ+δ[ F W0 x) dx. In order to anel π 0 = P [W δ = 0], the Poisson parameter λ 0 has been modified into λ = λ 0 1 π 0 ), and the π i have been hanged into P [W = i] = π i /1 π 0 ). All amounts, u, W ) are expressed in δ money unit, in order to get integer-valued amounts. This disretization proedure is fully desribed in De Vylder 1999). The interest of suh a distribution is that some results for ontinuous exponential laims distribution may be obtained as δ tends to 0. Consider first the influene funtion of the probability that the aggregate laim amount reahes a value u at time t. Figure 1 illustrates the non-monotoniity of this funtion. In this partiular example, hanging some laim amounts into some of value 1 or 2 may inrease the probability that the aggregate laim amount is 10. For other values, like 5, it may derease this probability. For perturbation points x > u, the influene funtion is obviously unhanged sine hanging only one laim into x implies that S t will not reah u. 21

23 Figure 2. Aspet of the influene funtion IF x [ψ 0, t)] as a funtion of x, for λ 0 = 1, = 1.1, t = 10, for disrete exponentially distributed laim amounts with δ = Figure 3. Aspet of the influene funtion IF x [ψ u, t)] as a funtion of x, for λ 0 = 1, = 1.1, t = 10, u = 10, for disrete exponentially distributed laim amounts with δ = 0.1. We have heked numerially that see equation 14)): IF x, j τ) = λτ P [ W + S τ/ = u ], x, j N, x > j. In this speial ase, where λ 0 = 1, = 1.1, t = 10, u = 10, for disrete exponential laims with δ = 0.1, with standard, 64-bit arithmeti preision, P [W + S t = u] = and IF x, u t) = From reursive shemes given in proposition 8 and 20, whih give similar results, we draw the influene funtion of ruin probability as a funtion of the ontamination point x. 22

24 We get as an example in Figures 2 and 3 the shape of the influene funtion of the ruin probability within finite time IF x [ψ 0, t)] and IF x [ψ u, t)]. We an verify that this influene funtion starts at a given negative value, is non-dereasing, bounded and onstant for x > u + t. We have heked numerially that see proposition 16): IF x [ϕ 0, n )] = λn [ E 1 W + S ) ] n/. n + In this speial ase where λ 0 = 1, = 1.1, t = 10, u = 10, for disrete exponentially distributed laim amounts with δ = 0.1, with standard 64-bit arithmeti preision, E [ 1 W + S ) ] t = and IF x [ϕ 0, t)] = t + [ )] We also heked in this ase that numerially see proposition 7) IF x ϕ 0, n = nj=0 n j IF n x, jn). At last, we heked that we retrieve numerially for small values of s: IF x [P [W + S t = j]] 1 [ [ P W s,x) + S s,x) t = j ] P [W + S t = j] ], s and IF x [ϕ u, t)] 1 [ ϕ s,x) u, t) ϕ u, t) ]. s 7.2 Impat of database size on ERSM We proved the onvergene to a entered Gaussian distribution of the resaled differene between the real finite-time ruin probability starting from zero and its empirial equivalent, we obtained formulas to ompute the asymptoti variane of this estimator both for null and positive initial reserves), but one pratial question immediately arises: how large should the size of the database be for the Normal approximation to be good enough? This is an important question to know from whih range of database size the Normal approximation enables us to orretly approximate the Estimation-Risk Solveny Margin. To takle this question, we plotted a few empirial distributions of the finite-time ruin probability for different values of database size N D and arried out several tests. Our finding is that the Normal approximation is of good quality for N D 1000 in our example, as the Gaussian hypothesis is not rejeted for N D 1000 see Tables 1 and 2 below). In the example of Table 2, the finite-time ruin probability is 3.7%, and the 95%-reliable finite-time probability is around 4.8% for N = 1000, whih orresponds to a signifiant inrease of 27%. In Table 3 we see that Ψ O N reliable 1 ɛ u, t) is a quite good approximation of the empirial quantile of ψ N u, t), as soon as N is greater than 100, whih is often true in pratie. 23

25 ,73 0,74 0,75 0,76 0,77 0,78 0,79 Q95 0,8 0,81 0,82 0,83 Figure 4. Histogram of empirial ruin probabilities ψ N 0, t) for N = 500 and Gaussian p.d.f. with mean µ = ψu, t) and variane V Y [λtϕ y) 0, t)]/n, λ 0 = 1, = 1.1, t = 10, δ = ,73 0,74 0,75 0,76 0,77 0,78 Q95 0,79 0,8 0,81 0,82 0,83 Figure 5. Histogram of empirial ruin probabilities ψ N 0, t) for N = 5000 and Gaussian p.d.f. with mean µ = ψu, t) and variane V Y [λtϕ y) 0, t)]/n, λ 0 = 1, = 1.1, t = 10, δ = 1. In Table 4 and Figure 8, we determine the smallest values of u η and u η,ɛ suh that ψu η, t) and ψ O N reliable 1 ɛ% u η,ɛ, t)) are less than η. Due to the Normal approximation of ψ N u, t) for N 1000 see Table 2), one an estimate here the ERSM η,1 ɛ by the differene u η,ɛ u η. We show that this margin is dereasing in the laim amount database size N. In Table 5 we determine the values of η that lead to values of u η,97.5% of the same magnitude as u 0.5%. As u 0.5% is the apital needed to ontrol a lassial 99.5% non-ruin probability, this gives us an idea of the onfidene level η that one should ontrol to get results of the same magnitude as in the lassial ase, but with a stronger robustness, and more onsistene with the impat of database size on estimation risk. 24

26 ,025 0,03 0,035 0,04 Q95 0,045 0,05 Figure 6. Histogram of 5000 empirial ruin probabilities ψ N u, t) for N = 5000 and Gaussian p.d.f. with mean µ = ψu, t) and variane V Y [[IF Y [ϕu, t)]] /N, λ 0 = 1, = 1.1, t = 10, u = 10, δ = ,025 0,03 0,035 0,04 Q95 0,045 0,05 Figure 7. Histogram of empirial ruin probabilities ψ N u, t) for N = 5000 and Gaussian p.d.f. with mean µ = ψu, t) and variane V Y [[IF Y [ϕu, t)]] /N, λ 0 = 1, = 1.1, t = 10, u = 10, δ = 1. 25

27 N µ E σ E µ σ µ % % % % % % % σ % % % % % % % D KS stat.) p-value <0.001 < >0.250 >0.250 >0.250 >0.250 Table 1 Empirial measures from 5000 values ψ N 0, t), and adequation to Gaussian distribution with parameters µ = ψ0, t) and σ 2 [ = V Y λtϕ Y ) 0, t) ] /N. u = 0, = 1.1, δ = 1, t = 10. N µ E σ E µ σ µ % % % % % % % σ % % % % % % % D KS stat.) p-value <0.001 <0.001 <0.001 >0.250 > >0.250 ψ reliable 5% Table 2 Empirial measures from 5000 values ψ N u, t), and adequation to Gaussian distribution with parameters µ = ψu, t) and σ 2 = V Y [IF Y [ϕu, t)]]/n. u = 10, = 1.1, t = 10, δ = 1. N empirial Ψ O N reliable 1 ɛ u, t) ΨO N reliable 1 ɛ u, t) relative error 1 0, , ,46% 10 0, , ,99% 100 0, , ,80% , , ,41% , , ,31% , , ,02% , , ,001% Table 3 Comparison between the 95% empirial quantile of ruin probability Ψ O N reliable 1 ɛ u, t) from 5000 values ψ N u, t)) and the quantile Ψ O N reliable 1 ɛ u, t) of the Gaussian asymptotial distribution, u = 10, = 1.1, t = 10, δ = 1, ɛ = 5%. 7.3 Convergene speed Here, we present results obtained by simulating ruin probabilities. Eah ruin probability ψ N u, t) is simulated as follows: first, we build one empirial distribution from N laim amounts drawn from distribution F. Seond, we ompute the exat ruin probability as a funtional of this empirial distribution. 26

28 N 1 ɛ = 95% 1 ɛ = 97.5% 1 ɛ = 99.5% Table 4 Different values of u η,ɛ suh that ψ O N reliable 1 ɛ u η,ɛ, t) = η = 0.5%, u = 10, = 1.1, t = 10, δ = 1, η = 0.5%., and for N = value of u η suh that ψu η, t) = η ERSM Figure 8. Values of u η and u η,ɛ > u η as funtions of N, suh that ψu η, t) = 2% and ψ O N reliable 1 ɛ% u η,ɛ, t) = 2%, λ 0 = 1, = 1.1, t = 10, δ = 1, ɛ = 5%. N η ratio η/0.5% % % % % % % 1.05 Table 5 Values of η suh that u η,97.5% = u 0.5%. = 1.1, t = 10, δ = 1. 27

29 Figure 9. Histogram of empirial ruin probability ψ N u, t) for N = 100, λ 0 = 1, = 1.1, t = 10, u = 20, δ = 1. Figure 10. Histogram of empirial ruin probabilities ψ N u, t) for N = 1000, λ 0 = 1, = 1.1, t = 10, u = 20, δ = Figure 11. Values of distane lnd KS ) as a funtion of log 10 N), from 5000 values ψ N u, t). λ 0 = 1, = 1.1, t = 10, u = 15, δ = 1. For small ruin probabilities, we see in Figures 9 and 10 that the asymmetry of the empirial distributions of ψ N u, t) would lead us to rejet Normality for N = 100 and 1000, and we an assume that the sample size needed to ensure the Gaussian hypothesis validation would be larger for smaller ruin probabilities. That is mainly what we try to quantify with further numerial analysis. We have tried to quantify the empirial size N from whih, in our simulations, random variable ψ N u, t) ould be onsidered as a Gaussian random variable. In a first step, we have simulated 5000 values of ψ N u, t), for various values of N. For eah sample, we have omputed the Kolmogorov-Smirnov distane D KS between the empirial distribution of the finite-ruin probability and its Gaussian asymptotial distribution. 28

30 0 6 6 u=25 u= u=1000 u=20 u=500 4 u=15 4 u=200 3 u=10 3 u=0 u=10 u=100 u=50 2 u=0 u= Figure 12. Values of log 10 N D ) as a funtion of log 10 ψu, t)). Exponential ase. λ 0 = 1, = 1.1, t = 10, δ = 1. Figure 13. Values of log 10 N D ) as a funtion of log 10 ψu, t)). Pareto1,1.2) ase. λ 0 = 1, = 1.1, t = 10, δ = 1. In Figure 11, we give the value of lnd KS ) as a funtion of N, for 5000 empirial ψ N u, t); we see for example that lnd KS ) reahes the partiular value 4 for every omputed N greater than a level N D. From 5000 empirial ruin probabilities, validating Normality with a 95% level signifiane level) leads to values of lnd KS ) approximately greater than 4, and here to an empirial size N D Of ourse, depending on simulations, this quantity may vary. As a first approah, we have hosen to draw some values of lnd KS ) for a set of different values of initial reserves u. Sine we did not observe situations where the barrier lnd K S) = 4 was rossed more than one, we ould determine one empirial N D by a dihotomi algorithm with a total of around 14 omputed points). We have then hosen to define N D as the first empirial value for whih lnd K S) was lose enough to the target value, whih gives an idea of the onvergene rate. More rigorous formalization of this value N D would require the determination and the validation of a preise regression model, but a suh model would require more simulations. We are just here trying to get rough indiations on onvergene speed. In Figure 12 we have omputed by this way the empirial size N D for whih the Normality is validated for different values of initial reserves u. Sine the ruin probability varies with u, we have given values of log 10 N D ) as a funtion of logψu, t)). As an example, for finite-time ruin probabilities of order 10 3, in this partiular model value 3 on horizontal axis), one may suppose that Normality is not validated for samples of size less than , whereas might be enough for ruin probabilities of order Note that data used for u = 1000 is the same as the one in Table 2, but in this last appliation the distane D KS has been omputed with unknown Gaussian parameters, ausing the small differene with the one indiated in Table 2. We finally insist on the fat that values of N D are just rough estimates and that this Figure only gives one empirial indiation of the global need of larger samples to validate Normality for smaller ruin probabilities. We also investigate the extreme ase where laim amounts are Pareto-distributed. Pareto parameters are a = 1 and α = 1.2 with mean 6 and undefined standard deviation), and 99% perentile around We always validate Normality in our simulations when N is large enough. Nevertheless, we empirially see that, for a given ruin probability, this 29