A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model

Transcription

1 A polynomial expansion to approximate the ltimate rin probability in the compond Poisson rin model - Pierre-Olivier GOFFARD (Université de Aix-Marseille, AXA France) - Stéphane LOISEL (Université Lyon 1, Laboratoire SAF) - Denys POMMERET (Université de Aix-Marseille) Laboratoire SAF 5 Avene Tony Garnier Lyon cedex 7

2 A polynomial expansion to approximate the ltimate rin probability in the compond Poisson rin model Pierre-Olivier GOFFARD*, Stéphane LOISEL** and Denys POMMERET*** * AXA France et Université de Aix-Marseille, pierreolivier.goffard@axa.fr. ** Université de Lyon, Université de Lyon 1, Institt de Science Financière et d Assrance, stephane.loisel@niv-lyon1.fr. *** Université de Aix-Marseille, denys.pommeret@niv-am.fr. Abstract A nmerical method to approximate rin probabilities is proposed within the frame of a compond Poisson rin model. The defective density fnction associated to the rin probability is projected in an orthogonal polynomial system. These polynomials are orthogonal with respect to a probability measre that belongs to Natral Exponential Family with Qadratic Variance Fnction (NEF-QVF). The method is convenient in at least for ways. Firstly, it leads to a simple analytical expression of the ltimate rin probability. Secondly, the implementation does not reqire strong compter skills. Thirdly, or approximation method does not necessitate any preliminary discretisation step of the claim sizes distribtion. Finally, the coefficients of or formla do not depend on initial reserves. Keywords: compond Poisson model, ltimate rin probability, natral exponential families with qadratic variance fnctions, orthogonal polynomials, gamma series expansion, Laplace transform inversion. 1 Introdction A non-life insrance company is assmed to be able to follow the financial reserves evoltion associated with one of its portfolios in continos time. The nmber of claims ntil time t is assmed to be an homogeneos Poisson process {N t } t, with intensity β. The sccessive claim amonts (U i ) i N, form a seqence of positive i.i.d. continos random variables and independent of {N t } t, with density fnction f U and mean µ. The initial reserves are of amonts, and the premim rate is constant and eqal to p. The risk reserve process is therefore defined as N(t) R t = + pt U i, the associated claims srpls process is defined as S t = R t. In this work, we focs on the evalation of ltimate rin probabilities (or infinite-time rin probabilities) defined as i=1 ( ) ( ) ψ() = P inf R t < R = = P sps t > S =. (1.1) t t 1

3 This model is called a compond Poisson model (also known as Cramer-Lndberg rin model) and has been widely stdied in the risk theory literatre. For general backgrond abot rin theory, we refer to [16], and [4]. There are several sal techniqes for calclation of ltimate rin probabilities. First, iterative methods with the so called Panjer s algorithm derived in [14] and applied to the ltimate rin probability comptation in [8]. Then, we have nmerical inversion of the Laplace transform sed for probability distribtions recovery. In a few particlar cases, the inversion is manageable analytically and leads to closed formla bt in most cases nmerical methods are needed. The nmerical inversion via Forier-series techniqes (Fast Forier Transform) received a great deal of interest. These techniqes have been presented in [2] in a qeing theory setting. For an application within the actarial framework, we refer to [1]. Recently, inversion techniqes via the scaled Laplace transform and exponential moments recovery has been performed in [12] for rin probabilities comptations. We also mention the nmerical inversion of Laplace transform sing Lagerre method described in [19] and [1]. The recovered fnction takes the form of a weighted sm of Lagerre fnctions derived thogh orthogonal projections, it can be viewed as both a polynomials and a gamma series expansion that have been commonly sed in the actarial litteratre. The expansion of probability density fnction as a sm of gamma densities with actarial applications has been first proposed in [7] and gave rise to the so-called Beekman-Bowers approximation for the ltimate rin probability, derived in [6]. The idea is to approximate the ltimate rin probability by the srvival fnction of a gamma distribtion sing moments fitting. Gamma series expansion has been employed in [18] and later in [3]. The athors highlight that it is sefl to carry ot both analytical calclations and nmerical approximations. They focs on the finite-time rin probability, injecting directly the gamma series expression into integro-differential eqations leading to reccrence relations between the expansion s coefficients. The reslts are valid in the infinite-time case by letting the time t tend to infinity. In addition, we can mention [15], in which rin probabilities expressions are derived sing generalized Appell polynomials. Or method is an expansion sing orthogonal polynomials and can be viewed as an extension of [7], based on Lagerre s polynomials. It can also be related to gamma series expansions and nmerical inversions of Laplace transform. In this paper, we provide a new way to both constrct and jstify expansions with orthogonal polynomials that leads to an approximation with good nmerical behavior. Or reslts rely on some properties of orthogonal polynomials with respect to probability measres in NEF-QVF. From a comptational point of view, no discretization of the claim sizes distribtion is needed and the coefficients that reqire a large part of the CPU time are the same for any vale of. Moreover, the accracy is not mch sensitive to initial reserves, even for large vale. In Section 2, we introdce a density expansion formla based on orthogonal projection within the frame of NEF-QVF. Or main reslts are developped in Section 3: the expansion for ltimate rin probabilities is derived and a sfficient condition of applicability is given. Section 4 is devoted to nmerical illstrations. 2

4 2 Polynomial expansions of a probability density fnction Let F = {P θ, θ Θ} with Θ R be a Natral Exponential Family (NEF), see [5], generated by a probability measre ν on R sch that P θ (X A) = exp{xθ κ(θ)}dν(x) A = f(x, θ)dν(x), where A R, κ(θ) = log ( R eθx dν(x) ) is the Cmlant Generating Fnction (CGF), A f(x, θ) is the density of P θ with respect to ν. Let X be a random variable P θ distribted. We have µ = E θ (X) = xdfθ (x) = κ (θ), V(µ) = Var θ (X) = (x µ) 2 df θ (x) = κ (θ). The application θ κ (θ) is one to one. Its inverse fnction µ h(µ) is defined on M = κ (Θ). With a slight change of notation, we can rewrite F = {P µ, µ M}, where P µ has mean µ and density f(x, µ) = exp{h(µ)x κ(h(µ))} with respect to ν. A NEF has a Qadratic Variance Fnction (QVF) if there exists reals v, v 1, v 2 sch that V (µ) = v + v 1 µ + v 2 µ 2. (2.1) The Natral Exponential Families with Qadratic Variance Fnction (NEF-QVF) inclde the normal, gamma, hyperbolic, Poisson, binomial and negative binomial distribtions. Define { } Q n (x, µ) = V n n (µ) f(x, µ) /f(x, µ), (2.2) µ n for n N. Each Q n (x, µ) is a polynomial of degree n in both µ and x. Moreover, {Q n } n N is a family of orthogonal polynomials with respect to P µ in the sense that < Q n, Q m >= Q n (x, µ)q m (x, µ)dp µ (x) = δ nm Q n 2, m, n N, where δ mn is the Kronecker symbol eqal to 1 if n = m and otherwise. For the sake of simplicity, we choose ν = P µ. Then, f(x, µ ) = 1 and we write { } Q n (x) = Q n (x, µ ) = V n n (µ ) f(x, µ). (2.3) µ n µ=µ For an exhastive review regarding NEF-QVF and their properties, we refer to [13]. We will denote by L 2 (ν) the space of fnctions sqare integrable with respect to ν. Proposition 1. Let ν = P µ be a probability measre that generates a NEF-QVF, with associated orthogonal polynomials {Q n, n N} given by (2.3). Let X be a random variable with density fnction dp X dν with respect to ν. If dp X dν L 2 (ν) then we have the following expansion dp + X dν (x) = E(Q n (X)) Q n(x) Q n 2. (2.4) 3

5 Proof. By constrction {Q n } n N forms an orthogonal basis of L 2 (ν), and by orthogonal projection we get It follows that < Q n Q n, dp X dν dp X dν (x) = + > Q n(x) Q n = = < Q n Q n, dp X dν Qn (y) Q n > Q n(x) Q n. dp X dν (y)dν(y) Q n(x) Q n Q n (y)dp X (y) Q n(x) Q n 2 = E(Q n (X)) Q n(x) Q n 2. 3 Application to the rin problem 3.1 General formla The ltimate rin probability in the Cramer Lndberg rin model is the srvival fnction of a geometric compond distribted random variable where M = N Ui I, i=1 N is an integer valed random variable having a geometric distribtion with parameter ρ = βµ p, (Ui I) i N is a seqence of independent and identically distribted nonnegative random variables having CDF F U I (x) = 1 x µ F U(y)dy. The distribtion of M has an atom at with probability mass P (N = ) = 1 ρ. The probability measre governing the vales of M is dp M (x) = (1 ρ) δ (x) + dg M (x), (3.1) where dg M is the continos part of the probability measre associated to M which admits a defective probability density fnction with respect to the Lebesge measre. The ltimate rin probability is then obtained by integrating the continos part as the discrete part vanishes ψ() = P (M > ) = dg M (x). Theorem 1. Let ν be an nivariate distribtion having a probability density fnction with respect to the Lebesge measre, and that generates a NEF-QVF. If dg M dν L 2 (ν) then ψ() = V (µ ) n { n } µ n e κ(h(µ)) Ĝ M (h(µ)) µ=µ Q n (x)dν(x) Q n 2, (3.2) where ĜM is the Laplace-Stieljes Transform of G M defined by ĜM(s) = e sx dg M (x). 4

6 Proof. We start by applying Proposition 1 to dg M dν dg M dν (x) = + = By the definition of Q n (x) as defined in (2.3), we obtain dg + M dν (x) = = { V n n (µ ) µ n e κ(h(µ)) which leads to < Q n Q n, dg M (3.3) dν Q n (x)dg M (x) Q n(x) Q n 2. (3.4) { V n n (µ ) µ n e κ(h(µ)) Ĝ M (h(µ)) > Q n(x) Q n } e h(µ)x Q n (x) dg M (x) µ=µ Q n 2 (3.5) } µ=µ Q n (x) Q n 2. (3.6) Integration of (3.6) between and + gives the expression (3.2) for the ltimate rin probability. Remark 1. Eqation (3.2) involves the Laplace-Stieljes transform of G M. The presented method cold also be related to Laplace transform inversion techniqes. 3.2 Approximation with Lagerre polynomials We derive an approximation for the ltimate rin probability, sing Theorem 1, combined with trncations of the infinite series (3.2). For K N, we will denote by ψ K () = K { } V n n (µ ) µ n e κ(h(µ)) Ĝ M (h(µ)) µ=µ the approximated rin probability with trncation order K. Remark 2. We can write (3.7) as Q n (x)dν(x) Q n 2 (3.7) ψ K () = K Q n (x)dν(x) a n Q n 2, where a n are independent of. Once the evalation of the a n for all n K is done, estimating the rin probability reqires one integral calclation. In practice, as the distribtion of M is spported on R +, we will choose the exponential distribtion with parameter ξ for ν, that is: dν(x) = ξe ξx 1 R +(x)dλ(x). The associated orthogonal polynomials are the Lagerre ones,see [17], satisfying L n (x)l m (x)e x dx = δ nm. The polynomials defined in (2.3) are the Lagerre polynomials ( ) with a slight change in n comparison to the definition given in [17]: Q n (x) = 1 ξ n!ln (ξx) and their norm is 5

7 Q n = n! ξ. As ν is the exponential probability measre with parameter ξ, the mean of n ν is µ = 1 ξ, the variance fnction is V (µ ) = 1, the cmlant generating fnction is ξ 2 ( ξ ξ θ ) κ(θ) = log and the inverse fnction of the first derivative of κ(.) is h(µ) = ξµ 1 µ. We can write the expression of the rin probability (3.2) in a more tractable way, that is: ψ() = { ( )} n 1 ξµ 1 ( ξ) µ n ξµĝm µ µ=µ n! n L n (ξx)dν(x). (3.8) Remark 3. The choice of ν is arbitrary in the sense that we cold have chosen a more general gamma distribtion dν(x) = ξα x α 1 e ξx α) 1 R +(x). The orthogonal polynomials wold have been the generalized Lagerre polynomials and, with ξ = 1, or expansion wold have been the same as in [7]. By taking a gamma distribtion with its two first moments eqal to those of M, the first term of the obtained expansion gives the Beekman-Bowers approximation. The se of a Normal distribtion wold have implied an expansion involving Hermite polynomials, bt it seems less intitive to approximate a probability density fnction spported on R + by a sm of probability density fnction spported on R. Remark 4. Lagerre polynomials analytical expression is n ( ) n ( x) k L n (x) =. (3.9) n k k! k= { ( )} Denoting by a n = n 1 ξµ 1 n ( ξ) µ n ξµĝm µ µ=µ n! for n N, the injection of Lagerre polynomials expression (3.9) into the rin probability expansion (3.8) gives ψ() = = = a n n k= k=1 n=k 1 k=1 ( n n k ) ( a n n ξ k+1 x k e ξx dx k! ) ξ k x k 1 e ξx dx n k + 1 Γ(k) ξ k x k 1 e ξx b k dx. (3.1) Γ(k) The right hand side of (3.1) is exactly a gamma series expansion as defined in [18]. The defective probability density fnction associated to G M has the following expression g M (x) = n=1 Remark 5. The Lagerre fnctions are defined in [1] as (1 ρ)ρ n f n U (x). (3.11) I l n (x) = e x/2 L n (x), x. (3.12) The application of the Lagerre method consists in representing g M as a Lagerre serie g M (x) = q n l n (t). (3.13) One can note that the representation (3.13) is really close to the expansion proposed in this paper. 6

8 By Taking the Laplace transform of (3.11), we get Ĝ M (s) = (1 ρ)ρ F U I (s) 1 ρ F U I (s), (3.14) with F U I (s) = e sx f U I (x)dx the Laplace Stieljes transform of F U I. The moment generating fnction of the claim size distribtion appears in the formla. This fact limits the application to claim sizes distribtions that admit a well defined moment generating fnction, namely light-tailed distribtions. 3.3 Integrability condition There exists a link between the choice of ξ and the adjstment coefficient γ. The adjstment coefficient is the only positive soltion of the so-called Cramer-Lndberg eqation, F U I (s) = 1 ρ. (3.15) The integrability condition dg M dν L 2 (dν) is eqivalent to In order to ensre this condition, we need the following reslts. g M (x) 2 e ξx dx <. (3.16) Theorem 2. Assme that U I admits a bonded density fnction and that the eqation (3.15) admits a positive soltion, then for all x g M (x) C(s )e s x, (3.17) with s [, γ) and C(s ), where γ is the adjstment coefficient. Proof. In order to prove the theorem we need the following lemma regarding the srvival fnction F U of the claim sizes distribtion. Lemma 1. Let U be a non-negative random variable with bonded density fnction f U. Assme there exists s > sch that F U (s ) < +. Then there exists A(s ) > sch that for all x F U (x) A(s )e s x. (3.18) Proof. As F U (s ) < +, we have F U (s ) 1 = = s (e s x 1)f U (x)dx x e s y f U (x)dydx = s e sy F U (y)dy x s e sy F U (y)dy F U (x)(e sx 1). ths, we dedce that x F U (x) ( F U (s ) 1 + F U (x))e sx. (3.19) 7

9 The eqation (3.15) is eqivalent to ρ F U (s) = 1 + sµ. (3.2) The fact that γ is a soltion of the eqation (3.15) implies that F U (s) < +, s [, γ) and by application of Lemma 1, we get the following ineqality pon the density fnction of U I f U I (x) = F U(x) B(s )e sx. (3.21) µ In view of (3.11), it is easily checked that g M satisfies the following defective renewal eqation, g M (x) = ρ(1 ρ)f U I (x) + ρ We can therefore bond g M as in (3.17), x f U I (x y)g M (y)dy. (3.22) g M (x) ρ(1 ρ)f U I (x) + ρ(1 ρ)b(s )e s x + B(s )e s x = (ρ(1 ρ) + ĜM(s ))B(s )e s x = C(s )e s x. f U I (x y)g M (y)dy e s y g M (y)dy The application of Theorem 2 yields a sfficient condition in order to se the polynomial expansion. Corollary 1. For ξ < 2γ, the integrability condition (3.16) is satisfied. We note the importance of the choice of the parameter ξ. The Lagerre method, briefly described in Remark 5, does not offer the possibility of changing some parameter. The expansion is still based on orthogonal projection permitted nder an integrability condition. However, if the fnction does not satisfy the integrability condition then a damped version of it is expanded. 4 Nmerical illstrations First, we analyse the convergence of the sm in or method toward known exact vales of rin probabilities with exponential, gamma and phase-type cases. For those claim sizes distribtion we have explicit formlas that allow s to assess the acracy of or approximated rin probabilities. The goodness of the approximation depends on the order of trncation K, and reslts show also a dependence on ξ. Or method also enables s to approximate rin probabilities in cases that are relevant for applications bt where no formlas are crrently available. We compare the reslts with Monte-Carlo simlations and discss the interest of or method in comparison to the widely sed Panjer s algorithm. First, we plot the difference between the exact rin probability vale and its approximation ψ() = ψ() ψ K (), (4.1) then we stdy the behavior of or approximation when changing the vale of ξ and finally we compare or approximations with the ones of Panjer s algorithm. In order to se 8

10 Panjer s algorithm, we need to discretize the integrated tail distribtion of the severities and choose a bandwidth h. The Ronded Method is employed to do the discretization as it seems to be the best way according to [9]. It consists in ronding the severities to the closest integer mltiple of h. We choose a bandwith arbitrary eqal to.1 as there is no tractable formla available for the discretization error. To prodce simlations of the compond geometric sm, we se the procedre described in [11]. An iterative method is given to simlate random vales from integrated tail distribtion. The nmber of simlations needed, 1, and the iterative component in the simlation procedre imply a significant CPU time that already jstify the se of nmerical techniqes. Confidence intervals are given in addition to the estimation. Regarding the rin model settings, we fix a safety loading at 2%. 4.1 Exponentially distribted claim sizes In the case of exponentially distribted claim sizes with parameter δ, the ltimate rin probability is ψ() = ρe δ(1 ρ), (4.2) where ρ = β/δp, β is the Poisson process intensity, p the premim rate and is the initial reserves. In this particlar case, reslts (3.2) or (3.8) can be sed as a tool for comptations. After some tedios calcls, we get ψ() = ρ ( ) δ(1 ρ) ξ n L n (ξx)dν(x). δ(1 ρ) We se a property pon the generating fnction of Lagerre polynomials, + ( ) wn L n (x) = (1 w) 1 exp xw 1 w, which gives after straightforward integration ψ() = ρe δ(1 ρ). (4.3) For nmerical illstrations, we set δ = 1. Reslts are displayed in Figre 1 and Table 1. 9

11 Figre 1: Difference between exact and approximated rin probabilities for exponentially distribted claim sizes ξ K γ/ γ/ γ γ/ γ/ Table 1: Difference between exact and approximated rin probabilities for exponentially distribted claim sizes, with = Phase-type distribted claim sizes A phase-type distribtion is the distribtion of the absorbtion time of some continostime absorbing Markov process with a finite states space. Many common distribtions are of phase-type, for instance exponential, hyperexponential or Erlang distribtions admit a phase-type representation. The exact rin probability is then given in the form of a Matrix- Exponential, see the book [4] for details. The entire chapter VIII is dedicated to phasetype distribtions. In this second example, we assme that the claim sizes distribtion is a mixtre of two Erlang distribtions. The associated density fnction is f(x) = qerlang(k 1, δ 1 ) + (1 q)erlang(k 2, δ 2 ), (4.4) where Erlang(k, δ) = δk x k 1 e δx (n 1)! 1 R + and q [, 1]. We set k 1 = 3, k 2 = 2, δ 1 = 1, δ 2 = 2/3, and q = 2/5. Reslts are displayed in Figre 2, and Tables 2, 3. 1

12 Figre 2: Difference between exact and approximated rin probabilities for phase-type distribted claim sizes ξ K γ/ γ/ γ γ/ γ/ Table 2: Difference between exact and approximated rin probabilities for phase-type distribted claim sizes for phase-type distribted claim sizes, with = 1 Exact Vale Polynomials expansion Panjer s algorithm ξ = γ, K=12 h= Table 3: Rin probabilities for phase-type distribted claims amonts approximated with polynomials expansions and Panjer s algorithm 11

13 4.3 Gamma distribted claim sizes We assme that the claim sizes are gamma distribted with a scale parameter which is not integer. The exact form of the rin probability has been derived in [2] for the Γ(1/2, 1/2) special case, nmerical reslts are displayed in Figre 3 and Tables 4, 5. We finally compare approximations for the Γ(1/3, 1) case to reslts obtained thogh Monte Carlo simlations, see Figre 4 and Table 6. Figre 3: Difference between exact and approximated rin probabilities for Γ(1/2, 1/2) distribted claim sizes ξ K γ/ γ/ γ γ/ γ/ Table 4: Difference between exact and approximated rin probabilities for Γ(1/2, 1/2) distribted claim sizes, and = 1 12

14 Exact Vale Polynomials expansion Panjer s algorithm ξ = γ, K=12 h= Table 5: Rin probabilities for Γ(1/2, 1/2) distribted claims amonts approximated with polynomials expansion and Panjer s algorithm Figre 4: Rin probabilities for Γ(1/3, 1) distribted claim sizes approximated with polynomials expansion (ble line), Panjer s algorithm (red line), and Monte-Carlo simlations: estimation (green line) and 99% confidence interval (black dashed line) Monte-Carlo simlations Polynomials expansion Panjer s algorithm ξ = γ, K=12 h= Table 6: Rin probabilities for Γ(1/3, 1) distribted claims amonts estimated with polynomials expansion and Panjer s algorithm 13

15 4.4 Discssion of the nmerical reslts Approximations seem to behave very well for every vale of the initial reserves. We did not pt the reslts here, bt when ξ is chosen greater than 2γ approximations seems to diverge qickly. The order of trncation needed to reach a certain level of accracy depends on the complexity of the claim sizes distribtion. This fact is clearly observed throgh simlations within the Γ(1/2, 1/2) case, in which a greater order a trncation is needed to reach an eqal level of accracy. In the exponential case, there is a symmetric pattern. ξ eqal to γ is the optimal choice in terms of order of trncation needed. In the other cases stdied, there might exist an optimal choice for ξ in the range [γ, 2γ). Panjer s algorithm performs well for small initial reserves, sometimes better than or method. Bt the rin probability approximation for large initial reserves, that are relevant for applications, is problematical in or opinion becase the comptation time is clearly increasing and the accracy is worsening. Or method prodces, in a reasonable time, an acceptable approximated rin probability for every vale of. 5 Conclsion Or proposed method provides a very good approximation of the rin probability when the claim sizes distribtion is light-tailed. We obtained a theoretical reslt that allows s to ensre the validity of or expansions. In addition, the repeatability of the coefficients a n (see Remark 2) makes or method very fast when changing the initial reserves, which makes it very convenient compared to Panjer s algorithm. The problem we dealt with is the approximation of a geometric compond distribtion density fnction. The reslts are qite promising and allow s to envisage an extension to more general compond distribtions or even finite-time horizon rin probabilities. In forthcoming research work, we wold like to stdy the possibility of a statistical extension when data are available. The explicit formla obtained involves qantities that can be estimated empirically and plgged in. Acknowledgement The athors wold like to thank H. Albrecher and X. Gerralt for sefl sggestions. This work is partially fnded by the Research Chair Actariat Drable sponsored by Miliman, and the insrance company AXA France. References [1] J. Abate, G.L. Chodhry, and W. Whitt. On the lagerre method for nmerically inverting laplace transforms. INFORMS Jornal on Compting, 8(4): , [2] J. Abate and W. Whitt. The forier-series method for inverting transforms of probability distribtions. Qeeing Systems, 1:5 88,

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