A polynomial expansion to approximate ruin probabilities

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Transcription:

A polynomial expansion to approximate ruin probabilities P.O. Goffard 1 X. Guerrault 2 S. Loisel 3 D. Pommerêt 4 1 Axa France - Institut de mathématiques de Luminy Université de Aix-Marseille 2 Axa France 3 Institut de Sciences financières et d assurance Université de Lyon, Université de Lyon 1 4 Institut de mathématiques de Luminy Université de Aix-Marseille September 2013 / CEQURA 2013 Junior Worshop 1/20 P.O. Goffard 1/20

Sommaire 1 Introduction 2 Compound Poisson ruin model 3 Natural Exponential Families with Quadratic Variance Function 4 Numerical illustrations 2/20 P.O. Goffard 2/20

Executive summary Main goal Work out a new numerical method to approximate ruin probabilities. main idea Polynomial expansion of a probability density function though orthogonal projection Change of measure via Natural Exponential Family with Quadratic Variance function Construction of a polynomial orthogonal system w.r.t this probability measure Achievement Approximation of the ultimate ruin probability in the compound Poisson ruin model with light tailed claim sizes 3/20 P.O. Goffard 3/20

Notations df is an univariate Probability Measure. Denote by F its Cumulated Distribution Function, f = F its Probability density Function w.r.t. a positive measure, F(θ) = e θx df(x) its Moment Generating Function, κ(θ) = ln( F(θ)) its Cumulant Generating Function, L 2 (F) is a function space such that : If f L 2 (F) then f 2 (x)df(x) <. L 2 (F) is a normed vector space : f 2 =< f, f >= f 2 (x)df(x). 4/20 P.O. Goffard 4/20

Definition and hypothesis Denote by {R(t); t 0} the Risk Reserve Process : N(t) R(t) = u + pt U i, where u is teh initial reserves, p is the constant premium rate per unit of time, N(t) is an homogeneous Poisson process with intensity β, {U i } i N are i.i.d. non-negative random variables, independant of N(t), with CDF F U and mean µ. Let {S(t); t 0} be the Surplus process : i=1 S(t) = u R(t). η > 0 is the safety loading and one had better make sure that : p = (1 + η)βµ. 5/20 P.O. Goffard 5/20

Risk and surplus processes visualization 6/20 P.O. Goffard 6/20

Ultimate ruin probability Denote by M = Sup{S(t); t > 0}, the ultimate ruin probability is defined as : Pollaczek-Khinchine formula ψ(u) = P(M > u) = F M (u). In the compound Poisson ruin model, the ruin probability can be written as : + ψ(u) = (1 ρ) ρ n FU n (u), I M D = n=0 N Ui I, F U I (x) = i=1 x 0 F U (y) µ dy, where N is geometric with parameter ρ = βµ p convolution of F U I. See Ruin probabilities par Asmussen et Albrecher (2001) [1]. < 1 and FU n denotes the nth I 7/20 P.O. Goffard 7/20

Numerical evaluation of ruin probability : a brief review Panjer s algorithm, Panjer (1981) [8] Laplace transform numerical inversion Fast Fourrier Transform, Embrecht et al. (1993) [5] Laguerre s method, Weeks (1966) [10] Weighted sum of Gamma densities, Bowers (1966) [4] Beekman-Bowers Approximation, Beekman (1969) [3] Monte-Carlo simulations method, Kaasik (2009) [6] 8/20 P.O. Goffard 8/20

Natural Exponential Families with Quadratic Variance Function Let df be a univariate probability measure possessing MGF in a Neighborhood of 0. {F µ ; µ M} is the NEF generated by df, with : df µ (x) = exp(φ(µ)x κ(φ(µ)))df(x). The variance function is said quadratic if : V(µ) = aµ 2 + bµ + c The NEF-QVF contain six distribution : Normal Gamma Hyperbolic Binomial Negative Binomial Poisson 9/20 P.O. Goffard 9/20

Orthogonal polynomials for NEF-QVF Distributions Define {F µ ; µ M} a NEF-QVF generated by df with mean µ 0. The PDF f (x, µ), of a F µ w.r.t. df is proportional to exp(φ(µ)x κ(φ(µ))). { } Q n (x, µ) = V n n (µ) f (x, µ) /f (x, µ), µ n is a polynomial of degree n in both µ and x. f (x, µ 0 ) = 1 et { } Q n (x) = Q n (x, µ 0 ) = V n n (µ 0 ) f (x, µ) µ n µ=µ 0. {Q n } is an orthogonal polynomials system such that : < Q n (x), Q m (x) >= Q n (x)q m (x)df(x) = Q n 2 δ nm. For a full description of the NEF-QVF see Barndorf-Nielsen (1978) [2] et Morris (1982) [7]. 10/20 P.O. Goffard 10/20

Polynomial Expansion and Truncations The polynomials are dense in L 2 (F). {Q n} is therefore an orthogonal basis of L 2 (F). Let df X be a probability measure associated to some random variable X. df X its density w.r.t. df df If dfx df L2 (F) we have : df X df (x) = < df X df, Q n Q n > Q n(x) Q n = E(Q n (X)) Q n(x) Q n 2. n N n N The CDF F X is then : F X (x) = n N E(Q n (X)) x Q n(y)df(y) Q n 2. Approximations are then obtained by truncation K x FX K (x) = E(Q n (X)) Q n(y)df(y) Q n 2. n=0 11/20 P.O. Goffard 11/20

Polynomial expansion for the ultimate ruin probability Recall that M = N i=1 UI i then : If dgm df L2 (F) then : + df M (x) = (1 ρ)δ 0 (dx) + (1 ρ) ρ n dfu n (x) I = (1 ρ)δ 0 (dx) + dg M (x). n=1 dg M df (x) = < dg M df, Q n Q n > Q n(x) Q n. n N Integration leads to the polynomial expansion of the ruin probability : ψ(u) = n N < dg M df, Q n Q n > + u Q n (y)df(y). Q n 12/20 P.O. Goffard 12/20

Approximation of the ruin probability though truncation of the polynomial expansion Approximation of the ultimate ruin probability {F µ ; µ M} is a NEF-QVF generated by F with µ 0, f (x, µ) exp(φ(µ)x κ(φ(µ))) is the PDF of F µ w.r.t. F. If dgm df L2 (df) then : ψ K (u) = K [ V F (µ 0 ) n n ( ) ] µ n e κ(φ(µ)) Ĝ M (φ(µ)) n=0 + Q u n (y)df(y) Q n (x) 2 µ=µ 0 13/20 P.O. Goffard 13/20

Choice of the NEF-QVF dg M is a defective probability measure supported on [0, + [. Among NEF-QVF, the only one suported on [0, + [ is generated by the exponential distribution. df(x) = ξe ξx dλ(x) The orthogonal polynomials linked to the exponential measure are the Laguerre ones, see Szegö (1939) [9] Which value for ξ to complete the integrability condition? ψ(u) e γu. where γ is the unique positive solution to the following equation : F U I (s) = 1 ρ ξ < 2γ 14/20 P.O. Goffard 14/20

Choice of the NEF-QVF dg M is a defective probability measure supported on [0, + [. Among NEF-QVF, the only one suported on [0, + [ is generated by the exponential distribution. df(x) = ξe ξx dλ(x) The orthogonal polynomials linked to the exponential measure are the Laguerre ones, see Szegö (1939) [9] Which value for ξ to complete the integrability condition? ψ(u) e γu. where γ is the unique positive solution to the following equation : F U I (s) = 1 ρ ξ < 2γ 14/20 P.O. Goffard 14/20

Choice of the NEF-QVF dg M is a defective probability measure supported on [0, + [. Among NEF-QVF, the only one suported on [0, + [ is generated by the exponential distribution. df(x) = ξe ξx dλ(x) The orthogonal polynomials linked to the exponential measure are the Laguerre ones, see Szegö (1939) [9] Which value for ξ to complete the integrability condition? ψ(u) e γu. where γ is the unique positive solution to the following equation : F U I (s) = 1 ρ ξ < 2γ 14/20 P.O. Goffard 14/20

Choice of the NEF-QVF dg M is a defective probability measure supported on [0, + [. Among NEF-QVF, the only one suported on [0, + [ is generated by the exponential distribution. df(x) = ξe ξx dλ(x) The orthogonal polynomials linked to the exponential measure are the Laguerre ones, see Szegö (1939) [9] Which value for ξ to complete the integrability condition? ψ(u) e γu. where γ is the unique positive solution to the following equation : F U I (s) = 1 ρ ξ < 2γ 14/20 P.O. Goffard 14/20

Choice of the NEF-QVF dg M is a defective probability measure supported on [0, + [. Among NEF-QVF, the only one suported on [0, + [ is generated by the exponential distribution. df(x) = ξe ξx dλ(x) The orthogonal polynomials linked to the exponential measure are the Laguerre ones, see Szegö (1939) [9] Which value for ξ to complete the integrability condition? ψ(u) e γu. where γ is the unique positive solution to the following equation : F U I (s) = 1 ρ ξ < 2γ 14/20 P.O. Goffard 14/20

Calibration des simulations Regarding the ruin model, we assume that : The premium rate p is equal to 1, The safety loading η is equal to 20%. A graphic visualisation is proposed, we plot the quantity : ψ(u) = ψ(u) ψ K (u), for an intitial reserves u and a truncation order K. Different values for ξ are tested with one equal to γ. 15/20 P.O. Goffard 15/20

Exponentially distributed claim sizes f U (x) = e x 1 R +(x) 16/20 P.O. Goffard 16/20

Γ(1/2, 1/2)distributedclaimsizes f U (x) = e x/2 Γ(1/2) 2x 1 R +(x) 17/20 P.O. Goffard 17/20

Γ(1/3, 1)distributedclaimsizes f U (x) = e x x 2/3 Γ(1/3) 1 R +(x) 18/20 P.O. Goffard 18/20

Comparison with Panjer s algorithm u Exact Value Polynomials expansion Panjer s algorithm ξ = γ, K=120 h=0.01 0.1 0.821313 0.821424 0.821356 1 0.736114 0.736238 0.736395 5 0.47301 0.472944 0.473757 10 0.274299 0.274252 0.275131 50 0.00352109 0.00352476 0.00357292 u Monte-Carlo simulations Polynomials expansion Panjer s algorithm ξ = γ, K=120 h=0.01 0.1 0.8 0.80505 0.805454 1 0.624 0.634979 0.636315 5 0.232 0.239601 0.241442 10 0.076 0.0712518 0.0723159 50 0 4.569555 10 6 4.686 10 6 19/20 P.O. Goffard 19/20

Conclusion + An efficient numerical method An approximation as precise as one might want + Easy to understand and to implement + No discretization of the claim sizes is needed Limited to light tailed distribution Outlooks : Theoritivcal sensitiveness study of the parameter ξ Agregate claim amounts distribution, more general compound distributions Statistical extension Extension statistique Finite time ruin probability 20/20 P.O. Goffard 20/20

S. Asmussen and H. Albrecher. Ruin Probabilities, volume 14 of Advanced Series on Statistical Science Applied Probability. World Scientific, 2010. O. Barndorff-Nielsen. Information and exponential Families in Statistical Theory. Wiley, 1978. J.A. Beekman. Ruin function approximation. Transaction of society of actuaries, 21(59 AB) :41 48, 1969. N.L. Bowers. Expansion of probability density functions as a sum of gamma densities with applications in risk theory. Transaction of society of actuaries, 18(52) :125 137, 1966. P. Embrechts, P. Grübel, and S. M. Pitts. Some applications of the fast fourrier transform algorithm in insurance mathematics. Statistica Neerlandica, 41 :59 75, March. 1993. 20/20 P.O. Goffard 20/20

A. Kaasik. Estimating ruin probabilities in the Cramér-Lundberg model with heavy-tailed claims. Mathematical statistics, University of Tartu, Tartu, October 2009. Carl N. Morris. Natural exponential families with quadratic variance functions. The Annals of Mathematical Statistics, 10(1) :65 80, 1982. H. H. Panjer. Recursive evaluation of a family of compound distributions. Astin Bulletin, 12(1) :22 26, 1981. G. Szegö. Orthogonal Polynomials, volume XXIII. American mathematical society Colloquium publications, 1939. W. T. Weeks. Numerical inversion of laplace transforms using laguerre functions. Journal of the ACM, 13(3) :419 429, 1966. 20/20 P.O. Goffard 20/20

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