The Influence of Extreme Claims on the Risk of Insolvency

Transcription

1 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 The Inflence of Extreme Claims on the Risk of Insolvency Valéria Skřivánková Abstract In this paper, the classical risk process with light-tailed distribtions (small claims) and its modifications for extreme claim amonts with heavy-tailed distribtions are stdied The risk of insolvency is measred here by the probability of rin We present some approximations and pper bonds for the probability of rin, and consider the conditions nder which these models rn correctly Keywords Risk process, Cramér-Lndberg model, insolvency, rin probability, approximations, pper bonds Introdction Risk is a key concept to characterize ftre ncertain otcomes in financial sphere The central problem of the insrance risk theory is modelling the probability distribtion of total (aggregate) ftre claims which is than sed to evalate rin probability (likelihood of insolvency) When the reserve of an insrer company falls below zero as a reslt of the last claim, we say that rin has occrred By the basic actarial principle, premims are calclated by relying on the mean vale of total claims increased by the relative safety loading (insolvency coefficient) Important decision variables are the initial capital (initial reserve) of the insrer and the risk premim inflenced by the safety loading Practical actarial approaches often ignore complex interdependences among timing of claims, their sizes (especially extreme claim amonts) and the possibility of rin In this paper, we will consider the general collective risk model with some modification, R(t) R() + C(t) S(t), () where R(t) is the risk reserve at time t, R() the initial reserve at time t The total risk premim is denoted by C(t), and S(t) represents the total claim amont p to time t We sppose that the individal claims X i are nonnegative independent identically distribted (iid) random variables and that they occre in random times T i, i, 2, 3 Frther, we sppose that the inter-arrival (waiting) times W i T i T i-, i 2, 3,, W T, are iid and the seqences {X i } and {W i } are independent Denote by N(t) the total nmber of claims p to time t, then {N(t), t } is a conting process (or claims arrival process), for which we have N(t) max { n; T n t } In general, {R(t)} and {S(t)} are random processes, and in some special cases also is {C t } The total claim S(t) X + X 2 ++ X N(t) has a compond Doc RNDr Valéria Skřivánková,CSc, Institte of Mathematics, Faclty of Science, PJ Šafárik University of Košice, Jesenná 5, 4 Košice, Slovakia, valeriaskrivankova@pjssk This work was spported by Slovak grant fondation VEGA No / 724/8 and /35/

2 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 distribtion N ( t ) n F S ( t ) P( S ( t) x) P X i x P( N ( t) n) F, i n (2) F x is the n-fold convoltion of the distribtion F where ( ) There exist different modifications of the general risk process, introdced in () We can consider different special distribtions for N(t) and X i For example, N(t) can be distribted Poissonial, binomial or negative binomial, and the claims X i may have light-tailed distribtion for small claims (exponential, normal, gamma), or heavy-tailed sbexponential distribtion for extremal claims ( Pareto, Brr, Weibll, loggamma, etc) The premim income rate c is sally constant (C(t) ct ), bt it can be also another non-random fnction of time, even random variable (as in bons-mals systems, see [6] ) In the following Section 2 we will consider the historical special case of the general risk process, called Cramér-Lndberg process (C-L model), and will deal with some approximations and pper bonds for the probability of rin in this case For this prpose we define ψ (, T ) P{ R( t) <, for some t T}, < T <, (3) the probability of rin in finite time (with finite horizon), and by ψ (, ),, the probability of rin in infinite time (with infinite horizon) 2 The standard Cramér-Lndberg model The historical version of the general risk process () was introdced by Lndberg (93) and Cramér (93), and is given by the following C-L conditions: C) The claim arrival process { N( t), t } is homogeneos Poisson process with constant intensity λ, E(N t ) λt C2) The claim sizes X i, i,2, are nonnegative iid random variables with common distribtion fnction F, finite mean E(X ) µ < and finite variance D(X ) < C3) The inter-arrival times W i are independent and exponentially distribted with finite mean E(W ) /λ C4) The net premim C t p to time t is considered to be payable at constant rate c per nit time, so that C t c t The risk process is now defined as R t + ct ( X + X X N(t) ), and its possible realization is on the following pictre The expected vale of risk reserves is

3 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 E( R( t)) + c t µ λt + ( c λµ ) t, t (2) Under the net profit condition, we reqire c-λµ >, more exactly c (+)λµ, where the relative safety loading is defined as c λµ > (22) λµ The initial reserve R() and the safety loading are important decision parameters for the risk management According to (2), the total claim amont S t has a compond Poisson distribtion given by F S(t) P( S( t) x) k ( λt) k! k e λ t F k, x, t (23) Under the net profit condition > one can show (see Embrechts [5]), that for the non-rin probability holds where FI µ F ( y) dy n - ( ) n ψ FI,, (24) + n + is the integrated tail distribtion of F and F ( y) - F( y), y > The relation (24) says, that the non-rin probability - represents a compond geometric distribtion This formla is the key tool for calclating rin probabilities, bt in general, it is difficlt to derive explicit expressions for ψ () by (24), except the case of x / µ exponentially distribted small claims with F( x) e, x, µ > In this case we get (see [9]) exp, (25) + ( + ) µ Grandell in 99 proved (see [8]) that the exact rin probability satisfies the following integral eqation of renewal type λ λ ψ ( ) ψ ( y) F ( y) dy + F ( y) dy (26) c c In general, again no explicit closed form soltion to the eqation (26) exists, except in the case where claims are mixtres of exponential distribtions (see []) In non-exponential cases, nder sitable conditions, one can obtain only some approximations to the rin probability The well known pper bond for the rin probability gives the Cramér-Lndberg ineqality as we can see in the following theorem Theorem 2 Let the net profit condition > hold Assme that there exists a positive constant L > (called adjstment coefficient or Lndberg coefficient), sch that λ L x e F dx (27) c Then for all holds L e (28)

4 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 Proof There exist several different proofs of the Lndberg ineqality (28) The original proof is based on Wiener-Hopf method, Gerber proved it by martingale method (see [7]), bt the most simply proof ses mathematical indction method ( see for example [2], [9]) Remarks to calclate the Lndberg coefficient L as a positive soltion of (27), in general is rather complicated The alternative version of Lndberg condition in terms of moment generating fnctions is c z e M ( z), (29) X z X where M X ( z) E( e ) is the moment generating fnction of individal claim X for exponentially distribted claims with mean vale µ, after ptting c (+ ) λµ, one can easy calclate that the soltion of (29) is: z L /( + ) µ (2) when the claims are not exponentially distribted, bt they have finite mean and variance, we get an approximation for safety loading sing the Taylor expansion of the moment generating fnction of individal claims in form: L 2E(X)/ D( X) (2) in the case when the claims are from interval (,M), we can derive the following pper or lower bonds for safety loading ( see [2], [9]) ln(+ ) 2E( X ) L (22) 2 M E( X ) Another general formla for calclation ψ () sing conditional mean vale as a fnction of the time of first rin, gives the following theorem ( for details see [5] or []) Theorem 22 For arbitrary holds L e ( ), (23) L R( T ) E e T < where T min{t; R(t) <, t > } is the time of the first rin 3 Modification of Cramér-Lndberg model for large claims When claims in the compond Poisson risk model are from heavy-tailed sbexponential distribtions ( Pareto, Brr, Weibll or lognormal), classical techniqes sed to compte the probability of rin in Section 2, are not sitable In this case, the individal claims have not finite exponential moments ( E( e zx ), for any z > ), so that the moment generating fnction does not exist Under high claims, the probability of rin is larger than in the classical case of finite moment generating fnction So, to approximate ψ () and constrct some non-exponential bond for large withot the Cramer-Lndberg condition, is very important problem In this Section 3 we will present some reslts of this type, bt at first, we define the sbexponential distribtions Definition 3 A distribtion fnction F with spport (, ) is called sbexponential, if for all n 2, F n lim n x F The class of sbexponential distribtion fnctions will be denoted as S (3)

5 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 Remark In the case of sbexponential distribtions, the tails of the sm S n X +X 2 + +X n and of the maximm M n max {X, X 2,, X n } are eqivalent The tail probabilities are P( S > x) P( M > x), for n n n Theorem 3 Consider the C-L risk model with net profit condition > and large claims with sbexponential distribtion F I S Then for FI (32) Proof Using formla (24) and the definition of sbexponential distribtion, we have n n ( + ) FI, + n n n FI ( + ) ( + ) FI + n FI + n From this we get the approximation (32) n n, for Theorem 32 Consider the C-L risk model with net profit condition > and large claims with sbexponential distribtion F I S Frther consider that for X i, i, 2, the first m moments exist (ie for all m, satisfying for < θ < the conditions ( + kx) k k k E( X ) x df < ) Let there exist a constant k > m dfi θ and I P( N > n + ) θ P( N > n) (33) Then the following ineqality holds for every m ( + k) (34) Proof For the proof see Conti, 29 [4] Theorem 33 Consider the C-L risk model with net profit condition > and large claims k x Assme that for any > there exists a constant k > so that e df + Then for the rin probability we obtain the following pper and lower bonds exp{ 2k} + + F F exp{ k} + F + F (35) Proof The proof one can find in Cai and Garrido, 999 [3] In the last decade, some reslts were pblished abot so called new worse than sed distribtions which are based on stochastic ordering and are sitable to generalize the C-L ineqality for the probability of rin Definition 3

6 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 A distribtion G spported on, ) is said to be new worse than sed ( NWU ), if for any x > and y >, G ( x + y) > G G ( y) (36) Theorem 3 4 Assme that there exists an NWU distribtion G so that for > ( ) df( x) + G Then Proof For proof see Willmot and Lin, 2 [2] G, for (37) 4 Conclsion The reslts, presented in Sections 2 and 3, have theoretical character, bt they can be sefl also in practical insrance management for calclating the probability of rin We dealt with statistical analysis of some insrance data inclding extreme claims already in [], bt this time we have obtained actal real non-life insrance data and or goal is their detailed analysis with emphasis on rin probability calclation Literatre [] BOWERS at al: Actarial Mathematics The Society of Actaries, Itasca, Illinois, 986 [2] BUHLMANN, H: Mathematical Methods in Risk Theory Springer, Berlin, 97 [3] CAI J, GARRIDO, J: Two-sided bonds for rin probabilities when the adjstment coefficient does not exist Scandinavian Actarial Jornal, (999),8-92 [4] CONTI, PL, MASIELO, E: Nonparametric statistical analysis of an pper bond of the rin probability nder large claims Extremes - Pblished online 9 November 29 [5] EMBRECHTS at al: Modelling Extremal Events Springer, Berlin, 997 [6] FECENKO, J: Neživotné poistenie Ekonóm, Bratislava, 28 [7] GERBER, HU: An Introdction to Mathematical Risk Theory Habner Fondation Monograph Series No 8, R Irvin, Homewood, 979 [8] GRANDELL, J: Aspects of Risk Theory Springer, New York, 99 [9] HORÁKOVÁ, G, MUCHA, V: Teória rizika II, Ekonóm, Bratislava, 28 [] SKŘIVÁNKOVÁ, V, TARTAĽOVÁ, A: Catastrophic Risk Management in Non-life Insrance E+M Economics and Management, 2/28, [] URBANIKOVÁ, M: Probability of Rin In: Applied Mathematics an Informatics at Universities, Bratislava, 22, 87-9 [2] WILLMOT, GE, LIN, XS: Lndberg Approximations for compond distribtions with insrance applications Springer, Berlin, 2

7 VŠB-TU Ostrava, Ekonomická faklta, katedra Financí 8-9 září 2 Smmary Vplyv extrémnych poistných plnení na riziko insolventnosti Príspevok sa zaoberá s klasickým procesom rizika, kde malé poistné plnenia majú rozdelenie s ľahkým chvostom, ako aj s modifikácio model pre extrémne veľké poistné plnenia s ťažkým chvostom Riziko insolventnosti je t vyjadrené pomoco pravdepodobnosti rinovania (krach) V príspevk prezentjeme niekoľko aproximácií a horných odhadov pre pravdepodobnosť krach a venjeme sa podmienkám, za ktorých važované modely sú korektné