How common sense can be misleading in ruin theory

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1 Annals of the University of Craiova, Mathematics and Computer Science Series Volume, 6, Pages 7 8 ISSN: -69 Dedicated to Marius Iosifescu on the occasion of his 8th anniversary How common sense can be misleading in ruin theory Anişoara Maria Răducan, Raluca Vernic, and Gheorghiţă Zbăganu Abstract. Some graphical representations of ruin probability computed mainly for Erlang type claims suggested an idea that intuitively seems to be true: if the first claims are small then the chance to get ruined is also small. However, for other claims this does not hold, as is shown by counterexamples. Key words and phrases. Ruin probability, nonhomogeneous claims, stochastic order.. The problem In the insurance field, the evaluation of ruin probabilities is of great importance since it influences the future financial politics of any insurance company. In this sense some recursive formulae were obtained in [], for the ruin probability evaluated at or before claim instants for the following risk model used to describe the evolution over time of the surplus of an insurance company n n U n u X i Y i u ξ i. i Here U n n denotes the remaining capital after paying the n-th claim, u is the initial capital, X n n are the claim sizes CSs assumed to be Erlang-distributed, independent and independent of Y n n, the nonnegative inter-claim revenues ICRs which themselves are assumed to be independent, identically distributed i.i.d., following an arbitrary distribution. Then ξ n X n Y n represents the loss increment between the n -th and the n-th claims. We recall [], [8] that the ruin probability at or before the n-th claim is j ψ n u P min U j < P max ξ i > u. j n j n The novelty of Răducan et al. [], [5] consists in assuming that the CSs are nonhomogeneous Erlang distributed, yielding a nonhomogeneous process. Motivated by many numerical examples, Răducan et al. [] stated the following conjecture relating the order in which the nonhomogeneous claims arrive to the magnitude of the corresponding ruin probability: if the claims arrive in the increasing stochastic order, then the ruin probability is smaller that if the same claims come under a different order. That seemed to be common sense: during the small claims period, the insurer s capital accumulates, hence it can face the hard claims period better than if a larger claim arrives sooner and decreases the insurer s capital. Received April, 6. i i 7

2 HOW COMMON SENSE CAN BE MISLEADING IN RUIN THEORY 7 However, this conjecture was proved only in the particular case when the CSs are exponentially distributed with all parameters distinct, see [5] and [6]. In this paper, we deal with a case which is more restrictive than the stated conjecture, and more general in the same time: it is more restrictive because we consider only two claims, and more general because we let them follow any distribution supported on positive values. More precisely, we study if there is a relation between the magnitude of the ruin probability and the arrival order of the two claims when these claims satisfy some stochastic order. In this sense, in Section we define a new stochastic relation which we call ruin domination. This relation is not always transitive, as we point out using a counterexample.. Various stochastic orderings Let X, X be two nonnegative random variables r.v, and let F, G be their distribution functions d.f. If the r.v are absolutely continuous either with respect to the Lebesgue measure, or with respect to the counting one, we will denote by p, q their densities. We recall the following stochastic orders see, for instance [7]:. The usual stochastic domination: X st X F G q G. The hazard rate domination: X hr X p F. The likelihood ratio domination: X like X p q is decreasing.. The increasing convex order: X icx X EuX EuX for any nondecreasing convex u. Remark.. It is known that the likelihood ratio domination implies the hazard rate one, which in turn, implies the stochastic one; see, e.g., [7]. Back to the conjecture stated by [5], if we deal with only two claims, it turns into the following claim: Conjecture. Let X, X, Y, Y be independent nonnegative r.v.s., withy and Y identically distributed. Let ξ i X i Y i, i,, L, max ξ, ξ + ξ +, L, max ξ, ξ + ξ +. If X st X, then L, st L, ; or, in other words, if ψ i,j t P L i,j > t, i j {, }, t >, represents the two ruin probabilities, then the claim is that ψ, ψ,. Or, in terms of survival probabilities denoted by φ i,j ψ i,j, the claim is that φ, φ,. We already know that this conjecture holds if the claims are exponentially distributed. More generally, the common sense says that if the ICRs are the same, then it is better if the smaller claim X comes first and the greater one X comes next than if the greater claim comes first and the smaller one afterwards i.e., in the first scenario one seems to be better prepared for the more dangerous claim than in the second one, provided that the incomes are identically distributed. Related to this situation, we define the following stochastic relation. Definition.. Let X, X be two independent non-negative r.v.s. We say that X dominates X in the ruin sense if, for any i.i.d. non-negative r.v.s Y, Y, independent of X, X it is true that L, st L, i.e., that max, X Y, X + X Y Y st max, X Y, X + X Y Y.

3 7 A.M. RĂDUCAN, R. VERNIC, AND G. ZBĂGANU We denote this relation by X ruin X. If the domination holds only for Y Y const we say that X dominates X in the weak ruin domination and write X wruin X. Otherwise written: two claims X, X are in the relation X ruin X if the scenario first comes claim X, then X is always better in stochastic order than first comes claim X, then X for any i.i.d. ICRs Y, Y, while X wruin X means that the first scenario is better only for constant ICRs. Thus, our conjecture says that X st X X ruin X. The current study was prompted by the unpleasant surprise that the conjecture is false. Here are two counterexamples. Example.. Suppose that X, X....,.... Y Y. The reader can check that X st X but it is not true that L, st L,. Precisely in the first scenario we have ψ, P L, >.6.6 while in the second one ψ, P L, >.9.6. However, it is true that when the initial capital u is greater than, then ψ, u ψ, u. The two scenarios are not stochastically comparable. Example.. Let X Uniform,, X exp, Y Y. Clearly X st X. Then ξ X, ξ X + X, and for t the corresponding survival probabilities are. φ t P L t P X + t, X + X + t P X min + t, + t X [ ] E P X min + t, + t X X [ ] E min, min + t, + t X + min, + t, + t x e x dx. + Therefore ϕ, t e + t + e t if t. Next, e t + e t if t > ϕ, t P L t P X + t, X + X + t E [P X min, + t X X ] e t. e Thus ϕ, + e.65 while ϕ, e.7. It is not e true that ϕ, ϕ,.

4 HOW COMMON SENSE CAN BE MISLEADING IN RUIN THEORY 75 It turns out that even the weaker conjecture X st X X wruin X fails to be true. Even the weaker implication X hr X X wruin X is false - as it is proved by the second example. However, in both examples it is true that the second scenario is worse in the increasing convex order meaning that L, icx L, but maybe this is only by chance; we do not dare to add a new false guess here.. Ruin domination in general case In order to see under which assumptions the ruin domination holds, we define the following notation: for two nonnegative r.v.s X, X and a, b, t let and a,b P X a, X + X a + b P X a, X + X a + b, D a,b t a+t,b + b+t,a. Lemma.. i. If a then a, and D a, t t,a. ii. Let X, X be non-negative independent r.v.s. and let F, F be their d.f.s. If they are absolutely continuous with respect to some measure µ, the densities are df x f x dµ x, df x f x dµ x and δ x, y f x f y f y f x then a,b +b x δ x, y dµ y dµ x... iii. If the d.f.s. F p p p... then for any a, b > positive integers, a,b a i a+b i j δ i,j min{a,b} a+b x, F min{a,b} i a... q q q... a+b i ja+ where δ i,j p i q j p j q i and, by convention, an empty sum is. δ x, y dµ y dµ x. 5 are discrete, δ i,j, 6 Proof. i. For any a we have a, P X a, X + X a P X a, X + X a P X + X a P X + X a and D a, t a+t, + t,a t,a. ii. The first equality is immediate. To prove the second one, we consider first the case a b, when we split a,b δ x, y dµ y dµ x + +b x a δ x, y dµ y dµ x. The first integral is since if we change the order of integration followed by interchanging x and y, we get δ y, x dµ y dµ x δ y, x dµ y dµ x δ y, x dµ x dµ y δ x, y + δ y, x dµ x dµ y.

5 76 A.M. RĂDUCAN, R. VERNIC, AND G. ZBĂGANU When a > b we split a,b b a δ x, y dµ y dµ x + b a+b x With a similar reasoning, we obtain that +b x a δ x, y dµ y dµ x +b x δ x, y dµ y dµ x. b a a+b x δ x, y dµ y dµ x δ x, y dµ y dµ x, hence the equality from 5 also holds. iii. Taking into consideration that δ i,j +δ j,i, this formula results after a reasoning similar with the one in the continuous case ii. Remark.. Let us notice that: i D a,b t D b,a t, a, b, t an obvious consequence of the definition, ii D a,b t, a, b, t a,b, a, b obvious from D a, t t,a. Proposition.. Let X, X be non-negative independent random variables. Then i X wruin X if and only if a,b for any b a. ii X ruin X if and only if D a,b t dh a dh b for any probability distribution H on the halfline [, and t. iii If X, X are discrete, then X wruin X if and only if m,n for all nonnegative integers m, n such that m n. Proof. i X wruin X P X x+t, X +X x+t P X x+t, X +X x + t x+t,x for all t, x. Put a x + t and b x. ii X ruin X P X Y +t, X +X Y +Y +t P X Y +t, X +X Y + Y + t for any i.i.d. nonnegative r.v.s. Y, Y, independent of X, X. Denoting by H the common distribution of Y, Y, we see that Next P X Y + t, X + X Y + Y + t E [P X Y + t, X + X Y + Y + t Y, Y ] P X y + t, X + X y + y + tdhy dhy P X y + t, X + X y + y + tdhy dhy P X y + t, X + X y + y + tdhy dhy P X Y + t, X + X Y + Y + t. P X Y + t, X + X Y + Y + t E [P X Y + t, X + X Y + Y + t Y, Y ] P X y + t, X + X y + y + tdhy dhy.

6 HOW COMMON SENSE CAN BE MISLEADING IN RUIN THEORY 77 Then P X Y + t, X + X Y + Y + t P X Y + t, X + X Y + Y + t [P X y + t, X + X y + y + t P X y + t, X + X y + y + t]dhy dhy t+y,y dh y dh y t+y,y + t+y,y dh y dh y t+y,y dh y dh y D a,b t dh a dh b. iii From i we know already that X wruin X x+t,x, x, t. But in the [x+t] [x+t] i discrete case x+t,x δ i,j, thus we have only to check that m n i j where m : [x + t] and m + n [x + t]. Here [x] is integer part of x. Thus the claim is that [a + t] [a + t] [a + t] + [a + t] [a + t] +. This is easy and it is left to the reader [a] [a + t]. Corollary.. Let X, X be nonnegative independent r.v.s, absolutely continuous with respect to some σ-finite measure µ having densities p, q. Then a sufficient condition in order that the relation X ruin X hold is that δ x, y p x q y p y q x for all x y. Proof. According to ii from Proposition. X ruin X if and only if D a,b t dh a dh b. But D a,b t a+t,b + b+t,a. We denote a a + t and b b + t. Using the second equality in 5, we note that mina,b a +b x a,b δ x, y dµ y dµ x a since in this integral x a and y a, and we know that x y implies δ x, y. Similarly, b,a, which completes the proof. The following corollary is a consequence of the above one. Corollary.. Let X, X be non-negative independent r.v.s. absolutely continuous with respect to some σ- finite measure. Then X like X X ruin X. Remark.. The like relation is too strong, it seems to be far away from the relation ruin. We consider another relation seemingly closer from the later one. Definition.. Denote by X Sruin X the condition D a,b t for all a, b and t. Remark.. According to remark. ii, one can also writex Sruin X a,b for all a, b. Proposition.5. Let X, X be non-negative independent r.v.s. absolutely continuous with respect to some σ- finite measure. The following implications hold: X like X X Sruin X X ruin X X wruin X Proof. The implications are obvious.

7 78 A.M. RĂDUCAN, R. VERNIC, AND G. ZBĂGANU. The comparison in the discret case Here we suppose that... X F p p p......, X F q q q... hence F m m p i, F m m q i. We shall compare these four orderings in the i i most simple cases. The most convenient comparisons are those which involve only the quantities i,j. They are easily to compare because, according to Proposition.. we have the following algorithm : X wruin X m,n for all positive integers m, n such that m n. X Sruin X m,n for all positive integers m, n. Proposition.. Let X, X be non-negative independent r.v.s. as above. Then i. If X wruin X then p pj sup. q j q j ii. If X Sruin X then X st X. iii. If X wruin X and X wruin X then X X. Proof. i. Note that for any n we have n, δ,n and we know that n, or, which is the same p q n p n q. ii. Let m, lim m,n. As m,n for all m, n it follows that m, n for all m. But it is easy to see that m, F m F m. iii. According to i. we see that X wruin X and X wruin X imply p sup pj j q j pj inf j q j and q p p i q i, i. sup j from where p q qj p j p q inf j pj q j. Therefore sup j qj p j, q p q p j for any j. As p i q i it follows that q j i i Here is a table with the first quantities m,n m,n. Here δ i,j:k means δ i,j + δ i,j δ i,k. m/n : δ, δ,: δ,: δ,: δ, δ,: + δ, δ,: + δ,: δ,:5 + δ,: δ, δ,: + δ, δ,:5 + δ,: + δ, δ,:6 + δ,:5 + δ,: δ, δ,:5 + δ, δ,:6 + δ,:5 + δ, δ,:7 + δ,:6 + δ,:5 + δ, So, if A. Supp F Supp F {, }, then the single interesting value of is, δ, and,. Thus X wruin X iff δ, p q p q p q q p p q X st X. Therefore we have Corollary.. If Supp F Supp F {, } then the relations st, wruin, ruin, Sruin and like are the same.

8 HOW COMMON SENSE CAN BE MISLEADING IN RUIN THEORY 79 B. Supp F Supp F {,, }. In this case only the matrix δ, δ i,j i, j,: δ, δ,: does matter. According to our algorithm X wruin X δ,, δ,, δ,: + δ,. Adding the first two inequalities we also get δ,: p q + q q p + p p q q p p q. The last one is p q + p q q p + q p q p + p p q + q q p p + p q + q. But p q, p + p q + q means that X st X. As the conditions for X Sruin X are the same, the conclusion is: Corollary.. If Supp F Supp F {,, } then i. X st X p q, p + p q + q p q, p q. ii. X wruin X X st X, p p, p p X Sruin X X ruin q q q q X. iii. X like X p p p. q q q Remark.. The answer to the question if the Sruin relation implies the like relation is negative, as we can see from the counterexample X and X in which X Sruin X but it is not true that X like X. 6 Remark.. This is the most simple case when one can deny the implication X st X X ruin X. For instance, if X and X, then X st X but it is not true that X ruin X since p q < p q. C. Supp F Supp F {,,, }. The interesting part of is δ, δ,: δ,: i,j i, j δ, δ,: + δ, δ,: + δ,:. δ, δ, + δ, δ, + δ, + δ, ThenX st X p q, p + p q + q, p + p + p q + q + q and X wruin X δ,, δ,, δ,, δ,: + δ,, δ, + δ,, δ, + δ, + δ,.adding the first three inequalities, we obtain p q ; the last one says that p q or, which is the same, that p + p + p q + q + q. The relation X sruin X adds to the inequalities for the weak comparison three more ones: δ,:, δ,: and δ,:, δ,:. The first two of them are superfluous since they are implied by δ,, δ,, δ,. The remaining one is not: written as p + p q + q q + q p + p p + p q + q q + q p + p it implies the stochastic order. To summarize: Corollary.. If Supp F Supp F {,,, } then i. X st X p q, p + p q + q, p + p + p q + q + q. ii. X wruin X p p max, p, p,p q,p q, δ,: + δ,, q q q q 9 9

9 8 A.M. RĂDUCAN, R. VERNIC, AND G. ZBĂGANU δ, + δ,. iii. X Sruin X X wruin X & X st X. iv. X like X p p p p. q q q q The relation X ruin X should be between the last ones. It is possible to coincide with the relation X Sruin X. Remark.. This is the first case when wruin does not imply the stochastic domination. For instance, if X and X, the reader can check that X wruin X but it is not true that X st X. Remark.. The notation attributed to ruin domination might not be the most proper one, since this is not an ordering relation as one can easily see from the following example: X, X, X for which X wruin X wruin X and yet it is not true that 7 7 X wruin X. 7 6 Acknowledgements Gheorghiţă Zbăganu acknowledges the support from a research grant of the Romanian National Authority for Scientific Research CNCS UEFISCDI, project number PN-II-ID-PCE References [] S. Assmussen, H. Albrecher, Ruin Probabilities, World Scientific Publishing Co., Singapore,. [] A. Castaner, M.M. Claramunt, M. Gathy, C. Lefevre, M. Marmol, Ruin problems for a disctere time risk model with non-homogeneous conditions, Scandinavian Actuarial Journal, 8. [] C. Lefevre, P. Picard, A nonhomogeneous risk model for insurance, Computers and Mathematics with Applications 6 5, 5. [] A.M. Răducan, R. Vernic, Gh. Zbăganu, Recursive calculation of ruin probabilities at or before claim instants for non-identically distributed claims, ASTIN Bulletin 5 5,. [5] A.M. Răducan, R. Vernic, Gh. Zbăganu, On the ruin probability for nonhomogeneous claims and arbitrary inter-claim revenues, Journal of Computational and Applied Mathematics 5 9, 9. [6] A.M. Răducan, R. Vernic, Gh. Zbăganu, On a conjecture related to the ruin probability for nonhomogeneous exponentially distributed claims, Submitted. [7] M. Shaked, J.G. Shanthikumar, Stochastic orders and their applications, Academic Press, New York, 99. [8] D.A. Stanford, K.J. Stroinski, K. Lee, Ruin probabilities based at claim instants for some non Poisson claim processes, Insurance: Mathematics and Economics 6, [9] D.A. Stanford, K. Yu, J. Ren, Erlangian approximation for finite time ruin probabilities in perturbed risk models, Scandinavian Actuarial Journal, [] R. Vernic, On a conjecture related to the ruin probability for nonhomogeneous insurance claims, Annals of the Ovidius University of Constanţa - mathematics series 5, 9.

10 HOW COMMON SENSE CAN BE MISLEADING IN RUIN THEORY 8 Anişoara Maria Răducan Institute of Mathematical Statistics and Applied Mathematics Gheorghe Mihoc - Caius Iacob of the Romanian Academy, Bucharest, Romania address: anaraducan@yahoo.ca Raluca Vernic Institute of Mathematical Statistics and Applied Mathematics Gheorghe Mihoc - Caius Iacob of the Romanian Academy, Bucharest, Romania Ovidius University of Constanţa Gheorghiţă Zbăganu Institute of Mathematical Statistics and Applied Mathematics Gheorghe Mihoc - Caius Iacob of the Romanian Academy, Bucharest, Romania University of Bucharest