University of Amsterdam. University of Amsterdam and K. U. Leuven

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1 BOUNDS ON STOP-LOSS PREMIUMS FOR COMPOUND DISTRIBUTIONS BY R. KAAS University of Amsterdam AND M. J. GOOVAERTS University of Amsterdam and K. U. Leuven ABSTRACT Upper and lower bounds are derived for the stop-loss premium of compound distributions with fixed claim number distribution and known mean, variance and range for the claim severity distribution. 1. INTRODUCTION In this paper we investigate bounds on stop-loss premiums for compound distributions (1) S = Xi-~- X2 +..-~- XN where the claim number distribution FN is fixed (e.g., Poisson (A)) and where the claim severity distribution Fx is restricted to have (2) F.(-0) =0 Fx(b) = 1 E[X]=~ for some finite b and /.,i. e [0, b]. The stop-loss premium of S with stop-loss point t will be denoted by (3) 7r(t; Fx)= f,~ (x-t) d[.~opr(n=n)f~'(x)] BOHLMANN et al. (1977), introducing the concept of stop-loss ordering, derived bounds for 7r(t; Fx). In fact, when the random variable X- equals tz with probability one, and X + has range {0, b} and mean /.z, (4) ~-(t; Fx-)~ ~(t; Fx)~ ~(t; Fx+) uniformly in t and for all X satisfying (2). ASTIN BULLETIN Vol 16, No I

2 14 KAAS AND GOOVAERTS (5) To prove that this is true first observe that cr and /3 exist such that Fx-(x)~Fx(x) forx<a Fx (x)~fx(x) forx~>a Fx (x)~fx(x) for x ~>/3 Fx*(x)~Fx(x) for x </3. As E[X-] = E[X]= E[X ], this means that X- is less dangerous than X and X is more dangerous. As a more dangerous distribution has higher stop-loss premiums, we have (6) X-oc X oc X + where oc denotes stop-loss order. Stop-loss order is preserved under compounding, so N N N (7) E X?oc E X, oc Z X, + i=l t=l I=1 which is equivalent to (4) holding for all real t. For a more detailed proof, see GOOVAERTS et al. (1984) Since X- and X satisfy all requirements for X, the bounds in (4) are best possible, and X- and X are extremal distributions. It is not possible to give such extreme distributions when the variance of X is also fixed, say (8) Var (X) = o.2 With the techniques of GOOVAERTS et al. (1984) one may compute extreme values of stop-loss premiums, but unfortunately the corresponding distributions depend on the value of the stop-loss point chosen. There is no severity distribution in this class that is smallest or largest in the sense of stop-loss order. In Section 2 we exhibit random variables Z- and Z + that give bounds like (4), uniformly in t. These bounds are not the best possible, since Z- and Z have variances different from o2. They are, however, the greatest lower and least upper bound with respect to dangerousness. In Section 3 we give a numerical illustration using the examples of GERBER (1982). 2. ANALYTICAL BOUNDS ON DISTRIBUTION FUNCTIONS In GOOVAERTS and KAAS (1986) extreme values are given for distributions Fx with range [0, b] and the first few moments fixed. When X has mean /.t and variance o.2, we have (10) F'(x)~ < Fx(x)~ F"(x) with the values of F t and F" given in the following table, where z = (x-,u,)/oand d = b/z -/z 2- o'2~ > 0 are used for notational convenience.

3 BOUNDS ON STOP-LOSS PREMIUMS 15 TABLE 1 BOUNDS FOR DISTRIBUTION FUNCTIONS WITH RANGE [0, b], MEAN ~ AND VARIANCE 0-2 x Ft(x) F"(x) d 1 0<x<~ - 0 b-/.t l+aa 2 d <~x~b -d-- 1 I.* d 1_~4 d b-~ I.~ b bx b b(b-x) d 1 b---~x~b /..t, 1+7, 2 Now define the following two severity distributions: (ll) Fz*(X)={F (a) ot~x~13 [F'(x) 13<~x~<b where a and 13 satisfy (12) Or a =/.z-io-2+mb-~.) x [cr(b - 2~) - 4cr=(b - 2~)=+ (Or=+/.~(b -/.~))2] and O r 13 =u.+ Or2+~(b_~ ) x [Or(b - 2/..t) + 4o'2(b - 2/x)2 + (ors+ p.(b - p.))2] I Ft(x) 0~<x</z (13) Fz-(X)=(F.(x) i.<~x~.b. With d as in table 1, we have a-~ d/(b-l*) and 13 I> b- d/tz. To check that Fz+ is well-defined and E[Z-]= E[Z +1= is a laborious process but involves only elementary calculus. Since the distribution G with dg(a)= F"(a)= l-rig(13) has mean t* and variance or2 we have Var (Z +) > or2. In fact, it may be shown that, writing t*=(t-t.~)/cr for all t, (14) Var(Z+)=or 2 ( l+ln \i~-7~7~i+~1 ) In the same way, considering the distribution H with dh(o) = F'(lx), dg(lz) = F"(tz) - F'(tz), dh(b) = 1 - F"(I*),

4 16 KAAS AND GOOVAERTS which has mean /.t and variance tr 2, one shows that Var (Z-)< cr 2. Because of (10) we have immediately that Z- is less dangerous than any X, and Z is more dangerous, so (15) zr(t; Z-)~< ~r(t; X)~< rr(t; Z ) uniformly for all t and for all feasible X. Now let W be a random variable with dfw(x)>o for some x where also F~(x) < Fw(x) < F"(x). It is easy to construct a feasible X with Fx(x) = Fw(x) and x outside the spectrum of X: dfx (x) = 0. But then either X is more dangerous than W, or X and W are not comparable because Fx and Fw have two more more sign changes. So to be more dangerous than all X, Fw must be first above F", then constant between F" and F ~, then below F t. But then it is easy to see that Fz and Fw have only one point of intersection, so Z is less dangerous than W. Reasoning along the same lines for Z- we may conclude that among the distributions more dangerous than any feasible X, Z + is the least dangerous, whereas Z- is the most dangerous less dangerous distribution. In this sense Z and Z- are optimal choices. 3. NUMERICAL ILLUSTRATION In order to assess the quality of the bounds derived in the previous section, we give a numerical example. In GERBER (1982) methods are described to bound as well as to approximate stop-loss premiums of compound Poisson distributions. His method to obtain a lower bound using mass concentration does not always give an arithmetic discrete distribution, so we used the method of matching (two) moments, which is much more exact with the same computational effort. To obtain Gerber's uniform (1, 3) claim severity distribution as a special case, we took b = 3, /x = 2 and cr 2= ½ in our examples, the claim numbers being Poisson (h) with h = 1, 10 and 100. TABLE 2 BOUNDS FOR STOP-LOSS PREMIUMS WITH CLAIM-RANGE [0,3], MEAN 2, VARIANCE 31- AND CLAIM NUMBER POISSON ( ) Stop-Loss Gerber's Upper Upper Lower Lower Point t Exact Value Bound (4) Bound (15) Bound (15) Bound (4) % 1000% 1000% 1000% x 10 -I 124 I x 10 -k x l x x x x I 366x x x

5 BOUNDS ON STOP-LOSS PREMIUMS 17 TABLE 3 BOUNDS FOR STOP-LOSS PREMIUMS WITH CLAIM RANGE [0,3], MEAN 2, VARIANCE 31 AND CLAIM NUMBER POISSON (10) Stop-Loss Gerber's Upper Upper Lower Lower Point t Exact Value Bound (4) Bound (15) Bound (15) Bound (4) % 103 0% 99 0% 99 0% x 10 -l x 10 -I x x I 289x I 4.49x 10 -'~ x I 067xl xl TABLE 4 BOUNDS FOR STOP-LOss PREMIUMS WITH CLAIM RANGE [0,3], MEAN 2, VARIANCE ~ AND CLAIM NUMaER POISSON (100) Stop-Loss Gerber's Upper Upper Lower Lower Point t Exact Value Bound (4) Bound (15) Bound (15) Bound (4) t 104 4% 102 2% 99.1% 99 1% I x 10 -I I 992 x x x REFERENCES BUHLMANN, H, GAGLIARDI, B, GERBER, H and STRAUB, E. (1977) Some Inequahttes for Stop-Loss Premmms, ASTIN-Bulletm IX, GERBER, H U. (1982) On the Numerical Evaluation of the Distribution of Aggregate Claims and its Stop-Loss Premiums Insurance Mathema,cs and Economics I, GOOVAERTS, M J. and KAAS, R (1986) Analytical Bounds on Distributions Under Integral Constraints, to be pubhshed GOOVAERTS, M J, DE VYLDER, F and HAEZENDONCK, J (1984) Insurance Premmms North- Holland Pubhshing Company Amsterdam R. KAAS and M. J. GOOVAERTS Universiteit van Amsterdam, Jodenbreestraat 23, NL-1011 Netherlands. WH Amsterdam, The

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