Articulo Sobre Riesgo

EXCESS OF LOSS REINSURANCE AND THE PROBABILITY

OF RUIN IN FINITE HORIZON

MARIA DE LOURDES CENTENO I

ISEG, Techmcal Untvelwity of Lt ~bon

July, 1995

ABSTRACT

The upper bound provided by Lundberg's inequahty can be improved for the probablhty

of rum in fimte horizon, as Gerber (1979) has shown This paper studies this upper

bound as a functmon of the retention hmH, for an excess of loss arrangement, and compares

it with the probability of ruin

KEYWORDS

Excess of loss, reinsurance; fimte time rum probability

1 INTRODUCTION

Several studies about the effect of reinsurance on the ultimate probability of ruin (for

example Gerber (1979), Waters (1979), Bowers, Gerber, HJckman, Jones and Nesbltt

(1987), Centeno (1986) and Hesselager (1990)) have concentrated their attention on

the effect of reinsurance on the adjustment coefficient.

Centeno (1986) has used an algorithm suggested by Panjer (1986) to calculate the

probabihty of ultimate ruin, incorporating reinsurance, to show with some examples

that the behavlour of this probability and Lundberg's inequality are very similar, both

considered as funcuons of the retention level, provided that the mltml reserve is not

too small This is consistent with the figures obtalnded more recently by Dickson and

Waters (1994) for some other examples and using a different algorithm for the probability

of ultimate rum In this paper, Dlckson and Waters have also calculated finite

horizon rum probabdmes, after reinsurance, by adapting the algorithm of De Vylder

and Goovaerts (1988) and by an approxmaauon provided by the translated Gamma

process Through an example they show that m continuous time for an excess of loss

arrangement, the optimal retention limit m finite horizon can be quite far from the

opumum value m infinite horizon. Of course, the sequence of optn'nal retention levels

Research perlormed under contract n ° SPES-CT91-O063

ASTIN BULLE'TIN, Vol 27. No I 1997 pp 59-70

60 MARIA DE LOURDES CENTENO

converges to the mfimte hor,zon opumal level as the t,me increases But, for a fin,te

horizon, Lundberg's inequality can be improved The purpose of this paper Is to show

how we can use this improvement to redefine the "opumal" retenuon hm~t for an

excess of loss arrangemenl, and to compare thls inequahty with the ruin probability in

finite horizon and continuous time for some examples Of course, the same methodology

can be applied to proport,onal reinsurance provided that, the moment generating

functmn of the individual cla,m amounts d~strlbuUon exists

2 ASSUMPTIONS AND PRELIMINARIES

In the classical r, sk process, the insurer's surplus at time t is denoted U(t), with

U(t) = u + ct - S(t),

where u ,s the Inlt,al surplus, c is the premmm income per umt of nine, assumed to be

received continuously, and S(t) ,s the aggregate claims occurred up to ume t. {S(t)}, >_ o

is assumed to be a compound Poisson process and without loss of general,ty the Poisson

parameter is assumed to be 1, which means that "trine t" IS the interval during

which t claims are expected Let G(x) denote the individual claim amount distribution

function and again without loss of generality, let us assume that this distribution has

mean I, wh,ch means that the monetary unit chosen ~s the expected amount of a claim

We further assume that G(0) = 0, with 0 < G(x) < 1 for x > 0 and also that G is such

that its moment generating function exists for x < T for some 0 < T < ~, and that

hm E[e rx ] = oo. (1)

r---~ 7

We assume that c is greater than 1, i.e. it is greater than the expected aggregate clmms

in each period. Let 0be such that c = I + 0

The ruin probability before time t is

~(u,t) = Pr{U(s) < 0 for some s,0 < s < t}.

Of course ~ (u, t) Is not greater than the ultimate probabfl W of ruin, denoted as ~ (u).

Therefore the upper bound given by Lundberg's mequahty is valid for finite horizon.

Gerber (1979), pp 139, has shown that this bound can be maproved in finite horizon

He proved that for u > 0 and t > 0

_< ,}, <2)

where Mx(r) Is the moment generating function of the individual claim amounts and R

denotes the adjustment coefficient, defined as the umque positive root of

Mx(r )- I = cr (3)

In the following we refer to express,on (2) as Gerber's inequality After an integration

by parts, inequality (2) can be written as

~(u,t) < mm,ie ?, (4)

"~R t J

EXCESS OF LOSS REINSURANCE AND THE PROBABILITY OF RUIN IN FINITE HORIZON 6 ]

and the equation defining the adJustment coefficient as

'~ e" (1 - G(a))dr = c (5)

Now suppose that the insurer has an excess of loss arrangement such that when a

clalnl X occurs he ~s responsible for rmn {X, M}, paying m return per unit of time a

reinsurance premium ~(M), which we assume to be calculated according to the expected

value principle with loading coetficlent ~, i.e.

c(M) = (I + ~)j'~(I - G(x))dx (6)

Assuming that the reinsurance premiums are prod continuously, the insurer's surplus

at time t is

N(t)

U(M;t) = u + (c- c(M))t - ~ mm {X~, M},

/I.=l

where N(t) denotes the number of clmms up to time t. The rum probabdlty before t~me

tis

tlt(M,u,t)= Pr{U(M,s)<O for some s,0 <,s _<t}.

After thin arragement Gerber's inequality becomes

f ...... t[J~te~(t-G(,))d~-(~-c(M))]]

~(M;u,t)_< mm ,ie ?, (7)

r->R(M)[ J

where R(M) denotes the adJustment coefficient after reinsurance, i e the umque positive

root of

j.~.t e p~ (I - G(x)) dx = c - c(M), (8)

when it exists or zero otherwise. Such a root exists if and only ff the expected profit

after reinsurance is posmve

We know that the value of M that maxlmises the adjustment coeffictent, when the

excess of loss reinsurance premmm is calculated according to the expected value principle

with ~ > O, is such that

1

M = --ln(1 + ~), (9)

R

(see for example Waters (1979)), nummmmg then the upper bound provided by Lundberg's

inequahty.

In the next section we wall study the problem that consists in choosing M m such a

way that the upper bound provided by (7) ts

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