Insurance: Mathematics and Economics 26 2000 239–250
Discounted probabilities and ruin theory in the compound binomial model
Shixue Cheng
a
, Hans U. Gerber
b,∗
, Elias S.W. Shiu
c,1
a
School of Information, People’s University of China, Beijing 100872, China
b
Ecole des hautes études commerciales, Université de Lausanne, CH-1015 Lausanne, Switzerland
c
Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA 52242-1409, USA Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999
Abstract
The aggregate claims are modeled as a compound binomial process, and the individual claim amounts are integer-valued. We study fx, y; u, the “discounted” probability of ruin for an initial surplus u, such that the surplus just before ruin is x and
the deficit at ruin is y. This function can be used to calculate the expected present value of a penalty that is due at ruin, and, if it is interpreted as a probability generating function, to obtain certain information about the time of ruin. An explicit formula
for fx, y; 0 is derived. Then it is shown how fx, y; u can be expressed in terms of fx, y; 0 and an auxiliary function hu that is the solution of a certain recursive equation and is independent of x and y. As an application, we use the asymptotic
expansion of hu to obtain an asymptotic formula for fx, y; u. In this model, certain results can be obtained more easily than in the compound Poisson model and provide additional insight. For the case u=0, expressions for the expected present value
of a payment of 1 at ruin and the expected time of ruin given that ruin occurs are obtained. A discrete version of Dickson’s formula is provided. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Risk theory; Ruin probability; Deficit at ruin; Time of ruin; Compound binomial model
1. Introduction
Traditionally, most results of risk theory are derived in a continuous-time model in which the aggregate claims are a compound Poisson process. The model appears to be more “realistic”, the methods are perceived as elegant
and sophisticated, and some results seem to be tied to the compound Poisson assumption. In contrast, this paper considers a discrete-time model, where the aggregate claims are modeled as a compound binomial process, and
where the possible claim amounts are integral multiples of the annual premium. With relative simple but perhaps also aesthetic methods, attractive results can be derived in this model. These results are of an independent interest,
and they provide a better understanding of the analogous results in the continuous-time model. In fact, the latter can be viewed as limiting cases of the former. In this sense, any given result in the discrete-time model is stronger than
the corresponding result in the continuous-time model.
∗
Corresponding author. Tel.: +41-21-692-3371; fax: +41-21-692-3305. E-mail addresses:
hgerberhec.unil.ch H.U. Gerber, eshiustat.uiowa.edu E.S.W. Shiu
1
Tel.: +319-335-2580; fax: +319-335-3017. 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 5 3 - 0
240 S. Cheng et al. Insurance: Mathematics and Economics 26 2000 239–250
In Section 3 we consider the discounted power martingale. For a given discount factor, the base must satisfy Lundberg’s fundamental equation, which has always one solution between zero and one, and a larger, second
solution under some regularity conditions for the claim amount distribution. The main goal of the paper is to study the function fx, y; u, which is the “discounted” probability of ruin, given an initial surplus of u, such that the
surplus just before ruin is x, and the deficit at ruin is y. With this function, the expected present value of a penalty due at ruin can be calculated. In Section 4, an explicit formula for fx, y; 0 is obtained. In Section 5, we show that
f
x, y; u can be expressed in a transparent fashion through a function hu, which does not depend on x or y, and is the solution of a certain recursive equation. In Section 6, an asymptotic formula for hu is derived and used to
obtain an asymptotic formula for fx, y; u. Applications include an explicit expression for the expected discounted value of a payment of 1 at the time of ruin, a discrete-time version of Dickson’s formula, formulas for the expected
time of ruin, as well as some asymptotic results.
2. The compound binomial model
