ABSTRACT

We consider a modified version of the classical Cramer-Lundberg risk model. In particular, we assume two classes of insurance business dependent through the claim number process: we consider that the number of claims is generated by a bivariate Poisson distribution. We also consider the presence of a particular kind of reinsurance contract, supposing that the first insurer concludes an Excess of Loss reinsurance limited by, with retention limits, for the respective classes of insurance business. The aim of this paper is to maximize the expected utility of the wealth of the first insurer, having the retention limits as decision variables. We assume an exponential utility function and, fixed, we discuss optimal.

1. Introduction

The classical Cramer-Lundberg risk model assumes that the number of claims up to time t is independent of the claim size and the claim sizes are independent and identically distributed. However, the independence assumption can be restrictive in practical applications. Several authors have therefore suggested models where the risks dependence is assumed. Within this models, we distinguish between risk models with dependence among claim size and among inter-occurrence time (see [1-5]) and risk models that consider aggregate claims amount processes generated by correlated classes of insurance business (see [6-13]). In [14] a wide set of dependent risk processes involving particular classes is presented with references therein. In some cases, the authors consider reinsurance contracts (see [9,10]) to improve the risk situation to which the company is subjected. In particular, [10] suggests an unlimited Excess of Loss reinsurance and maximizes the expected utility of the wealth of the insurer or the adjustment coefficient. In this paper, we consider a risk model involving two dependent classes of insurance business and a limited Excess of Loss reinsurance with the aim to maximize the expected utility of the first insurer wealth, having the retention limits as decision variables. The paper is organized as follows. Section 2 presents the risk model and the reinsurance contract. In Section 3 the maximization problem is proposed and in Section 4 the solution is discussed.

2. The Model

We consider a risk model involving two dependent classes of insurance business. Let be the claim size random variables for the first class with common distribution function and let be those for the second class with common distribution function. We assume that and have continuous and positive first derivative, with if Their expected values are denoted by respectively. Then, the aggregate claims amount process generated from the two classes of business, in a given period of time, is with

, (1)

where is the number of claims, for classes, in the given period of time.

We assume that and are independent, and they are independent of and. The claim number processes are correlated by means of the following relationships (see [10]):

(2)

where and K are independent Poisson random variables with parameters and, respectively.

As usual, we define the surplus process in the given period of time:

(3)

where, is the insurance premium of risk i and u is the value of the surplus at the beginning of the time period.

We assume the following exponential utility function:

(4)

then, in the absence of reinsurance, the expected utility of wealth is

(5)

We instead assume that the first insurer concludes an Excess of Loss (XL) reinsurance contract limited by, with retention limits, for the respective classes of insurance business. In [10], an unlimited XL reinsurance is considered.

In the following, we will consider the variables, identically distributed to. In practical, XL reinsurance contracts with retention limit are limited by some constant, which leads to the following division of claim size.

The reinsurer pays

and the first insurer pays what is left:, (see [15]). That is:

(6)

and

(7)

Observe that if the limited XL reinsurance becomes unlimited. We fix with, for the two classes of insurance business, respectively, and we make use of the retention limits, as decision variables. We refer to (6) and (7) and we set

(8)

identically distributed to the j-th payment of the reinsurer and

(9)

identically distributed to the j-th payment of the first insurer.

Then, the aggregate claims amount process for the insurer generated from the two classes of business after reinsurance, in a given period of time, is with

. (10)

As usually, the reinsurance premiums are evaluated as follows:

where, is the corresponding safety loading. We therefore have

(11)

Then, after reinsurance, the surplus process in the given period of time is

(12)

and the expected utility is

(13)

In the following Section, we will take the problem of the maximization of the wealth expected utility, having the retention limits, as decision variables.

3. The Problem

As already mentioned, our problem is to find the pair, with, that maximizes (13). We observe that from the moment generating function of the bivariate Poisson distribution (see [10,16]) it follows that

(14)

where we assume that the moment generating function

, exists. In particular, we assume that the variables have a limited distribution or that the following relationship is true:

(15)

It therefore results

(16)

We put:

(17)

and we straightaway observe that

(18)

since it is and by assumption.

Because of (14), (16) and (18), (13) can be written as

(19)

that is, putting

we have

(20)

Therefore, maximizing (13) is equivalent to minimizing it follows that, since

, our problem consists of

(21)

We proceed in a similar way to that followed in [10]. The Kuhn-Tucker conditions for the P problem are

(22)

where, remembering (11) and (18):

(23)

and

(24)

Let us start by proving the following theorem.

Theorem 1.

The Hessian matrix of is negative definite, whenever the gradient is null.

Proof. Considering (23) and (24), it results

and

where

4. The Khun-Tucker Conditions

We look for the points that fulfill the KuhnTucker conditions. To this purpose, we put

(25)

(26)

(27)

and we prove the following theorem where and, are defined by (25), (26) and (27), respectively.

Theorem 2.

1) The point satisfies the system (22) if and only if it is

(28)

2) The point, with, satisfies the system

(22) if and only if it is

(29)

3) The point, with, satisfies the system

(22) if and only if it is

(30)

4) If it is

(31)

and it is

(32)

or

(32')

then it exists the point, with and, satisfying the system (22), and it is

(33)

Proof.

1) It results

the first of (28) holds and

the second of (28) holds.

2) We recall that is defined by (26). From (29) it follows that the first of (29) holds.

Furthermore, remembering (25) and (26),

and, remembering (27)

the second of (29) holds.

3) We recall that is defined by (26). From (29) it follows that the second of (30) holds.

Furthermore, remembering (25) and (26),

and, remembering (27)

the first of (30) holds.

4) Because of (31) it results

(34)

We put

and

and we note that it results

(35)

We solve the system (35). We have:

(36)

To solve the equation, we make use of a similar procedure to that in [10]. We note that g₂ is a continuous function of and that, by (34), it results

If (32) holds, remembering (27) and (25), and being an increasing function, it results

and

it therefore exists, satisfying the second of (36). Substituting in (36), it results

(37)

with, by (31) and (32), since is an increasing function: (38)

therefore, satisfying (33) fulfills the system (22).

Similarly, if (32’) holds, remembering (27) and (25), and being an increasing function, it results

and

it therefore exists, satisfying the second of (36). Substituting in (36) we obtain (37), (38) and the point fulfilling the system (22).

REFERENCES