Journal of Mathematical Finance
Vol.07 No.04(2017), Article ID:80712,17 pages
10.4236/jmf.2017.74053
Effect of an Excess of Loss Reinsurance on Upper Bounds of Ruin Probabilities
Nguyen Quang Chung1,2
1Department of Basic Sciences, Hungyen University of Technology and Education, Hung Yen, Vietnam
2Applied Mathematics and Informatics School, Hanoi University of Science and Technology, Hanoi, Vietnam
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 29, 2017; Accepted: November 26, 2017; Published: November 29, 2017
ABSTRACT
In this paper, discrete time risk models under an excess of loss reinsurance are studied. Adjustment coefficients of the cedent and the reinsurer are established as functions of quota share level and retention level. By the martingale method, ruin probabilities of the cedent and the reinsurer still have exponential form. Finally, numerical examples are provided to illustrate the results obtained in this paper.
Keywords:
Excess of Loss Reinsurance, Ruin Probability, Quota Share Level, Retention Level, Interest Rate, Markov Chain, Martingale Process
1. Introduction
We consider the insurer’s surplus in period denoted as is defined by:
(1.1)
where:
・ is the insurer’s initial surplus;
・ denotes the premium income in period n (i.e., from time to time n), is a sequence of independent and identically distributed (i.i.d.) non-negative random variables;
・ denotes the claim amount in period n, is a sequence of i.i.d. non-negative random variables and is independent of Y.
The process defined by (1.1) is called a surplus process (see [1] ). Yang [1] gave the upper bounds of ruin probabilities of the insurer by using the martingale method. Cai and Dickson [2] extended the surplus process in (1.1) by including the interest rate. Then, the surplus process can be written as following
(1.2)
where and denotes the interest rate of the insurer in period n. The sequence is assumed to be a Markov chain and independent of X and Y. With surplus process (1.2), the upper bound of the insurer’s ruin probability was established by the martingale and inductive methods in [2] .
In the classical risk model, claims are assumed to be paid by one insurer. However, insurers can transfer risks from one primary insurer (the ceding company or cedent) to another one (the reinsurance company) through reinsurance contracts. For that reason, some authors extended the classical surplus process in a consideration of an excess of loss reinsurance. For example: the various articles [3] [4] [5] investigated the effect of the reinsurance contract on the upper bound of the cedent’s ruin probability. The upper bounds of ruin probabilities of the cedent and the reinsurer were estimated in [6] [7] where Dam and Chung considered the risk model under quota share reinsurance. The explicit expression was given for finite-time joint survival probability of the cedent and the reinsurer in [8] . An optimal reinsurance retention was studied under ruin-related optimization criteria in [9] .
This paper investigates the effect of an excess of loss reinsurance on the ultimate ruin probabilities of the cedent and the reinsurer in the discrete-time model. The risk models are investigated in two cases without interest rate and with homogeneous Markov chain interest rate. The premium income is assumed as a sequence of independent and identically distributed random variables. In particular, the author shows that for given value then there exists a quota share level and a retention level so that both the ruin probabilities of the cedent and the reinsurer are less than value .
The content of this paper is organized as follows: A brief description of the models and some notions are presented in Section 2. Section 3 is devoted to the construction of the ruin-related problems in the risk model without an interest rate. The upper bounds of ruin probabilities in the risk model with interest rate are given in Section 4. Finally, numerical illustrations are given.
2. The Risk Models
In this paper, we investigate the effect of an excess of loss reinsurance on the surplus processes (1.1) and (1.2). First, the cedent and the reinsurer arrange an excess of loss reinsurance that we denote as the quota share level and is retention level. The premiums are calculated according to the expected value principle. i.e. for each insurance company, the premium income expectation is greater than the claim expectation.
Proposition 1 shows that there exists such that the premiums satisfy the expected value principle. We will denote the probability space as a triple
, and if .
Proposition 1. Assuming that
(2.1)
for any given M there exists such that:
(2.2)
and
(2.3)
Proof. For any M, we denote and .
We have
(2.4)
(2.5)
Using (2.4) and (2.5), we imply that
From , thus
We have
Since, we imply the existence such that
(2.6)
where and . □
We now consider the surplus process defined by (1.1) with the excess of loss reinsurance. Then, the cedent’s surplus and the reinsurer’s surplus in period are denoted by and , respectively. Surpluses and can be expressed as
(2.7)
and
(2.8)
where u and v are initial surpluses of the cedent and the reinsurer. The processes (2.7) and (2.8) are called surplus processes.
The finite-time and ultimate ruin probability of the cedent with initial surplus u are respectively defined by
(2.9)
Similarly, the finite-time and ultimate ruin probability of the reinsurer with initial surplus v are denoted by and . The probabilities are defined by
(2.10)
Obviously: and
.
Let and be the interest rate sequences of the cedent and the reinsurer, respectively. The interest rates satisfy assumptions (2.1) and (2.2).
・ Assumption 2.1. The cedent’s interest rate sequence is a homogeneous Markov chain, takes the values in a finite set of positive and
(2.11)
where for any and for all .
・ Assumption 2.2. The reinsurer’s interest rate sequence is a homogeneous Markov chain, takes the values in a finite set of positive and
(2.12)
where for any and for all .
Then, the cedent’s surplus and the reinsurer’s surplus in period are denoted by and .
(2.13)
and
(2.14)
where and are the cedent’ initial surplus and the reinsurer’s initial surplus, respectively.
It is easy to see that (2.13) and (2.14) are equivalent to
(2.15)
and
(2.16)
The finite-time and ultimate the cedent’s ruin probabilities with surplus process (2.15), initial surplus u and a given are respectively defined by
(2.17)
and
(2.18)
Similarly, the finite-time and ultimate the reinsurer’s ruin probabilities with surplus process (2.16), initial surplus v and a given are
(2.19)
and
(2.20)
Clearly: and
.
3. The Ruin Probabilities in the Risk Model without Interest Rate
The adjustment coefficients, which depend on quota share level and retention level, are established in the following lemmas.
Lemma 2. If , , and for any then there exists the unique such that
(3.1)
Proof. We set and . We have
Therefore, there exists the expectation value of for all .
For any , if we set for then
(3.2)
Differentiating the above function, we get
(3.3)
Let , be an arbitrary real-valued sequence as . Using the Mean Value Theorem for , we obtain
Moreover, for any , there exists a natural number such that for all , where .
Thus, we have
and
Applying Lebesgue’s Dominated Convergence Theorem, we imply that
(3.4)
From (3.3) and (3.4) function is differentiable
So,
(3.5)
It means that the is decreasing at .
Since there exists so that . We have
(3.6)
The right side of (3.6) tends to infinity as . It implies that
(3.7)
Combining (3.2), (3.5) and (3.7), function must intersect the x-axis. In other words, there exists a positive x-intercept of . Let’s denote it . Apparently, is a root of the following equation
(3.8)
Similarly, function is twice differentiable. Hence,
That means is strictly convex for . Thus, is the unique positive of Equation (3.8)
□
The proof of Lemma 3 is similar to Lemma 2 and we omit the proof here.
Lemma 3. If
and
for any then there exists the unique such that
(3.9)
The following theorem provides the exponential upper bounds of and .
Theorem 4. Assuming that the surplus processes given in (2.7) and (2.8) satisfy assumptions in Lemma 2 and Lemma 3. Then,
(3.10)
and
(3.11)
for any .
Proof. In order to prove (3.10), we set the stochastic process :
, for and
the filtration where , ;
The stochastic process is a martingale with respect to the filtration .
Indeed
(3.12)
We now consider for
(3.13)
Moreover
(3.14)
Combining (3.13) and (3.14), thus
(3.15)
Since the stochastic process is a martingale with respect to the filtration . Let . Then is a finite stopping time. Thus, using the optional stopping theorem for martingale , (see [10] ) we get
This deduces that
(3.16)
From (3.16) and , we obtain
(3.17)
Therefore, inequality (3.10) is followed by letting in (3.17).
The proof of inequality (3.11) is similar to the one for inequality (3.10).
□
In reinsurance businesses, evaluation the two ruin probabilities of the cedent and the reinsurer are crucial. Because the insurers based on the ruin probabilities to determine so that and are decreased. However, the issue is a difficult topic. The following theorem shows us how to determine so that and are less than a given value .
Theorem 5. Assuming that the surplus processes given in (2.7) and (2.8) satisfy the following assumptions:
1) Random variable takes values in a finite set of non-negative numbers where and
, , ;
2)
For any given satisfies
(3.18)
there exists such that
(3.19)
and
(3.20)
Proof. By , we imply that there exists such that
(3.21)
Obviously .
Expression (3.21) is equivalent to
For any given satisfies (3.18). We have
Since, .
Moreover, Expression (3.21) can be written
Hence
(3.22)
Using (3.22) and , thus
□
Li [9] investigated the optimal M to maximize the joint survival probability for the cedent and the reinsurer in one period insurance. If both companies don’t occur ruin at certain period , and will be initial surpluses of the insurance companies before period n, respectively. Therefore, we apply Theorem 5 to estimate the probabilities of the insurance companies to period from period n.
4. The Ruin Probabilities in the Risk Model with Interest Rate
In the section, we consider surplus processes (2.15) and (2.16). The proofs of Lemma 6 and Lemma 7 are similar to the one for Lemma 2.
Lemma 6. If
, , and for any then there exists the unique such that
(4.1)
for any .
Proof. For any and , we set
(4.2)
for .
Similarly, we show the expectation value existence in (4.1). In particular
(4.3)
(4.4)
(4.5)
Moreover
(4.6)
Combining assertions to (4.3) from (4.6), we deduce that function must intersect the x-axis. Let’s denote it . Apparently, is the unique intersection.
□
For any . We set
(4.7)
The function is strictly convex for , . Since, this is equivalent to
(4.8)
for all .
We have
(4.9)
Hence
(4.10)
Lemma 7. If
and
for any then there exists the unique such that
(4.11)
for any .
Proof. The proof of Lemma 7 is similar to the one for Lemma 2. □
If
(4.12)
then
(4.13)
where
Using martingale method, we present the exponential upper bounds of and .
Theorem 8. Assuming that the surplus processes given in (2.15) and (2.16) satisfy assumptions in Lemma 6 and Lemma 7. For any then
(4.14)
and
(4.15)
for all and .
Proof. We first consider the stochastic process and the filtration where
and
We have
(4.16)
For , we get
We set . According to Jensen’s Inequality, it
implies that
(4.18)
Combining (4.17) and (4.18), we obtain
Hence, the stochastic process is a supermartingale with respect to the filtration .
Let . Then
is a finite stopping time. Thus, by the optional stopping theorem for supermartingale , (see [10] ), we get
This implies that
(4.19)
By and (4.19), we have
(4.20)
By letting in (4.20), we obtain inequality (4.14).
The proof of inequality (4.15) is similar to the one for inequality (4.14). □
5. Numerical Illustrations
5.1. Example 5.1
Suppose that sequences and satisfy the conditions in Theorem 5. Initial surpluses and . The distribution functions of and are defined in Table 1 and Table 2, respectively:
Inequality (3.18) implies for all there exists such that and . E.g. from (3.21) we chose
this follows then couple is the solution of Theorem 5.
If then , and . Hence, couple also satisfy Proposition 1.
5.2. Example 5.2
In this example, let and take the same structure and values as the ones in Example 5.1. Initial surpluses and .
The interest rate sequence of the cedent is a homogeneous Markov chain, takes values: , and . The transition probability matrix of the process is
Similarly, the interest rate sequence of the reinsurer is a homogeneous Markov chain, takes values: , and . The transition probability matrix of the process is
Let and , we have Table 3 and Table 4.
We denote the moment-generating functions of and are
Table 1. Distribution function of .
Table 2. Distribution function of .
Table 3. Distribution function of .
Table 4. Distribution function of .
(5.1)
and
(5.2)
for .
Using Matlab software, we obtain and which are the solutions of equations and , respectively.
If then Equation (4.1) can be written
(5.3)
Similarly, for and Equation (4.1) is equivalent to, respectively,
(5.4)
and
(5.5)
Combining the solutions of Equations (5.3), (5.4) and (5.5), we have
Similarly, for and Equation (4.11) can be written, respectively,
(5.6)
(5.7)
and
(5.8)
From the solutions of Equations (5.6), (5.7) and (5.8), this implies .
Other couples then , , and are defined as the ones above. Table 5 gives some numerical results of the upper bounds of , , and for all and . Note couples in Table 5 satisfy Proposition 1.
In Table 5, the upper bounds of (4.14) and (4.15) are tighter than the ones (3.10) and (3.11), respectively. This is in a good accordance with [1] [2] . If and increase then the cedent’s upper bounds of the ruin probabilities increase while the reinsurer’s upper bounds of the ruin probabilities decrease.
6. Conclusions and Suggestions
・ The surplus processes given by (2.7) and (2.13) can be viewed as an extension of the ones (1.1) and (1.2);
・ By martingale method, the author obtains the upper bounds of the ultimate ruin probabilities of the cedent and the reinsurer in the risk models under excess of loss reinsurance;
Table 5. The upper bounds of the ruin probabilities for other couples .
・ There remain many open issues, e.g.
- building the upper bounds of the ultimate ruin probabilities in the risk model under combination of quota share and excess of loss reinsurance;
- investigating the joint ruin probability of the cedent and the reinsurer in the risk model under excess of loss reinsurance;
- establishing the optimality problems under ruin-related optimization criteria.
Further research in some of these directions is in progress.
Acknowledgements
The author wishes to thank professor Bui Khoi Dam for his helpful suggestions and many valuable comments while the author makes this paper.
Cite this paper
Chung, N.Q. (2017) Effect of an Excess of Loss Reinsurance on Upper Bounds of Ruin Probabilities. Journal of Mathematical Finance, 7, 958-974. https://doi.org/10.4236/jmf.2017.74053
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