Ruin Probabilities in the Mixed Claim Frequency Risk Models

Abstract

We consider two mixed claim frequency risk models. Some important probabilistic properties are obtained by probability-theory methods. Some important results about ruin probabilities are obtained by martingale approach.

1. Introduction

Mixture models are a fundamental tool in applied statistics, for most mixture models, including the widely used mixtures of Gaussians and hidden Markov models (HMMs); the current practice relies on the Expectation-Maximization (EM) algorithm, a local search heuristic for maximum likelihood estimation; an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components was developed [1–3]. Finite mixture models have a long history in statistics; a detailed review of mixture models and models-based clustering was provided in [4], for a finite mixture of regressions model, [5, 6] develop an efficient EM algorithm for numerical optimization with provable convergence properties.

In this paper we consider two mixture models. In Section 2 we set up the binomial-Poisson model, and some important results about ruin probabilities are obtained by martingale approach.

In Section 3 we also set up another Poisson-dualistic model; in this section we obtain some important probabilistic properties and estimates for ruin probability.

In the classical risk model, the surplus of an insurance company over the interval is , which is defined by where is to be interpreted as the number of claims on the company during the interval ; assume that is an homogeneous Poisson process with intensity . In the complex problems of actual operation in insurance business, insurance company classifies the risk by its some characteristics, but the claim frequency of the individual policy which has been classified into the same kind of portfolio may be different; that is, this is the so-called nonhomogeneity. For a nonhomogeneous portfolio, we can assume that is a random variable; thus the mixed Poisson distribution model can be derived.

In general, just as reported in [6] if the number for claim is a discrete distribution with parameter and the distribution sequence is and the parameter is random variable or random vector, its probability distribution function is Then the corresponding risk model is a mixed claim frequency risk model.

2. Binomial-Poisson Model

2.1. The Setting-Up of the Model

Definition 1. Suppose is an homogeneous Poisson process with intensity . Consider ; then the conditional distribution of is a binomial distribution with parameters , . A sequence is an independent and identically distributed nonnegative random variable, having the common distribution function , with , mean value , and variance ; the above random process and random sequence are mutually independent; are both constants.
Let Then the process defined by (4) is a binomial-Poisson mixed claim frequency risk model.
The conditional moment generating function of is Therefore Then is a homogeneous Poisson process with intensity . Thus the model defined by (4) is a classical risk model.

2.2. The Meaning of the Model in the Insurance Practice

Suppose that is the number of accidents during the interval ; the number of claims per accident is distribution with ; that is, , (for example, in the deductible insurance, the probability of loss exceeding the amount of deductible is ); then the number of claims during the interval is a conditional binomial distribution with parameters . Obviously the process is a -sparse process of the process . is a homogeneous Poisson process with intensity , if only is a homogeneous Poisson process with intensity . is the premium rate, is the size of claim amount per accident, and is the total claim process.

2.3. Several Conclusions about the Ultimate Ruin Probability

We can get easily In order to stabilize the operation of company, we should ensure that premiums received in a unit of time meet .

The relative safety loading is defined by

We can now define the ruin probability of a company facing the risk process (4) and having initial capital . Consider

Let

We assume that there exists such that when , that is, a light-tailed distribution .

Theorem 2. If , .

Theorem 3. Let amount claimed sequence with exponential distribution with mean ; then where is given by (8).

Theorem 4. Consider where is the positive solution of , which is called adjustment coefficient. And is given by (8).

Theorem 5. Ruin probability: where is the positive solution of , which is called adjustment coefficient.

The above theorem can be derived directly by the corresponding results in classical risk theory [7].

3. Poisson-Dualistic Model

3.1. The Setting-Up of the Model

Definition 6. The random variable follows two-point distribution; that is, , , . The conditional distribution of about is a homogeneous Poisson process with intensity . A sequence is independent and identically distributed nonnegative random variable, having the common distribution function , with , mean value , and variance ; and are mutually independent; are both positive real constants.
Let Then the random process defined by (14) is a Poisson-dualistic mixed claim frequency risk model.
Obviously, is a mixed Poisson process.

3.2. The Meaning of the Model in the Insurance Practice

We assume that a portfolio is composed of high risk and low risk insurance policy, where high risk policy is accounting for and low risk policy is accounting for ; the Poisson parameter of these two kinds of policy, respectively, is , (corresponding high risk cover for , obviously ). is the premium rate, is the size of claim amount per accident, and is the total claim process.

3.3. Some Probabilistic Properties of Model

The probabilistic properties of the number of claims are as follows.

Property 1. Claim frequency distribution is

Proof. The above distribution can be derived by The proof is ended.

Property 2. The mean and variance of are

Proof. Consider The proof is ended.

Since the mean and variance of the Poisson distribution are always equal, if the sample variance of the portfolio’s claim of a random variable is greater than the number of the sample mean, we can conclude that the existence of this policy combination of a degree of nonhomogeneity, and because can reflect the variance of the degree of nonhomogeneity of portfolio, if the variance is much more greater than the mean, that is, the bigger is, the more serious the nonhomogeneity is.

Property 3. The moment generating function of is

Proof. Consider The proof is ended.

Theorem 7. Assume that the random variable follows two-point distribution; that is, , , and . The conditional distribution of about is a homogeneous Poisson process with intensity . Then the process has stationary increments.

Proof. Consider The proof is ended.

Theorem 8. Assume that the random variable follows two-point distribution; that is, , , . The conditional distribution of about is a homogeneous Poisson process with intensity if and only if the interval of about has conditional independence and is the same as the exponential distribution with parameter ; thus follow the following mixed exponential distribution:

Proof. We only need to prove that when the conditional distribution of about is exponentially distributed, follow the mixed exponential distribution. Then, when , the distribution function of is Thus the density function of is The proof is ended.

Theorem 9. It occur claims over ; then the occurrence probability of intensity for claim is

Proof. It can be derived by The proof is ended.

Thus we can get

Theorem 10. Assume that is the time interval from the moment t to the next claim; then the conditional distribution of is

Proof. Obviously is the occurrence time after the moment , when is a constant; then follows the exponential distribution with . Then the result can be derived by The proof is ended.

The probabilistic properties of total amount of claims are as follows.

Property 4. The mean and variance of total amount of claims are

Proof. Consider The proof is ended.

Property 5. The moment generating function of total amount of claims is where is the moment generation function of the individual claim amount.

Proof. We can get the following from (19): The proof is ended.

3.4. Estimation of Lundberg Exponential Upper Bounds for the Ultimate Ruin Probability

Obviously, .

It seems very natural to assume the premium rate per unit of time . Further, let , in order to stabilize the operation of the company.

The relative safety loading: Obviously,

can be seen as the corresponding relative safety loading of portfolio consisted by high risk policy. Obviously in terms of premium rate and the average individual claim amount are equal, the relative safety loading of the corresponding portfolio of model (14) should be greater.

Let be the initial capital; then the ruin moment is defined as Obviously is a -stopping time.

The finite time ruin probability is The ultimate ruin probability is Let We assume that there exists , such that when . It is easy to be seen that , , and and that is continuous on (where ).

Easy to verify, the risk process defined by (14) is a right continuous process and has the following properties:(i) ;(ii) has stationary and independent increments about ;(iii).

Let Note that .

We can easily get From the hypothesis of function , we know , s.t. .

From property (ii) of , we have Thus

Theorem 11. Let Thus is an -martingale.

Proof. We get from (43) that The proof is ended.

Choose and consider which is a bounded -stopping time. We get from Doob’s stopping theorem that Using on , the lower bound was shown to be given by Thus we have and, by taking expectation, When in the above equation, we get

We now want to choose as large as possible under the restriction . Let denote that value, named adjustment coefficient of surplus process (14).

Since where , , the two terms of the right side in the above equality are both positive, which corresponds to restrict Since thus we just need to restrict Then we have

Theorem 12. The ultimate ruin probability meets the inequality where is the only positive solution of , named adjustment coefficient.

Proof. For , we have Therefore Then there exists small enough such that . Further when and .
When , since , where is the th moment of the claim amount , then Thus there must exist such that . And from , we know is a lower convex function.
Then for and for .
Therefore there exists the only positive solution of ; let denote that value. From (55) we know , so we get .
The proof is ended.

3.5. Boundary of the Adjustment Coefficient

The exact value of adjustment coefficient generally cannot be determined by , but because of its great significance to estimate the upper bound of the ruin probability, we estimate the boundary of the adjustment coefficient .

Theorem 13. The adjustment coefficient meets the inequality

Proof. From [7] we know ; then we can get the conclusion.
The proof is ended.

Form the above theorem and the expression of the relative safety loadings , , we get

The result shows that the upper bound of can be only defined by the one- or two-order moment of individual claim amount and the relative safety loading; while having nothing to do with the Poisson parameter, this case have its convenience in use.

Theorem 14 (see [7]). If the individual claim amount has the upper bound , that is, , we have

The result indicates that, under the conditions of the theorem, the lower bound of can be only defined by the relative safety loading and the upper bound of individual claim amount.

When the exact value of cannot be determined by the equation , we can solve its approximate solution by the numerical method, and using these two boundaries as the initial values for iteration, we can quickly find the approximate solution satisfying requirements of certain accuracy.

3.6. The Probability of Survival

Except for a few special cases, generally, complicated calculation is needed to get the adjustment coefficient values. Through the following discussion on integrodifferential equations satisfied by survival probability, we can avoid the computation of the adjustment coefficient.

Let ; then indicates survival probability.

Since the number of claim process about is a conditional renewal process and will not be ruined during the period of , then we get

Let Then we have Let two terms of the hand side in the above equality be, respectively, and ; then .

Theorem 15. The integrodifferential equations assured by and are

Proof. From (65) and the hypothesis of and , by taking derivation, we can get the above result.
The proof is ended.

and are calculated using the above equations; thus the expression of can be obtained. According to the above method to obtain the survival probability, we can immediately get the expression of ruin probability, and then the calculation of the adjustment coefficient can be avoided.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 11101053, the China Postdoctoral Science Foundation under Grant no. 20140550097, the Key Project of Chinese Ministry of Education under Grant no. 211118, the Hunan Provincial NSF under Grant no. 11JJ1001, and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grant no. 2013SK3143.