Journal of Applied Mathematics

Volume 2019, Article ID 2191509, 9 pages

https://doi.org/10.1155/2019/2191509

## An Approximation of Minimum Initial Capital of Investment Discrete Time Surplus Process with Weibull Distribution in a Reinsurance Company

^{1}Department of General Science, Faculty of Science and Engineering, Kasetsart University, Chalermphrakiat Sakon Nakhon Province Campus, Sakon Nakhon 47000, Thailand^{2}Mathematics and Statistics Program, Faculty of Science, Sakon Nakhon Rajabhat University, Sakon Nakhon 47000, Thailand

Correspondence should be addressed to Soontorn Boonta; ht.uk@ob.nrotnoos

Received 31 January 2019; Accepted 23 May 2019; Published 11 June 2019

Academic Editor: Ying Hu

Copyright © 2019 Soontorn Boonta and Somchit Boonthiem. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Catastrophe is a loss that has a low probability of occurring but can lead to high-cost claims. This paper uses the data of fire accidents from a reinsurance company in Thailand for an experiment. Our study is in two parts. First, we approximate the parameters of a Weibull distribution. We compare the parameter estimation using a direct search method with other frequently used methods, such as the least squares method, the maximum likelihood estimation, and the method of moments. The results show that the direct search method approximates the parameters more precisely than other frequently used methods (to four-digit accuracy). Second, we approximate the minimum initial capital (MIC) a reinsurance company has to hold under a given ruin probability (insolvency probability) by using parameters from the first part. Finally, we show MIC with varying the premium rate.

#### 1. Introduction

The risks of an insurance company can be assessed based on disasters of varying severity. The insurance company evaluates its risks in order to maintain consistency. Insolvency cannot occur if the company knows how to manage the risk process. For instance, if an insurance company does not have sufficient initial capital to pay some claims, then the company can share some of the risks by transferring them to reinsurance. Parameter estimation is an important method to construct a risk model in an insurance business. The well-known methods are the least squares method, the maximum likelihood estimation, and the method of moments. This research is interested in the estimation of the parameters of a Weibull distribution which are represented by fire accident data. Many authors have studied the different aspects of Weibull parameters. Bergman [1] and Sullivan and Lauzon [2] proposed four probability estimators which were frequently applied in the least squares method. Based on a Monte Carlo simulation, Khalili and Kromp [3] and Trustrum and Jayatilaka [4] compared an estimation of Weibull parameters by using the least squares method, the maximum likelihood estimation, and the method of moments. Boonta et al. [5] proposed a direct search technique to estimate the parameters of Weibull distribution. They compared the Chi-squared value of the direct search technique to the least squares method, the maximum likelihood estimation, and the method of moments. The results showed that the direct search technique gave a more precise estimation than the least squares method, the maximum likelihood estimation, and the method of moments.

In this paper, we start by introducing the surplus of nonlife insurance. The surplus can be described as

Lundberg [6] was the first actuary who considered the surplus process of nonlife insurance under three assumptions in his model.

(1) Claims happening at times satisfying are called* claim arrivals* or* claim times*.

(2) The -th claim arriving at time causes the* claim size* or* claim severity *. The sequence of claim sizes constitutes an independent and identically distributed (i.i.d.) sequence of nonnegative random variables.

(3) The claim severity process and the claim arrival process are mutually independent.

Next, we define the* claim number process*Thus, is the number of claims in .

Next, we denote as the i.i.d. process of the claims, as the initial capital, and as the premium rate for one unit of time. There are many research studies of ruin probability in terms of initial capital (see in Pavlova and Willmot [7], Dickson [8] Li [9, 10], and Rongming and Haifeng [11]). Chan and Zhang [12] have studied a discrete time surplus process such as claim time (claims which occur every day). They proposed recursive and explicit formulae of the ruin probabilities which are in the form

Sattayatham et al. [13] generalized the results of Chan and Zhang for claim times which do not occur every day. Their model is of the form

This can be rewritten aswhere is the inter arrival time process, assuming i.i.d. such that Since the formula of ruin probability is difficult to find explicitly, they proposed the ruin probability in the recursive form:where . Chengguo Weng et al. [14] studied a model of ruin probability by adding some investments as where is the constant interest rate for a period of time . They considered as a sequence of dependent individuals that have a regular variation distribution and zero index of upper tail dependence. They also established some asymptotic results for both finite ruin probability and ultimate ruin probability.

Reinsurance and investment are a normal activity of insurance companies because reinsurance can reduce the risk (ruin probability) arising from claims, and the investment can make more profit for the company. The process can be controlled by reinsurance, i.e., by choosing the* retention level* (or risk exposure) of a reinsurance for one period. The (measurable) function specifies the part of the claim paid by the insurer. Then depends on the retention level (fixed in the risk sharing contract) at the beginning of the respective period where . Hence is the part paid by the reinsurer. It is natural to assume that is increasing in .

In the case of an* excess of loss reinsurance* we havewith retention level .

In case of a* proportional reinsurance* we havewith retention level 1.

Therefore, the retention level stands for the control action “no reinsurance” which explains the property “”. The smallest retention level may be chosen in such a way that the condition is satisfied. Then may be calculated according to the* expected value principle *with* safety loading * of the reinsurer:

Recently, Luesamai and Chongcharoen [15] expanded the risk model by adding proportional reinsurance and investment. Insurers can invest in the bond and stock markets, and they assume that the interest rates of the bonds have a finite number of possible values and follow a time-homogenous Markov chain. Moreover, they assume that the controlling reinsurance and stock investment values in each time period are constant values. For every time period unit , the risk model is formulated aswhere is computed by a proportional reinsurance function, and the sequences , and are mutually independent. The interest rate is assumed to follow a time-homogeneous Markov chain . is the gross return and is assumed to be a sequence of i.i.d. nonnegative random variables. is the amount of stock investment. The results of the study led to the proposal of two upper bounds of ruin probability under a discrete time risk model for reinsurance by generalizing the classical model in terms of two controlling factors: proportional reinsurance and investment.

In this research, we study a risk model of reinsurance by adding investment (buying bonds or fixed accounts). We present two parts consisting of an approximation of the parameters of Weibull distribution and the calculation of the minimum initial capital of investment discrete time surplus process with Weibull distribution. The first part is the estimation of the Weibull parameters using the direct search technique, the least squares method, the maximum likelihood estimation, and the method of moments. The selected method gives the minimum KS statistic value (to four-digit accuracy). The second part is a simulation to calculate the ruin probability of the surplus process under the condition that a reinsurance company can invest in risk-free assets (bonds or fixed accounts). The surplus process is of the formwhere is the i.i.d. process of the claims, is the initial capital, and is the daily interest rate which can calculated by per annum. is the premium rate for one unit of time which can be computed by where is a safety loading and is the inter arrival time process, and assuming i.i.d. such that . We approximate the minimum initial capital (MIC) an insurance company has to hold under a given ruin probability (insolvency probability) by using parameters from the first part.

#### 2. Materials and Methods

##### 2.1. Weibull Distribution

The classical Weibull distribution is useful for reliability engineering. Moreover, it can be extended to the various families of probability distributions which deal with the estimation of model parameters by maximum likelihood and it can also be used to illustrate the potentiality of the extended family with two applications to real data [16]. Furthermore, a nonclassical Weibull distribution can be used to estimate the statistical characteristics in a cellular automaton such that a cell’s yield stress is assumed to be a Weibull distribution [17].

Normally, claims that occur infrequently but have high costs will be called catastrophe losses. For example, a fire accident is a type catastrophe loss. Furthermore, a Weibull distribution that shape parameter being less than one and scale parameter being greater than zero is also an example of catastrophe loss. The probability density function of three parameters of a Weibull distribution is of the formand the cumulative distribution function is of the formfor all , where is a positive shape parameter, is a positive scale parameter, and is a positive location parameter, respectively.

In our work, the costs of all claims are greater than twenty million Baht. We establish and . Thus we revise (13) and (14) as (15) and (16)and

##### 2.2. Weibull Parameter Estimation

###### 2.2.1. Least Squares Method

Bergman, Sullivan, and Lauzon proposed the probability estimator for the th ranked and is the sample size as shown in the following equations:

We take the natural logarithm to (16),Then we set and . Therefore, (18) is the linear equation which is of the form .

###### 2.2.2. Maximum Likelihood Estimation

Let be the samples from a Weibull distribution. A log-likelihood function is defined by

By partial derivative the log-likelihood function , , and setting to zero, we have

Thus

###### 2.2.3. Method of Moments

The* k* th moment of the Weibull distribution, , is defined bywhere is a gamma function which is given by

Setting , we obtain

Since ,

We calculate a coefficient of variation (CV) of the Weibull distribution from the formula

If we apply the bisection method to (27), we get the parameter . Thereafter, we substitute in (25) which obtains the parameter .

###### 2.2.4. Kolmogorov-Simirnov Test

The Kolmogorov-Simirnov (KS) test is a distance test. The KS test works well with small samples. Let be the order of statistics of a random sample . The empirical distribution function is of the form

Let be a hypothesized distribution function, so the KS statistic is defined by .

Throughout this research, we use the data of the cost of all claim sizes that are greater than twenty million Baht from the fire insurance of Thai Reinsurance Public. The amount is shown in Table 1 which is greater than twenty million Baht, i.e., . Boonta et al. [5] estimated the shape parameters and scale parameters from a variety of estimation methods by means of a minimum Chi-squared test as shown in Table 2 and using the data according to Table 1.