Upper Bound of Ruin Probability for an Insurance Discrete-Time Risk Model with Proportional Reinsurance and Investment

Two upper bounds for ruin probability under the discrete time risk model for insurance controlled by two factors: proportional reinsurance and surplus investment are presented. The latter is of interest because of the assumption that insurers invest some or their entire financial surplus on both the stock and bond markets, for which bond interest rates follow a time – homogeneous Markov chain. In addition, the control of reinsurance and stock investment in each time period are assumed to be constant values. The first upper bound for finite time ruin probability and ultimate ruin probability was derived under the condition that the Lundberg coefficient exists. The second upper bound is for finite time ruin probability and was developed from a new worse than used function. Numerical examples are used to illustrate these results, and the upper bound of ruin probability using real-life motor insurance claims data from a broker is also presented.


Introduction
There is increasing attention on ruin probability for insurance discrete time risk models with reinsurance and investment of financial surplus over the last decade due to the fact that insurance companies can purchase reinsurance, invest in the stock market, and receive dividends, among other transactions.However, obtaining an explicit solution of a company's ruin probability is actually a difficult task.One alternative method commonly used in ruin theory is deriving bounds for the ruin probabilities (Diasparra andRomera, 2009, andLin et al., 2015) thus the focus in this paper is on upper bounds for ruin probability.
Lundberg's inequality provides a well-known upper bound for the probability of ultimate ruin in the classical risk model when the moment generating function of the claim size random variable exists.However, in many practical distributions, the moment generating function does not exist, so the Lundberg inequality is not available in these cases (Cai andWu, 1997, andCai andGarrido, 1999).Thus, there are many researches in which the upper bound of ruin probability has been derived (see, for example, Dickson (1994), Willmot (1994), Kalashnikov (1999), and other researchers) and can be applied to more general claim size distributions.
In this paper, two upper bounds of ruin probability for a discrete time risk model controlled by reinsurance and investment are presented.The former was derived under the condition that the moment generating function of a claim size exists (the Lundberg coefficient exists).This upper bound can be viewed as an extension of the results from the studying of Diasparra and Romera (2009), and Jasiulewicz and Kordecki (2015) by adding investment to a risk model.The latter was developed from the idea of Willmot (1994) by providing the upper bound in terms of the new worse than used (NWU) function.This upper bound can be applied for more general claim size distributions.

Model description
The typical discrete time risk model for insurance can be written as where n U denotes the insurer's surplus at the end of time period n with initial constant 0 U u = , n X being the total premiums amount during time period n (i.e. from time 1 n − to n ), and n Y is the total claims amount during n .We assume that this sequence consists of independent and identically distributed (i.i.d.) random variables with a common distribution function ( ) ( ) n P y Pr Y y = ; 0 y  .In this study, the above risk model is expanded upon by adding proportional reinsurance and investment.
Under proportional reinsurance contracts, the reinsurer agrees to cover a fraction of each claim equal to the fraction of premiums that it receives from the insurer.Throughout this study, (  .Let  be the safety loading factor added by the reinsurer and re c be the premium constant for the reinsurer.Thus, by the expected value principle, the constant premium for reinsurer in a unit period is given by Next, the constant premium which is retained by the insurer in a unit period denoted by ( ) n cb , when ( ) For the effect of an investment on a risk model, we assume that the insurer can invest in two assets.One is a bond with a known interest rate at the initial time ( 0 I ); the interest rate at time n ( ) ), and we assume that n dd = for all n throughout this dissertation.In addition, n I is assumed to follow a time-homogeneous Markov chain, i.e. both the transition probabilities and the time are independent, and are denoted by with the assumption that the sequences   From the risk model in Equation ( 6), if we replace the values of n as 12 n , ,..,m = , then the output from this action is another form of the former model as follows: ( ) ( ) ( )  The ruin probability for finite time is given by and the ultimate ruin probability is also given by Consider that the ruin probabilities are the cumulative probability from Equations ( 8) and (9), then Subsequently, the famous Lundberg inequality for the ultimate ruin probability, ( ) for Equation (12) states that if ( ) ) 2) If we omit the investment factor (  6), the model is reduced to ( ) ( ) Again, the ultimate ruin probability, ( )

Recursive and Integral Equations form for Ruin Probability
The recursive form of ruin probability for finite and ultimate time under discrete time risk model for insurance as in Equation ( 6), in which ruin probabilities are defined as in Equations ( 8) and ( 9), are derived as follows.
Lemma 1: The recursive form of ruin probability for finite time and the integral equation of the ultimate ruin probability under the discrete time insurance risk model as in Equation ( 6) are given as ( ) ( ) where ( ) ( ) ( ) ( ) Proof: = .Thus, ( ) ( ) ( ) where However, if ( ) implying that for ( ) Consider ( ) 8) is as follows: where From Equation (21), consider that ruin will occur in the first period if ( ) is defined at the beginning of the proof, we now define ( ) ( ) for the short term.In order to use Equations ( 22) and (23) to derive the recursive form, we need to rewrite ( ) , then we can rewrite 1 ( , ) By using Equation ( 11) and the Lebesgue dominated convergence theorem, the result of taking n → in Equation (25) becomes 1 ( , ) lim ( , ) Furthermore, following on from Equation (18), we obtain

Upper bounds for ruin probability
In this section, two upper bounds for ruin probability are derived.The first is the case when the Lundberg coefficient exists, and the second is based on the NWU function.( ) Proof:  ( ) From (18) in Lemma 1, we have ( ) ( ) (from Equation( 18)) (29) By using the inductive method, we obtain From Equation (19) in Lemma 1, we obtain   ( )

The upper bound for finite time ruin probability based on the NWU function.
The upper bound of ruin probability in Theorem 1 is derived using the condition that "the Lundberg coefficient, 0 R exists, which satisfies Equation (16)."However, this condition is not true for some distributions of n Y claims, especially in heavy-tailed distributions such as Pareto and Weibull, among others, because 0 R cannot be found due to the fact that the moment generating function is not present in these distributions.Hence, the next theorem for the upper bound of ruin probability is derived based on another term of 0 R , the NWU function.However, our derivation of this upper bound only restricts the results where n Y is a compound distribution in order to use the outcome of Willmot (1994) to support this procedure.Therefore, the additional assumptions for the next theorem are as follows.
Let ( ) Bx be the distribution of a non-negative random variable and ( ) ( )  From the afore mentioned additional assumptions, Willmot (1994) shows us that if the non-negative, non-increasing function ( ) and in addition, then the upper bound for ( ) ( ) , where ( ) ( ) where 0 j is defined as in Equation (37).
Since the total claims is assumed and the fraction of the total claims paid by insurer when the company signed the reinsurance contract is ( ) where ( ) Similar to Equations (37) -(39), if the non-negative, non-increasing function ( ) and in addition, then the upper bound for ( ) ( ) , where The next theorem is derived from the above information.44) and (45), then the upper bound for the finite time ruin probability in Equation ( 19) can be written as
By using the inductive method, we arrive at ) From Equation (19) in Lemma 1, we obtain ( ) ( ) By considering ( ) ( ) And by replacing Equation (51) in Equation (50), we can achieve

Numerical example
The upper bound of ruin probability in Theorems 1 and 2 are illustrated in Examples 1 and 2, respectively, using R programming.u is sufficient large and the time period n should not be set too many.In other cases, overrated values of upper bound might lead to a misunderstanding of the risk level that makes insurers more nervous about the risk than is really necessary.

Real-life example
Data from 334 real-life motor insurance claims for a broker with three branches in the year 2012 were used to analyze the upper bound for ruin probability.The real-life claims dataset were fitted to a lognormal distribution with maximum likelihood estimation of log data parameter 2 467120 . = (thousand baht) and 1 039443 . = (thousand baht).The moment generating function of the lognormal distribution was infinite at any positive number, thus the upper bound of ruin probability in Theorem 2 was appropriate in this situation.We used the dataset in December to find the upper bound of ruin probability for the next 5 months.This dataset fit to lognormal distribution with maximum likelihood estimation of log data parameter 2 385506 . = (thousand baht) and 1 089433 . = (thousand baht).The number of claims occurring in each month was assumed to be i.i.d. with a Poisson distribution for which the mean was estimated as the average value of 12 months of real-life claims data (the result was 27.75).The other factors for finding the upper bound of ruin probability for this broker were an initial surplus of 5 million baht and an initial bond interest rate at 0.03 (based on Example 1), and the NWU function used was ( ) ( ) Assuming the dealer kept 0.6 of the reinsurance retention level and invested 500,000 baht on the stock market under previous assumptions, then the upper bounds of ruin probability in the next 5 months were 0.670366, 0.670384, 0.670401, 0.670419, and 0.670436 respectively.High values of the upper bound for ruin probability indicate that there is a high risk (value of ruin probability) of the company going bust under these conditions.The amount and frequency of claims with respect to the initial capital seem to be the main cause of this situation.

Conclusions
In this study, we propose two upper bounds of ruin probability under a discrete time risk model for reinsurance by generalizing the classic model for two controlling factors: proportional reinsurance and investment.The insurer can invest in the bond and stock markets, and we assume that the interest rates of the bond have a finite number of possible values and follow a time-homogenous Markov chain.Moreover, we assume that the controlling reinsurance and stock investment values in each time period are constant values.
The ruin probability for finite time is presented in a recursive form while the ultimate ruin probability is given as integral equations.The first upper bound for finite time and ultimate ruin probability is derived under the condition that the Lundberg coefficient exists.This upper bound can be view as an extension of the ideas of Diasparra and Romera (2009) and Jasiulewicz and Kordecki (2015).The second upper bound for finite time ruin probability is developed from the idea of Willmot (1994) in terms of NWU.Numerical examples show the results for the two proposed upper bounds.In the first example, the total claims amount in each time period were assumed to follow an exponential distribution so that the Lundberg coefficient can be found in this case, thus Theorem 1 was applied.In the second example, the claims amount in each time period was assumed to be an i.i.d.Pareto distribution, under which circumstances the Lundberg coefficient does not exist, thus, Theorem 2 was applied in this case.
the retention level of a reinsurance contract for time period n .This means that the insurer pays nn bY of total claim amount this means that there is no reinsurance.Let ( ) nn h b ,Y denote the fraction of the total claim n Y paid by the insurer, is the case throughout this paper).In addition, the insurance premium during time period n, n X , is assumed to be a fixed constant c for all n.Subsequently, by the expected value principle with safety loading factor 0   , the premium constant is calculated as c = ( ) ( ) is called the gross return.Throughout this paper, n W is assumed to be a sequence of i.i.d.nonnegative random variables with the distribution function ( ) on the risk model in Equation (1), if at the beginning of th n period the insurer has the chance to decide the amount of stock investment n ...,n =− , and the retention level n b of a reinsurance contract, our risk model is finally formulated as

Remark 1 :
In the case where the value of k is greater than n, of ruin probability in the article of Cai and Dickson (2004) and Jasiulewicz and Kordecki (2015), and given the initial values 0 Uu = and 0s Ii = , the ruin probability for the insurance risk model can be written as follows.

1 )
When the insurance risk model in Equation (6) is neither reinsurance ( from the insurance risk model in Equation ( (15) with constant values of reinsurance in each time period, i.e. and Romera, 2009, p. 102).

4. 1 Theorem 1 :
The upper bound for ruin probability when the Lundberg coefficient exists If the Lundberg coefficient, 0 0 R  satisfies (16), suppose that the reinsurance and investment in stock in each time period are controlled to be constant values, i.e. upper bound of ruin probability for finite time and the ultimate ruin probability from Lemma1 is ( )

Theorem 2 :
Let the total claims satisfy Equations (33) and (39); suppose that the reinsurance and investment in stock in each time period are controlled to be constant values, i.e. suppose that the non-negative, non-increasing function ( ) 1 ( )D x D x =− for 0 x  exists, inwhich () Dx is NWU, ( ) 01 D = , and ( ) Dx satisfies Equations ( at the right-hand side of Equation (49) in theorem 2 affects the variation of the upper bound values for finite time ruin probability in this theorem as the time period n increases, the upper bound values increase.The values of the upper bound from Theorem 2 depend on not only the change of factors in the risk model mentioned in Example 2, but also the NWU function selected.Based on running the results from data in Example 2 and resembling data, we found that the upper bound from Theorem 2 is appropriate when the initial surplus 1, 2,3...
n kd 