Measuring Durability of Insurance Company Based on RuinProbability

Sukanya SompromDepartment of Mathematics

Khon Kean UniversityKhon Kean 40002

Thailandemail:sukanya.so@srru.ac.th

Watcharin KlongdeeDepartment of Mathematics

Khon Kean UniversityKhon Kean 40002

Thailandemail:kwatch@kku.ac.th

Abstract: In this paper, we present the process of the measuring durability of insurance company, in which, thisstudy focus on the discrete-time under the limited time the company must reserve sufficient initial capital to ensurethat probability of ruin does not exceed the given quantity of risk. Therefore the illustration of the minimum initialcapital under the specified period for the claim size process to the exponential distribution has explained.

Key–Words: Measuring Durability, minimum initial capital, probability of ruin.

Received: January 14, 2020. Revised: May 12, 2020. Accepted: May 28, 2020. Published: May 29, 2020.

1 IntroductionAccording to the economic and world situation, busi-ness companies have been fluctuating. Consequently,business companies will face loss at a high risk if theydo not pay attention on it. Currently, businesses arenow considering insurance a help which tends to be avery popular investment for business company own-ers. On the other hand, in the insurance business, theinitial capital, claim severities and premium rate con-trolled the fluctuation of surplus process:

Surplus = Initial surplus + Inflow - Outflow.

In this paper, we denote the probability space(Ω, F, P ). Let Un be the surplus process for the claimhappen at time n ∈ N. Consequently, the surplus pro-cess at time n as the following expression,

Un = u+ c0n−n∑

k=1

Yk ; U0 = u, (1)

where u ≥ 0 the initial capital, c0 ≥ 0 is the pre-mium rate for one time unit and the claim size processYn;n ∈ N is the set of independent and identicallydistributed (i.i.d) random variables. The probabilityof ruin of one unit time is defined by

Φn(u) = P (Uk < 0 for some k = 1, 2, 3, ..|U0 = u)(2)

Thus, the probability of survival for the time n ∈ N isdefined by

φn(u) = P (Uk ≥ 0 for all n = 1, 2, 3, ..|U0 = u)(3)

Many researchers are interested to study the sur-plus process in model (1). The general approachfor studying the probability of ruin in the discrete-time surplus process is called Gerber-Shiu discountedpenalty function; For instances, Pavlova and Will-mot [3] , Li [4,5] and Dickson [6]. They studied theprobability of ruin as a function of the initial capitalu ≥ 0. Moreover, Chan and Zhang [1] consideredin a case of discrete time surplus process obtainingthe closed form for the probability of ruin with ex-ponential claim distribution. Sattayatham et al.[2] in-troduced the closed form of the probability of ruin,and studied the minimum initial capital problem con-trolled the probability of ruin so that it was not greaterthan a given quantity of risk. Klongdee et al. [7] stud-ied the model (1) for the minimum initial capital underα-regulation of the discrete-time risk process in thecase of motor insurance which separated two claimseverities and calculated the regression analysis. Inthis paper, we focus on the discrete-time under thelimited time of the company which must reserve suf-ficient initial capital to ensure that the probability ofruin does not exceed the given quantity of risk by con-sidering the relationship of the ruin probability, safetyloading and initial capital.

2 Materials and MethodsIn this section, the premium rate c0 of the surplus pro-cess (1) can calculate by the expected value principleas the following expression,

c0 = (1 + θ)E[Y ],

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS DOI: 10.37394/23209.2020.17.11 Sukanya Somprom, Watcharin Klongdee

E-ISSN: 2224-3402 97 Volume 17, 2020

where θ > 0 is the safety loadings of the insurer. Thesurvival duration is defined by

D = minn|Un ≥ 0 and Un+1 < 0,

for n ∈ N, and set P (D = n) is the probabilityof survival duration at the time n. Next, we inves-tigate the probability of the survival duration greaterthan the specified period L at the time n, denoted byP (D > L). After that, we study the discrete-timesurplus process under the regulation that the insurancecompany has to reserve sufficient initial capital of L toensure that the probability of ruin does not exceed thequantity α. Consequently, the minimum initial capitalcorresponding to (u, c0, L, α, Yn;n ∈ N) and canbe calculated as the following expression,

MIDα(u, c0, L) = minu|P (D > L) ≥ α. (4)

In this paper, we consider the appropriated andedited the data of the claim severities of motor insur-ance in Thailand provided by an insurance company.We assume that the claim size has exponential distri-bution and weibull distribution. The claim sizes dataconsist of 103 days as shown in Fig.1.

Figure 1: The claim sizes data consist of 103 days.

Next, we introduce the probability density func-tion and the cumulative distribution function of thesedistributions which are described as shown below

The exponential distributionA random variable Y is said to have exponential

distribution has a positive scale parameter λ, denotedby Y ∼ Exp(λ), if its probability density function is

fX(x;λ) =1

λe−

xλ , x ≥ 0.

The cumulative distribution function is of the form

FX(x;λ) = 1− e−xλ , x ≥ 0.

The weibull distributionA random variable Y is said to have weibull dis-

tribution has a positive real scale parameter λ and apositive real shape parameter α, if its probability den-sity function is

fX(x;α;λ) =αxα−1

λαe−( x

λ)α , x ≥ 0.

The cumulative distribution function is of the form

FX(x;α;λ) = 1− e−( xλ)α , x ≥ 0.

Maximum Likelihood EstimatorLet X be a population random variable with a pa-

rameter θ. Assuming X is continuous, the densityfunction as fX(x; θ) in order to emphasize the depen-dency onθ as well as X . The likelihood function isgiven by

L(x; θ) = L(θ;x1, x2, ..., xn) =

n∏i=1

fX(x; θ). (5)

Taking logarithm in (5), lnL(x; θ), is called log-likelihood function that solved by taking derivative oflnL(x; θ) with respect to the parameter θ, and settingit equal to zero in order to solve for θ. The value of θthat obtained from maximizing the function lnL(x; θ)is exactly that the same value of θ, is called maximumlikelihood estimator (MLE).

Next, we approximate the minimum initial capitalunder the specified period L based on the probabilityof ruin as mentioned in (4). Firstly, we shall approx-imate its parameter by using the maximum likelihoodestimation (MLE). We obtain an appropriate value ofthe parameter of the dataset as

Table 1: Parameters of distributions of experiment

Distributions ParametersScale Shape

Exponential 6.805 ×105 -Weibull 1.5642× 105 1.0084

For the second experiment, we perform thesimulation for computing the minimum initial capitalunder the specified period L as mentioned in (4).The simulation method can proceed as the followingflowchart in Fig. 2-3

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS DOI: 10.37394/23209.2020.17.11 Sukanya Somprom, Watcharin Klongdee

E-ISSN: 2224-3402 98 Volume 17, 2020

Figure 2: The flowchart of algorithm of simmulation

Figure 3: The flowchart of algorithm of simmulation(cont.)

3 Simulation Results

We recall the discrete-time surplus process Un at timen as the following expression

Un = u+ c0n−n∑

k=1

Yk ; U0 = u,

where u ≥ 0 is initial surplus, c0 is the premium rateand Yn;n ∈ N is the independent and identicallydistribution claim size random variables. Thus, therecursive formula of the probability of ruin of the dis-crete times surplus process is the form

Φn(u) =n∑

k=1

(α(u+ kc))k−1e−α(u+kc)

(k − 1)!· u+ c

u+ nc

(6)

for all n = 1, 2, 3, ...[1].We consider the finite ruin probability of the sur-

plus process Un;n ∈ N which is driven by i.i.d.claim size process Yn;n ∈ N and the premium ratec0 > 0. Let u ≥ 0 be an initial capital. For eachn ∈ N, we let

D = minn|Un ≥ 0 and Un+1 < 0, (7)

denote the survival duration and P (D) is the proba-bility of survival duration. Then, we have

Theorem Let n ∈ N, c0 > 0 and u ≥ 0 begiven. If Yn;n ∈ N are i.i.d. claim sizes process,

P (D) = φn(u)− φn+1(u).

Proof We set Sn = ω : Un(ω) ≥ 0 and Sn = ω :Un(ω) < 0. Consider,

P

(n∩

k=1

Sk

)= P

(n∩

k=1

Sk ∩

n+1∩k=1

Sk ∪Rn+1

)

= P

(n+1∩k=1

Sk ∪

n∩

k=1

Sk ∩Rn+1

)

= P

(n+1∩k=1

Sk

)+ P

(n∩

k=1

Sk ∩Rn+1

)Then, we have P (D) = φn(u)− φn+1(u).

Moreover, the case of probability of ruin at timen, we have

P (D) = Φn+1(u)− Φn(u).

Consequently, the closed from of the probability ofruin for the discrete time surplus process in the caseof exponential distribution is of the form

P (Dn) =n∑

k=1

(α(u+ kc))k−1e−α(u+kc)

(k − 1)!· u+ c

u+ nc

(8)

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS DOI: 10.37394/23209.2020.17.11 Sukanya Somprom, Watcharin Klongdee

E-ISSN: 2224-3402 99 Volume 17, 2020

for all n.Next, we investigate the probability of the sur-

vival duration greater than the specified period L atthe time n, denoted by P (D > L). Next, we studythe discrete-time surplus process under the regula-tion that the insurance company has to reserve suffi-cient initial capital of L to ensure that the probabil-ity of ruin does not exceed the quantity α. Conse-quently, the minimum initial capital corresponding to(u, c0, L, α, Yn;n ∈ N) and can be calculated asmentioned (4).

The simulation results for MIDα(u, c0, L) of ex-ponential distribution are achieved with 1000 pathsand set α = 0.1, α = 0.05 and θ = 0.1, 0.2, ..., 0.9 asshown in the following Tables 2 and Fig.4.

Table 2: MIDα(u, c0, L) of exponential distribution

sefety MIDα(u, c0, L)loading(θ) α = 0.1 α = 0.05

0.1 151969 1935570.2 74471 959100.3 52000 699900.4 35108 522130.5 30023 419290.6 23522 360330.7 18761 302880.8 14196 250510.9 11259 21984

Table 2 shows simulation of MIDα(u, c0, L) forexponential distribution in case of α = 0.1 and α =0.05 which has the details as follows: the first col-umn shows the safety loading θ from 0.1 to 0.9, col-umn 2-3 shows the value of minimum initial capitalunder the specified period L = 1000. The illustrate ofthe numerical results as shown in Table 2 of column 2can describe that under the regulation of the insurancecompany, L = 1000, θ = 0.4. The insurance com-pany must reserve the minimum initial capital equals35108 to ensure that probability of ruin does not ex-ceed the given quantity α = 0.1.

Figure 4: The simulation of MIDα(u, c0, L) for expo-nential distribution.

Fig.4 shows the comparing of the experimentalresults between α = 0.1 and α = 0.05 for thesafety loading and the minimum initial capital underthe specified period L = 1000 and we choose param-eter of the exponential distribution.

The simulation results for MIDα(u, c0, L) ofweibull distribution are achieved with 1000 paths andset α = 0.1, α = 0.05 and θ = 0.2, 0.25, 0.3, ..., 0.9as shown in the following Tables 3 and Fig.5.

Table 3: MIDα(u, c0, L) of weibull distribution

sefety MIDα(u, c0, L)loading(θ) α = 0.1 α = 0.05

0.2 721579 9373650.25 572315 7848090.3 481011 635944

0.35 398394 4967100.4 343263 489305

0.45 288246 4064230.5 253247 374708

0.55 210309 3476540.6 198513 310613

0.65 165323 2912160.7 150659 251883

0.75 124120 2363640.8 107203 214019

0.85 89370 2140190.9 86319 177166

Table 3 shows simulation of MIDα(u, c0, L) forweibull distribution in case of α = 0.1 and α = 0.05which has the details as follows: the first column

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS DOI: 10.37394/23209.2020.17.11 Sukanya Somprom, Watcharin Klongdee

E-ISSN: 2224-3402 100 Volume 17, 2020

shows the safety loading θ from 0.2 to 0.9, column2-3 shows the value of minimum initial capital underthe specified period L = 1000.

Figure 5: The simulation of MIDα(u, c0, L) forweibull distribution.

The illustrate of the numerical results as shown inTable 3 of column 3 can describe that under the regu-lation of the insurance company, L = 1000, θ = 0.25.The insurance company must reserve the minimuminitial capital equals 784809 to ensure that probabilityof ruin does not exceed the given quantity α = 0.05.

4 ConclusionThe mentioned results can be suggest the insurancecompany for planning to reserve the minimum initialcapital under the limited regulation to ensure that theprobability of ruin does not exceed the given quantityof risk.

References:

[1] W.S. Chan and L. Zhang,”Direct derivation offinite-time ruin probabilities in the discrete riskmodel with exponential or geometric claims”,North American Actuarial Journal, vol.10, no.4,pp. 269-278, 2006.

[2] P. Sattayatham, K. Sangaroon and W. Klongdee,”Ruin Probability Base Initial Capital of the dis-crete Time Surplus Process”, Variance 3, vol.7,no.1, pp.74-81, 2013.

[3] K.P. Pavlova and G.E. Willmot, ”The Dis-crete Stationnary Renewal Risk Model and theGerber-Shiu Discounted Penalty Function”, In-surance: Mathematics and Economics, vol.35,no.2, pp.267-277, 2004.

[4] S. Li, ”On a class of discrete Time Renewal RiskModels”, Scandinavian Actuarial Journal, vol.4,pp.271-284, 2005a.

[5] S. Li, ”Distributions of the Surplus before Ruin,the Deficit at Ruin and the claim Causing Ruinin a class of Discrete Time Risk”, ScandinavianActuarial Journal, vol.2, pp.271-284, 2005b.

[6] D. C. M. Dickson, Insurance Risk and Ruin,New York: Cambridge University Press, 2005.

[7] W. Klongdee, P. Sattayatham and S. Boonta,”On approximating the ruin probability the min-imum initial capital of the finite-time risk pro-cess by separated claim technique of motorinsurance”, Far East Journal of MathematicalSciences (FJMS), (Spacial Volume 2013) (IV)pp.597-604, 2013.

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS DOI: 10.37394/23209.2020.17.11 Sukanya Somprom, Watcharin Klongdee

E-ISSN: 2224-3402 101 Volume 17, 2020