Ruin Probabilities in a Risk Model with Two Types of Claims

Han, Ji-Yeon,Choi, Seung-Kyoung,Lee, Eui-Yong The Korean Statistical Society 2012 응용통계연구 Vol.25 No.5

A surplus process with two types of claims is considered, where Type I claims occur more frequently, however, their sizes are smaller stochastically than Type II claims. The ruin probabilities of the surplus caused by each type of claim are obtained by establishing integro-differential equations for the ruin probabilities. The formulas of the ruin probabilities contain an infinite sum and convolutions that make the formulas hard to be applicable in practice; subsequently, we obtain explicit formulas for the ruin probabilities when the sizes of both types of claims are exponentially distributed. Finally, we show through a numerical example, that Type II claims have more impact on the ruin probability of the surplus than Type I claims.

UNIFORM ASYMPTOTICS FOR THE FINITE-TIME RUIN PROBABILITY IN A GENERAL RISK MODEL WITH PAIRWISE QUASI-ASYMPTOTICALLY INDEPENDENT CLAIMS AND CONSTANT INTEREST FORCE

Gao, Qingwu,Yang, Yang Korean Mathematical Society 2013 대한수학회보 Vol.50 No.2

In the paper we study the finite-time ruin probability in a general risk model with constant interest force, in which the claim sizes are pairwise quasi-asymptotically independent and arrive according to an arbitrary counting process, and the premium process is a general stochastic process. For the case that the claim-size distribution belongs to the consistent variation class, we obtain an asymptotic formula for the finite-time ruin probability, which holds uniformly for all time horizons varying in a relevant infinite interval. The obtained result also includes an asymptotic formula for the infinite-time ruin probability.

자산배분 전략에 따른 자가연금화의 재원부족확률 분석

정도영 ( Doyoung Cheong ),이동화 ( Donghwa Lee ) 한국연금학회 2016 연금연구 Vol.6 No.2

본 연구는 은퇴시점에서 종신연금상품 구매 시의 연금액을 벤치마크로 가정하여 4가지 자산배분전략(샤프지수 극대화, 동일가중, 리스크 페러티, 재원부족확률 최소화)을 통한 자가연금화(self anutization)의 재원부족확률을 비교분석하였다. 각 자산배분 전략 별 재원부족확률을 살펴본 결과, 위험자산 투자비중이 높은 포트폴리오가 상대적으로 낮은 재원부족확률을 보였다. 연금화 시점 및 연금화 비중에 대한 의사 결정은 은퇴 이후에 동적으로 결정해야 하는 것이 보다 바람직한 전략이 되며, 연금 구입을 통한 연금화 전략과 자가연금화 전략을 혼합하여 추구하는 것이 은퇴재원 확보에 더욱 효율적인 대안으로 제시되었다. 은퇴자의 노후재원이 충분히 확보되기 위하여 자가연금화 실행시 장기적인 자산배분전략을 명확히 설정해야 하며, 정부 및 금융회사는 다양한 연금화 전략을 추구할 수 있는 제도 개선 및 상품개발이 요구된다. This paper analyse the probability of ruin in self annuitization which invests the lump-sum wealth into 4 different asset allocation strategies(Maximum sharpe ratio, Equal weight, Risk parity, Minimum probability of ruin)and withdraws fixed amount of annuity. The benchmark annuity amount is assumed as the fully annuitized amount in whole life annuity which retirees buy from insurance company. The equal weight portfolio and minimum probability of ruin portfolio which have more risk asset exposure shows lower probability of ruin. The mixture of annuitization as a form of whole life annuity and self annuitization is recommended as an alternative annuity strategy for retirees. This paper indicate that the timing and portion of annuitization should be dynamically decided after retirement. It`s also recommended that government and financial institutions should consider various annuity products and payout types.

혼합지수분포를 이용한 파산확률의 새로운 근사방법

정대현(Daehyeon Jung),이지연(Jiyeon Lee) 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.1

보험상품의 파산확률이란 보험상품에 대한 연속시간 잉여금 과정의 상태가 0 아래로 떨어지는 사건의 확률로서 보험상품의 리스크를 판단할 수 있는 중요한 척도 중의 하나이다. 일반적으로 파산확률 값은 보험청구가 발생하는 확률과정과 보험청구액의 확률분포에 따라 그 계산 과정이 매우 복잡하며 경우에 따라 정확한 확률값 대신에 근사값으로 해결하려는 노력이 병행되어 왔다. 본 논문에서는 기존의 다양한 파산확률에 대한 근사방법을 설명하고, 보험상품의 손실 과정과 1차부터 4차까지의 적률이 동일한 혼합지수 보험청구액 분포의 손실 과정을 소개하고 이 손실 과정에 대응되는 잉여금 과정에서 얻어지는 정확한 파산확률 값을 이용하여 근사시키는 새로운 방법을 제안하고 기존의 파산확률에 대한 근사방법 들과 비교한다. The ruin probability, one of the important criteria for determining the risk of an insurance product is defined as the probability that the continuous-time surplus process falls below zero. Generally, it is very complicated to calculate the exact ruin probability which depends on the arrival process of the claims and the distribution of the insurance claim amount. Therefore, much efforts have been made to get the approximate value instead of the exact value for the ruin probability. In this paper, we propose a new method to approximate the ruin probability by fitting the first four moments of the loss process with claims which are distributed by the mixture of two exponentials and we compare it with the other approximation methods.

ASYMPTOTIC RUIN PROBABILITIES IN A GENERALIZED JUMP-DIFFUSION RISK MODEL WITH CONSTANT FORCE OF INTEREST

Qingwu Gao,Di Bao 대한수학회 2014 대한수학회지 Vol.51 No.4

This paper studies the asymptotic behavior of the finite-time ruin probability in a jump-diffusion risk model with constant force of in- terest, upper tail asymptotically independent claims and a general count- ing arrival process. Particularly, if the claim inter-arrival times follow a certain dependence structure, the obtained result also covers the case of the infinite-time ruin probability.

ASYMPTOTIC RUIN PROBABILITIES IN A GENERALIZED JUMP-DIFFUSION RISK MODEL WITH CONSTANT FORCE OF INTEREST

Gao, Qingwu,Bao, Di Korean Mathematical Society 2014 대한수학회지 Vol.51 No.4

This paper studies the asymptotic behavior of the finite-time ruin probability in a jump-diffusion risk model with constant force of interest, upper tail asymptotically independent claims and a general counting arrival process. Particularly, if the claim inter-arrival times follow a certain dependence structure, the obtained result also covers the case of the infinite-time ruin probability.

The finite-time ruin probability in two non-standard renewal risk models with constant interest rate and dependent subexponential claims

Yang Yang,Jinguan Lin,Chao Huang,Xin Ma 한국통계학회 2012 Journal of the Korean Statistical Society Vol.41 No.2

This paper considers an ordinary renewal risk model and a compound renewal risk model with constant interest rate, subexponential claims and a general premium process. We derive some asymptotic results on the finite-time ruin probabilities.

A m-type risk model with Markov-modulated premium rate

Wen-Guang Yu 한국전산응용수학회 2009 Journal of applied mathematics & informatics Vol.27 No.5

In this paper, we consider a m-type risk model with Markovmodulated premium rate. A integral equation for the conditional ruin probability is obtained. A recursive inequality for the ruin probability with the stationary initial distribution and the upper bound for the ruin probability with no initial reserve are given. A system of Laplace transforms of non-ruin probabilities, given the initial environment state, is established from a system of integro-differential equations. In the two-state model, explicit formulas for non-ruin probabilities are obtained when the initial reserve is zero or when both claim size distributions belong to the Kn-family, n ∈ N+. One example is given with claim sizes that have exponential distributions. In this paper, we consider a m-type risk model with Markovmodulated premium rate. A integral equation for the conditional ruin probability is obtained. A recursive inequality for the ruin probability with the stationary initial distribution and the upper bound for the ruin probability with no initial reserve are given. A system of Laplace transforms of non-ruin probabilities, given the initial environment state, is established from a system of integro-differential equations. In the two-state model, explicit formulas for non-ruin probabilities are obtained when the initial reserve is zero or when both claim size distributions belong to the Kn-family, n ∈ N+. One example is given with claim sizes that have exponential distributions.

은퇴자가계의 지출수준과 노후자금 고갈가능성에 관한 연구

김민정(Minjeung Kim) 한국FP학회 2010 Financial Planning Review Vol.3 No.1

본 연구는 은퇴자들이 남은 은퇴기간동안 노후자금으로 생활하면서도 노후자금이 고갈될 가능성을 최소화하기 위해 지출수준을 결정할 때 적용가능한 가이드라인을 제공하는 것을 목적으로 하였다. 이에 몬테카를로 시뮬레이션 방법을 적용하여 노후자금 고갈가능성에 대한 허용수준별로 지속가능한 최대초기인출율(MaxSIWR)과 그 때의 포트폴리오 구성을 찾고 초과지출로 인한 고갈가능성의 변화를 살펴보았다. 노후자금은 주식과 채권으로 구성된 포트폴리오로 관리된다고 가정하였고, 몬테카를로 시뮬레이션에 적용한 주식과 채권수익률은 각각 14.2%와 5.6%, 표준편차는 34.9%와 1.4%였다. 은퇴기간은 30년, 물가상승률은 3%로 가정하였고 인출은 매년 초에 발생하고, 인출 후 남은 자금은 재투자된다고 가정하였다. 주요결과로는 첫째, 노후자금 고갈가능성에 대한 허용수준이 0%, 5%, 10%인 경우 MaxSIWR은 각각 4.2%, 4.6%, 4.9%였고 주식비중은 각각 0%, 10~30%, 20~40%로 실패허용수준이 클수록 MaxSIWR과 그때의 주식투자비중은 높아지는 것으로 나타났다. 둘째, 각 실패허용수준에서 제시된 MaxSIWR에서 0.2%p, 0.4%p, 0.6%p 초과하여 지출할 경우 노후자금이 고갈될 가능성을 살펴본 결과, 실패허용수준이 낮은 경우 더 민감하게 증가하는 것으로 나타났다. 셋째, 각 실패허용수준에서의 MaxSIWR을 0.2%p 초과하여 지출할 경우 실패허용수준을 만족할 수 있는 은퇴기간이 약 1~2년 정도씩 감소되는 것으로 나타났다. 마지막으로 MaxSIWR과 이를 위한 위험자산의 비중, 그리고 결정된 지출수준과 희망하는 지출수준 사이의 불균형이 발생할 경우 은퇴자에게 발생하는 문제를 예방하기 위해 은퇴자를 대상으로 하는 전문적인 은퇴설계와 투자상담이 이루어져야 할 것이다. This study indicated guidelines for deciding upon a standard of living during retirement without ruining a retirement portfolio. The purpose of financial planning for retirees is minimizing the ruin probability of their retirement assets. For this purpose, we employed the Monte Carlo Simulation (MCS), which considers the volatility of returns, but the equation does not consider it. Thus, the Initial Withdrawal Rates (IWRs), ranging from 0% to 10% in steps of 0.1%, were inputted into the model for the MCS. We also indicated the Sustainable Maximum Initial Withdrawal Rate (MaxSIWR) and the portfolio allocation in each MaxSIWR at acceptable levels of 0%, 1%, 5%, and 10% as the probability of failing. For this study, we assumed that the retirement portfolio was composed of stocks and bonds, the retirement period was 30 years, and the annual inflation rate remained at 3%. The mean returns on stocks (KOSPI) and bonds (3-year government bond) were 14.2% and 5.6%, respectively, and the standard deviation was 34.9% and 1.4%, respectively. We also assumed that retirees had tried to withdraw some funds at the beginning of the year and that assets left after withdrawing were reinvested into stocks and bonds according to the allocations. As our main results, the first with 0% probability of failing, the MaxSIWR was 4.2% with a 0/100 bond/stock allocation. For the 1%, 5%, and 10% probabilities, the MaxSIWR was 4.4%, 4.6%, and 4.9% with 10/90, 20/80, and 30/70 bond/stock allocations, respectively. First, the MaxSIWR and the weight of stocks in the MaxSIWR increased with higher acceptable levels for the ruin probabilities of the retirement portfolio. Second, the probability of exhausting retirement assets that resulting spending more than the MaxSIWR increased more sensitively in the case of a retiree who had a low acceptable level. Third, the retirement period that satisfies the acceptable level for failing in case of spending that exceeds the MaxSIWR by 0.2%p at each level decreased for about two to three years. Finally, professional planning for retirement and investment consulting for retirees are required in order to prevent problems that can develop from the mismatch of MaxSIWR and the weight of risky assets in the MaxSIWR, as well as the imbalance between the new spending level and the desired spending level.

Ruin probabilities in the risk model with two compound Binomial processes

Mao-Jun Zhang,Jiang-Xia Nan,Sen Wang 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.1

In this paper, we consider an insurance risk model governed by a compound Binomial arrival claim process and by a compound Binomial arrival premium process. Some formulas for the probabilities of ruin and the distribution of ruin time are given, we also prove the integral equation of the ultimate ruin probability and obtain the Lundberg inequality by the discrete martingale approach.