RECURSIVE CALCULATION OF RUIN PROBABILITIES AT OR BEFORE CLAIM INSTANTS FOR NON-IDENTICALLY DISTRIBUTED CLAIMS

2014 ◽  
Vol 45 (2) ◽  
pp. 421-443 ◽  
Author(s):  
Anisoara Maria Raducan ◽  
Raluca Vernic ◽  
Gheorghita Zbaganu
Keyword(s):  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Recursive Algorithm ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Numerical Examples ◽  
Claim Number ◽  
Recursive Calculation ◽  
The Impact

AbstractIn this paper, we present recursive formulae for the ruin probability at or before a certain claim arrival instant for some particular continuous time risk model. The claim number process underlying this risk model is a renewal process with either Erlang or a mixture of exponentials inter-claim times (ICTs). The claim sizes (CSs) are independent and distributed in Erlang's family, i.e., they can have different parameters, which yields a non-homogeneous risk process. We present the corresponding recursive algorithm used to evaluate the above mentioned ruin probability and we illustrate it on several numerical examples in which we vary the model's parameters to assess the impact of the non-homogeneity on the resulting ruin probability.

scholarly journals Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance

2012 ◽  
Vol 49 (04) ◽  
pp. 954-966
Author(s):  
R. Romera ◽  
W. Runggaldier
Keyword(s):  
Financial Market ◽  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Finite Horizon ◽  
Ruin Probabilities ◽  
Optimal Solutions ◽  
Semi Markov Process ◽  
Insurance Model ◽  
Recursive Relations

A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

scholarly journals Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance

2012 ◽  
Vol 49 (4) ◽  
pp. 954-966 ◽  
Author(s):  
R. Romera ◽  
W. Runggaldier
Keyword(s):  
Financial Market ◽  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Finite Horizon ◽  
Ruin Probabilities ◽  
Optimal Solutions ◽  
Semi Markov Process ◽  
Insurance Model ◽  
Recursive Relations

A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

scholarly journals The Perturbed Dual Risk Model with Constant Interest and a Threshold Dividend Strategy

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Fanzi Zeng ◽  
Jisheng Xu
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Present Value ◽  
Numerical Examples ◽  
Threshold Dividend Strategy ◽  
Absolute Ruin ◽  
Dual Risk Model ◽  
The Moment ◽  
The Impact ◽  
Constant Interest

We consider the perturbed dual risk model with constant interest and a threshold dividend strategy. Firstly, we investigate the moment-generation function of the present value of total dividends until ruin. Integrodifferential equations with certain boundary conditions are derived for the present value of total dividends. Furthermore, using techniques of sinc numerical methods, we obtain the approximation results to the expected present value of total dividends. Finally, numerical examples are presented to show the impact of interest on the expected present value of total dividends and the absolute ruin probability.

scholarly journals Ruin Probabilities of a Discrete-time Multi-risk Model

2015 ◽  
Vol 44 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Andrius Grigutis ◽  
Agneška Korvel ◽  
Jonas Šiaulys
Keyword(s):  
Discrete Time ◽  
Ruin Probability ◽  
Arrival Time ◽  
Risk Model ◽  
Recursive Formula ◽  
Ruin Probabilities ◽  
Numerical Examples ◽  
Premium Rate ◽  
Insurance Business ◽  
Independent Series

In this work,  we investigate a  multi-risk model describing insurance business with  two or more independent series of claim amounts. Each series of claim amounts consists of independent nonnegative random variables. Claims of each series occur periodically with some fixed   inter-arrival time. Claim amounts occur until they   can be compensated by a common premium rate and the initial insurer's surplus.  In this article, wederive a recursive formula for calculation of finite-time ruin probabilities. In the case of bi-risk model, we present a procedure to calculate the ultimate ruin probability. We add several numerical examples illustrating application  of the derived formulas.DOI: http://dx.doi.org/10.5755/j01.itc.44.4.8635

scholarly journals On a discrete-time risk model with claim correlated premiums

2015 ◽  
Vol 9 (2) ◽  
pp. 322-342 ◽  
Author(s):  
Xueyuan Wu ◽  
Mi Chen ◽  
Junyi Guo ◽  
Can Jin
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Insurance Industry ◽  
Joint Probability ◽  
Ruin Probabilities ◽  
Deficit At Ruin ◽  
Probability Of Ruin ◽  
Numerical Examples ◽  
Car Insurance ◽  
The Impact

AbstractThis paper proposes a discrete-time risk model that has a certain type of correlation between premiums and claim amounts. It is motivated by the well-known bonus-malus system (also known as the no claims discount) in the car insurance industry. Such a system penalises policyholders at fault in accidents by surcharges, and rewards claim-free years by discounts. For simplicity, only up to three levels of premium are considered in this paper and recursive formulae are derived to calculate the ultimate ruin probabilities. Explicit expressions of ruin probabilities are obtained in a simplified case. The impact of the proposed correlation between premiums and claims on ruin probabilities is examined through numerical examples. In the end, the joint probability of ruin and deficit at ruin is also considered.

Ruin Probabilities of Continuous-Time Risk Model with Dependent Claim Sizes and Interarrival Times

2020 ◽  
Vol 28 (Supp01) ◽  
pp. 69-80
Author(s):  
Nguyen Huy Hoang ◽  
Bao Quoc Ta
Keyword(s):  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Random Variables ◽  
Ruin Probabilities ◽  
Dependent Random Variables ◽  
Arrival Times ◽  
Exponential Bound ◽  
Interarrival Times

In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability.

Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums

2009 ◽  
Vol 39 (1) ◽  
pp. 117-136 ◽  
Author(s):  
Lourdes B. Afonso ◽  
Alfredo D. Egídio dos Reis ◽  
Howard R. Waters
Keyword(s):  
Gamma Distribution ◽  
Continuous Time ◽  
Finite Time ◽  
Ruin Probability ◽  
Numerical Evaluation ◽  
Risk Process ◽  
Ruin Probabilities ◽  
Aggregate Claims ◽  
Classical Risk Process ◽  
Major Consideration

AbstractIn this paper we present a method for the numerical evaluation of the ruin probability in continuous and finite time for a classical risk process where the premium can change from year to year. A major consideration in the development of this methodology is that it should be easily applicable to large portfolios. Our method is based on the simulation of the annual aggregate claims and then on the calculation of the ruin probability for a given surplus at the start and at the end of each year. We calculate the within-year ruin probability assuming a translated gamma distribution approximation for aggregate claim amounts.We illustrate our method by studying the case where the premium at the start of each year is a function of the surplus level at that time or at an earlier time.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (03) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

scholarly journals Ruin in the perturbed compound Poisson risk process under interest force

2005 ◽  
Vol 37 (3) ◽  
pp. 819-835 ◽  
Author(s):  
Jun Cai ◽  
Hailiang Yang
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Process ◽  
Integrodifferential Equations ◽  
Ruin Probabilities ◽  
Compound Poisson ◽  
Interest Force ◽  
Stochastic Interest ◽  
Constant Interest Force ◽  
Constant Interest

In this paper, we study ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, we show that the ruin probability in the perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. We decompose the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by oscillation. Integrodifferential equations for these ruin probabilities are derived when the interest force is constant. When the claim sizes are exponentially distributed, explicit solutions of the ruin probabilities are derived from the integrodifferential equations. Numerical examples are given to illustrate the effects of diffusion volatility and interest force on the ruin probabilities.

scholarly journals Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 982
Author(s):  
Yujuan Huang ◽  
Jing Li ◽  
Hengyu Liu ◽  
Wenguang Yu
Keyword(s):  
Fourier Series ◽  
Ruin Probability ◽  
Risk Model ◽  
Expansion Method ◽  
Nonparametric Estimator ◽  
Insurance Risk ◽  
Numerical Examples ◽  
Complex Fourier Series ◽  
Premium Income ◽  
Insurance Risk Model

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Sign in / Sign up

Export Citation Format

Share Document