A Functional Approach for Ruin Probabilities

On the probability of ruin in a Markov-modulated risk model

The ruin probability of an insurance company is a central topic in risk theory. We consider the classical Poisson risk model when the claim size distribution and the Poisson arrival rate are unknown. Given a sample of inter-arrival times and corresponding claims, we propose a semiparametric estimator of the ruin probability. We establish properties of strong consistency and asymptotic normality of the estimator and study bootstrap confidence bands. Further, we present a simulation example in order to investigate the finite sample properties of the proposed estimator.

In this paper we present a stable recursive algorithm for the calculation of the probability of ultimate ruin in the classical risk model. We also present stable recursive algorithms for the calculation of the joint and marginal distributions of the surplus prior to ruin and the severity of ruin. In addition we present bounds for these distributions.

of thesis entitled STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES Submitted by CHEN YIQING for the degree of Doctor of Philosophy at The University of Hong Kong in August 2009 In recent years, insurance risk studies in the presence of dependence structure and heavy-tailed claims are popular topics in actuarial science. In this thesis, we consider the renewal risk model with certain dependence structures, namely negative dependence and quasi-asymptotic independence. We also assume that claim sizes follow a heavy-tailed distribution, in particular, a subexponential distribution. We focus on studying the impact of heavy tails and dependence structures on ruin probabilities and the tail probabilities of aggregate claims. The asymptotic behavior of the ultimate ruin probability in the renewal risk model was studied in the presence of negatively dependent claim sizes and a constant force of interest. A purely probabilistic approach was employed to establish a simple asymptotic formula for the ruin probability. This result coincides with some known results in the recent literature, and discovers a phenomenon that heavy tails can offset the impact of the dependence structure of claim sizes on the ruin probability. A two-dimensional renewal risk model was considered in which the insurance company consists of two dependent classes of business. The dependence between the two classes comes from the fact that their claims arrive according to a common renewal counting process. For this model, several ruin probabilities were defined and simple asymptotic formulae for these ruin probabilities were derived. In particular, the obtained asymptotic formulae hold uniformly for all time horizons. This uniformity greatly enhances the theoretical value of the results. The next problem was to study precise estimates of the large deviation probabilities of aggregate claims with sizes which are negatively dependent. Specially, the problem for the so-called quasi renewal risk model was investigated in which inter-arrival times are also negatively dependent. A new dependence structure was considered and was termed “quasi-asymptotic independence” structure, which is more general and verifiable than some commonlyused dependence structures. The proposed dependence structure covers a wide range of positive and negative dependence. For the case of Pareto-like tails, the tail probability of the sum was proved to be asymptotically equivalent to the sum of tail probabilities. This confirms again that heavy tails offset the impact of dependence between summands. This result was then extended to randomly weighted sums and random sums. All the obtained results have obvious applications to concrete actuarial problems. In the last part of the thesis, each primary random variable representing the net loss during a period was associated with a random weight or a stochastic discount factor. Therefore, the weighted sum represents the stochastic present value of aggregate losses. Suppose that the primary random variables are long tailed, the weights are positive and bounded, and the primary random variables and their weights are independent. The tail probability of the maximum of the weighted sums was shown to be asymptotically equivalent to that of the last weighted sum. STUDY ON INSURANCE RISK MODELS WITH SUBEXPONENTIAL TAILS AND DEPENDENCE STRUCTURES

Ruin theory plays a crucial role in risk measurement and risk regulation. Bankruptcy claim is a major focus of ruin theory and the distribution of the aggregate amount of claim can well describe the risk of insurance portfolio. According to the distribution characteristics, we can adopt such means as capital injection and premium re-adjustment to regulate risk. In the classical risk model, the priorities are given to four ruin-related variables: the time of ruin, the aggregate claim amount up to ruin, the total number of claims up to ruin and the deficit at ruin. In this paper, we mainly consider the joint probability density function of the aggregate claim amount up to ruin with other ruin related quantities. The explicit expressions are given for the joint densities when the individual claim follows exponential distribution. In addition, when the individual claim follows a particular decomposition form, the joint density can also be obtained in a decomposition form.

The Sparre Andersen model in the collective risk theory is investigated. We obtain the rate of convergence for the ruin probability after thenth payoff in the case where claim sizes are heavy tailed (say, subexponential).

We consider the classical risk model with subexponential claim size distribution. Three methods are presented to simulate the probability of ultimate ruin and we investigate their asymptotic efficiency. One, based upon a conditional Monte Carlo idea involving the order statistics, is shown to be asymptotically efficient in a certain sense. We use the simulation methods to study the accuracy of the standard Embrechts-Veraverbeke [16] approximation for the ruin probability and also suggest a new one based upon ideas of Hogan [21].

Research on Ruin Problems with Two-type by-claims and Delay Period Based on Compound Binomial Risk Model

In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinnStieltjes convolution P*G of a phase-type distribution G and a nonnegative but otherwise ar-bitrary distribution P. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y s G and a nonnegative random variable S s P. We call S the scaling random variable and its corresponding distribution P the scaling distribution. We investigate conditions for such a class of distributions to be either light- or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support m such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical CramernLundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phase-type scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a MellinnStieltjes convolution P*Gmnof a discrete nonnegative distribution P with an Erlang distribution Gmn. A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions. We suggest a simple methodology for constructing a sequence of distributions having the form P*Gmnwith the purpose of approximating the integrated tail distribution of the claim sizes. Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

Calculation of finite time ruin probabilities for some risk models

On the classical risk model with credit and debit interests under absolute ruin

This study examines the implementation of the ruin probability model in risk management in life insurance companies. The main focus of this study is to evaluate how factors such as initial surplus, premium revenue level, and claim frequency affect the ruin probability of insurance companies. Using the collective risk model approach and relevant claim distribution, this study develops two methods to calculate the ruin probability: an analytical approach and a Monte Carlo simulation. The simulation results show that increasing the initial surplus and premium level significantly reduces the ruin risk, while increasing the claim frequency increases the ruin probability. In addition, the gamma claim distribution is more suitable for modeling claims in life insurance than the exponential distribution. Model validation is carried out by comparing the prediction results with historical data of insurance companies, which shows a high level of accuracy. This study provides important insights for insurance companies in designing more effective and optimal risk management strategies

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the ith arrival is at level u, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$ . The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arrival process of proportional gains. In the dividend calculations, we also consider a random perturbation of our basic risk process modeled by an independent Brownian motion with drift.

Upper bounds for ultimate ruin probabilities in the Sparre Andersen risk model with interest and a nonlinear dividend barrier