Some Methods Used in the Estimation of the Ruin Probability

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

Four methods of estimation of the ruin probability in the presence of heavy tails are compared in accordance with their effectiveness on the Pareto-distributed claim sizes. The first method, proposed by Embrechis and Veraverbeke, provides an asymptotic expression when the initial capital tends to infinity. The second method, proposed by Goovaerts and De Vylder, provides an algorithm for two-sided estimation based on the solution of the renewal equation through discretization. Its advantage is the perfect applicbility in cases with small initial capital. The third method, proposed initially by Willmoi and improved by Kalashnikov, provides an upper bound of the ruin probability with the help of a test function. The last method, proposed by Kalashnikov, provides two-sided bounds for the ruin probability using truncation of random variables to avoid the Cramer condition. The last two methods enable one to handle cases with intermediate initial capital.

In this paper, we consider a risk model in which each main claim induces a delayed claim called a by-claim. The time of delay for the occurrence of a by-claim is assumed to be exponentially distributed. From martingale theory, an expression for the ultimate ruin probability can be derived using the Lundberg exponent of the associated nondelayed risk model. It can be shown that the Lundberg exponent of the proposed risk model is the same as that of the nondelayed one. Brownian motion approximations for ruin probabilities are also discussed.

ABSTRACTIn this paper we consider a discrete time insurance risk model with interest income. Using the recursive calculation method of De Vylder & Goovaerts (1988), recursive equations for the finite time ruin probabilities and the distribution of the time of ruin are derived. Fredholm type integral equations for the ultimate ruin probability, the distribution of the severity of ruin, the joint distribution of surplus before and after ruin, and the probability of absolute ruin are obtained. Numerical results are included.

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.

According to the characteristics of risk model with perturbed by diffusion, a new kind of stochastic process— F-B process is proposed following from Fuzzy Markov Skeleton process. Then the ruin probabilities (caused by oscillation or by a claim) are discussed by F-B process. We derive a distribution of the ruin probability and a distribution of the surplus immediately before the time of ruin. Then continuous model and discrete model, risk model with compound Poisson process and risk model with double Poisson process, are used to illustrate the implementation of the model. Next limit theorems of two continuous risk models perturbed by diffusion are discussed. Finally simulation experiment is introduced.

Optimal reinsurance under the mean–variance premium principle to minimize the probability of ruin

Kluppelberg and Stadtmuller (1998) proved a precise asymptotic formula for the ruin probability of the classical interest force and regularly varying tailed claims when the initial capital u tends to infinity . This paper extends their results in several aspects as follows: First, the risk models are the Conditional Poisson and Non-Qici Poisson Process respectively; Second, the ruin probability is replaced by the finite time ruin probability within time T and the claimsize is of Subexponential family which is more wide then that of regularly varying's family. At last, the diffusion term of Brownian Motion is considered. construct et ψ(u;T) be the finite time ruin probability in the renewal risk model, where u is the initial capital of the company and $T$ denotes some given time bound. Under the assumption that the distribution of the claim size belongs to the Extended regular variation class, this paper obtains an asymptotic formula for ψ(u;T). This result improves the related works in the recent literature.

Ruin probabilities and penalty functions with stochastic rates of interest

Ruin problems with compounding assets

Markovian risk process

In this paper the probability of ruin is investigated under the influence of a premium rate which varies according to the intensity of claims, and the occurrence of claims is described by a Cox process in the considered risk model. The idea is originally enlightened by Jasiulewicz (2001). We make a slight modification on the model and a generalization on the intensity process. By a “backward differential argument” and the Markov property of the intensity process we strictly derive the integral equation satisfied by the probability of ruin. Further, we solve the equation when the intensity process is a homogeneous n-state Markov process by Laplace transforms. At the end of the paper, an example is given.

This paper examines the impact of the parameters of the distribution of the time at which a bank’s client defaults on their obligated payments, on the Lundberg adjustment coefficient, the upper and lower bounds of the ruin probability. We study the corresponding ruin probability on the assumption of (i) a phase-type distribution for the time at which default occurs and (ii) an embedding of the stochastic cash flow or the reserves of the bank to the Sparre Andersen model. The exact analytical expression for the ruin probability is not tractable under these assumptions, so Cramér Lundberg bounds types are obtained for the ruin probabilities with concomitant explicit equations for the calculation of the adjustment coefficient. To add some numerical flavour to our results, we provide some numerical illustrations.

This paper studies the optimal excess-of-loss reinsurance contract between an insurer and a reinsurer in a dynamic risk model. The risk process is assumed to be a diffusion approximation process of the classical Cramer-Lundberg model which is perturbed by a Brownian motion. In addition to reinsurance, we assume that the insurer is allowed to invest his/her surplus into a financial market containing one risk-free rate of return and determines the reinsurance strategy by a self-reinsurance function. Our aim is to obtain the simultaneous equilibrium strategy in this reinsurance dynamic risk setting using the objective functions of insurer and reinsurance. By employing the dynamic programming approach, we derive the minimization of insurer’s ruin probability and maximization of reinsurance’s expected aggregate discounted net profits to have the optimal portfolio for the two parties treaties in a fixed term insurance contract. In order to provide a more explicit reinsurance contract and to facilitate our quantitative analysis, we study the case when the reinsurance premium function is based on the standard-deviation principle from the integro-differential equations. A numerical example is given to investigate the effects of model parameters on the equilibrium strategy.

Calculation of finite time ruin probabilities for some risk models