Individual Claim Amount Research Articles

Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion

In this paper, we consider the compound Poisson process that is perturbed by diffusion (CPD). We derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD and also present some examples of the CPD with an exponential individual claim amount distribution and a mixture exponential individual claim amount distribution.

The joint distributions of several important actuarial diagnostics in the classical risk model

In this paper we examine the joint distributions of several actuarial diagnostics which are important to insurers’ running in the classical risk model. They include the time of the surplus process leaving zero ultimately (simply, the ultimately leaving-time), the number of zero, the surplus immediately prior to ruin, the deficit at ruin, the supreme and minimum profits before ruin, the supreme profits and deficit until it leaves zero ultimately and so on. We obtain explicit expressions for their joint distributions mainly by strong Markov property of the surplus process—a technique used by Wu et al. (2002) [J. Appl. Math., in press], which is completely different from former contributions on this topic. Further, we give the exact calculating results for them when the individual claim amounts are exponentially distributed.

Discounted probabilities and ruin theory in the compound binomial model

The aggregate claims are modeled as a compound binomial process, and the individual claim amounts are integer-valued. We study f(x, y; u), the “discounted” probability of ruin for an initial surplus u, such that the surplus just before ruin is x and the deficit at ruin is y. This function can be used to calculate the expected present value of a penalty that is due at ruin, and, if it is interpreted as a probability generating function, to obtain certain information about the time of ruin. An explicit formula for f(x, y; 0) is derived. Then it is shown how f(x, y; u) can be expressed in terms of f(x, y; 0) and an auxiliary function h(u) that is the solution of a certain recursive equation and is independent of x and y. As an application, we use the asymptotic expansion of h(u) to obtain an asymptotic formula for f(x, y; u). In this model, certain results can be obtained more easily than in the compound Poisson model and provide additional insight. For the case u=0, expressions for the expected present value of a payment of 1 at ruin and the expected time of ruin (given that ruin occurs) are obtained. A discrete version of Dickson’s formula is provided.

Ruin Problems for Phase-Type(2) Risk Processes

In this paper we consider a risk process in which claim inter-arrival times have a phase-type(2) distribution, a distribution with a density satisfying a second order linear differential equation. We consider some ruin related problems. In particular, we consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. We also consider explicit solutions for the infinite time ruin probability in the case where the individual claim amount distribution is phase-type.

Predictive Aggregate Claims Distributions

In the collective risk model we use the objective Bayesian approach to calculate predictive aggregate claims distributions. Very often this is equivalent to the profile predictive likelihood approach, but the former is more straightforward to apply, and accordingly we use it as a pragmatic device. We compare the predictive distributions with fitted distributions which take no account of parameter uncertainty and show that actuarial functions such as premiums can be substantially understated if parameter uncertainty is ignored. We illustrate the situation when the moments of the predictive individual claim amount distribution do not exist and we discuss ways of applying such distributions to insurance problems.

Ruin probabilities for Erlang(2) risk processes

In this paper we consider a risk process in which claim inter-arrival times have an Erlang(2) distribution. We consider the infinite time survival probability as a compound geometric random variable and give expressions from which both the survival probability from initial surplus zero and the ladder height distribution can be calculated. We consider explicit solutions for the survival/ruin probability in the case where the individual claim amount distribution is phase-type, and show how the survival/ruin probability can be calculated for other individual claim amount distributions.

An upper bound for the probability of ultimate ruin

Abstract In the classical compound Poisson risk model, Lundberg's inequality provides both an upper bound for, and an approximation to, the probability of ultimate ruin. The result can be applied only when the moment generating function of the individual claim amount distribution exists. In this paper we derive an upper bound for the probability of ultimate ruin when the moment generating function of the individual claim amount distribution does not exist.

How long is the surplus below zero?

Assuming the classical compound Poisson continuous time surplus process, we consider the process as continuing if ruin occurs. Due to the assumptions presented, the surplus will go to infinity with probability one. If ruin occurs the process will temporarily stay below the zero level. The purpose of this paper is to find some features about how long the surplus will stay below zero. Using a martingale method we find the moment generating function of the duration of negative surplus, which can be multiple, as well as some moments. We also present the distribution of the number of negative surpluses. We further show that the distribution of duration time of a negative surplus is the same as the distribution of the time of ruin, given ruin occurs and initial surplus is zero. Finally, we present two examples, considering exponential and Gamma(2,β) individual claim amount distributions.

An Analysis of Life Insurer Retention Limits

This study examines the relationship between ceding life insurers' retention limits on an insured life basis and certain operational and financial characteristics of the companies. Analysis is performed using 97 major U.S. insurers with 1987 assets ranging from $108 million to $32 billion. Retention limits for these insurers vary from $25,000 to $20,000,000 per insured life. Five factors are shown to have the greatest impact on life insurer retention limits. Not unexpectedly, firm size is found to be the most important factor. Four other explanatory factors, each statistically significant at the 0.01 level, provide important additional insight into this relationship. These factors are (1) form of insurer organization (i.e., mutual or stock), (2) the firm's emphasis on new business, (3) average policy size, and (4) the company's emphasis on term insurance. Collectively, the principal components regression model explains nearly 90 percent of the total variation in retention limits for the sample insurers. The life reinsurance decision centers on the establishment of retention limits. Retention is the amount of any insured loss to be borne by the primary (or ceding) company. Life retention limits normally are expressed as an amount per insured life. Claim amounts in excess of the insurer's retention are the responsibility of its reinsurers. This study examines the relationship between the size of ceding companies' retention limits and certain insurer characteristics. The setting of an appropriate retention limit is a non-trivial decision for an insurer. It ultimately affects both the risk exposure and the profitability of the ceding company. It establishes the maximum individual claim amount that the insurer believes it can tolerate, given its financial and operating characteristics. Further, the dollar retention limit is directly related to other elements in the total reinsurance process. For example, an automatic binding

Risk theory for the compound Poisson process that is perturbed by diffusion

The classical model of collective risk theory is extended in that a diffusion process is added to the compound Poisson process. It is shown that the probabilities of ruin (by oscillation or by a claim) satisfy certain defective renewal equations. The convolution formula for the probability of ruin is derived and interpreted in terms of the record highs of the aggregate loss process. If the distribution of the individual claim amounts are combinations of exponentials, the probabilities of ruin can be calculated in a transparent fashion. Finally, the role of the adjustment coefficient (for example, for the asymptotic formulas) is explained.

Calculation of the probability of eventual ruin by Beekman's convolution series

This paper presents an algorithm for evaluating the probability of eventual ruin in a collective risk model. The method is based on Beekman's convolution series for the ruin probability. It is assumed that the claim number process is Poisson and the individual claim amounts take on integer values only.

From Aggregate Claims Distribution to Probability of Ruin

When the distribution of the number of claims in an interval of time of length t is mixed Poisson and the moments of the independent distribution of individual claim amounts are known, the moments of the distribution of aggregate claims through epoch t can be calculated (O. Lundberg, 1940, ch. VI). Several approximations to the corresponding distribution function, F(·, t), are available (see, e.g., Seal, 1969, ch. 2) and, in particular, a simple gamma (Pearson Type III) based on the first three moments has proved definitely superior to the widely accepted “Normal Power” approximation (Seal, 1976). Briefly,where the P-notation for the incomplete gamma ratio is now standard and α, a function of t, is to be found fromthe kappas being the cumulants of F(·, t). An excellent table of the incomplete gamma ratio is that of Khamis (1965).The problem that is solved in this paper is the production of an approximation to U(w, t), the probability of non-ruin in an interval of time of length t, by using the above mentioned gamma approximation to F(·, t).

The ruin problem in case the tail of the claim distribution is completely monotone

Summary In this paper the case is considered in which the distribution function, P(y), of the individual claim amounts has the form where V(α) is a distribution function with V(O) = 0. Formulas for the ruin probabilities for a finite or infinite time period are derived which in the latter case require no further Fourier inversion and in the former only one such inversion. We assume that the interoccurrence times between successive claims are independent and equidistributed according to a distribution function K(t), t⩾0, K(0)=0. A particular case is thus the Poisson process.