On the Class of Erlang Mixtures with Risk Theoretic Applications

Abstract

A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.     

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.     

Upper bounds on stop-loss premiums under constraints on claim size distribution

Abstract

The problem of maximal stop-loss premium under prescribed constraints on claim size distribution is taken up again. The methods of linear programming are used to show that the recent results of others are intuitively obvious. These results are then extended by the linear programming technique to cases of more general constraints, e.g. prescribed claim size variance, or prescribed minimum frequency of excess claims. In particular it is shown that, typically, the upper bound on stop-loss premiums is generated by a claim size distribution which has all its mass concentrated at very few points. In contrast with the results obtained by others recently, it is seen that the claim size distribution which produces the maximal stop-loss premium is not generally independent of the excess. Some numerical examples are given showing that the methods used here can sometimes improve considerably the recent results of others. The case of a compound Poisson distribution is treated briefly.     

Probability of ruin with variable premium rate

Abstract

The probability of ruin is investigated under the influence of a premium rate which varies with the level of free reserves. Section 4 develops a number of inequalities for the ruin probability, establishing upper and lower bounds for it in Theorem 4. Theorem 5 gives an expression for the ruin probability, and it is seen in Section 5 that this amounts to a generalization of the ruin probability given by Gerber for the special case of a negative exponential claim size distribution. In that same section it is shown the Lundberg's inequality is not derivable from the generalized theory of Section 4, and this is seen as a drawback of the methods used there. Sections 6 and 7 deal with some special cases, including claim size distributions with monotone failure rates. Section 8 shows that, in contrast with the result for a constant premium that the probability of ruin for zero initial reserve is independent of the claim size distribution, the same result does not hold when the premium rate is allowed to vary. Section 9 gives some comments on the possible effect of “dangerousness” of a claim size distribution on ruin probability.     

Asymptotics for large claims reinsurance in a time-dependent renewal risk model

Abstract

We study the asymptotic tail behaviour of reinsured amounts of the LCR and ECOMOR treaties under a time-dependent renewal risk model, in which a dependence structure is introduced between each claim size and the interarrival time before it. Assuming that the claim size distribution has a subexponential tail, we derive some precise asymptotic results for both treaties.     

Authors’ Reply: Corporate Hedging in the Insurance Industry: The Use of Financial Derivatives by U.S. Insurers - Discussion by L. Lee Colquitt; Arlette C. Wilson; Gray G. Venter; Morton Lane; Joan Lamm-Tennant,January 1997

Abstract

This paper studies the joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin. The time of ruin is analyzed in terms of its Laplace transforms, which can naturally be interpreted as discounting. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. The expected discounted penalty, considered as a function of the initial surplus, satisfies a certain renewal equation, which has a probabilistic interpretation. Explicit answers are obtained for zero initial surplus, very large initial surplus, and arbitrary initial surplus if the claim amount distribution is exponential or a mixture of exponentials. We generalize Dickson’s formula, which expresses the joint distribution of the surplus immediately prior to and at ruin in terms of the probability of ultimate ruin. Explicit results are obtained when dividends are paid out to the stockholders according to a constant barrier strategy.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.     

Moments of Discounted Dividends for a Threshold Strategy in the Compound Poisson Risk Model

Abstract

We consider a compound Poisson risk model in which part of the premium is paid to the shareholders as dividends when the surplus exceeds a specified threshold level. In this model we are interested in computing the moments of the total discounted dividends paid until ruin occurs. However, instead of employing the traditional argument, which involves conditioning on the time and amount of the first claim, we provide an alternative probabilistic approach that makes use of the (defective) joint probability density function of the time of ruin and the deficit at ruin in a classical model without a threshold. We arrive at a general formula that allows us to evaluate the moments of the total discounted dividends recursively in terms of the lower-order moments. Assuming the claim size distribution is exponential or, more generally, a finite shape and scale mixture of Erlangs, we are able to solve for all necessary components in the general recursive formula. In addition to determining the optimal threshold level to maximize the expected value of discounted dividends, we also consider finding the optimal threshold level that minimizes the coefficient of variation of discounted dividends. We present several numerical examples that illustrate the effects of the choice of optimality criterion on quantities such as the ruin probability.     

On The Expected Discounted Penalty function for Lévy Risk Processes

Abstract

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes.

In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.     

Semiparametric Estimation for Non-Ruin Probabilities

We consider the classical risk model with unknown claim size distribution F and unknown Poisson arrival rate u . Given a sample of claims from F and a sample of interarrival times for these claims, we construct an estimator for the function Z ( u ), which gives the probability of non-ruin in that model for initial surplus u . We obtain strong consistency and asymptotic normality for that estimator for a large class of claim distributions F . Confidence bounds for Z ( u ) based on the bootstrap are also given and illustrated by some numerical examples.     

“Relative Importance of Risk Sources in Insurance Systems”, Edward W. Frees,April 1998

Abstract

This paper considers a Sparre Andersen collective risk model in which the distribution of the interclaim time is that of a sum of n independent exponential random variables; thus, the Erlang(n) model is a special case. The analysis is focused on the function φ(u), the expected discounted penalty at ruin, with u being the initial surplus. The penalty may depend on the deficit at ruin and possibly also on the surplus immediately before ruin. It is shown that the function φ(u) satisfies a certain integro-differential equation and that this equation can be solved in terms of Laplace transforms, extending a result found in Lin (2003). As a consequence, a closed-form expression is obtained for the discounted joint probability density of the deficit at ruin and the surplus just before ruin, if the initial surplus is zero. For this formula and other results, the roots of Lundberg’s fundamental equation in the right half of the complex plane play a central role. Also, it is shown that φ(u) satisfies Li’s (2003) renewal equation. Under the assumption that the penalty depends only on the deficit at ruin and that the individual claim amount density is a combination of exponential densities, a closed-form expression for φ(u) is derived. In this context, known results of the Cauchy matrix are useful. Surprisingly, certain results are best expressed in terms of divided differences, a topic deleted from the actuarial examinations at the end of last century.     

Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

In this paper, we derive two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist. These bounds also apply directly to the tails of compound geometric distributions. The upper bound is tighter than that of Dickson (1994). The corresponding lower bound, which holds under the same conditions, is tighter than that of De Vylder and Goovaerts (1984). Even when the adjustment coefficient exists, the upper bound is, in some cases, tighter than Lundberg's bound. These bounds are applicable for any positive distribution function with a finite mean. Examples are given and numerical comparisons with asymptotic formulae for the ruin probability are also considered.     

A solvency study in non-life insurance

Abstract

This is the first of two papers which report on a solvency study. The study is based on statistical analyses of policy and claims data of a portfolio of single-family houses and dwellings. This paper deals mainly with analyses of fire, windstorm, and glass liabilities. Claim frequencies and claim size distributions are estimated, and the results are used to derive moments of the annual claim amounts and to provide examples of solvency margin requirements for different classes of business. The second paper is devoted to a broader discussion of solvency margin requirements in non-life insurance.     

On the Moments of the Time of Ruin with Applications to Phase-Type Claims

Abstract

We describe an approach to the evaluation of the moments of the time of ruin in the classical Poisson risk model. The methodology employed involves the expression of these moments in terms of linear combinations of convolutions involving compound negative binomial distributions. We then adapt the results for use in the practically important case involving phase-type claim size distributions. We present numerical examples to illuminate the influence of claim size variability on the moments of the time of ruin.     

Exact solutions for ruin probability in the presence of an absorbing upper barrier

Abstract

An explicit solution for the probability of ruin in the presence of an absorbing upper barrier was developed by Segerdahl (1970) for the particular case in which both the interoccurrence times between successive claims and the single claim amounts follow an exponential distribution with unit mean. In this paper we show that his method of solution may be extended to produce explicit solutions for two more general types of single claim amount distribution. These are the gamma distribution, denoted γ(a), where a is an integer, and the mixed exponential distribution. Comparisons are drawn between this approach when the upper barrier tends to infinity, and the classical solution for ruin probability in these particular cases given in Cramér (1955).     

Stationarity aspects of the sparre andersen risk process and the corresponding ruin probabilitles

Abstract

The Sparre Andersen risk model assumes that the interclaim times (also the time between the origin and the first claim epoch is considered as an interclaim time) and the amounts of claim are independent random variables such that the interclaim times have the common distribution function K(t), t|>/ 0, K(O)= 0 and the amounts of claim have the common distribution function P(y), - ∞ < y < ∞. Although the Sparre Andersen risk process is not a process with strictly stationary increments in continuous time it is asymptotically so if K(t) is not a lattice distribution. That is an immediate consequence of known properties of renewal processes. Another also immediate consequence of such properties is the fact that if we assume that the time between the origin and the first claim epoch has not K(t) but as its distribution function (k^b1 denotes the mean of K(t)) then the so modified Sparre Andersen process has stationary increments (this works even if K(t) is a lattice distribution).

In the present paper some consequences of the above-mentioned stationarity properties are given for the corresponding ruin probabilities in the case when the gross risk premium is positive.     

Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.     

On the complete monotonicity of the compound geometric convolution with applications in risk theory

We prove that the complete monotonicity is preserved under mixed geometric compounding, and hence show that the ruin probability, the Laplace transform of the ruin time, and the density of the tail of the joint distribution of ruin and the deficit at ruin in the Sparre Andersen model are completely monotone if the claim size distribution has a completely monotone density.