Mean and dispersion modelling for policy claims costs

An application of time series methods to financial guarantee insurance

On the number of near-maximum insurance claim under dependence

Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform

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.

This chapter complements the material of Chap. 4 in the sense that explicit identification of requisite quantities is possible with stronger parametric assumptions about either the (marginal) distribution of the claim sizes or that of the interclaim times. Convenient parametric assumptions involve Erlang (or Coxian) distributions, due to the generality and mathematical tractability inherent in the use of such models. Sect. 5.1 directly generalizes the classical Poisson model of Chap. 3 by essentially assuming a (marginal) Erlang form for the interclaim times. A Lagrange polynomial approach together with Laplace transform arguments is used to explicitly identify all components of the Gerber-Shiu function. The independent model with an arbitrary interclaim time distribution and exponential claim sizes is then considered in Sect. 5.2. The conditioning argument of Sect. 5.2 is then extended in Sect. 5.3 to a model where the (marginal) claim size distribution is assumed to be of Coxian form. In particular, identification of the Gerber-Shiu function is shown to be possible for special choices of the penalty function, and more generally for an arbitrary penalty function of the deficit. Finally, the Coxian assumption of Sect. 5.3 is replaced by a (possibly infinite) mixture of Erlang distributions, resulting in tractable results for ruin probabilities, the deficit at ruin, and related quantities.

Alternative dispute resolution ~ADR! is a common term in the twenty-first century and especially for those involved in construction projects. Commercial construction projects are also on the rise nationwide as developers acquire more property and transform once barren land into large arenas, skyscrapers, and office buildings. Due to the increase in the demand for general contractors to compete and ‘‘win’’ the best job for the lowest price, dispute resolution is gaining importance in maintaining success for the general contractor by reducing the chance of claims for the contractor and the owner. A form of dispute resolution for construction, which is quite prevalent today, is partnering. Partnering is a process that brings the parties of the construction project together to resolve potential issues. The hypothesis of this paper is that partnering will reduce the overall number and size of claims on a construction project. By providing a framework for resolving the issues early in the construction process, it becomes more likely that conflict will not arise within the parties. The objective of this paper was to determine whether the relationship between partnering in construction projects influences the number and magnitude of claims that are filed or paid by the contractor and owner. Before one can develop an effective partnering methodology, one must understand the major reasons relating to employee dissatisfaction with current partnering methods. This study will provide an insight into partnering and its effects on the number and magnitude of claims as well as suggestions for more effective partnering methodology.

Insurance is a contractual agreement between two parties, namely the insured party (customer) and the insurer (insurance company), in which the insured party pays a premium to the insurer, then in return, the insurer will provide compensation (claim) to the insured party if an insured event occurs. Each customer is required to pay apremium as an obligation stated in the insurance agreement by paying a premium, the customer fulfills his obligations and is entitled to the benefits stated in the policy. Therefore, the Insurance Company needs to carry out a scheme in the process of paying pure premiums for the sustainability of the insurance company. When determining the premium, it is done by estimating the aggregate loss distribution. This research will calculate thepure premium at the Purwakarta Branch of Jasa Raharja Insurance Company. The model used in this study is the distribution of aggregate loss with a compound distribution of claim frequency and claim size. Many claims follow the Poisson distribution and large claims follow the Lognormal distribution. In the process of estimating the probability of aggregate loss with the compound distribution model, the Inverse method with the Fast Fourier Transform (FFT) algorithm is used. This research will provide understanding and insight to insurance companies in determining the amount of premium that must be charged to customers.

In this paper, we investigate some ruin problems for risk models that contain uncertainties on both claim frequency and claim size distribution. The problems naturally lead to the evaluation of ruin probabilities under the so-called G-expectation framework. We assume that the risk process is described as a class of G-compound Poisson process, a special case of the G-Lévy process. By using the exponential martingale approach, we obtain the upper bounds for the two-sided ruin probability as well as the ruin probability involving investment. Furthermore, we derive the optimal investment strategy under the criterion of minimizing this upper bound. Finally, we conclude that the upper bound in the case with investment is less than or equal to the case without investment.

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution

This paper analyses the stop-loss premium II(c, t) for the standard risk model in which the number of claims in (0, t] is Poisson with intensity 1 and the claimsize distribution F is on (-∞, +∞) with mean µ>0. It is shown that, under typical conditions on F, II(c, t)=µt-c+R(c, t) where R(c, t) tends to zero exponentially fast as t tends to infinity. The precise behaviour of R(c, t) is established.

In this paper, the absolute ruin in the compound Poisson risk model with interest and a constant dividend barrier is investigated. First, integro-differential equations satisfied by the expected discounted dividend payments are derived. The explicit expressions are obtained when the individual claim size is exponential distributed. Second, the moment generating function of the discounted dividends is considered, and integro-differential equations satisfied by the moment generating function of the discounted dividends are derived. Third, by a “differential” argument, the time to recovery to zero from a given negative surplus is considered. Finally, how long it takes for the surplus process to reach the dividend barrier is discussed.

In this paper, we investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

In this paper, we give a result on the local asymptotic behaviour of the probability of ruin in a continuous-time risk model in which the inter-claim times have an Erlang distribution and the individual claim sizes have a distribution that belongs toS(v) withv ≥ 0, but where the Lundberg exponent of the underlying risk process does not exist.

Practical aspects of stop-loss calculations

Asymptotic and numerical analysis of the optimal investment strategy for an insurer