Hanspeter Schmidli

Optimal Proportional Reinsurance Policies in a Dynamic Setting

We consider dynamic proportional reinsurance strategies and derive the optimal strategies in a diffusion setup and a classical risk model. Optimal is meant in the sense of minimizing the ruin probability. Two basic examples are discussed.

RUIN ESTIMATION FOR A GENERAL INSURANCE RISK MODEL

The theory of piecewise-deterministic Markov processes is used in order to investigate insurance risk models where borrowing, investment and inflation are present.

Cramer-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion

Abstract In the present paper risk processes perturbed by diffusion are considered. By exponential tilting the processes are inbedded in an exponential family of stochastic processes, such that the type of process is preserved. By change of measure techniques asymptotic expressions for the ruin probability are obtained. This proves that the coefficients obtained by Furrer and Schmidli (1994) are the adjustment coefficients.

Modelling of extremal events in insurance and finance

Extremal events play an increasingly important role in stochastic modelling in insurance and finance. Over many years, probabilists and statisticians have developed techniques for the description, analysis and prediction of such events. In the present paper, we review the relevant theory which may also be used in the wider context of Operation Research. Various applications from the field of insurance and finance are discussed. Via an extensive list of references, the reader is guided towards further material related to the above problem areas.

Pricing catastrophe insurance products based on actually reported claims

Abstract This paper deals with the problem of pricing a financial product relying on an index of reported claims from catastrophe insurance. The problem of pricing such products is that, at a fixed time in the trading period, the total claim amount from the catastrophes occurred is not known. Therefore, one has to price these products solely from knowing the aggregate amount of the reported claims at the fixed time point. This paper will propose a way to handle this problem, and will thereby extend the existing pricing models for products of this kind.

Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion

Abstract A class of diffusion processes following locally a vector field is constructed and the extended generator is computed for a subset of the domain of the generator. Using this theory, martingales for risk processes perturbed by diffusion are obtained. This leads to exponential bounds for the ruin probability in infinite as well as in finite time.

Finite-time Lundberg inequalities in the Cox case

Abstract We consider an insurance model, where the underlying point process is a Cox process. Using a martingale approach, applied to piecewise-deterministic Markov processes, finite-time Lundberg inequalities are obtained.

On the Distribution of the Surplus Prior and at Ruin

Consider a classical compound Poisson model. The safety loading can be positive, negative or zero. Explicit expressions for the distributions of the surplus prior and at ruin are given in terms of the ruin probability. Moreover, the asymptotic behaviour of these distributions as the initial capital tends to infinity are obtained. In particular, for positive safety loading the Cramer case, the case of subexponential distributions and some intermediate cases are discussed.

ON THE MAXIMISATION OF THE ADJUSTMENT COEFFICIENT UNDER PROPORTIONAL REINSURANCE

In this note we consider how to maximise the adjustment coefficient in the case of proportional reinsurance. This complements some work of Waters (1983), where it was shown that there is a unique retention level maximising the adjustment coefficient. The advantage of our method is that only one implicit equation has to be solved.

Minimising expected discounted capital injections by reinsurance in a classical risk model

In this paper we consider a classical continuous time risk model, where the claims are reinsured by some reinsurance with retention level , where means ‘no reinsurance’ and b=0 means ‘full reinsurance’. The insurer can change the retention level continuously. To prevent negative surplus the insurer has to inject additional capital. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We show that an optimal reinsurance strategy exists. For some special cases we will be able to give the optimal strategy explicitly. In other cases the method will be illustrated only numerically.