The Effect of a Threshold Proportional Reinsurance Strategy on Ruin Probabilities

In the context of a compound Poisson risk model, we define a threshold proportional reinsurance strategy: A retention level k1 is applied whenever the reserves are less than a determinate threshold b, and a retention level k2 is applied in the other case. We obtain the integro-differential equation for the Gerber-Shiu function (defined in Gerber and Shiu (1998)) in this model, which allows us to obtain the expressions for ruin probability and Laplace transforms of time of ruin for several distributions of the claim sizes. Finally, we present some numerical results.time of ruin, threshold proportional reinsurance strategy, ruin probability, gerber-shiu function

Ruin models with investment income

This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].Comment: Published in at http://dx.doi.org/10.1214/08-PS134 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ruin probability and time of ruin with a proportional reinsurance threshold strategy

In this paper, we present a threshold proportional reinsurance strategy and we analyze the effect on some solvency measures: ruin probability and time of ruin. This dynamic reinsurance strategy assumes a retention level that is not constant and depends on the level of the surplus. In a model with inter-occurrence times being generalized Erlang(n)-distributed, we obtain the integro-differential equation for the Gerber-Shiu function. Then, we present the solution for inter-occurrence times exponentially distributed and claim amount phase-type(N). Some examples for exponential and phase-type(2) claim amount are presented. Finally, we show some comparisons between threshold reinsurance and proportional reinsurance

Sharing Risk - An Economic Perspective

We revisit the relative retention problem originally introduced by de Finetti using concepts recently developed in risk theory and quantitative risk management. Instead of using the Variance as a risk measure we consider the Expected Shortfall (Tail-Value-at-Risk) and include capital costs and take constraints on risk capital into account. Starting from a risk-based capital allocation, the paper presents an optimization scheme for sharing risk in a multi-risk class environment. Risk sharing takes place between two portfolios and the pricing of risktransfer reflects both portfolio structures. This allows us to shed more light on the question of how optimal risk sharing is characterized in a situation where risk transfer takes place between parties employing similar risk and performance measures. Recent developments in the regulatory domain (‘risk-based supervision') pushing for common, insurance industry-wide risk measures underline the importance of this question. The paper includes a simple non-life insurance example illustrating optimal risk transfer in terms of retentions of common reinsurance structure

A characterisation of optimal strategies to deal with extreme events in insurance.

This thesis looks at the Actuarial area of risk, and more specifically Ruin Theory. In the ruin model the stability of an insurer is studied. Starting from capital u at time t=0, his capital is assumed to increase linearly in time by fixed annual premiums, but it decreases with a jump whenever a claim occurs. Ruin occurs when the capital is negative at some point in time. The probability that this ever happens, under the assumption that the premium, as well as the claim generating process remains unchanged, is a good indication of whether the insurer's assets are matched to his liabilities sufficiently well. If not, the insurer has a number of options available to him such as reinsuring the risk, raising the premiums or increasing the initial capital. Analytical methods to compute ruin probabilities exist only for claims distributions that are mixtures and combinations of exponential distributions. Algorithms exist for discrete distributions with few mass points. Also, tight upper and lower bounds can be derived in most cases. This thesis explores a topic of particular practical interest in queuing and insurance mathematics, namely the analysis of extreme events leading to the financial ruin of an insurance company. The phrase 'extreme events' here, means an unusually high number of claims and/or unexpectedly high claim sizes. However, similar problems also appear naturally in the context of communication networks, where extreme events are responsible for delays to messages. The proper mathematical framework for this analysis is the theory of Large Deviations, one of the most active and dynamic branches of modern applied probability. This framework provides powerful tools for computing the probability of extreme events when the more conventional approaches like the law of large numbers and the central limit theorem fail. The overall objective of this thesis is to study the linking of Large Deviation techniques with elements of control and optimisation theory. After covering the background theory required for the exploration of the ruin model, and the application of Large Deviations, we explore previous work, with a strong emphasis on methods used to calculate the ruin probability for more realistic distributions. Next, we start to explore some of the options available to the insurer should he wish to reduce his risk (but ultimately retain high profits). The first option we cover is that of taking on new business with the aim of increasing premium income to offset immediate liabilities. In doing so, we produce a simulation package that is able to compute ruin probabilities for many complicated and more realistic situations. The claims on an insurance company must be met in full, but to protect itself from large claims the company itself may take out an insurance policy. We study a combination of both proportional and excess of loss reinsurance in a Large Deviations Regime and examine the results for both the popular exponential distribution and the more realistic 'heavy tailed' gamma distribution. Finally, we discuss the findings of our work, and how our results could be beneficial to the Actuarial profession. Our investigations, although based on limited parameter values, illustrate useful conclusions on the use of alternative distributions and, consequently, are of potential value to a practitioner who, prior to making a decision about his risk, would like to know what type of new business to take on, or how much business to reinsure in order to minimise his probability of ruin, whilst maximising profit. After summarising our results and conclusions, some ideas for future research are detailed

Dependent risk modelling in (re)insurance and ruin

The work presented in this dissertation is motivated by the observation that the classical (re)insurance risk modelling assumptions of independent and identically distributed claim amounts, Poisson claim arrivals and premium income accumulating linearly at a certain rate, starting from possibly non-zero initial capital, are often not realistic and violated in practice. There is an abundance of examples in which dependence is observed at various levels of the underlying risk model. Developing risk models which are more general than the classical one and can successfully incorporate dependence between claim amounts, consecutively arriving at the insurance company, and/or dependence between the claim inter-arrival times, is at the heart of this dissertation. The main objective is to consider such general models and to address the problem of (non-) ruin within a finite-time horizon of an insurance company. Furthermore, the aim is to consider general risk and performance measures in the context of a risk sharing arrangement such as an excess of loss (XL) re insurance contract. There are two parties involved in an XL re insurance contract and their interests are contradictory, as has been first noted by Karl Borch in the 1960s. Therefore, we define joint, between the cedent and the reinsurer, risk and performance measures, both based on the probability of ruin, and show how the latter can be used to optimally set the parameters of an XL reinsurance treaty. Explicit expressions for the proposed risk and performance measures are derived and are used efficiently in numerical illustrations