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

Swansea University Author: Adam Shore

Abstract

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 cla...

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Published: 2008
Institution: Swansea University
Degree level: Master of Philosophy
Degree name: M.Phil
URI: https://cronfa.swan.ac.uk/Record/cronfa42373
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Abstract: 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.
Keywords: Economic theory.
College: School of Management