This thesis constitutes a research work on Bonus-Malus (BM) systems in insurance portfolios, featuring designing pricing strategies and examining associated solvency risks. The first piece of work proposed two different pricing models via the Bayesian approach. Results imply adverse attitudes towards policyholders having a history of many small claims, when the modelling for claim severities takes different forms. On the other hand, the rest of the work dedicated to embedding a BM structure under a risk analysis framework, where the focus lies in measuring the underlying ruin probabilities. It was necessary to initially investigate a discrete model where such probability could be obtained through recursions. As for a continuous model, BM feature was reflected by a Bayesian estimator for premium adjustment. Such construction normally brings in a dependence structure to the risk model thus violating classical assumptions. One way was to inspect how different it is from a classical risk model. Then through some conditional arguments one could find accordingly a solution based on results in literature. From another perspective, it has been found that for a No Claim Discount (NCD) or a Bonus system, an alteration in premium rates could be transformed equivalently to an interchange of distribution between inter-claim times. Then some Markov properties were able to be diagnosed under higher dimensions, which leads to a further possibility of computations. Results can be found in the form of simulations.

Severe weather events, for instance, heavy rainfall, snow-melt or droughts, cause large losses of lives and money every year. Insurance companies offer some form of protection against such undesirable outcomes, and decision makers want to take precautions to prevent future catastrophes. Both, decision makers and insurance companies, are hence interested to understand which weather events induce a high risk. This information then allows the insurance companies to set premiums for their policies by predicting future losses. Further, the relationship between damages and weather is also important to assess the impact of climate change. Several aspects have to be considered in the statistical modelling of this relationship. For instance, some regions in the world are more used to severe rainfall events than others and, hence, presumably less vulnerable to small amounts of rainfall than others. Spatial statistics provides a statistical framework which allows for a spatially varying relationship while accounting for certain similarities for areas which are geographically close. Further, damages, especially large losses, are rather rare and the statistical analysis is hence usually based on a low number of observations. Methods from extreme value theory consider the modelling of such events and may hence be beneficial. This thesis aims to develop statistical models for the relationship between damages, in particular property insurance claims, and weather events, based on daily Norwegian insurance and weather data. To improve existing models, new methodology is introduced which allows for substantial flexibility of the statistical model. The risk induced by certain weather events is assumed to be spatially varying across Norway but with neighbouring regions exhibiting similar vulnerability. To account for certain non-linear effects, the class of monotonic regression functions is considered. Specifically, this work is the first to de- fine flexible dependence structures for such functions. In particular, the first approach considers a Bayesian framework and estimates are obtained by Markov chain Monte Carlo algorithms while the second approach is optimization-based. The last part of the thesis derives extreme value models for discrete data and estimates them in a Bayesian framework. In particular, a mixture model which allows for a flexible tail behaviour is motivated by an exploratory analysis of the highest claims in the data. Additionally, the data are restructured based on spatial and temporal patterns and then combined with the proposed extreme value mixture model. All these approaches, monotonic regression and extreme value analysis, lead to an improved model fit and a better understanding of the relationship between insurance claims and weather events.

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The 20th century has witnessed some of the largest and most widespread gains in human longevity ever witnessed, which show no sign of slowing down during the early years of the 21st century. The risk of further, higher than anticipated improvements in life expectancy - known as longevity risk - is now a major and growing field of study. This thesis investigates a number of theoretical and practical problems within the field of longevity risk relating to the structure and identifiability issues within many of the most common models used to study mortality rates, the construction of new mortality models, the projection of these models into the future, the impact of differences in the level and evolution of mortality rates in different populations (such as pension schemes) and the market-consistent valuation and measurement of risk in longevity-linked liabilities and securities.

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Over the last ten years there has been a renewed interest in providing agricultural insurance in developing countries. However, voluntary demand for unsubsidised insurance products has been low, particularly from the poorest farmers. Chapter One presents a model of rational demand for hedging products, where there is a risk of contractual nonperformance. Demand is characterised and bounded for risk averse and decreasing absolute risk averse decision makers. For constant absolute and relative risk averse utility functions, demand is hump-shaped in the degree of risk aversion when the price is actuarially unfair, first increasing then decreasing, and either decreasing or decreasing-increasing-decreasing in risk aversion when the price is actuarially favourable. The apparently low level of demand for consumer hedging instruments, particularly from the most risk averse, is explained as a rational response to deadweight costs and the risk of contractual nonperformance. A numerical example is presented which suggests that some of the unsubsidised weather derivatives currently being designed for and marketed to poor farmers may in fact be poor products. Chapter Two presents experimental evidence collected from a framed microinsurance lab experiment using poor subjects in rural Ethiopia. In line with the theoretical model of Chapter One, demand for actuarially unfair index insurance is hump-shaped in wealth, first increasing then decreasing. In contrast with recent field experiments where it is not possible to demonstrate that low demand for indexed insurance is `too low', use of a laboratory experiment with an objectively known joint probability distribution allows normative statements to be made about the observed level of demand. The observed level of demand for index insurance in the experiment is higher than the decreasing absolute risk averse upper bound of Chapter One, suggesting that subjects bought `too much' index insurance. Chapter Three presents a vision of insurance design for the poor. Technically optimal arrangements involve insurance providers, such as microinsurers or governments, acting as reinsurer to groups of individuals who have access to cheap information about each other, such as extended families or members of close-knit communities, who in turn offer mutual insurance to each other.

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In most developed economies, the insurance sector earns premiums that amount to around eight percent of their GNP. In order to protect both the financial market and the real economy, this results in strict regulations, such as the Solvency II Directive, which has monitored the EU insurance sector since early 2016. The largest item on general insurers’ balance sheets is often liabilities, which consist of future costs for reported claims that have not yet been settled, as well as incurred claims that have not yet been reported. The best estimate of these liabilities, the so-called reserve, is given attention to in Article 77 of the Solvency II Directive. However, the guidelines in this article are quite vague, so it is not surprising that modern statistics has not been used to a great extent in the reserving departments of insurance companies. This thesis aims to combine some theoretical results with the practical world of claims reserving. All results are motivated by the chain ladder method, and provide different reserving methods that will be introduced thoughout four separate papers. The first two papers show how claim estimates can be embedded into a full statistical reserving model based on the double chain ladder method. The new methods introduced incorporate available incurred data into the outstanding liability cash flow model. In the third paper a new Bornhuetter-Ferguson method is suggested, that enables the actuary to adjust the relative ultimates. Adjusted cash flow estimates are obtained as constrained maximum likelihood estimates. The last paper addresses how to consider reserving issues when there is excess-of loss reinsurance. It provides a practical example as well as an alternative approach using recent developments in stochastic claims reserving.

This thesis is concerned with the projection and valuation of the Egyptian social security pension system which represents the first and main pillar in pension provision in Egypt. The system is officially a funded defined benefit one and is managed by two public Funds on behalf of the state. As a result of the pre-funding strategy, the two Funds have been accumulating a large amount of assets, which makes them important institutional investors with certain characteristics. Larger contributions from employees and/or employers or cutting back some benefits cannot be recommended by the system's actuary in the event of an actuarial deficit for many political, economic and social reasons. Actuarial deficits can be dealt with by two methods, the first is higher interest rates on the invested funds from the National Investment Bank (NIB). The second is a transfer from the Treasury to shoulder the actuarial deficit alongside the annual subsidy given to improve the level of benefits. This strategy raises four very important questions. The first is whether the system's expected annual cash flow is sustainable under different demographic and economic scenarios, particularly whether the system will face any cash flow liquidity shortage in the near future. The second is how much the expected annual subsidy will be. The third is what is the required rate of interest on the invested funds to achieve the funding objective of covering 100% of the liabilities. The fourth is whether the current contribution rates are fair and adequate for new entrants at certain ages. In answering these questions a pension projection and valuation model is developed. This involves analysing and modelling the relevant demographic and economic factors in order to project them. It is found that the system will face cash flow deficits unless it liquidates some of its assets over the projection period. It is also found that the current contribution rates are more than enough to cover the cost of new entrants, even with delays in starting work as a result of the high unemployment. It is also found that a moderate rate of interest of around 6-7% per annum with salary growth of around 8-9% per annum can keep the funding level at 100% of the liabilities. Finally, a set of recommendations are made for reforming the system to enable it to survive the changes it faces in an uncertain economic and demographic environment. Some suggestions for further work are also discussed.