Every portfolio of insurance policies incurs losses of greater and smaller amounts at irregular intervals. The sum of all the losses (paid&reserved) in any one-year period is described as the annual loss burden or as simply total incurred claims amount. The future annual loss burden is estimated on the basis of predicted claims frequency and predicted average individual claim amount, but what actually happens considerably deviates from forecasts on a purely random basis. The economic risk capital represents what needs to be stored as a hedge against adverse developments in the future. There can be relevant and extreme events that will require significant new capital infusions if they occur and this new capital will be additional to what is needed for daily operations or investment.
The occurrence of these events also impacts on the credit profile of an insurance company, thus affecting the associated credit spread dynamics. Sudden losses may cause extreme cash-flow stress and investors may require more favorable terms when offering new lines of financing as addressed by Dembo, Deuschel and Duffie. A large class of event-based credit risks cannot be hedged and handled with standard credit instruments but reinsurers have been dealing with event-based risks all along. From this point of view, re cent activity in credit-related insurance contracts could be the beginning of a new class of credit derivatives called contingent-capital instruments providing new capital if one or more events occur as outlined by Neftci.
An insurance company generally decides to transfer the high cost of contingent capital to a third party, a reinsurance company. The foremost goal of reinsurance is to absorb the impact of unexpected losses on the economic risk capital and from this point of view reinsurance coverage is one of the tool available for capital management. This capability critically depends on the characteristics of the contract and the nature of the underlying risk. By transferring the risk to reinsurer, the insurance company will lower the economic risk capital allocated for the retained risk at the cost of a reinsurance premium to be paid for the transferred risk. A pure comparison of the price of the reinsurance structure versus the cost of capital could lead to wrong conclusions and inappropriate company decisions. It is necessary to take into account not only the spread but also the allocated capital that reinsurance enables to “save” and how this structure affects the economic risk capital whose goal is to cover the volatility of probable annual claims amount not transferred to the reinsurer. The solvency and viability of an insurance company depends on probabilistic calculations of risk and critically on the size and frequency of large claims.
In this paper we are specifically interested in modelling the tail of loss severity distribution according to the extreme value theory (EVT) and the frequency distribution of claims according to both a Negative Binomial and a Poisson distribution. The extreme value theory has found more application in hydrology and climatology than in insurance and finance. This theory is concerned with the modelling of extreme events and recently has begun to play an important methodological role within risk management for insurance, reinsurance and finance. Various authors have noted that the theory is relevant to the modelling of extreme events for insurance finance through the estimation of Value at Risk and reinsurance and financial engineering of catastrophic risk transfer securization products. The key result in EVT is the Picklands-Blakema-de Haan theorem proving that for a wide class of severity distributions which exceed high enough thresholds the Generalized Pareto distribution (GPD) holds true.
We examined a large data set representing a portion of the fire claims of RAS, an italian insurance company, analyzing the losses larger than 100 thousand euro from 1990 to 2000. To protect the confidentiality of the source, the data are coded so that the exact time frame and frequency of events covered by data are not revealed. The figures and the comments do not represent the views and/or opinion of the RAS management and risk capital measures are purely indicative.
We calculate the economic risk capital defined as the difference between the expected loss, the expected annual claims amount, and the 99.93th quantile of the distribution corresponding to a Standard&Poor’s A rating. Then we simulate the impact of a quota share and an excess of loss, analyzing the reinsurance effect on total claims distribution and consecutively on the economic risk capital. We graphically represent the kernel density distributions of total claims. We develop two approaches to derive the total claims amount. A traditional approach is followed assuming that the number of big claims made on a stable portfolio during a fixed premium period is negative binomial distributed. We are concerned with fitting the Generalized Pareto distribution (GPD) to data on exceedances of high thresholds to calculate claim severity. Finally we consider the development of scenarios for loss frequency and for loss severity separately as usual in reinsurance practice. We then develop a second approach according to McNeil and Saladin’s work, extending the Peak over Thresholds method to obtain a joint description of both the severity and frequency of losses exceeding high thresholds. We also analyze and model the time dependence of our data looking for an effective model for large losses in the past so that we can extrapolate the claims process into the future.
Section 2 discusses why and how we model the probability distribution of the total large claims and how we measure the impact of the reinsurance structures on economic risk capital. Section 3 describes the dataset used and the hypothesis adopted to trend and adjust the data. We provide exploratory graphical methods to detect heavy tail behavior and dependence on time. Section 4 discusses the approaches followed to derive the total claims amount. In subsection 4.1 we introduce a traditional approach, as usual in reinsurance practice, fitting frequency distribution with particular reference to Binomial Negative distribution and separately fitting the severity distribution with Pareto Generalized distribution. In subsection 4.2 we expose the Peak over Thresholds approach: we derive a model for point process of large losses exceeding a high threshold and we obtain a joint description of frequency and severity considering the issue of trend of them. In section 5 we estimate a pure price of an excess of loss treaty adopting the family of fitted Generalized Pareto distributions. Section 6 concentrates on the simulation approach to model the total claims amount and to assess the impact of different reinsurance structures on economic risk capital. Finally, Section 7 presents conclusions and possible future extensions.
