Modeling issues in longevity and bonus-malus system - Tan, Chong It (2015)
This thesis explores several modeling issues in longevity and bonus-malus system. On one hand, research in the areas of longevity and mortality are closely related. Mortality is one of the oldest research topics in actuarial science, dates back to year 1825 when Benjamin Gompertz published the law of human mortality. On the other hand, bonus-malus systems are widely used in motor insurance to relate premium amounts to individual past claims experience. In the study of human longevity, both period and cohort survival curves have their own advantages and shortcomings. We construct a new type of survival curves called hybrid survival curve by combining the strengths of these two types of survival curves. We then propose two stochastic survival models for modeling the evolution of a hybrid survival curve. The estimated time-varying parameters are highly interpretable and their respective trends can be used as an indicator for the rectangularization of survival curve. On top of that, we demonstrate how the time-varying parameters can be extrapolated into the future to obtain projected hybrid survival curves. In recent years, a collection of stochastic mortality models were developed to forecast the uncertainties in mortality projections. In particular, the time-varying parameters in common stochastic mortality models can be used to construct mortality indexes. We study how existing mortality models can be adapted to satisfy the new-data-invariant property, a property that is required to ensure the resulting mortality indexes are tractable by market participants. We find that the adapted Model M7 (the Cairns-Blake-Dowd model with cohort and quadratic age effects) is the most suitable for constructing mortality indexes. Based on the indexes created from this model, one can write a standardized mortality security called K-forward, which can be used to hedge longevity risk exposures. We also contribute a hedging method called key K-duration that permits one to calibrate a longevity hedge formed by K-forward contracts. Our numerical illustrations indicate that a K-forward hedge has a potential to outperform a q-forward hedge in terms of the number of hedging instruments required. Traditionally, when working with cross-sectional data, the motor insurance ratemaking process is separated into two steps -- a priori ratemaking and a posteriori ratemaking. When a bonus-malus system (BMS) with a single set of optimal relativities and a set of simple transition rules is implemented, two inadequacy scenarios are induced because all policyholders are subject to the same a posteriori premium relativities independent of their a priori characteristics and the same level transitions independent of their current levels occupied. We develop alternatives to alleviate these two inadequacy scenarios under the BMS framework developed for cross-sectional data. In recent times, a panel data structure is available for the ratemaking process. For this kind of data, Boucher and Inoussa (2014) proposed a single-step ratemaking approach to estimate a priori premiums and a posteriori relativities simultaneously to tackle the first inadequacy scenario. Using our proposed varying transition rules, the second inadequacy scenario can be mitigated under the BMS framework developed for panel data.