Stochastic mortality in a complex world: methodologies and applications within the affine diffusion framework

In this Thesis, we address the modelling of stochastic mortality, a key issue for life insurance, pension funds, public policy and fiscal planning. Indeed, the prospective increase of longevity can be an advantage for individuals, but it represents also a relevant social achievement. The stability and consistency of social welfare systems are put in danger worldwide due to the combined phenomenon of the progressive increase in life expectancy, along with the reduction of birth-rates in industrialized Countries. This phenomenon needs to be interpreted in the context of the connected world in which we live, where the multiple networks arising from the globalization, the Internet communication and the global economic development propagate any event in a very short time, making risks more complex. Due to their very nature, insurance and reinsurance deal with several risks on their balance sheet and, when determining the total risk of a portfolio, they need to establish the rules for aggregating the various risks that compose it. The introduction of market-consistent accounting and risk-based solvency requirements has called for the integration of mortality risk analysis into stochastic valuation models; moreover mortality-linked securities have attracted the interest of capital market investors, who in turn demand transparent tools to price demographic and financial risks in an integrated fashion. Accordingly, a coherent mathematical framework for studying the changes in financial and demographic conditions over time, is suitable. The class of the affine processes has been used in a wide range of applications in financial and actuarial sciences, thanks to its computational tractability and flexibility. For instance, affine processes have been extensively used in modelling the term structure of interest rates, that underpin extensive literatures on the pricing of bonds and interest-rate derivatives and are also at the basis of many of the pricing systems used by the financial industry. Affine models for the force of mortality have been developed in the literature under the assumption of both dependence and independence between mortality and interest rate dynamics. The core of this Thesis are the affine models and their properties for modelling the evolution of mortality. We propose and discuss two contributions: (i) we fit and compare past mortality trends among different Countries under the mathematical framework of the Feller process; (ii) we design a multiplicative affine model for the future evolution of mortality, by combining two components: the forecast provided by any existing mortality model, representing the deterministic baseline, and an affine driving process that stochastically affects the baseline over the forecasting time horizon. The so structured model not only is affine, thus fitting well our targets, but, when assessing its forecasting performance, it proves to be parsimonious and to provide a more accurate forecast with respect to the baseline. Within such a model, the affine driving factor is tasked with describing the dynamics over time of a measure of the fitting error of the existing mortality model providing the baseline and it is stochastically described by a Cox-Ingersoll-Ross process. For our numerical application, we choose, as the existing mortality model giving the baseline, the Cairns-Blake-Dowd (or M5) model, that is combined with the CIR process describing the stochastic factor affecting the baseline in a multiplicative way. The resulting model is called mCBD. Using the Italian females mortality data, for fixed ages, and implementing the backtesting procedure, over both a static time horizon and fixed-length windows rolling one-year ahead through time, we empirically test the performance of the CBD and the mCBD models in forecasting death rates. On the basis of average measures of forecasting errors and information criteria, we demonstrate that the mCBD model is a parsimonious model providing better results in terms of predictive accuracy than the CBD model and showing a stronger potential to gain accuracy in the long-run when a rolling windows analysis (dynamic approach) is performed. To conclude, in the Thesis, we explore and test the properties and capabilities of some affine models in fitting and forecasting mortality data both by themselves and as dynamic driving processes multiplying a deterministic baseline. Combining models and mixing techniques prove to give satisfactory results and show a concrete potential to bring the research forward. Our future research is thus oriented to use approaches that combine Monte Carlo simulations and benefit from the synergy between different techniques.