A multivariate claim number process with simultaneous claim arrivals and its application to insurance modeling
Daniela Selch (Technische Universität München, Germany)
Joint work with Matthias Scherer

Wednesday June 4, 14:30-15:00 | session 5.7 | Insurance | room I

Recent events like floods, hurricanes, and other environmental catastrophes have shown the importance to account for dependence between different types of risks in insurance modeling. Neglecting dependence can lead to severe underestimation of risk in a portfolio perspective. We present a realistic, yet mathematically tractable model to describe the joint behavior of multiple claim arrival processes. The processes are derived from independent Poisson processes by introducing a Lévy subordinator as common stochastic clock. The model supports simultaneous claim arrivals and captures the often observable phenomenon of overdispersion in claim count data. There is a very efficient simulation routine available and distributional properties like Laplace transform, probability mass function, and (mixed) moments can be derived in closed form. A convenient approximation for the loss in a large portfolio is given as well. Furthermore, it is studied how the model affects pricing and risk management of (re-)insurance products.

Designing longevity-indexed annuity products
Kees Bouwman (Cardano Risk Management, The Netherlands)

Wednesday June 4, 15:00-15:30 | session 5.7 | Insurance | room I

Longevity risk -- the risk of unexpected increases in the systematic life expectancy of a population -- poses a great challenge to pension plans and annuity providers. This risk cannot be diversified away over a large pool of annuitants as it is systematic. It is also difficult and expensive to reinsure this risk or hedge it in the financial markets. As a result, standard life annuities have become increasingly expensive.
We explore different annuity designs where the longevity risk is shared between the insurer and the annuitant by indexing the annuity income to shocks to systematic longevity. Recently, different designs have been proposed in the literature, such as the Group self-annuitization (GSA) by Richter and Weber (2011) and the design by Denuit, Haberman, and Renshaw (2011). We consider different design elements, such as: i) the definition of the reference population and the resulting basis risk, ii) the potential to smooth longevity shocks in the annuity income and iii) limiting longevity via caps and floors on the indexation.
An important contribution of our work is that we compare different designs on their pricing and risk implications. Longevity risk is modelled by the Lee-Carter model and pricing implications are analysing by considering different assumptions for the risk premium for longevity risk.

Optimal timing for annuitization, with a jump diffusion fund and stochastic mortality
Griselda Deelstra (Université libre de Bruxelles, Belgium)
Joint work with Donatien Hainaut

Wednesday June 4, 15:30-16:00 | session 5.7 | Insurance | room I

This paper contributes in several directions to the existing research on optimal timing for annuitization. First, the mutual fund in which the individual invests before annuitization is modeled by a jump diffusion process. Second, instead of maximizing an economic utility, the stopping time maximizes the market value of future cash-flows. Third, a semi-closed form solution is proposed in terms of Expected Present Value (EPV) operators and shows that the non annuitization (or continuation) region is either delimited by a lower or an upper boundary, in the space time versus realized returns. Necessary conditions are given under which these mutually exclusive frontiers exist. Further, a method is proposed to compute the probability of annuitization. Finally, as a case study, the mutual fund is fitted to the S\&P500 and the mortality is modeled by a Gompertz Makeham law. Several numerical scenarios are discussed.