Exploring the longevity risk using statistical tools derived from the Shiryaev–Roberts procedure

Abstract

Longevity and mortality risks are daily issues for the actuarial community. To monitor this risk, various, accurate and efficient tools have been developed (e.g. Shiryaev in Theory Probab Appl 8(1):22–46, 1963; Roberts in Technometrics 8(3):411–430, 1966; Polunchenko and Tartakovsky in Ann Stat 38(6):3445–3457, 2010). A particular attention is usually spent on the detection of a change-point, i.e. the time where the level of mortality changes (El Karoui et al. in Minimax optimality in robust detection of a disorder time in poisson rate (working paper or preprint), 2015; Croix et al. in Bulletin Francais d’Actuariat 15(29):75–112, 2015). A common assumption in all these works is that the distribution of the deaths is well known not only before the change-point but also after. In the present paper, we consider a parametric framework for the distribution after the changer and we suppose that we do not know its parameter after the change-point. Thus we focus on its estimation. Our method is derived from the sequential Shiryaev–Roberts procedure. The paper starts with a presentation of this procedure and our methodology in a general framework. We provide a specific Poisson model, designed here for the study of the mortality as in Rhodes and Freitas (Advanced statistical analysis of mortality. AFIR papers. Boston Colloquia. MIB inc., Westwood, 2004) and Planchet and Tomas (Insur Math Econ 63:169–190, 2015). Two versions are analysed: in the first one, we assume that the current mortality is stable and we look for a sudden but persistent change of level. In the second model, we introduce a new set-up: the mortality evolves at a steady pace and we look for a change of the trend. Variants of these approaches are also widely expressed and are compared to benchmark methodologies. An important part of this work is devoted to the application of our methodology on real data, in a context where the change is obvious, using specific methodologies to adjust the data as in Mei et al. (Stat Sin 597–624, 2011). We also study a real insurance portfolio where no specific information might help us to understand the change, and where the change itself does not seem perceptible. For the given examples, the main results allow us to identify the change-points of the mortality when they happen and to measure the lag before clear identification of the phenomena.