Continuous time semi-Markov inference of biometric laws associated with a Long-Term Care Insurance portfolio

Guillaume Biessy (1)

1 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry

Unlike the mortality risk on which actuaries have been working for more than a century, the long-term care risk is young and as of today hardly mastered. Semi-Markov processes have been identified as an adequate tool to study this risk. Nevertheless, access to data is limited and the associated literature still scarce. Insurers mainly use discrete time methods directly inspired from the study of mortality in order to build experience tables. Those methods however are not perfectly suited for the study of competing risk situations. The present paper aims at providing a theoretical framework to estimate biometric laws associated with a long-term care insurance portfolio. The presented method relies on a continuous-time semi-Markov model with three states: autonomy, dependency and death. The dependency process is defined using its transition intensities. We provide a formula to infer the mortality of autonomous people from the general population mortality, on which we ought to have more reliable knowledge. We then propose a parametric expression for the remaining intensities of the model. Incidence in dependency is described by a logistic formula. Under the assumption that the dependent population is a mixture of two populations with respect to the category of pathology that caused dependency, we show that the resulting intensity of mortality for dependent people takes a very peculiar form, which is semi-Markov. Estimation of parameters relies on the maximum likelihood method. A parametric approach eliminates issues related to segmentation in age categories, smoothing or extrapolation at higher ages. While creating model uncertainty, it proves very convenient for the practitioner. Finally, we provide an application using data from a real long-term care insurance portfolio.

Mots clés: Long-Term Care Insurance; continuous time semi-Markov process; competing risks; maximum likelihood; mixture model; parametric model

Domaines: Applications [stat.AP]

Guillaume Biessy. Continuous time semi-Markov inference of biometric laws associated with a Long-Term Care Insurance portfolio. 2015. ⟨hal-01220564v2⟩