Actuarial Mathematics. Time-Continuous Models for Life Insurance

The aim of these Lecture Notes is to complement the material, concerning Life Insurance, presented in the textbook by A. Olivieri and E. Pitacco, “Introduction to Insurance Mathematics. Technical and Financial Features of Risk Tansfers”, by providing the basics of a time-continuous setting of actuarial models for life insurance products.A time-continuous setting can extend the reader knowledge by offering a theoretical framework, however encompassing issues of interest also from a practical perspective, for example providing help in understanding approximations frequently adopted in the actuarial practice.These Lecture Notes are organized as follows. Chapter 1 aims at explaining the advantagesof a time-continuous setting with respect to a time-discrete one. Time-continuous biometric models are described in Chapter 2, where special attention is placed on parametric modelling and various mortality “laws”. Approximations adopted in the actuarial practice are also presented. Finally, an introduction to parametric models in a dynamic mortality context is provided. To a large extent, life insurance calculations are based on the concept of “actuarial value”, that is, expected present value of random cash flows. This topic is addressed in Chapter 3, where calculation of premiums for various insurance products is also presented. Despite the prominent role of expected present values in traditional actuarial calculations, a sound stochastic approach calls for analysis of probability distributions of present values and related dispersion measures.This topic concludes the chapter. Mathematical reserves are presented in Chapter 4. Thanks to the reserving concept, the fundamental components of the life insurance activity, that is, “risk” and “savings”, can be singled-out, together with a corresponding splitting of the expected profits