Fundamentals of Actuarial Mathematics, 3rd Edition

16 Simplifications under level benefit contracts

   

16.1 Introduction

The calculation of variances and other distributional features simplifies considerably when we have level benefits and constant interest. In fact, we can write down formulas for exact distributions of the major random variables of interest. Throughout this chapter, we consider the following setup. We have a general failure time T. We will consider insurances paying a level amount upon failure, either at the end of the year of failure or at the moment of such, and we will consider annuities paid prior to the failure of T with either a level payment or continuous payments at a level rate. In addition, we assume a constant force of interest δ.

By taking T to be T(x), this will apply to level benefit, whole-life insurances and to level benefit whole-life annuities. By taking T = min{T(x), n}, this will apply to level benefit n-year endowment policies and to level benefit n-year temporary life annuities. Our assumption does not apply to term insurance, even when there is a level benefit during the term, since the benefit drops to zero after the expiration of the contract. However, in Section 16.5 we do illustrate that the calculation of exact distributions is possible for term or deferred insurances with a level death benefit paid over the benefit period.

16.2 Variance calculations in the continuous case

It is convenient to begin with a continuous failure time T.

16.2.1 Insurances

Consider an insurance policy ...