Ph.D Student Levy Alon A New View on Insurance and Pension Department of Applied Mathematics Professor Emeritus Abraham Zaks

Abstract

In Actuary, the force of mortalityx and the expected number of survivors lxas given in a life table play a key role in describing mortality and in calculating premiums of various insurance plans. By statistical methods one evaluatesx, which is used to construct a life table, whose entries are lx Both x and lx have no simple formulae.

Investigating the behavior of ex, we noticed the existence of reasonable approximation by low degree polynomials. Furthermore, we noticed that the differential equation for ex may serve to evaluate the force of mortality, x.

We study this differential equation, and we derive new mortality laws. We obtain a Gompertz-like mortality law with known functions for qx, lx and ex. We also derive mortality laws related to the normal distribution and the exponential one.

Most mortality laws were formed using the expression for the force of mortality in terms of the expectation of life. We derive a condition for a function to describe the expectation of life of some population.  In particular we determine terms under which the expectation of life increases with age. We find that the exponential distribution law may serve to separate between the cases of increasing and decreasing expectation of life with age. We study the cases in which the expectation of life is given by a polynomial. In this case, we find a mortality law for which the expectation of life is increasing in every age. For the case of polynomial expectation of life, we are able to calculate both lx and qx. Using a polynomial expression we were able to approximate the expectation of life for various populations. These approximations all share a typical root structure. Using these approximations we were able to generate approximations to lx and qx. We compared these approximations to direct polynomial least square approximation of the same order. In all cases the approximation based on polynomial approximation for the expectation of life was better. Moreover, approximation of lx based on polynomial approximation of the expectation of life did not show a relative difference of more then 2% up to ages between 75 and 80. Similar results were found for the approximation of qx. Although the least square approximation fails to follow the pattern of qx at small ages, the approximations based on polynomial approximation of the expectation of life do that reasonably well.