Several expressions of the net single premiums under the constant force of mortality

TL;DR

This paper derives formulas for calculating net single premiums under the assumption of constant mortality force, verifies them with real data, and compares different mortality assumptions and their impact on premium calculations.

Contribution

It introduces new formulas for moments of future lifetime variables under constant mortality and compares these with other mortality assumptions.

Findings

Formulas for moments of future lifetime variables are validated with real data.

Comparison shows differences between constant force, UDD, and Balducci assumptions.

Constant force assumption simplifies premium calculations and provides bounds for other models.

Abstract

In this article, we present several formulas that make it easier to compute the net single premiums when the mortality force over the fractional ages is assumed to be constant (C). More precisely, we compute the moments of the random variables $\nu^{T_x}$, $T_x$, $T_x\nu^{T_x}$, etc., where $T_x$ denotes the future lifetime of a person who is $x\in\{0,\,1,\,\ldots\}$ years old, and $\nu$ is the annual discount multiplier. We verify the obtained formulas on the real data from the human mortality table and the Gompertz survival law. The obtained numbers are compared with the corresponding ones when the survival function over fractional ages is interpolated using the uniform distribution of deaths (UDD) and Balducci's (B) assumptions. We also formulate and prove the statement on the comparison of the moments of the mentioned random variables under assumptions (C), (UDD), and (B).