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

TL;DR

The paper addresses pricing net single premiums under fractional ages by deriving explicit $m$-th moment formulas for present-value lifetimes under a constant-force mortality model (C) and comparing them to (UDD) and (B) via two lemmas and $j$-partition refinements. It presents closed-form expressions for $\mathbb{E}[\nu^{mT_x}]$, $\mathbb{E}[T_x^m \mathbb{1}_{\{l\le T_x<l+n\}}]$, and related quantities, with extensions to partitions of each year and to $[jT_x+1]$-type claims, all validated against real mortality data and Gompertz models. The authors illustrate the impact of fractional-age interpolation on net premiums using the Lithuanian data and Gompertz benchmarks, providing practical formulas for actuarial pricing across product types. Overall, the work offers actionable, data-backed tools for actuaries to incorporate fractional ages into premium calculations while clarifying the trade-offs among interpolation schemes.

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 $ν^{T_x}$, $T_x$, $T_xν^{T_x}$, etc., where $T_x$ denotes the future lifetime of a person who is $x\in\{0,\,1,\,\ldots\}$ years old, and $ν$ 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).

Figures (9)

• Figure 1: Example of $s(x)$ interpolation according to (UDD), (B), and (C).
• Figure 2: View of the conditional density $f_x(k+t)$ under (UDD), (C), and (B) assumptions when when $s(0)=1,\,s(1)=0.8,\,s(2)=0.7,\,s(3)=0.5,\,s(4)=0.2$.
• Figure 3: The values of ${}^{1}_{2\mid}\,\overline{A}_{x:\actuarialangle{7}}$ for $x \in \{18,\dots,70\}$ under the assumption (UDD).
• Figure 4: The values of ${}^{1}_{2\mid}\,\overline{A}_{x:\actuarialangle{7}}$ for $x \in \{18,\dots,70\}$ under the assumption (C).
• Figure 5: The values of ${}^{1}_{2\mid}\,\overline{A}_{x:\actuarialangle{7}}$ for $x \in \{18,\dots,70\}$ under the assumption (B).

Theorems & Definitions (19)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 1
• Lemma 2
• Proposition 5
• Proposition 6
• Example 1
• Example 2