Several facts about Theodor Wittstein, Gaetano Balducci, and some expressions of the net single premiums under their mortality assumption

TL;DR

This paper explores historical mortality assumptions by Wittstein and Balducci, and develops formulas for calculating moments of key life insurance variables under Balducci's survival assumption, verified with hypothetical data.

Contribution

It introduces new formulas for computing moments of life insurance variables assuming Balducci's survival function, linking historical assumptions with practical calculations.

Findings

Formulas for moments of lifetime variables under Balducci's assumption

Verification of formulas using hypothetical mortality data

Historical insights into Wittstein and Balducci's mortality assumptions

Abstract

The mathematical essence in life insurance spins around the search of the nu\-me\-ri\-cal characteristics of the random variables $T_x$, $\nu^{T_x}$, $T_x\nu^{T_x}$, etc., where $\nu$ (deterministic) denotes the discount multiplier and $T_x$ (random) is the future lifetime of an in\-di\-vi\-dual being of $x\in\{0,\,1,\,\ldots\}$ years old. This work provides some historical facts about T. Wittstein and G. Balducci and their mortality assumption. We also develop some formulas that make it easier to compute the moments of the mentioned random variables assuming that the survival function is interpolated according to Balducci's assumption. Derived formulas are verified using some hypothetical mortality data.