Life Annuities Payable Several Times a Year¶

Definition¶

A life annuity payable $m$ times a year pays instalments of $1/m$ at intervals of $1/m$ year while the annuitant survives. The due version starts at time zero and has actuarial present value (APV) $\displaystyle \ddot a_x^{(m)}=m^{-1}\sum_{k=0}^\infty v^{k/m}{}_{k/m}p_x$ $m^{- 1} k = 0 \sum \infty v^{k / m}_{k / m} p_{x}$ .

Interpretation¶

The annual benefit rate stays one as frequency changes. Fractional-age survival must be calculated from a continuous mortality model or an explicit table interpolation rule.

Two timelines for a life annuity-due, with annual payments of 1 and quarterly payments of 1/4; the instalments after the death time T_x are drawn in grey because they are not paid.

The figure compares $m=1$ and $m=4$ for an annuitant who dies at time $T_x=2.4$ (an illustrative value). Both pay at annual rate 1, but the annual annuity-due pays 1 at times $0,1,2$ , a total of 3, while the quarterly one pays $1/4$ at times $0,1/4,\ldots,9/4$ , ten instalments totalling $2.5$ . Grey arrows are instalments that are not paid because the annuitant has died.

Raise $m$ : instalments of $1/m$ every $1/m$ year bring $\ddot a_x^{(m)}$ down towards the continuous $\bar a_x$ in both survival models offered here. The constant- $q$ model uses ${}_tp_x=p^t$ with $p=1-q$ $q$ , the CFM interpolation of the annual table ${}_kp_x=p^k$ $p^{k}$ . The uniform-deaths model uses ${}_tp_x=1-t/(\omega-x)$ for $0\le t\le\omega-x$ $x$ ; it is linear within each year, so it satisfies UDD for its own annual table. The two models have different annual tables, so their values differ even at $m=1$ $1$ .

The whole life annuity-due is the annual-payment case $m=1$ $1$ . Fractional payment dates require survival probabilities between integer ages: uniform distribution of deaths (UDD) interpolates these linearly, whereas constant force of mortality (CFM) within each year of age interpolates exponentially. Thus the same annual table can give different fractional-payment values.

Key Results¶

The immediate version starts at time $1/m$ , so $a_x^{(m)}=\ddot a_x^{(m)}-1/m$ $1/ m$ .
A temporary due version with integer term $n$ stops its sum at $mn-1$ $1$ .
Under suitable finiteness conditions, the limit as $m\to\infty$ is the continuous life annuity.
UDD and CFM can produce different values from the same annual table. Woolhouse's formula, an approximation of $\ddot a_x^{(m)}$ from annual annuity values, is outside the course scope.

Examples¶

An annuity paying monthly at annual rate 1200 pays 100 at each payment date. Its APV is $\displaystyle 100\sum_{k\ge0}v^{k/12}{}_{k/12}p_x$ for payments in advance.

Prerequisites¶

Sources¶

Life Insurance · Lecture 5

Aliases: m-thly life annuity, fractional-year annuity payments, m-thly annuity, monthly life annuity, quarterly life annuity Keywords: ä x (m), instalments of 1/m, a x (m) = ä x (m) - 1/m, Woolhouse's formula Symbols: infinity, omega, sum

Last updated: 27 October 2026 11:52