Annuity Certain¶

Definition¶

An annuity certain makes specified payments independently of survival. For $n$ annual unit payments, the annuity-due pays at $0,\ldots,n-1$ $1$ , whereas the annuity-immediate pays at $1,\ldots,n$ . With annual discount factor $v=(1+i)^{-1}$ at interest rate $i$ , their present values are $\displaystyle \ddot a_{\actuarialterm{n}}=\sum_{k=0}^{n-1}v^k$ and $\displaystyle a_{\actuarialterm{n}}=\sum_{k=1}^nv^k$ $k = 1 \sum n v^{k}$ .

Interpretation¶

This is the deterministic building block for life annuities, such as the whole life annuity-due. The due/immediate distinction describes payment timing, not uncertainty.

Two timelines at v = 0.9: an annuity-due with unit payments at times 0, 1, 2 worth 2.71, and an annuity-immediate at times 1, 2, 3 worth 0.9 times as much, 2.439.

The figure uses the Examples case with $v=0.9$ and $n=3$ $3$ . Each red arrow is a unit payment, and the number below it is its present value $v^k$ . The annuity-due pays at times $0,1,2$ and is worth $1+0.9+0.81=2.71$ $2.71$ . The annuity-immediate makes the same three payments one year later, so every present value, and hence the total, is multiplied by $v$ : $a_{\actuarialterm{3}}=0.9(2.71)=2.439$ $2.439$ .

Change $v$ and $n$ : each arrow shrinks to its present value $v^k$ . The immediate row is the due row moved one year later, so $a_{\actuarialterm{n}}=v\,\ddot a_{\actuarialterm{n}}$ $v \overset{a}{¨}_{n}$ .

Because these payments are certain, their actuarial present value equals their ordinary discounted value. The geometric series formula evaluates the finite sums of powers of $v$ and gives the closed forms below.

Key Results¶

With $d=1-v\ne0$ $0$ , $\ddot a_{\actuarialterm{n}}=(1-v^n)/d$ and $a_{\actuarialterm{n}}=v\ddot a_{\actuarialterm{n}}$ $v \overset{a}{¨}_{n}$ .
A continuous unit-rate annuity has value $\bar a_{\actuarialterm{n}}=(1-\mathrm{e}^{-\delta n})/\delta$ $e^{- δ n}) / δ$ , with limit $n$ at $\delta=0$ $0$ .
Paying $1/m$ in advance every $1/m$ year gives $\displaystyle m^{-1}\sum_{k=0}^{mn-1}v^{k/m}$ .
At positive interest, the infinite-duration versions ( $n \to$ $n\to\infty$ $\infty$ ) are called perpetuities, with $\ddot a_{\actuarialterm{\infty}}=1/d$ and $a_{\actuarialterm{\infty}}=v/d=1/i$ $1/ i$ .

Examples¶

At $v=0.9$ $0.9$ , three unit payments in advance have value $1+0.9+0.81=2.71$ $2.71$ . Moving all payments one year later gives $0.9(2.71)=2.439$ $2.439$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 4

Aliases: annuity due certain, annuity immediate certain, continuous annuity certain, certain annuity Keywords: perpetuity, present value of n level payments, (1 - v^n)/d, a angle n = v times ä angle n, geometric series of discount factors Symbols: delta, infinity, sum

Last updated: 14 October 2026 08:21