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$$1$, whereas the annuity-immediate pays at $1,\ldots,n$. With annual discount factor $v=(1+i)^{-1}$ at interest rate $i$, their present values are $\displaystyle \ddot a_{\actuarialterm{n}}=\sum_{k=0}^{n-1}v^k$ and $\displaystyle a_{\actuarialterm{n}}=\sum_{k=1}^nv^k$$\sum _{k=1}^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.

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$.

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{\ddot{a}}_{\overline{n\hspace{.2em}|}}$.
• A continuous unit-rate annuity has value $\bar a_{\actuarialterm{n}}=(1-\mathrm{e}^{-\delta n})/\delta$${\mathrm{e}}^{-\delta n})/\delta$, with limit $n$ at $\delta=0$$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¶

• Actuarial Present Value
• Geometric Series
• Interest and Discounting

Related Concepts¶

• Whole Life Annuity Due
• Guaranteed Life Annuity

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