Exam FMAnnuitiesFree to read

Annuity-immediate

n payments of 1 at the END of each period: the workhorse annuity, and the form every other annuity is written in terms of.

The formulas

Present value

$$a_{\overline{n}|i}=\frac{1-v^n}{i}$$

Accumulated value

$$s_{\overline{n}|i}=\frac{(1+i{)}^n-1}{i}$$

Link

$$s_{\overline{n}|}=a_{\overline{n}|}(1+i{)}^n$$

Recursion

$$a_{\overline{n}|}=v\left(1+a_{\overline{n-1}|}\right)$$

Where it comes from

1. $a_{\overline{n}|}=v+v^2+\cdots +v^n$ is a geometric series with first term $v$ and ratio $v$.
2. Summing: $a_{\overline{n}|}=v\frac{1-v^n}{1-v}$, and since $1-v=d=iv$ this simplifies to $\frac{1-v^n}{i}$.
3. Accumulating the same payments to time $n$ multiplies by $(1+i{)}^n$, giving $s_{\overline{n}|}=\frac{(1+i{)}^n-1}{i}$.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• (1 − vⁿ)/i: the numerator is 'how much of the perpetuity you did NOT buy', the denominator turns it into an annuity.
• aₙ is a PRESENT value at time 0; sₙ is the SAME payments valued at time n.

Traps

• Using aₙ when the first payment is immediate — that is an annuity-due.
• Valuing an annuity-immediate at time 1 instead of time 0. aₙ is already one period before the first payment.

Related

• Annuity-due
• Deferred annuities
• Perpetuities
• Loan amortisation

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.