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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

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

Accumulated value

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

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

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

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