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Annuity-due

n payments of 1 at the BEGINNING of each period: an annuity-immediate valued one period later, which is where the (1 + i) factor comes from.

The formulas

Present value

$$\ddot{a}_{\overline{n}|} = \dfrac{1 - v^{n}}{d} = a_{\overline{n}|}(1+i)$$

Accumulated value

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

Relations

$$\ddot{a}_{\overline{n}|} = 1 + a_{\overline{n-1}|}, \qquad \ddot{s}_{\overline{n}|} = s_{\overline{n+1}|} - 1$$

Where it comes from

1. Each of the $n$ payments arrives one period EARLIER than in the immediate case, so every term is multiplied by $(1+i)$.
2. Hence $\ddot{a}_{\overline{n}|} = (1+i)a_{\overline{n}|} = \frac{1-v^n}{i}(1+i) = \frac{1-v^n}{d}$, using $d = i/(1+i)$.
3. Alternatively, an annuity-due is a payment now plus an $(n-1)$-payment annuity-immediate: $1 + a_{\overline{n-1}|}$.

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

• Two dots means paid early, so multiply by (1 + i). One factor, once.
• ä uses d in the denominator; a uses i. The dots and the d go together.

Traps

• Multiplying by (1 + i)ⁿ instead of (1 + i) - the shift is one period, not n.
• Reading 'payments at the start of each year for 8 years' as an immediate annuity.

Related

• Annuity-immediate
• Perpetuities
• Annuities payable m-thly

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