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Annuities payable m-thly

One unit per year paid in m instalments: same total, paid sooner, so the value sits between the annual immediate and due forms.

The formulas

Present value

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

Conversion

$$a_{\overline{n}|}^{(m)}=\frac{i}{i^{(m)}}{a}_{\overline{n}|}$$

Due version

$${\ddot{a}}_{\overline{n}|}^{(m)}=\frac{1-v^n}{d^{(m)}}$$

Ordering

$$a_{\overline{n}|}<a_{\overline{n}|}^{(m)}<{\bar{a}}_{\overline{n}|}<{\ddot{a}}_{\overline{n}|}^{(m)}<{\ddot{a}}_{\overline{n}|}$$

Where it comes from

1. The total paid per year is 1 either way, so only the TIMING differs; the same numerator $1-v^n$ appears, divided by whichever rate matches the payment frequency.
2. Since $i^{(m)}<i$, dividing by the smaller number gives a LARGER present value — exactly what paying sooner should do.

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

• Same numerator 1 − vⁿ every time; only the denominator changes with the payment frequency.
• The ordering a < a⁽ᵐ⁾ < ā < ä⁽ᵐ⁾ < ä is just 'paid sooner is worth more'.

Traps

• Using the monthly PAYMENT with the ANNUAL annuity factor (or vice versa) — a⁽ᵐ⁾ values one unit per YEAR.
• Dividing by i^(m)/m instead of i^(m).

Related

• Annuity-immediate
• Continuously payable annuities
• Nominal rates convertible m-thly

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