Annuities Payable m-thly and Continuously

A pension pays $2,000 at the start of every month for 25 years. Value it as an annual annuity, or with the wrong frequency factor, and the liability swings by tens of thousands.

There are two equivalent methods. Method 1, the effective rate per period: convert to the effective rate per $m$-th of a year, $j=(1+i{)}^{1/m}-1$, count the $mn$ payments, and run an ordinary annuity at $j$. Method 2, the conversion factor: keep the annual count and scale by $i/i^{(m)}$:

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

Both methods give the same answer. The coefficient of an $m$-thly factor is the total paid per year, so $500 per month means a coefficient of $6,000 against $a_{\overline{n|}}^{(12)}$, not $500.

Accumulated values swap only the numerator:

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

HIGH-FREQUENCY: Forgetting to scale the per-payment amount to an annual total is the most common m-thly error.

Common mistakes

• Not scaling to the annual total. $500 per month is a coefficient of $6,000 against $a_{\overline{n|}}^{(12)}$, not $500. Using $500 understates the value by a factor of 12.
• Using $i$ instead of $\delta$ for a continuous annuity. The continuous denominator is $\delta =\ln (1+i)$, not $i$. At $i=5\%$, $\delta =0.04879$, so $\bar{a}=(i/\delta )a=1.0248a$. Using $i$ understates the continuous value.
• Swapping $i^{(m)}$ and $d^{(m)}$. End-of-period payments use $i^{(m)}$; beginning-of-period use $d^{(m)}$. The wrong one shifts every payment by one sub-period.

Bottom line

• m-thly: $a_{\overline{n|}}^{(m)}=(1-v^n)/i^{(m)}=(i/i^{(m)})a_{\overline{n|}}$; due uses $d^{(m)}$. AV swaps the numerator to $(1+i{)}^n-1$ with the denominator unchanged.
• Coefficient = total annual payment. A payment of $1/m$ made $m$ times per year totals 1 annually, so $500/month means $6,000 against $a_{\overline{n|}}^{(12)}$.
• Continuous: ${\bar{a}}_{\overline{n|}}=(1-v^n)/\delta =(i/\delta )a_{\overline{n|}}$; perpetuity ${\bar{a}}_{\overline{\infty |}}=1/\delta$, with $\delta =\ln (1+i)$.
• Continuously increasing: $(\bar{I}\bar{a}{)}_{\overline{n|}}=({\bar{a}}_{\overline{n|}}-n{v}^n)/\delta$; an exponentially increasing rate forever is worth $1/(\delta -g)$.

Exam shortcut

Convert the rate first. For $m$-thly, decide immediate ($i^{(m)}$) or due ($d^{(m)}$), then remember the coefficient is the total paid per year. For continuous, swap $i$ for $\delta$ in the denominator, or multiply the level annuity by $i/\delta$. "Denominator sets the type": $i,d,i^{(m)},d^{(m)},\delta$. "Continuous is the i-over-delta annuity." Rate proportional to $a(t)$? The integrand is a constant.

The full lesson (about 2,513 words, 17 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b

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