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Continuously payable annuities

Payment flows at a constant rate rather than in instalments; the annuity factor is the same numerator over the force of interest.

The formulas

Present value

$${\bar{a}}_{\overline{n}|}={\int }_0^n{v}^{t}dt=\frac{1-v^n}{\delta }$$

Accumulated value

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

Conversion

$${\bar{a}}_{\overline{n}|}=\frac{i}{\delta }{a}_{\overline{n}|}$$

Where it comes from

1. A payment rate of 1 per unit time over $[t,t+dt]$ contributes $v^{t}dt$ of present value.
2. ${\int }_0^n{v}^{t}dt={\int }_0^n{e}^{-\delta t}dt=\frac{1-e^{-\delta n}}{\delta }=\frac{1-v^n}{\delta }$.
3. This is the $m\to \infty$ limit of the m-thly annuity, since $i^{(m)}\to \delta$.

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

• Continuous means divide by δ. That is the whole change.
• ā sits strictly between a and ä, closer to the middle than either.

Traps

• Using i instead of δ in the denominator.
• Forgetting that the payment given is a RATE per year, not an amount per payment.

Related

• Force of interest
• Annuities payable m-thly
• Annuity-immediate

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