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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

$$\bar{a}_{\overline{n}|} = \int_0^{n} v^{t}\,dt = \dfrac{1-v^{n}}{\delta}$$

Accumulated value

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

Conversion

$$\bar{a}_{\overline{n}|} = \dfrac{i}{\delta}\,a_{\overline{n}|}$$

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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