Continuous Annuities

A continuous annuity pays at a continuous rate of 1 per period instead of in discrete lumps. Its present value \(\bar a_n\) and accumulated value \(\bar s_n\) replace the interest rate \(i\) with the force of interest \(\delta = \ln(1+i)\) — the limit of the m-thly annuity as \(m\to\infty\).

By the end you'll be able to compute \(\bar a_n=(1-v^n)/\delta\) and \(\bar s_n=((1+i)^n-1)/\delta\), and explain why continuous payment is worth slightly more than the same total paid once a year.

Predict: paying continuously instead of once a year (same annual total) — is the present value larger or smaller? Slide m up to continuous and check.

Fixed at \(i=6\%\) over \(n=4\) years, this pays a total of 1 per period, split into m equal pieces. Each bar's height sits on the flat rate-1 line — its shading fades to show how much less a later piece is worth once discounted. Slide m from 1 up through 2, 4, 12, 52, then all the way to continuous, and watch the bars merge into one smooth payment density. The present value readout climbs toward the continuous limit \(\bar a_4\) as it does.

\(a_4^{(1)}\) and \(s_4^{(1)}\) = 3.4651, 4.3746 · \(\delta=\ln 1.06=0.05827\) · continuous limit \(\bar a_4 = 3.5680\) · accumulated limit \(\bar s_4 = 4.5046\)

Payment stream over [0, n] — flat rate-1 line, shaded by discount weight
payment density (shaded by discount weight) rate = 1

A continuous annuity is the m-thly annuity pushed to its limit: as the number of payments per period \(m\to\infty\), the nominal rate \(i^{(m)}\) converges to the force of interest \(\delta\), and the annuity's value converges to \(\bar a_n\).

Formal

The present value is an integral of the discount factor over the payment window. Writing the discount factor at time \(t\) as \(v^t=e^{-\delta t}\), \[\bar a_n = \int_0^n v^t\,dt = \int_0^n e^{-\delta t}\,dt = \frac{1-e^{-\delta n}}{\delta} = \frac{1-v^n}{\delta}.\] The accumulated value follows the same pattern: \(\bar s_n = ((1+i)^n-1)/\delta\). So the family slots together cleanly: annual uses \(i\), m-thly uses \(i^{(m)}\), continuous uses \(\delta\) — and since both \(i^{(m)}\) and \(d^{(m)}\) converge to \(\delta\) as \(m\to\infty\), \(\bar a_n\) is exactly that limit of \(a_n^{(m)}\). Converting from the annual annuity: \(\bar a_n=(i/\delta)\cdot a_n\).

Applied

Picture a fund paying out income smoothly all year — rent, salary, or a trust distribution — rather than in one lump at year-end. Because \(\delta = \ln(1+i) < i\), the factor \(i/\delta > 1\): spreading the same annual total across every instant moves cash flows earlier on average, so continuous payment is worth slightly more than the same total paid once a year.

Worked example

A fund pays out continuously at 50,000 per year for 10 years; \(i=5\%\). \(\delta=\ln 1.05=0.048790\). \(v^{10}=1.05^{-10}=0.613913\), so \(1-v^{10}=0.386087\). \(\bar a_{10}=0.386087/0.048790=7.91321\). \(PV=50{,}000\times7.91321\approx\) 395,660.6. Compare the ordinary annual annuity: \(a_{10}=7.721735\to 386{,}086.7\) — continuous is larger by the factor \(i/\delta=0.05/0.048790=1.02480\).

Your turn

Same setup, different numbers: a fund pays continuously at 30,000 per year for 7 years, \(i=4\%\). \(\delta=\ln 1.04=0.039221\). \(v^7=1.04^{-7}=0.759918\), so \(1-v^7=0.240082\). Now finish it: \(\bar a_7 = 0.240082/0.039221 = \) ____, and \(PV = 30{,}000\times \bar a_7 = \) ____.

Reveal the answer

\(\bar a_7 = 0.240082/0.039221 = \) 6.1213, so \(PV = 30{,}000\times6.1213 \approx\) 183,639. Set the slider above to continuous with these numbers in mind — the same integral is what's converging on the diagram.

More info — why "rate of 1" isn't "amount of 1"

"Pays at a continuous rate of 1 per period" means the annual total is 1, not that 1 arrives at every instant. Over an infinitesimal slice of time \(dt\), the payment is \(dt\) — a vanishingly small amount — and it's the discount factor \(v^t\,dt\) summed (integrated) over every slice from 0 to \(n\) that builds up to \(\bar a_n\). This is the same bookkeeping the m-thly annuity uses with \(m\) finite pieces of size \(1/m\); continuous just lets \(m\to\infty\). See the Finan guide in Dive deeper below for the full integration.

Check your understanding

Question 1 of 4

A trust distributes income continuously at a rate of 8,000 per year for 6 years, with \(i=8\%\). Find the present value using \(\bar a_6\).

Question 2 of 4

For any interest rate \(i>0\), how does \(\bar a_n\) compare to the ordinary annuity-immediate value \(a_n\), and why?

Question 3 of 4

As the payment frequency \(m\) in an m-thly annuity increases without bound (\(m\to\infty\)), what happens to \(i^{(m)}\) and the annuity value \(a_n^{(m)}\)?

Question 4 of 4

A sinking fund receives continuous deposits at a rate of 5,000 per year for 8 years, \(i=7\%\). Find the accumulated value at time 8 using \(\bar s_8\).

Recap

  • \(\bar a_n = (1-v^n)/\delta\) and \(\bar s_n = ((1+i)^n-1)/\delta\), where \(\delta=\ln(1+i)\) is the force of interest.
  • The continuous annuity is the \(m\to\infty\) limit of the m-thly annuity: \(i^{(m)} \to \delta\) and \(a_n^{(m)} \to \bar a_n\).
  • \(\bar a_n = (i/\delta)\cdot a_n\); since \(\delta < i\), continuous payment is worth slightly more than the same total paid once a year.
  • "Rate of 1 per period" is an annual total — over an instant \(dt\) the payment is \(dt\), not 1.

Dive deeper

Sources

  • The Continuous Annuity