Continuous Annuity

Study guide: Exam FM-2

The continuous annuity \bar{a}_{\overline{n}|} is the Present Value of a payment stream flowing continuously at a constant rate of 1 per unit time over [0, n], discounted at a constant Force of Interest \delta.

\bar{a}_{\overline{n}|} = \int_0^n e^{-\delta t}\,dt

= \frac{1 - v^n}{\delta}

\bar{s}_{\overline{n}|} = \int_0^n e^{\delta(n-t)}\,dt

= \frac{(1+i)^n - 1}{\delta}

• Here v = e^{-\delta} is the annual discount factor and \delta = \ln(1+i) links the force of interest to the effective annual rate i.
• Equivalent forms: \bar{a}_{\overline{n}|} = \dfrac{1 - e^{-\delta n}}{\delta} and \bar{s}_{\overline{n}|} = \dfrac{e^{\delta n} - 1}{\delta}.
• Compared with the annuity-immediate a_{\overline{n}|} = (1 - v^n)/i, the interest rate i in the denominator is replaced by \delta. Since \delta < i, we have \bar{a}_{\overline{n}|} > a_{\overline{n}|} — continuous receipt is worth more than end-of-period receipt.
• Accumulated Value and present value are linked by \bar{s}_{\overline{n}|} = (1+i)^n\,\bar{a}_{\overline{n}|}.

Present Value of a Continuous Annuity Payments flow continuously at a rate of $1{,}000 per year for 10 years at a force of interest \delta = 0.06. Find the present value.

Answer

\begin{align*}
\bar{a}_{\overline{10}|} &= \frac{1 - e^{-0.06 \times 10}}{0.06} \\
&= \frac{1 - 0.5488116}{0.06} \\
&= \frac{0.4511884}{0.06} \\
&= 7.519807 \\
PV &= 1{,}000 \times 7.519807 \\
&= \$7{,}519.81
\end{align*}

Accumulated Value of a Continuous Annuity A fund receives contributions continuously at $500 per year for 8 years. The annual effective interest rate is i = 5\%. Find the accumulated value at time 8.

Answer First convert to the force of interest: \delta = \ln(1.05) = 0.04879016.

\begin{align*}
\bar{s}_{\overline{8}|} &= \frac{(1.05)^8 - 1}{0.04879016} \\
&= \frac{1.4774554 - 1}{0.04879016} \\
&= \frac{0.4774554}{0.04879016} \\
&= 9.785895 \\
AV &= 500 \times 9.785895 \\
&= \$4{,}892.95
\end{align*}