Payable Continuously

Study guide: Exam FM-2

An annuity payable continuously pays at a constant rate of 1 per year, with payments flowing continuously. The present value of a continuous n-year annuity is denoted \bar{a}_{\overline{n}|}:

\bar{a}_{\overline{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}

• Here \delta is the Force of Interest.
• This is the limiting case of a Payable m-thly annuity as m \to \infty.

Continuous Annuity Present Value Find the present value of a 5-year continuous annuity paying at rate 1000 per year, at \delta = 0.07.

Answer \text{PV} = 1000 \cdot \bar{a}_{\overline{5}|} = 1000 \cdot \frac{1-e^{-0.35}}{0.07} = 1000 \cdot \frac{1-0.70469}{0.07} = 1000 \times 4.219 = 4219