Notes › MATH 5441: Financial Mathematics Lecture 25

Continuous Annuities

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calendar_today November 1, 2025 article 65 words access_time 1 min replay Modified November 1, 2025

Table of Contents

Concept

Annuities that accrue interest compounded continuously for recurring payments (using the force of interest, $\delta$)
Notation: $\bar{a}_{\left< n \right>}$ or $\bar{s}_{\left< n \right>}$

Solution

Constant Force of Interest

Since $\delta$ is constant, then $i = e^{\delta} - 1$

$$\bar{a}_{\left< n \right>} = \frac{1 - e^{-\delta n}}{\delta}$$$$\bar{s}_{\left< n \right>} = \frac{e^{\delta n} - 1}{\delta}$$

Varying Force of Interest

$$\bar a_{\langle n\rangle}=\int_{0}^{n}e^{-\int_{0}^{t}\delta_{\tau}\,\mathrm{d}\tau} \, \mathrm{d}t$$$$\bar s_{\langle n\rangle}=\int_{0}^{n} e^{\int_{t}^{n}\delta_{\tau}\,\mathrm{d}\tau}\,\mathrm{d}t$$

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