Notes › MATH 5441: Financial Mathematics Lecture 14

Annuities Intro

calendar_today October 25, 2025 article 164 words access_time 1 min replay Modified October 26, 2025

Table of Contents

Annuity

Fixed payments at specific time intervals
Objective is to find the present value of the $n$’th payment

Geometric Series

The $n$’th partial sum is

$$S_{n} = a \frac{1-r^{n+1}}{1-r}$$

Annuity Into Geometric Series

$a = P(1-i)^{-1}$
$r = (1-i)^{-1}$
The formula becomes:

$$ \begin{align} \text{PV}_{n} &= P(1+i)^{-1} \frac{1-((1+i)^{-1})^{(n-1)+1}}{1-(1+i)^{-1}} \\ &= P \frac{1-(1+i)^{-n}}{i} \end{align} $$

The shorthand notation for the present value of an annuity with 1 dollar invested for $n$ time periods at an interest $i$ is $a_{\langle n \rangle}$

Accumulated Value of Annuity

Denoted $s_{\langle n \rangle}$
Just accumulating the present value over $n$ payment periods (multiplying by $(1+i)^n$)

$$s_{\langle n \rangle} = \frac{(1+i)^n - 1}{i}$$

Sum and Difference Identities

$$ \begin{aligned} a_{\langle m+n \rangle} &= a_{\langle m \rangle} + \nu^{m} a_{\langle n \rangle} \\ a_{\langle m-n \rangle} &= a_{\langle m \rangle} - \nu^{m-n} a_{\langle n \rangle} \\ s_{\langle m+n \rangle} &= s_{\langle m \rangle} + (1+i)^{m} s_{\langle n \rangle} \\ s_{\langle m-n \rangle} &= s_{\langle m \rangle} - (1+i)^{m-n} s_{\langle n \rangle} \end{aligned} $$

Graph

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