Notes › MATH 5441: Financial Mathematics Lecture 26

Varying-Payment Annuities

calendar_today November 1, 2025 article 127 words access_time 1 min replay Modified November 1, 2025

Table of Contents

Concept

Compute the present and future values of annuities with varying payment amounts

Solution

Arithmetic Progression

Payment installments increase by a fixed amount $Q$

$$\text{PV} = Pa_{\left< n \right> } + Q\left( \frac{a_{\left< n \right>} - n(1+i)^{-n}}{i} \right)$$$$\text{FV} = Ps_{\left< n \right> } + Q \left( \frac{s_{\left< n \right> } - n}{i} \right)$$

In perpetuity:

$$\text{PV}_{\infty} = \frac{P}{i} + \frac{Q}{i^2}$$

Geometric Progression

Payment installments increase by $k$%

$$\text{PV} = \begin{cases} P\left( \frac{1- \left( \frac{1 + k}{1 + i} \right)^n }{i-k} \right), \ i \ne k \\ \frac{Pn}{1+i}, \ i = k \end{cases}$$$$ \text{FV} = \begin{cases} P\,\dfrac{(1+i)^n - (1+k)^n}{\,i - k\,}, & i \ne k,\\[6pt] Pn(1+i)^{n-1}, & i = k. \end{cases} $$

In perpetuity (valid iff $i \gt k$):

$$\text{PV}_{\infty} = \frac{P}{i-k}$$

Annuity-Due

If it’s annuity-due, just multiply by $(i+1)$

Graph

mindmap ↗