Geometric Sequences and Series

A popular mathematical concept which recurs in finance is the idea of an infinite geometric sequence. Any term Un in a geometric sequence is defined as follows:

Un+1 = Un * r

The parameters to define any sequence is the first term (denoted as 'a') as well as the ratio between terms (denoted as 'r'). So an infinite geometric sequence is [a, ar, ar^2, ar^3, ...]

A geometric sequence value converges to a finite value IFF r is less than 1 as n approaches ∞. Therefore, the sequence can be summed to create a series (the addition of all terms) which is denoted as 'S'.

By definition:

S = a + ar + ar^2 + ar^3 + ...
rS = ar + ar^2 = ar^3 + ar^4 + ...
(1 - r)S = a
∴S = a / [1 - r]

Does this look familiar? It's the underlying math for many assumptions in finance including discounting growing cash flows such as the Dividend Discount Model or some other perpetuity.

An annuity (a perpetuity with a termination point) can be calculated as the subtraction of the value from 0 to infinity to the value at termination, n, to infinity (at which point the numerator of the equation for S would be (ar^n).

Annuity = [a - ar^n] / [1 - r]
= a (1 - r^n) / [1 - r]

This formula is useful for computing the value of coupons in a bond (which terminates at maturity) or securities which pay dividends are sold, called or otherwise terminated.