Sinking Fund (Future Value)

A sinking fund is a stream of equal payments (or deposits) at regular intervals typically annually or semi-annually.
This is an example of a Geometric Sequence where a series of numbers are multipled by a fixed amount.

Future Value of a Sinking Fund

An ordinary sinking fund is a sinking fund in which the first payment occurs at the end of the first period.
The following formula can be used to calculate the future value of an ordinary sinking fund.

Example 1

What is the future value of your investment if you deposit £100 every 4 months for 3 years in an account that pays 5% interest every year compounded quarterly.

Example 2

You want to put some money away every 6 months and then after 15 years have a lump sum of £20,000.
How much do you need to save every month if you put the money into an account that pays 5.6% interest compounded semi-annually.

Proof - Geometric Series

We know the formula for a single future value
Therefore the future value of a sinking fund would be the sum of all the individual future values.
This sum can be represented as a Geometric Series.
We know the sum of a geometric series so we can make the necessary substitution.

Proof - Derived from PV

An alternative way of arriving at this equation is to substitute the value of PV in the equation.
Substituting PV for the Present Value of an Annuity formula.