### 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.

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