Perpetuities are annuities where a sum of money is permanently invested which provides regular payments that continue indefinitely.

## The Formula

You will be asked to find the regular payment ($Q$), the principal ($P$) and the interest rate ($r$).

$Q=\dfrac {Pr}{100}$
$P=\dfrac {100Q}r$
$r=\dfrac {100Q}P$

These formulas can be used for both simple and compounding interest rates and will only work if $Q$ and $r$ are in the same time unit.

### Example 1

A wealthy business leader wishes to provide a scholarship to assist disadvantaged students attend secondary school. To cover the fees, the scholarship would need to provide $1,500 per term. They decide to invest with an institution that offers an interest rate of 7% p.a. How much would the business leader to need invest? We are told the values of $Q$ and $r$. $Q=1,500$ per term $r=12%$ p.a. $=3%$ per term Use the perpetuity formula to find $P$. $P=\dfrac {100\times 1,500}3$ $P=50,000$ The principal required to provide$1,500 per term at a rate of 7% per annum is \$50,000.

## Use of Calculator

The TVM solver on your calculator can calculate the answer to most questions. Make sure to check the amount of marks a question is worth and show the appropriate working.

Use the formula (not the TVM solver) if the principal is unknown.