An annuity is an investment where an initial amount is invested and regular deposits are made.

## How Does an Annuity Work?

In an annuity, the interest earned is calculated regularly on the balance of the investment which increases with each regular deposit. As long as withdrawals are equal to or less than the interest gained, the annuity should last indefinitely. Superannuation is an example of an annuity.

## The Formula

The money that accumulates in an annuity is: $A_n=PR^n+\frac {Q(R^n-1)}{(R-1)}$ where $R=1+\frac r{100}$

### Example 1

Amanda wants to retire in 25 years and estimates that she will need $520,000 for her retirement. Presently, her superannuation fund has$80,000 which compounds monthly at 9% per annum.

Find the formula that could best be used to calculate the monthly contributions required.

(A) $80,000=520,000(1.0075^{300})+\frac {Q(1.0075^{300}-1)}{(1.0075-1)}$
(B) $520,000=80,000(1.0075^{25}+\frac {Q(1.0075^{25}-1)}{(1.0075-1)}$
(C) $520,000=80,000(0.09^{300}+\frac {Q(0.09^{300}-1)}{(0.09-1)}$
(D) $520,000=80,000(1.0075^{300}+\frac {Q(1.0075^{300}-1)}{(1.0075-1)}$

We are told the initial amount ( $P$) and the desired (final) amount ( $A$). $P=80,000$ $A=520,000$

Now calculate $n$ $r$ and $R$ from the information provided. $n=25\times 12=300$ $r=9%$ p.a. $=\frac 9{12}$ per month $=0.75$ per month $R=1+\frac r{100}=1+\frac {0.75}{100}=1.0075$

Now that we have each of the values, we just need to check which of the above formulas is correct by comparing them to the annuities formula. $A_n=PR^n+\frac {Q(R^n-1)}{(R-1)}$

(A) is wrong as the values of $A$ and $P$ have been reversed.

(B) is incorrect as the value of $n$ has not been multiplied by the amount of repayments.

(C) is wrong as the value of $r$ has been used instead of $R$.

(D) is correct.

## Use of Calculator

The TVM solver on your calculator is a quick way to calculate questions relating to annuities. Ensure that you double check that you have the values of $n$ and $R$ correct and that all values are correctly entered. Make sure that you show enough working to gain all of the possible marks.