As you keep making payments, in Reducing Balance Loans, the amount you owe is reduced and therefore you pay less interest.

## The Annuities Formula

The annuities formula can be used to find the amount of the loan that this still owing at any point in time during the life of the loan.

To calculate the amount of the loan:

$A=PR^n-\dfrac {Q(R^n-1)}{R-1}$

To calculate the repayment value:

$Q=\dfrac {PR^n(R-1)}{R^n-1}$

Where $Q$ is the amount of regular repayments per period.

### Example 1

Alana wishes to go on an international holiday. Alana decides to borrow $6,000 which she agrees to repay over three years at 6.6% p.a. (adjusted monthly). Find the instalment value. Write down the information we have. $P=6,000$ $n=3\times 12=36$ $r=\dfrac {6.6}{12}=0.55$ $R=1+\dfrac {0.55}{100}=1.0055$ Now, substitute these values into the annuities formula to find the monthly payment ($Q$). $Q=\dfrac {PR^n(R-1)}{R^n-1}$ $Q=\dfrac {6,000 (1.0055)^(36)(1.0055-1)}{1.0055^{36}-1}$ $Q=184.1672$ The value of the instalment is$184.17.

## Use of Calculator

Although you should know how to use the formulas, the easiest way to calculate the answer is to use the TVM solver on your calculator. When doing this, double check that you have entered the correct values. Bad data in $=$ bad data out!