# AN3 Annuity Examples

### Problem

A business borrows 100,000 now and repays it in 10 equal annual payments made at the end of each year. If the annual interest rate is 6%, how much will the annual payments be?

### Solution

The loan has a present value of 100,000 (PV) which is repaid over the next 10 years (n) with interest being compounded annually at 6% (i).

As the payments (Pmt) are made at the end of each year the problem is solved using the present value of an ordinary annuity formula as follows:

```PV = 100,000
n = 10
i = 6%
PV = Pmt x (1 - 1 / (1 + i)n) / i
```

Rearranging the formula to solve for Pmt

```Pmt = PV x i / (1 - 1 / (1 + i)n)
Pmt = 100,000 x 6% / (1 - 1 / (1 + 6%)10)
Pmt = 13,586.80
```

### Explanation

For the loan balance to be cleared at the end of 10 payments, the present value of the payments must be equal to the present value of the loan.

The solution to this problem simply uses the present value of an annuity formula to find the 10 payments which, at a discount rate of 6%, will give a present value of 100,000.

### AN3 Annuity Examples

This is one of many time value of money examples, discover another at the links below.