# AN4 Annuity Examples

### Problem

A bank informs you that if you invest 5,000 a year for 8 years into one of its investment accounts you will have accumulated 50,000 at the end of the 8 years. Assuming the payments are made at the end of each year, what is the annual interest rate on the account?

### Solution

The payments of 5,000 (Pmt) are made for 8 years (n), and accumulate to a future value 50,000 (FV)

As the payments (Pmt) are made at the end of each year and you know the future value, the problem is solved using the future value of an annuity formula as follows:

```FV = 50,000
n = 8
Pmt = 5,000

FV = Pmt x ( (1 + i)n - 1 ) / i
```

Rearranging to solve for i gives

```
((1 + i)n - 1 ) / i = FV / Pmt
((1 + i)8 - 1 ) / i = 50,000 / 5,000
((1 + i)8 - 1 ) / i = 10
```

Due to the exponent, the solution for i can only be found by trial and error or by using the RATE function in Excel

Guessing at a value of 6%

```((1 + i)8 - 1 ) / i = ?
((1 + 6%)8 - 1 ) / 6% = 9.90
```

As this is less than the value of 10 required a higher value of i is needed

Guessing at a value of 6.5%

```((1 + i)8 - 1 ) / i = ?
((1 + 6.5%)8 - 1 ) / 6.5% = 10.08
```

As this is higher than the value of 10 required a lower value of i is needed

Guessing at a value of 6.3%

```((1 + i)8 - 1 ) / i = ?
((1 + 6.3%)8 - 1 ) / 6.3% = 10.00
```

The solution chosen is 6.3%.

Using the excel RATE function

```i = RATE(n, pmt, PV, FV, type, guess)
i = RATE(8,5000,,-50000)
i = 6.29%
```

The PV type, and guess arguments have not been used in this problem.

### Explanation

The regular investments made at the end of each year represent an annuity. As we know the future value, the future value of an annuity formula can be used to solve for the discount rate i, which is equivalent to the interest rate paid on the account.

Because of the exponent of i (sometimes called power or index) involved in the formula, the solution can only be found by trial and error by guessing an initial rate.

The RATE function in Excel effectively performs this trial and error calculation for you until it arrives at the answer (this is why the function sometimes needs a guess entered to start the process).

### AN4 Annuity Examples

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