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When a value appreciates, it increases in value. If it is said to depreciate, it is decreasing in value.

When calculating compound interest, it is important to understand that the interest earned is added on at the end of the year. The next year the interest is calculated on this new amount. This means that going into the following year the value is higher and so the interest earned will be larger.

Example

Frederick invests £400 into his savings account with interest paid at 2% per annum into the same account. Calculate how much he has after three years.

Solution 1

Start of year 1: £400.

£400 ÷ 100 × 2 = £8 interest.

Start of year 2: £400 + £8 = £408.

£408 ÷ 100 × 2 = £8.16 interest.

Start of year 3: £408 + £8.16 = £416.16.

£416.16 ÷ 100 × 2 = £8.3232 interest = £8.32 (to the nearest penny).

At the end of three years, Frederick has £416.16 + £8.32 = £424.48.

Solution 2

At the end of each year, Frederick will have 100% + 2% = 102% of what he had at the start of the year. Finding 102% of the value means that we don’t have to keep adding the interest on.

£400 ÷ 100 × 102 = £408.

£408 ÷ 100 × 102 = £416.16.

£416.16 ÷ 100 × 102 = £424.4832 = £424.48 (to the nearest penny).

Solution 3

As the end of year value is 102% of the start of year value, we can represent this using the multiplier 1.02.

To find the value after three years of compound interest, we can calculate as follows:

400 × 1.02³ = £424.4832 = £424.48 (to the nearest penny).

This method is particularly useful when we need to calculate interest after a large number of years.

Example

A new car is bought for £45,000. Every year it depreciates by 12.5%. Calculate how much it is worth after two years.

Solution 1

Year 1: £45,000.

£45,000 ÷ 100 × 12.5 = £5,625 depreciation.

Year 2: £45,000 - £5,625 = £39,375.

£39,375 ÷ 100 × 12.5 = £4,921.875 depreciation.

End of year 2:

£39,375 - £4,921.875 = £34,453.125 = £34,453.13 (to the nearest penny).

Solution 2

The car loses value by 12.5% so at the end of the year it is with 100% - 12.5% = 87.5%.

As a multiplier, this is 87.5 ÷ 100 = 0.875.

To find the value after two years of depreciation, we can calculate:

£45,000 × 0.8752² = £34,453.125 = £34,453.13 (to the nearest penny).