Compound interest

Compound interest is worked out on the amount in the account each year, including the interest already added, so it grows faster than simple interest. Multiply by the same multiplier every year: A = P(1 + r)ⁿ. Values that fall by a percentage each year (depreciation) work the same way with (1 − r).

Key facts

  • Compound interest: A = P(1 + r)ⁿ.
  • Depreciation (value falls by r each year): A = P(1 − r)ⁿ.
  • Compounded monthly: use the monthly rate (r ÷ 12) and the number of months (12n).
  • The multiplier for a 4% rise is 1.04; for a 15% fall it is 0.85.
  • Inflation raises prices like compound interest.

Worked examples

Example 1

£2,000 is invested at 5% compound interest per year. Find its value after 2 years.

  1. Year 1: multiply by 1.05. 2,000 × 1.05 = 2,100
  2. Year 2: multiply by 1.05 again. 2,100 × 1.05 = 2,205

Answer: £2,205

Example 2

£2,000 is invested at 3% compound interest per year. Find its value after 5 years.

  1. Use A = P(1 + r)ⁿ with r = 0.03 and n = 5. A = 2,000 × 1.03⁵
  2. Work it out. A = 2,318.548… ≈ 2,318.55

Answer: £2,318.55

Example 3

A house is worth £200,000. Its value rises by 4% each year. What is it worth after 3 years?

  1. A 4% rise each year means multiplying by 1.04 each year. 200,000 × 1.04³
  2. Work it out. = 224,972.80

Answer: £224,972.80

Example 4

£1,000 is invested for 3 years at 5% per year. How much more is it worth with compound interest than with simple interest?

  1. Simple interest: 1,000 + 3 × 50 = 1,150
  2. Compound interest: 1,000 × 1.05³ = 1,157.625
  3. Difference: 1,157.63 − 1,150 = 7.63

Answer: £7.63

Example 5

A car bought for £18,000 loses 15% of its value each year. What is it worth after 3 years?

  1. Down 15% → × 0.85 each year. 0.85³ = 0.614125
  2. Multiply. 18,000 × 0.614125 = 11,054.25

Answer: £11,054.25

Example 6

£10,000 is invested at 9% per year, compounded monthly, for 2 years. Find the final amount.

  1. Monthly rate: i = 0.09 ÷ 12 = 0.0075
  2. Number of months: n = 24
  3. Use the formula. A = 10,000 × 1.0075²⁴ ≈ 11,964.14

Answer: £11,964.14

Example 7

£6,000 is invested at 2.5% compound interest per year for 4 years. Find its value, to 2 decimal places.

  1. Use the power of the multiplier. 6,000 × 1.025⁴ = 6,622.88

Answer: £6,622.88

Example 8

How many whole years does it take £4,000 to grow to more than £5,000 at 3.5% compound interest?

  1. Work out the value after different numbers of years. 4,000 × 1.035⁶ = 4,917.02
  2. 7 years is the first to go over. 4,000 × 1.035⁷ = 5,089.12

Answer: 7 years