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Unit A2 Section 8
Compound Interest and Depreciation

Simple interest is where interest is paid at the end of the first year of an investment, that is, on the initial amount invested (the principal). This interest, though, is not re-invested and so the principal remains the same and, if interest rates do not change, the same amount of interest is paid again at the end of the next year.

Compound interest is when the interest paid each year is not paid out but is added to the principal. So the value of the account increases each year and the amount of interest earned each year will also increase. We say that the interest is compounded.

Worked Examples

1

A person invests £200 in a building society account which pays 4% interest at the end of each year.

Find the value of the investment after 3 years if the interest is compounded.

Interest of 4% will be added at the end of each year by multiplying by 1.04.

So, value of account after 1 year: £200 × 1.04 = £208
value of account after 2 years: £208 × 1.04 = £216.32
value of account after 3 years: £216.32 × 1.04 = £224.97

Note that the amount of interest added increases each year.

The final value could have been found in one calculation:

£200 × 1.043 = £224.97
2

When Gemma was born, her grandmother invested £200 in a building society for her. Find the value of this investment after 18 years if the interest rate is 6% per year and the interest is compounded.

Final value = £200 × 1.0618
= £570.87

Problems with depreciation can be tackled in a similar way.
3

A boat is bought for £14 000. Its value decreases by 8% each year. Find the value of the boat after:

Decreasing the value by 8% leaves 92% of the original value.

(a)

1 year

Value after one year = £14 000 × 0.92
= £12 880
(b)

5 years

Value after 5 years = £14 000 × 0.925
= £9227.14
(c)

10 years.

Value after 10 years = £14 000 × 0.9210
= £6081.44
4

A public address system has a cash price of £14 000. Under a hire purchase arrangement a 20% deposit is required, plus monthly instalments of £650 for two years.

How much money is saved by paying cash?

Deposit = × £14 000
= £2800
Total instalments = 24 × £650
= £15 600
Total hire purchase price = £2800 + £15 600
= £18 400
Amount saved by paying cash = £18 400 − £14 000
= £4400

Note

You can see from these worked examples that the total amount in an account after n years, An, with interest of r % is given by

An = ⎛
⎝1 + ⎞
⎠n A0

where A0 is the initial sum invested.

Exercises

Jill invests 1200 euros in a bank account which earns compound interest at the rate of 6% per annum. Find the value of her investment after:

(a)
1 year euros
(b)
2 years euros
(c)
5 years. euros

A sum of £5000 is to be invested for 10 years. What is the final value of the investment if the annual compound interest rate is:

(a)
5% £
(b)
4.8% £
(c)
7.2%? £

Which of the following investments would earn most interest?

A £300 for 5 years at 2% compound interest per annum,

B £500 for 1 year at 3% compound interest per annum,

C £200 for 3 years at 8% compound interest per annum

The value of a computer depreciates at a rate of 25% per annum. Sam bought a computer in the USA for $1600. What will be the value of the computer after:

(a)
2 years $
(b)
6 years $
(c)
10 years? $

A couple borrow £1000 to furnish their new home. They have to pay interest of 18% each year on the amount they owe.

(a)

Find the amount of interest which would be charged at the end of the first year.

£
(b)

If they repay £300 at the end of each year, how much do they owe at the end of the third year of the loan?

£