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Higher Applications of Mathematics

Present value and future value

Comparing money values across time.

Formula referenceFinance mixed quiz

Formula reference

Check the key Finance formulae and when to use each one.

Mixed revision quiz

Review the whole Finance section with adaptive feedback.

Before you start

  • Understand compound growth notation.
  • Know that present value means today's equivalent value.
  • Use division by a growth factor when discounting.
  • Interpret whether a question moves forwards or backwards in time.

Method chooser

Which method do I use?

Are you finding interest on the original amount only?

Use simple interest.

Is the amount increasing by a percentage repeatedly?

Use compound interest.

Are you finding what future money is worth today?

Use present value.

Are you finding what today's money will be worth later?

Use future value.

Are there regular payments, deposits or withdrawals?

Use a payment timeline.

Are you tracking a loan balance after repayments?

Use a repayment schedule.

Are you comparing insurance policies or risk?

Use expected value and excess comparisons.

Are you working with gross pay, deductions and take-home pay?

Use net pay.

Are you comparing prices over time with inflation?

Use inflation / real value.

Finance lesson

Key idea

  • Future value tells you what an amount will grow to. Present value tells you what a future amount is worth today. These are opposite directions on a financial timeline.
  • Discounting is used when you move backwards in time. Accumulating is used when you move forwards in time. The same percentage rate can be used, but the calculation direction changes.
  1. Place the amount on a timeline.
  2. If moving forwards, multiply by the growth factor.
  3. If moving backwards, divide by the growth factor.
  4. Use the number of years or periods as the power.
  5. Compare values only when they are measured at the same point in time.

Key formulae

  • Future value: FV = PV(1 + r)n
  • Present value: PV = FV / (1 + r)n
  • Discount factor: 1 / (1 + r)n

Worked examples

Worked example 1

Finding a future value

A family invests £8,000 for a future house deposit at 3.8% for 6 years.

  1. Use FV = PV(1 + r)n.
  2. FV = 8000(1.038)⁶
  3. FV = 8000 × 1.251164... = 10009.31..

So: The future value is £10,009.31.

Worked example 2

Discounting a future payment

A grant of £12,000 will be paid in 4 years. Use a discount rate of 5% to find its present value.

  1. Use PV = FV / (1 + r)n.
  2. PV = 12000 / (1.05)⁴
  3. PV = 12000 / 1.21550625 = 9872.43..

Final step: The present value is £9,872.43.

Worked example 3

Choosing between two future payments

A business can receive £18,000 now or £21,000 in 4 years. Use a discount rate of 4%.

  1. Find the present value of the future payment.
  2. PV = 21000 / 1.04⁴ = 17948.46
  3. Compare £17,948.46 with £18,000 now.

So: £18,000 now is worth slightly more today by £51.54.

Worked example 4

Finding the required investment today

A family wants £25,000 in 7 years for university costs. The account is expected to grow at 3.5%.

  1. This is a present value question because the future target is known.
  2. PV = 25000 / 1.035⁷
  3. PV = 25000 / 1.272279... = 19649.77

Final step: They need to invest about £19,649.77 today.

Watch out

  • Multiplying when the question asks for present value.
  • Dividing when the question asks what a value grows to.
  • Forgetting that n is the number of compounding periods, not always the number of years.
  • Comparing a value today with a value in the future without discounting or accumulating.

Spreadsheet connection

Spreadsheet connection

Build investment growth or discounting tables and change the rate to test scenarios.

Open spreadsheet skill

Next step

Move into practice

Use the method notes to choose the correct financial model, then try varied rates, time periods, tables and decision contexts.

Finance mixed quiz