Compound Interest - Present Value & Rate
Work backwards with compound interest for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-02). The present value is the amount you need to invest now to reach a future value later, so it answers the question “how much should I put away today?”
You will find the present value needed to reach a compound interest savings goal, work out the interest rate from the growth over one year, and read a balance table to see when a savings target is passed — useful when saving for a car, a holiday or a house deposit.
Theory
The present value is the amount you must invest now to reach a future amount. Work backwards by dividing the future value by the growth factor \((1+r)^n\). This NSW Year 12 Mathematics Standard 1 guide shows how to calculate present value with compound interest, find the interest rate over one year, and read a balance table to see when a target is passed.
The present value \(PV\) is the amount you must invest now to reach a future value \(FV\) after \(n\) years of compound interest. It works backwards from \(FV=PV(1+r)^n\): divide the future value by the growth factor \((1+r)^n\).
For a positive rate the present value is always less than the future value. A longer time or a higher rate means a smaller present value is needed now. Each drop is smaller than the last, so the graph of \(PV\) against \(n\) is a decreasing curve that flattens out.
Over one year, \(FV=PV(1+r)\), so \(\dfrac{FV}{PV}=1+r\). Subtract \(1\) to find the interest rate. A year-by-year balance table shows when a balance first passes a target amount.
Divide both sides of \(FV=PV(1+r)^n\) by \((1+r)^n\):
For the rate over one year:
How to find a present value or a rate
- Write the future value, the rate as a decimal and the number of years.
- Divide the future value by \((1+r)^n\) — calculate the power first, using brackets on the calculator.
- For a rate over one year, divide \(FV\) by \(PV\) to get \(1+r\), subtract \(1\) and change to a percentage.
Divide the future value by the growth factor.
| \(PV\) | \(=\) | \(\dfrac{FV}{(1+r)^n}\) |
| \(\) | \(=\) | \(\dfrac{6500}{(1.05)^3}\) |
| \(\) | \(=\) | \(\dfrac{6500}{1.157625}\) |
| \(\) | \(=\) | \(5614.94\) |
The present value is \(\$5614.94\).
Find \(PV\), then interest \(=FV-PV\).
| \(PV\) | \(=\) | \(\dfrac{12{,}000}{(1.045)^4}\) |
| \(\) | \(=\) | \(\dfrac{12{,}000}{1.1925186}\) |
| \(\) | \(=\) | \(10{,}062.74\) |
| \(I\) | \(=\) | \(12{,}000-10{,}062.74\) |
| \(\) | \(=\) | \(1937.26\) |
Invest \(\$10{,}062.74\) now; it earns \(\$1937.26\) interest.
For one year \(FV=PV(1+r)\); divide to find the growth factor.
| \(2587.50\) | \(=\) | \(2500(1+r)\) |
| \(1+r\) | \(=\) | \(\dfrac{2587.50}{2500} = 1.035\) |
| \(r\) | \(=\) | \(0.035 = 3.5\%\) |
The annual interest rate is \(3.5\%\).
| Year | Balance |
|---|---|
| 1 | \(\$3240.00\) |
| 2 | \(\$3499.20\) |
| 3 | \(\$3779.14\) |
| 4 | \(\$4081.47\) |
Read down the balance column.
| \(\text{Year 2: } 3499.20\) | \(<\) | \(3700\) |
| \(\text{Year 3: } 3779.14\) | \(>\) | \(3700\) |
The balance first exceeds \(\$3700\) after \(3\) years.
Common pitfalls
Frequently asked questions
What is present value?
Present value is the amount you need to invest now so that, with compound interest, it grows to a set future value. For a positive interest rate it is always less than the future value.
How do you calculate present value with compound interest?
Divide the future value by the growth factor: PV equals FV divided by (1 plus r) to the power n. For example, $6500 in 3 years at 5% needs 6500 divided by 1.157625, which is $5614.94 now.
Why is present value less than future value?
The money invested now earns interest, so it grows to the future value. You therefore need to start with a smaller amount. A longer time or a higher rate makes the present value even smaller.
How do you find the interest rate from the present and future value?
Over one year, divide the future value by the present value to get 1 plus r, then subtract 1. For example, $2500 growing to $2587.50 gives 1.035, so the rate is 0.035, which is 3.5% per annum.
How do you find when a balance first passes a target?
Use a year-by-year balance table and read down the balance column until it first goes above the target. For $3000 at 8%, the balance is $3499.20 after 2 years and $3779.14 after 3 years, so it first exceeds $3700 after 3 years.