Compound Interest & Future Value
Master compound interest and future value for NSW Year 12 Mathematics Standard 2. In this topic you use the formula \(FV = PV(1+r)^n\) to find how much an investment grows to, because compound interest earns interest on the interest already added.
You will learn to calculate a future value, find the interest earned and the present value, handle monthly and quarterly compounding periods, and compare simple interest (a straight line) with compound interest (an upward-bending curve) — a core Standard 2 skill for savings accounts, term deposits and comparing real-world investment options.
Theory
Compound interest is interest earned on your interest, so an investment grows by \(FV = PV(1+r)^n\). This Year 12 Standard 2 (NSW) guide shows how to find the future value, the interest earned and the present value, how to handle monthly and quarterly compounding, and how compound interest compares with simple interest.
Compound interest is interest calculated on the original amount invested (the principal or present value \(PV\)) and on the interest already added. Because each period's interest is left in the account, the next period earns interest on a larger balance — interest earning interest.
The future value \(FV\) after \(n\) compounding periods at a rate \(r\) per period is \(FV = PV(1+r)^n\). The interest earned is \(FV - PV\). If the money compounds \(m\) times a year for \(t\) years, use the periodic rate \(r/m\) and \(n = m\times t\) periods.
Unlike simple interest, which adds a fixed amount each period and grows as a straight line, compound interest grows as an upward-bending curve. Over time the compound balance pulls further and further ahead of the simple-interest balance.
The future value of a compound-interest investment is:
where \(PV\) is the present value, \(r\) the rate per period (a decimal) and \(n\) the number of compounding periods. The interest earned is:
For interest compounding \(m\) times a year for \(t\) years, use the periodic rate and period count:
How to find a compound future value
- Identify the present value \(PV\), the rate \(r\) and the term.
- Match the period. If it compounds \(m\) times a year, use rate \(r/m\) and \(n = m\times t\) periods; for yearly compounding \(n\) is just the number of years.
- Substitute into \(FV = PV(1+r)^n\).
- Evaluate in one go (do not round part-way through), then round the final amount to the nearest cent.
- Answer the question: interest earned is \(FV - PV\); to compare with simple interest, work out both and find the difference.
Substitute into \(FV=PV(1+r)^n\) and evaluate.
| \(FV\) | \(=\) | \(2500(1+0.045)^6\) |
| \(\) | \(=\) | \(2500(1.045)^6\) |
| \(\) | \(=\) | \(2500 \times 1.302260\) |
| \(\) | \(=\) | \(3255.65\) |
The future value is \(\$3255.65\).
Find the future value, then subtract the principal.
| \(FV\) | \(=\) | \(3200(1.048)^5\) |
| \(\) | \(=\) | \(3200 \times 1.264173 = 4045.35\) |
| \(I\) | \(=\) | \(FV - PV\) |
| \(\) | \(=\) | \(4045.35 - 3200 = 845.35\) |
The interest earned is \(\$845.35\).
Use the periodic rate \(r/m\) with \(m=12\) and \(n = 12\times 5 = 60\).
| \(FV\) | \(=\) | \(6000\left(1+\dfrac{0.036}{12}\right)^{60}\) |
| \(\) | \(=\) | \(6000(1.003)^{60}\) |
| \(\) | \(=\) | \(6000 \times 1.196895\) |
| \(\) | \(=\) | \(7181.37\) |
The future value is \(\$7181.37\).
Simple interest is linear; compound interest is exponential.
| \(\text{Simple: } FV\) | \(=\) | \(4000(1+0.06\times 8) = 5920.00\) |
| \(\text{Compound: } FV\) | \(=\) | \(4000(1.06)^8 = 6375.39\) |
| \(\text{Difference}\) | \(=\) | \(6375.39 - 5920.00 = 455.39\) |
Compound interest earns \(\$455.39\) more over the \(8\) years.
Common pitfalls
Frequently asked questions
What is the compound interest formula?
The future value is FV = PV(1 + r) to the power n, where PV is the amount invested now, r is the interest rate per compounding period written as a decimal, and n is the number of compounding periods. The interest earned is FV minus PV.
What is the difference between simple and compound interest?
Simple interest is calculated only on the original principal, so the balance grows by the same amount each period and forms a straight line. Compound interest is calculated on the principal plus the interest already earned, so it grows by a larger amount each period and forms an upward-bending curve. For the same rate and time, compound interest always earns more.
How do you work out compound interest that compounds monthly or quarterly?
Divide the annual rate by the number of compounding periods per year, m, to get the rate per period, and multiply the number of years by m to get n. For example, 6% per annum compounded monthly for 3 years uses r = 0.06/12 and n = 12 times 3 = 36 in FV = PV(1 + r) to the power n.
How do you find the present value from a future value?
Rearrange the formula to make PV the subject: PV = FV divided by (1 + r) to the power n. This tells you how much you would need to invest now to reach a given future value, which is called discounting.
Does compounding more often earn more interest?
Yes. For the same annual rate and term, compounding more frequently (monthly rather than yearly, say) earns slightly more, because interest is added to the balance sooner and then itself earns interest. The extra amount is usually small but it grows with a higher rate or a longer term.
What does the graph of compound interest look like?
It is an upward-bending curve that gets steeper over time, because each year's interest is larger than the last. A simple-interest graph, by contrast, is a straight line with a constant slope.