Compound Interest
Theory
Compound interest earns interest on interest: each period's interest is added to the principal, and the next period earns interest on the new (larger) total. The balance grows exponentially, not linearly. Use \(A = P\left(1 + \dfrac{r}{n}\right)^{nt}\), where \(n\) is the number of compounding periods per year. The more often it compounds, the more interest earned.
With compound interest, each period's interest is added to the principal, and the next period earns interest on the new (larger) total. This is "interest on interest" — balances grow exponentially, not linearly.
The compounding frequency (\(n\)) is how many times per year interest is added: annually (\(n=1\)), half-yearly (\(n=2\)), quarterly (\(n=4\)), monthly (\(n=12\)), or daily (\(n=365\)).
The rate \(r\) is always the annual rate as a decimal — the formula divides it by \(n\) for you. The exponent \(n \cdot t\) is the total number of compounding periods over the whole investment.
The compound interest formula:
Interest earned:
Compounding frequency values for \(n\):
| Compounded | \(n\) (per year) |
|---|---|
| Annually | 1 |
| Half-yearly | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
Compound depreciation (loses \(r\%\) annually):
How to calculate compound interest
- Convert the rate to a decimal (\(6\% \to 0.06\)).
- Identify \(n\) — the compounding frequency per year (annual = 1, quarterly = 4, monthly = 12).
- Compute the per-period rate \(r/n\) and the total number of periods \(n \cdot t\).
- Substitute into \(A = P(1 + r/n)^{nt}\) and evaluate.
- For interest only, subtract: \(I = A - P\).
\(n = 1\), so \(r/n = 0.05\) and \(n \cdot t = 4\).
| \(A\) | \(=\) | \(8{,}000 \times 1.05^4\) |
| \(A\) | \(=\) | \(8{,}000 \times 1.21551\) |
| \(A\) | \(=\) | \(\$9{,}724.05\) |
The final value is \(\textbf{\$9{,}724.05}\).
\(n = 4\), so \(r/n = 0.01\) and \(n \cdot t = 12\).
| \(A\) | \(=\) | \(10{,}000 \times 1.01^{12}\) |
| \(A\) | \(=\) | \(10{,}000 \times 1.12683\) |
| \(A\) | \(=\) | \(11{,}268.25\) |
| \(I\) | \(=\) | \(11{,}268.25 - 10{,}000\) |
| \(I\) | \(=\) | \(\$1{,}268.25\) |
The interest earned is \(\textbf{\$1{,}268.25}\).
\(n = 12\), so \(r/n = 0.005\) and \(n \cdot t = 60\).
| \(A\) | \(=\) | \(5{,}000 \times 1.005^{60}\) |
| \(A\) | \(=\) | \(5{,}000 \times 1.34885\) |
| \(A\) | \(=\) | \(\$6{,}744.25\) |
The final value is \(\textbf{\$6{,}744.25}\).
Each year multiplies value by \(0.85\); over \(3\) years that's \(0.85^3\).
| \(A\) | \(=\) | \(25{,}000 \times 0.85^3\) |
| \(A\) | \(=\) | \(25{,}000 \times 0.614125\) |
| \(A\) | \(=\) | \(\$15{,}353.13\) |
After \(3\) years the car is worth \(\textbf{\$15{,}353.13}\).
Common pitfalls
Frequently asked questions
What is compound interest?
Interest calculated on both the original principal and any interest already earned. The balance grows exponentially because each period's interest earns interest itself.
What is the compound interest formula?
\(A = P\left(1 + \dfrac{r}{n}\right)^{nt}\), where \(P\) is the principal, \(r\) is the annual rate as a decimal, \(n\) is the number of compounding periods per year, and \(t\) is time in years.
What does "compounded quarterly" mean?
Interest is added \(4\) times per year, so \(n = 4\). For \(3\) years compounded quarterly, the total number of periods is \(12\).
Why does compounding more often give more interest?
Because new interest is added back into the balance sooner and starts earning interest itself sooner. Monthly beats quarterly beats annual, at the same headline rate.
What is the difference between simple and compound interest?
Simple interest is only on the original principal — the balance grows in a straight line. Compound interest is on principal plus accumulated interest — the balance grows exponentially.
How do you calculate compound depreciation?
\(A = P\left(1 - \dfrac{r}{100}\right)^t\). It's compound interest with a negative rate, applied annually.
Practice Questions
11 questions available.
Practice Questions