Formula & Calculator
Compound Interest
The future value of an investment with interest compounded periodically.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| A | Future value | |
| P | Principal | |
| r | Annual interest rate | |
| n | Compounding periods per year | |
| t | Time | years |
What it means
The compound interest formula gives the accumulated amount A of a principal P after t years at an annual interest rate r (as a decimal), compounded n times per year. It is derived from exponential growth. This formula is the basis for most savings and loan calculations. For continuous compounding (n→∞), it becomes A = P e^(rt). The formula is used in personal finance for mortgage payments, credit card balances, investment growth, and in corporate finance for discounting cash flows. Understanding compound interest is essential for financial literacy and for economic modeling. It illustrates the power of compounding, where interest earns interest over time, leading to exponential growth of savings or debt. It is also used in calculating the effective annual rate (EAR) and in comparing different investment options.
Worked example
Compound Interest – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| P | 1000 |
| r | 0.05 |
| n | 1 |
| t | 10 |
| Parameter | Value |
|---|---|
| P | 5000 |
| r | 0.06 |
| n | 4 |
| t | 5 |
Common mistakes
- Interest rate r: Must be in decimal form (e.g., 5% = 0.05) – do not use percentage directly.
- Compounding periods n: The number of times interest is compounded per year – ensure it matches the time period.
- Time t: In years – if given in months, divide by 12.
- Future value A: Includes principal and interest – do not confuse with interest alone.
- Continuous compounding: For continuous, use A = P·e^(rt) – not this formula.
Applications
The compound interest formula, A = P(1 + r/n)^(nt), calculates the future value of an investment or loan when interest is compounded at regular intervals. This is the foundation of modern finance, used to estimate growth of savings, to calculate loan costs, and to value investments. Banks, investment managers, and individuals use it to plan for retirement, to compare financial products, and to understand the effect of compounding frequency on returns. The formula shows that higher interest rates, longer time horizons, and more frequent compounding lead to greater growth. By mastering this formula, professionals can make informed financial decisions, evaluate the impact of fees, and project future wealth. It is essential for personal finance, corporate finance, and portfolio management, enabling accurate planning and forecasting.
- Savings account growth and retirement planning
- Loan amortisation and mortgage cost estimation
- Investment product comparison (interest rate vs. compounding frequency)
- Corporate financial forecasting and capital budgeting
- Education on time value of money