Formula & Calculator
Compound Interest Formula
Calculates the future value of an investment or loan that accrues interest compounded periodically over time.
Interpretation
A = P(1 + r/n)^(nt). Future value of an investment with compound interest. P principal, r annual rate, n compounding frequency, t years. Used in finance.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| A | Final amount after interest | |
| P | Principal (initial amount) | |
| r | Annual interest rate (as a decimal) | |
| n | Number of times interest compounds per year | |
| t | Time in 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.
Worked example
Compound Interest – Two Examples
Real‑World| Parameter | Value |
|---|---|
| P | 1000 |
| r | 0.05 |
| n | 1 |
| t | 10 |
| Parameter | Value |
|---|---|
| P | 5000 |
| r | 0.04 |
| n | 4 |
| t | 5 |
Common mistakes
- Compound interest: A = P(1 + r/n)^(nt) – where P is principal, r annual rate (decimal), n compounding periods per year, t in years.
- Rate r: Must be in decimal form (e.g., 5% = 0.05) – do not use percentage directly.
- n: The number of compounding periods per year (e.g., monthly = 12, daily = 365).
- Time t: In years – if given in months, convert to years.
- Continuous: For continuous compounding, use A = P·e^(rt).
Applications
The compound interest formula, A = P(1 + r/n)^(nt), computes the future value of an investment or loan with compounding. This is essential in finance for calculating returns on investments, mortgage payments, and loan amortisation. Engineers use it in project cost estimation and life‑cycle cost analysis. In economics, it models inflation and savings growth. By understanding the effect of compounding frequency, professionals can make informed financial decisions. This formula is widely used in banking, personal finance, and business planning, enabling accurate projections of future values under various interest and compounding scenarios.
- Personal financial planning (savings, retirement funds)
- Loan and mortgage amortisation calculations
- Investment portfolio analysis and forecasting
- Engineering economics – project cost estimation and life‑cycle costing
- Economic modelling of growth and inflation
Frequently Asked Questions
The compound interest formula calculates the future value (A) of an investment or loan: A = P (1 + r/n)^(nt), where P is the principal, r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the time in years.
Using the full annual rate for each compounding period instead of dividing it by the compounding frequency first.
Simple interest is calculated only on the principal: I = Prt. Compound interest is calculated on the principal and accumulated interest.
More frequent compounding (larger n) gives a higher final amount, assuming the same annual rate, due to interest on interest.
As n → ∞, the formula becomes A = P e^(rt). This is the limiting case of continuous compounding.
The EAR is the actual annual rate accounting for compounding: EAR = (1 + r/n)^n – 1.
Use the rule of 72: t ≈ 72 / (r × 100) for annual compounding. More precisely, solve for t.
PV = FV / (1 + r/n)^(nt).
- Savings and investment growth.
- Loan repayment calculations.
- Inflation adjustments.
A higher rate leads to faster growth, especially with compounding.