Compound Interest Calculator
Compute the future value of an investment with compound interest: A = P·(1 + r/n)n·t. Solve for final amount A, principal P, annual rate r, compounding frequency n, or time t — with a live growth curve, contribution breakdown, rate sensitivity chart, and step-by-step algebra.
Exponential vs linear growth
Compound interest grows exponentially — each period's interest is added to the base, so the next period earns more. Simple interest grows linearly. Over 10 years the difference is modest; over 30 years the compound curve is dramatically steeper. This is why starting early matters more than any other factor.
Frequency matters, but with diminishing returns
Going from annual to monthly compounding adds about 0.5% to the effective annual rate at r = 10%. Going from daily to hourly adds far less. The continuous-compounding limit A = P·ert is the theoretical maximum, and at typical rates it is only slightly above daily compounding.
The rule of 72
A useful mental shortcut: doubling time ≈ 72 / (r × 100). At 6% annually, money doubles in about 12 years. At 8%, about 9 years. At 12%, about 6 years. This works because ln(2) ≈ 0.693, and 72 is close to 0.693 × 100 with a small correction for higher rates.
Inflation and taxes erode returns
The formula gives the nominal amount, but real purchasing power is reduced by inflation and taxes. At 3% inflation, $20,000 in 10 years buys what $14,900 buys today. Taxes on interest further reduce the effective rate. For real planning, use rnet = r × (1 − tax) − inflation.
Compound Interest Formulas
For discrete periodic compounding.
| Quantity | Formula | Notes |
|---|---|---|
| Final amount | A = P·(1 + r/n)n·t | The master formula |
| Principal | P = A / (1 + r/n)n·t | Present value |
| Annual rate | r = n·[(A/P)1/(n·t) − 1] | From A, P, n, t |
| Compounding frequency | n = ln(A/P) / [t·ln(1 + r/n)] (implicit) | Solve numerically |
| Time | t = ln(A/P) / [n·ln(1 + r/n)] | In years |
| Interest earned | I = A − P | Compound interest |
| Effective annual rate | APY = (1 + r/n)n − 1 | Also called EAR |
| Continuous limit | A = P·er·t | As n → ∞ |
| Simple interest | A = P·(1 + r·t) | No compounding |
| Doubling time (Rule of 72) | t₂ ≈ 72 / (r × 100) | Approximation |
Effective Annual Yield vs Compounding Frequency
For a nominal rate r = 7% (0.07), what you actually earn per year.
| Frequency n | Label | APY at r = 7% | Growth factor per year |
|---|---|---|---|
| 1 | Annual | 7.000% | 1.070000 |
| 2 | Semiannual | 7.122% | 1.071225 |
| 4 | Quarterly | 7.186% | 1.071859 |
| 12 | Monthly | 7.229% | 1.072290 |
| 52 | Weekly | 7.246% | 1.072458 |
| 365 | Daily | 7.250% | 1.072501 |
| 8760 | Hourly | 7.251% | 1.072508 |
| ∞ | Continuous | 7.251% | e0.07 = 1.072508 |
Frequently Asked Questions
Compound interest is interest earned on both the original principal and on previously accumulated interest. Unlike simple interest, which grows linearly (A = P + P·r·t), compound interest grows exponentially: A = P·(1 + r/n)n·t. The more frequently interest is compounded, the more often the accumulated interest itself earns interest, and the faster the balance grows.
APR (Annual Percentage Rate) is the nominal annual rate r — it does not account for compounding within the year. APY (Annual Percentage Yield, also called EAR or Effective Annual Rate) is the rate you actually earn after compounding: APY = (1 + r/n)n − 1. At r = 7% compounded monthly, APR = 7.000% but APY = 7.229%. When comparing accounts or loans, always use APY.
Only modestly. Going from annual (n = 1) to daily (n = 365) compounding at r = 7% adds about 0.25 percentage points to the APY — from 7.000% to 7.250%. Beyond daily, the gains are negligible: hourly compounding is only 0.0007 percentage points above daily, and continuous compounding (the theoretical maximum) is essentially indistinguishable from hourly for practical purposes. Frequency matters much less than the rate itself and the time horizon.
Because the growth is exponential. Adding 10 more years to a 20-year investment does far more than adding 10 years to a 5-year investment. For example, $10,000 at 7% monthly grows to $20,097 in 10 years but to $40,388 in 20 years and $81,165 in 30 years. Each decade roughly doubles the balance at this rate. This is why the standard advice — start early, invest consistently, don't withdraw — is so powerful.
As the compounding frequency n approaches infinity, the term (1 + r/n)n·t converges to er·t, where e ≈ 2.71828. This gives the theoretical maximum: A = P·er·t. Real accounts can never exceed this, though daily compounding gets very close. At r = 7% and t = 10 years, the continuous limit is $20,137 vs $20,097 for monthly compounding — a difference of just $40, or 0.2%. Continuous compounding is mostly useful as a mathematical convenience for calculus and for pricing certain financial derivatives.
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
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