Understanding your compound interest results
The table below illustrates how compounding frequency affects the effective annual yield (EAY) of a nominal 6% annual rate. Higher compounding frequency produces incrementally more interest.
| Compounding frequency | Effective annual yield (at 6% nominal) |
|---|---|
| Annually (n=1) | 6.000% |
| Quarterly (n=4) | 6.136% |
| Monthly (n=12) | 6.168% |
| Daily (n=365) | 6.183% |
| Continuous | 6.184% |
- This calculator models contributions as fixed monthly amounts. Real investment returns are variable; using a single average return rate is a simplification that may not reflect actual portfolio volatility.
- The Rule of 72 is most accurate for annual compounding and interest rates between 4% and 12%. For rates outside this range or for more frequent compounding, the exact doubling time should be calculated as ln(2)/ln(1+r).
- Inflation is not subtracted from the future value shown. To estimate real (purchasing-power-adjusted) growth, subtract the expected inflation rate from the nominal interest rate before running the calculation.
- Investment returns from stocks, bonds, and other assets are not guaranteed and can be negative. Past returns are not indicative of future results.
What is compound interest?
Compound interest is the process by which interest earned in one period is added to the principal, so that interest in subsequent periods accrues on the larger combined amount. The more frequently interest compounds, the faster the balance grows, because earned interest begins earning its own interest sooner. Albert Einstein is often (though apocryphally) credited with calling compound interest the 'eighth wonder of the world'; regardless of the source, the mathematical phenomenon is well-established.
The compounding frequency — annual, quarterly, monthly, or daily — determines how many times per year the interest is calculated and added to the balance. Monthly compounding (n = 12) is the most common frequency for savings accounts and many investment accounts. Daily compounding (n = 365) is used by some high-yield savings accounts and money market funds. The difference between monthly and daily compounding is small for most practical purposes.
The Rule of 72 is a well-known approximation in finance: dividing 72 by the annual interest rate gives the approximate number of years required for an investment to double in value. At 6% per year, 72 ÷ 6 = 12 years. The rule is most accurate for rates between 4% and 12% and for annual compounding. This calculator shows the Rule of 72 estimate whenever a positive interest rate is entered.
How to use this compound interest calculator
- Enter the initial principal — the starting balance or lump-sum investment.
- Enter any regular monthly contribution amount. Set to zero for a single lump-sum calculation.
- Enter the annual interest rate as a percentage.
- Enter the time period in years.
- Select the compounding frequency from the dropdown (annually, quarterly, monthly, or daily).
- Read the future value, total invested, total interest earned, and estimated doubling time.
The compound interest formula
The future value of a lump sum compounded at rate r, n times per year, over t years is given by the formula below. The future value of regular monthly contributions is added using a separate annuity formula compounded at the monthly rate (r/12), because contributions are assumed to occur monthly regardless of the selected compounding frequency for the lump sum.
The Rule of 72 approximation divides 72 by the annual interest rate (expressed as a percentage, not a decimal). The approximation works because ln(2) ≈ 0.693 and the rule replaces the natural log with 0.72/rate for easy mental arithmetic.
Frequently asked questions
What is the compound interest formula?
The compound interest formula is A = P(1 + r/n)^(nt), where A is the future value, 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. For example, $10,000 invested at 5% compounded monthly for 10 years grows to: 10,000 × (1 + 0.05/12)^(12×10) = $16,470.
What is the Rule of 72?
The Rule of 72 is a mental-arithmetic approximation that estimates how many years an investment takes to double: divide 72 by the annual interest rate expressed as a percentage. At 8% per year, 72 ÷ 8 = 9 years to double. The rule is derived from the exact formula t = ln(2)/ln(1+r) and is most accurate for rates between roughly 4% and 12%.
How does compounding frequency affect growth?
More frequent compounding produces higher effective yields, because earned interest is added to the principal sooner and begins earning its own interest. However, the difference between monthly and daily compounding is small for most practical purposes. At a 6% nominal rate, monthly compounding yields 6.168% effectively while daily compounding yields 6.183% — a difference of 0.015 percentage points.
What is the difference between APY and APR?
APR (Annual Percentage Rate) is the nominal annual interest rate that does not account for compounding. APY (Annual Percentage Yield, also called effective annual rate or EAR) reflects the actual return after compounding within the year. APY = (1 + APR/n)^n − 1, where n is the number of compounding periods per year. Savings accounts advertise APY because it shows the true return earned.
Does compound interest apply to debt?
Yes. Credit cards, mortgages, student loans, and many other forms of debt accrue compound interest — meaning unpaid interest is added to the outstanding balance, and subsequent interest charges are applied to that larger amount. This is why carrying a credit card balance at high APR for a long period results in paying far more than the original borrowed amount.
References
- Brealey RA, Myers SC, Allen F. Principles of Corporate Finance (13th ed.). McGraw-Hill, 2020. Chapter 2: Discounted Cash Flow Analysis.
- Ross SA, Westerfield R, Jordan BD. Fundamentals of Corporate Finance (12th ed.). McGraw-Hill, 2019. Chapter 5: Introduction to Valuation.
- Consumer Financial Protection Bureau (CFPB). What is APY? consumerfinance.gov.
- Federal Deposit Insurance Corporation (FDIC). Understanding deposit insurance and savings account yields. fdic.gov.