Compound Interest Rate Converter
Convert interest rates between compounding frequencies (daily, monthly, quarterly, annual). Find the effective annual rate (EAR) and compare APR vs APY
Compound Interest Rate Converter
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Formula: EAR = (1 + r/n)^n − 1
Worked example — Future Value: $691,150 | Contributed: $190,000 | Interest: $501,150 (264%)
Formula
EAR = (1 + r/n)^n − 1
EAR = (1 + r/n)^n − 1 — Where EAR = Effective Annual Rate (APY), r = Nominal annual interest rate (decimal), and n = Number of compounding periods per year (365 for daily, 12 for monthly, 4 for quarterly, 1 for annually). This formula converts any nominal rate into its true annual equivalent so you can compare rates across different compounding frequencies on an apples-to-apples basis. To convert between two compounding frequencies, first solve for EAR, then solve the inverse: r_new = n_new × ((1 + EAR)^(1/n_new) − 1). The future value of monthly contributions uses FV = PMT × ((1 + r/12)^(12t) − 1) / (r/12) since contributions are made monthly.
Worked Examples
Example 1: Retirement Savings Growth
Problem:You invest $10,000 today and add $500/month at 7% annual return for 30 years. How much will you have?
Solution:FV of initial $10,000 = $10,000 × (1 + 0.07/12)^(12×30) = $10,000 × 8.116 = $81,165 FV of $500/month = $500 × ((1.005833)^360 - 1) / 0.005833 = $500 × 1,219.97 = $609,985 Total = $81,165 + $609,985 = $691,150 Total contributed = $10,000 + $500 × 360 = $190,000 Interest earned = $691,150 - $190,000 = $501,150
Result:Future Value: $691,150 | Contributed: $190,000 | Interest: $501,150 (264%)
Example 2: Early vs Late Start Comparison
Problem:Person A starts at 25, invests $300/month for 40 years. Person B starts at 35, invests $300/month for 30 years. Both earn 7%.
Solution:Person A (40 years): FV = $300 × ((1.005833)^480 - 1) / 0.005833 = $791,957 Total contributed: $300 × 480 = $144,000 Interest: $647,957 Person B (30 years): FV = $300 × ((1.005833)^360 - 1) / 0.005833 = $365,991 Total contributed: $300 × 360 = $108,000 Interest: $257,991
Result:10 years earlier = $425,966 MORE (2.16x) with only $36,000 extra invested
Frequently Asked Questions
What is compound interest and how does it work?
APR (Annual Percentage Rate) is the nominal rate lenders quote — it does not account for compounding within the year. APY (Annual Percentage Yield), also called the Effective Annual Rate (EAR), is what you actually earn or pay once compounding is factored in. For example, a 6% APR compounded monthly produces an APY of (1 + 0.06/12)^12 − 1 = 6.168%. The gap between APR and APY widens as compounding becomes more frequent. This distinction matters enormously when comparing a savings account that compounds daily against a CD that compounds quarterly — the quoted rates can look similar but deliver meaningfully different real returns. This converter exists precisely to make those comparisons accurate.
How does compounding frequency affect returns?
The Effective Annual Rate (EAR) is the single standardized number that lets you compare any two interest rates regardless of how often they compound. Formula: EAR = (1 + r/n)^n − 1. Examples for a 6% nominal rate: compounded annually = 6.000% EAR; quarterly = 6.136%; monthly = 6.168%; daily = 6.183%. To convert a monthly rate back to an equivalent quarterly rate, first compute EAR from the monthly rate, then solve: r_quarterly = 4 × ((1 + EAR)^(1/4) − 1). The EAR is also what regulators require banks to disclose as APY, which is why comparing APY figures across institutions is the safest way to shop for savings products.
What is the Rule of 72?
Nominal rate and effective rate describe the same underlying rate from different angles. The nominal rate (APR) is the stated periodic rate multiplied by compounding periods — it ignores intra-year compounding. The effective rate (APY/EAR) is what you actually experience over a full year. The spread between them is called the compounding premium: at 12% nominal, monthly compounding yields a 12.68% effective rate — a 0.68% premium. This premium is not trivial on large balances: on a $500,000 mortgage, a 0.68% difference in true cost equals $3,400 per year. When lenders advertise a rate, always convert to EAR before comparing across products with different compounding schedules.
How much should I save monthly to reach my goal?
When accounts use different compounding frequencies, direct rate comparisons are misleading. The correct approach is to convert all rates to the same basis — usually EAR — before comparing. Example: Bank A offers 5.10% APR compounded monthly (EAR = 5.224%). Bank B offers 5.15% APR compounded quarterly (EAR = 5.263%). Despite Bank A's lower APR, Bank B's quarterly-compounded rate actually delivers a slightly higher effective yield. Step-by-step: (1) Identify each account's nominal rate and compounding frequency. (2) Compute EAR = (1 + r/n)^n − 1 for each. (3) Compare EARs directly. (4) If you need to re-express the winner in a different frequency (e.g., for a monthly-pay instrument), use r = n × ((1 + EAR)^(1/n) − 1).
What is a realistic rate of return to use?
Different financial products use different compounding conventions, and knowing them helps you interpret rates correctly. Savings accounts and money-market accounts: typically compound daily, advertise APY. U.S. mortgages and most consumer loans: compound monthly, advertise APR. Canadian mortgages: compound semi-annually by law. U.S. Treasury bonds and most government securities: compound semi-annually. Corporate bonds: compound semi-annually, quoted as a semi-annual rate × 2. Credit cards: compound daily, quoted as an annual APR. Certificates of deposit (CDs): vary — daily, monthly, or at maturity. Knowing the convention for each product lets you use this converter to produce a true apples-to-apples EAR comparison before committing funds.
What is the difference between the 30/360 and actual/365 day-count conventions when converting a daily rate?
A day-count convention decides what one period means before any compounding arithmetic runs. Under actual/365 the daily periodic rate is the nominal rate divided by 365 and is applied to the real number of elapsed days. Under 30/360 every month is treated as thirty days and the year as 360, so the periodic rate is larger but the accrual calendar is shorter. Put both quotes on the same convention before computing an effective annual rate, or the comparison reflects calendar mechanics rather than a genuine rate difference.
What is continuous compounding and where does it sit in the conversion table?
Continuous compounding is the limit as the number of periods per year grows without bound, and its effective rate is e raised to the nominal rate, minus one. For 6 percent nominal that is 6.1837 percent, barely above the 6.1831 percent produced by daily compounding. It therefore acts as the ceiling on the conversion table: no compounding schedule can push a given nominal rate to a higher effective rate than the continuous case.
Why is the equivalent nominal rate lower when compounding is more frequent?
Hold the effective annual rate fixed and the nominal quote must fall as periods are added, because each period now contributes growth that the earlier periods will themselves compound. Solving the inverse, nominal equals n times the nth root of one plus EAR, minus one, makes this explicit. It is the reason a savings account compounding daily can advertise a lower stated rate than a rival compounding annually and still pay the saver more.
How do you get the correct monthly periodic rate from a quoted APY?
Take the twelfth root of one plus the APY and subtract one, rather than dividing the APY by twelve. An APY of 6.1678 percent yields a monthly periodic rate of exactly 0.5 percent, whereas dividing gives 0.514 percent and overstates the growth. The distinction matters whenever you build a monthly cash-flow model from an advertised yield, because using the wrong periodic rate compounds the error across every period of the projection.
Why can two loans quoting the same APR still cost different amounts?
APR standardises the disclosure but not everything behind it. Two contracts can share an APR yet differ in compounding basis, in payment frequency, in how fees are amortised over the assumed term, and in whether the loan is repaid early, which changes how prepaid fees are spread. Converting both to an effective annual rate on the same compounding basis, and comparing total finance charges over the term you actually expect to hold the debt, exposes the real difference.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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