Interest Rate Converter

Imagine you lend someone $1,000 and they agree to pay 6% interest. At the end of the year they owe you $60. Simple enough. But what if interest is calculated monthly? After January, you are owed $5 — and in February, that $5 itself earns interest. By December, you have collected slightly more than $60 even though the stated rate never changed. That gap between the number on the page and what you actually receive is the entire point of this tool.

The effective annual rate (EAR), also called the annual percentage yield (APY), tells you what a nominal rate actually produces in a full year after compounding. The formula is EAR = (1 + r/n)^n - 1, where r is the nominal annual rate and n is the number of compounding periods per year. For continuous compounding, this becomes EAR = e^r - 1. The periodic rate — r divided by n — is what actually gets applied each period; understanding it helps you see why daily-compounding credit cards feel so aggressive even at rates that sound comparable to mortgages.

Converting between compounding frequencies matters when you need to compare two products on equal footing. A mortgage compounded semi-annually (common in Canada) and a savings account compounded monthly cannot be compared by their stated rates alone. You solve by finding the EAR of each, which removes the compounding from the equation entirely. The equivalent nominal rate section of this tool goes the other direction: given a target compounding frequency, it finds the nominal rate that produces the same EAR. This is how banks price swaps, how actuaries compare annuities, and how you figure out which of two savings accounts actually pays more.

Use this tool when you are comparing any two financial products that quote rates using different compounding conventions. Mortgages, credit cards, savings accounts, GICs, bonds, and personal loans all potentially use different conventions. Before you sign anything, convert everything to EAR. The 30 seconds this takes has saved people thousands of dollars on a single loan decision.

This tool is also appropriate when you are building a spreadsheet model and need to translate an annual rate into a per-period rate, or when you are pricing an instrument in a different market convention. If you receive a yield in semi-annual terms (bond market) and need to express it as a monthly nominal rate (swap market), this conversion is the right step.

This tool is not appropriate for calculating the total cost of a loan — it does not account for fees, points, or the amortization schedule. A mortgage with a low effective rate but high origination fees may cost more total than one with a slightly higher rate and no fees. Similarly, this tool does not handle variable rates: if your rate resets annually, the effective rate of the next period is unknown and cannot be modeled here. For total loan cost, use an amortization calculator. For variable-rate products, model each period separately.

Mistake 1: Multiplying the monthly rate by 12 and calling it annual. This gives you the nominal rate, not the effective rate. A 1% monthly rate is not 12% annually — it is 12.68% (EAR). The difference matters: on a $50,000 balance that is $340 per year. The cause is that simple multiplication ignores the compounding of interest on interest. The consequence is systematically underestimating the cost of debt or overestimating what a savings rate will produce.

Mistake 2: Comparing a loan APR to a savings account APY as if they are the same metric. Lenders quote APR (nominal); savings institutions quote APY (effective). If a car loan says 7.9% APR and a savings account says 5.2% APY, the loan's true cost is higher than 7.9% while the savings account really does return 5.2%. Comparing these numbers directly makes the loan look cheaper than it is. Always convert both to the same metric — preferably EAR — before comparing.

Mistake 3: Assuming a higher stated rate always means a higher return. A 5.10% rate compounded annually pays exactly 5.10%. A 5.00% rate compounded daily pays 5.127%. The lower-stated product wins. This happens because compounding frequency amplifies the effective rate, and the amplification grows with the base rate. At low rates (under 3%) the difference is small enough to ignore; at high rates (credit cards, payday loans) it is the dominant factor in the true cost.

The core relationship is: EAR = (1 + r/n)^n - 1, where r is the nominal annual rate (as a decimal) and n is compounding periods per year. For the continuous case, the limit as n approaches infinity gives EAR = e^r - 1. Both formulas assume reinvestment of interest at the same rate — a condition that holds in deposit accounts but not always in bond portfolios where coupons must be reinvested at prevailing market rates.

To find the periodic rate from a nominal rate: periodic rate = r/n. This is the number that appears in your monthly credit card statement as the daily periodic rate (DPR) multiplied by the days in the billing cycle. A 24% nominal rate compounded daily has a daily periodic rate of 24/365 = 0.06575%, which compounds 365 times to give an EAR of 27.11%.

The reverse conversion — finding a nominal rate for a target compounding frequency that is equivalent to a known EAR — uses: equivalent nominal rate = m * ((1 + EAR)^(1/m) - 1), where m is the target number of compounding periods. For continuous: equivalent nominal = ln(1 + EAR). This identity is what lets a corporate treasurer translate a semi-annual bond yield into a monthly swap rate without either side of the trade gaining an economic advantage from the convention difference.

The EAR formula assumes that each compounding period is identical in length, which is false in practice. Credit card issuers use actual day counts in each billing cycle — February has fewer days than March, so the effective rate in February is slightly lower than in March, even with the same DPR. Bond math uses day-count conventions (Actual/360, Actual/365, 30/360) that affect how many periods a year actually contains. The standard EAR formula overstates the effective rate slightly for any instrument that uses a 360-day year, and understates it for those using Actual/365 in a leap year. In most retail contexts this is negligible; in a $100 million swap, it is a material pricing difference.