The plain-language guide
Why APR and APY are two numbers, not one
In 1968 Congress told lenders to disclose borrowing costs as an Annual Percentage Rate. 23 years later it told banks to advertise deposit returns as an Annual Percentage Yield. Two Acts, two rate conventions, one underlying interest rate. That legislative split, not the math, is the reason a converter between APR and APY needs to exist at all.
The statutory origin
Two laws, fifty-three years apart, split a single interest rate in two
The Truth in Lending Act arrived as Title I of the Consumer Credit Protection Act, Public Law 90-321, signed in 1968. Its implementing rule is Regulation Z, today codified at 12 CFR Part 1026, with the APR computation rules at §1026.22 and Appendix J. Reg Z forces a lender to quote an APR: the nominal yearly rate, the per-period rate multiplied by the number of periods, with compounding deliberately left out of the headline figure.
The Truth in Savings Act came in 1991, carried inside the Federal Deposit Insurance Corporation Improvement Act of that year. Its rule is Regulation DD, codified at 12 CFR Part 1030, with the APY formula spelled out at §1030.7 and Appendix A. Reg DD requires the opposite convention: deposit accounts must be advertised in APY, the effective yearly rate, with compounding already folded in.
Both rules were originally written and enforced by the Federal Reserve Board. After the 2010 Dodd-Frank reforms created the Consumer Financial Protection Bureau, rulewriting authority for both moved to the CFPB, which is why the Part numbers shifted from the old 12 CFR 226 and 230 into the 1026 and 1030 ranges. One bureau now owns both halves, but they still speak different dialects of the same arithmetic, because they descend from two separate Acts written 23 years apart. That is the accident this tool reconciles. Nothing in the math requires two numbers. The law requires two numbers.
| Truth in Lending Act, 1968 | Truth in Savings Act, 1991 | |
|---|---|---|
| Implementing rule | Regulation Z (12 CFR Part 1026) | Regulation DD (12 CFR Part 1030) |
| Applies to | Consumer loans and credit | Deposit accounts and CDs |
| Rate it mandates | APR (nominal) | APY (effective) |
| Original enforcer | Federal Reserve Board | Federal Reserve Board |
| Current enforcer | CFPB (since 2011) | CFPB (since 2011) |
| Why the mandated number looks the way it does | APR omits compounding, so the lender's headline rate is the lower of the pair | APY includes compounding, so the bank's headline yield is the higher of the pair |
The quiet asymmetry
Why lenders quote the smaller number and banks quote the bigger one
The two statutes produce a quiet asymmetry. For any rate that compounds more than once a year, APY is the larger figure and APR is the smaller one. Compounding can only add to an effective return, never subtract from it.
Reg Z hands lenders the smaller-looking APR. Reg DD hands banks the larger-looking APY. In both cases the regulator chose the number that flatters the advertiser: a credit card issuer leads with the lower nominal rate, a savings bank leads with the higher effective yield, and both comply fully with federal law while doing it. The legislative intent in each case was disclosure that lets a consumer compare like with like, a card against a card and a savings account against a savings account. The side effect is that comparing a loan's APR against a deposit's APY, or against each other, means converting one convention into the other first.
A worked case
A credit card with a 24.99% purchase APR that compounds daily, which most cards do, carries an effective cost of about 28.38% APY on a balance you revolve. The 3.39-point spread between the disclosed APR and the real annualized cost is exactly the territory Reg Z's nominal convention leaves unlit.
The bridge
What the compounding clock does between those two quoted figures
The bridge between the two regulatory conventions is one formula:
APY = (1 + APR ÷ n)n − 1
The variable n is the number of times interest is calculated and added back to the balance in a year. Parify offers eight values for it: daily (365), weekly (52), bi-weekly (26), semi-monthly (24), monthly (12), quarterly (4), semi-annually (2), and annually (1). Each compounding event earns interest on interest already credited, so a larger n pushes APY further above APR.
Two edges of that formula are worth naming. Set n to 1 and the expression collapses to APY equal to APR: with annual compounding the two conventions agree and the converter is a no-op. The rate magnitude also matters more than people expect. A 4.75% APR compounded monthly yields 4.855% APY, a tenth-of-a-point lift, while a 24.99% APR compounded daily lifts more than three full points. Same formula, same mechanism, wildly different visible gap, because compounding scales with the rate it is applied to.
The reverse direction inverts the algebra:
APR = n × ((1 + APY)1/n − 1)
You cannot run that reversal without knowing n, because a single APY maps to a different APR at every compounding frequency. That dependency is also why Parify clears the active input the instant you flip conversion direction, rather than inverting the prior result and pre-filling it: a carried-over value silently re-interpreted under the other convention is precisely the misrepresentation Reg Z and Reg DD were written to prevent. The labels flip, the field empties, and you re-enter the number you actually hold. The converter runs in double precision and rounds only at the display boundary, to 3 decimal places, so a 5.00% APY at daily compounding reads as 4.879% APR rather than a figure smeared by intermediate rounding.
The boundary
Where the legislative split breaks down: mortgages, credit cards, and daily compounding
The neat lenders-quote-APR, banks-quote-APY division frays at the edges, and the fraying is where most real confusion lives.
Mortgages
A mortgage APR under Reg Z is not a pure nominal rate at all; it rolls certain finance charges, points, and fees into the disclosed figure, which is why a mortgage's quoted APR usually sits above its note rate. Feed a 6.5% mortgage rate compounding monthly into the converter and you get roughly 6.70% APY as the effective interest cost, but that figure is not the same animal as the fee-loaded TILA APR printed on the closing disclosure. The converter handles the interest-compounding half honestly; the fee-bundling half is a separate Reg Z definition the math here cannot reconstruct.
Daily compounding
Daily compounding hides its own ambiguity: n of 365 ignores leap years, and some card issuers and the underlying account agreements use a 360-day or actual-day convention instead, so a "daily" APY can shift in the third decimal depending on whose day count you trust. Credit cards add a further wrinkle, because the APR you are quoted is often a variable rate pinned to the prime rate, so the single number you convert today is a snapshot, not a constant.
None of these break the formula. They mark the boundary of what a conversion can tell you, which is the real limit of any tool that turns one regulatory convention into the other. The split is statutory, the bridge is arithmetic, and the residue is fees and day-counts the statutes define separately.
Common questions
APR vs APY FAQ
Is APY always higher than APR?
APY is higher than APR whenever interest compounds more than once a year, and the gap widens as the rate gets bigger. When interest compounds exactly once a year, or when the rate is zero, APR and APY are identical. They are never lower than APR, because compounding can only add to the effective return, never subtract from it.
Which rate should I compare when shopping for a savings account?
Compare APY. APY already folds the compounding schedule into one number, so two accounts quoted in APY are directly comparable even if one compounds daily and the other monthly. The Truth in Savings Act of 1991, through Regulation DD, requires banks to advertise deposit accounts in APY for exactly this reason.
Why do credit cards quote APR instead of APY?
The Truth in Lending Act of 1968 and its Regulation Z require lenders to disclose the cost of borrowing as an APR, and a lower-sounding number is easier to advertise. A 24.99% card APR that compounds daily costs you about 28.38% APY on a carried balance, so the real cost of revolving debt is higher than the headline rate suggests.
Does the compounding frequency really change the math that much?
At small rates the difference is a few hundredths of a percent. At large rates it is meaningful. A 4.75% rate compounded monthly lifts to 4.855% APY, barely a tenth of a point, while a 24.99% rate compounded daily lifts more than three full points to 28.38% APY. Same formula, same mechanism, very different visible gap.
What is the n in the APR to APY formula?
n is the number of compounding periods per year. Parify offers eight values: daily (365), weekly (52), bi-weekly (26), semi-monthly (24), monthly (12), quarterly (4), semi-annually (2), and annually (1). The formula is APY = (1 + APR ÷ n) raised to the power n, minus 1. As n grows, each period earns interest on the interest already credited, which is what pushes APY above APR.
Can I convert an APY back into an APR?
Yes. The conversion runs both ways. Parify reverses it with APR = n × ((1 + APY) raised to the power 1 ÷ n, minus 1). You need to know the compounding frequency to do it, because the same APY maps to a different APR depending on how often the interest compounds.
Run your own numbers
Type any rate, pick the compounding frequency, and see the honest equivalent with the formula shown. Both directions on one screen.
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