To find the nominal rate behind an advertised APY, use the reverse formula nominal rate = n x ((1 + APY)1/n - 1), where APY is the annual percentage yield as a decimal and n is the number of compounding periods per year. A 4.30% APY compounded monthly, for example, comes from a nominal rate of 4.2175%. APY is the number banks advertise; the nominal (stated) rate is the smaller number hiding underneath it. The quickest way to check your work is our APY calculator, but the algebra below takes one line.
Most rate math runs forward - you start with a nominal rate and a compounding frequency and solve for APY. This guide runs it backward. You already know the APY (because that is what gets advertised) and you want the nominal rate that produced it, or the periodic rate the bank actually applies each month. That matters for verifying disclosures, reverse-engineering a quote, and converting a loan's APY back to a stated APR. This is a rate conversion, not a balance projection; to project a balance once you have the rate, use a savings calculator or a compound interest calculator.
Why you would back-solve a nominal rate at all
You back-solve the nominal rate when you have an advertised APY but need the underlying stated rate - to verify a disclosure, compare against a loan APR, or find the periodic rate a bank uses each compounding step. Three situations come up most often.
- Verifying a bank's disclosure. A bank advertises a 4.30% APY but quietly states a 4.21% "interest rate" in the fine print. Back-solving confirms those two numbers are consistent with daily compounding, so the disclosure is honest rather than a typo or a trick.
- Finding the periodic rate. If you want to know how much interest posts each month or each day, you need the nominal rate, because dividing the APY by 12 gives the wrong answer. The periodic rate is nominal/n, not APY/n.
- Comparing deposits to loans. Deposit accounts advertise APY; many loans advertise APR (a nominal-style rate). To compare them on equal footing, you sometimes need to move a number from APY language into nominal/APR language.
The reverse formula, step by step
The reverse conversion is nominal rate = n x ((1 + APY)1/n - 1). It undoes the forward APY formula by taking the n-th root instead of the n-th power. Here is a 4.30% APY (APY = 0.043) compounded monthly (n = 12), worked out:
- Add 1 to the APY: 1 + 0.043 = 1.043.
- Take the n-th root: 1.0431/12 = 1.0035146. This recovers the monthly growth factor.
- Subtract 1: 1.0035146 - 1 = 0.0035146. That is the periodic (monthly) rate, about 0.35146%.
- Multiply by n: 0.0035146 x 12 = 0.042175, or a 4.2175% nominal rate.
To confirm, run it forward: (1 + 0.042175/12)12 - 1 = 0.0430, which is exactly 4.30% APY. The check matching is the whole point - if your back-solved nominal rate does not reproduce the original APY, you used the wrong frequency.
APY-to-nominal reference table
This table back-solves the nominal rate for two common advertised APYs across every standard frequency. The more often an account compounds, the lower the nominal rate it needs to hit the same APY - which is why a daily-compounding account can advertise a slightly smaller "interest rate" and still match a monthly account's yield.
| Advertised APY | Compounding | Periods (n) | Required nominal rate | Periodic rate |
|---|---|---|---|---|
| 4.30% | Annual | 1 | 4.3000% | 4.3000% / year |
| 4.30% | Quarterly | 4 | 4.2324% | 1.0581% / quarter |
| 4.30% | Monthly | 12 | 4.2175% | 0.3515% / month |
| 4.30% | Daily | 365 | 4.2104% | 0.01154% / day |
| 4.30% | Continuous | infinite | 4.2101% | n/a |
| 5.00% | Monthly | 12 | 4.8889% | 0.4074% / month |
| 5.00% | Daily | 365 | 4.8793% | 0.01337% / day |
The continuous-compounding case has its own shortcut: nominal rate = ln(1 + APY). For a 4.30% APY, ln(1.043) = 4.2101%, matching the table. That is the theoretical floor - the smallest nominal rate that could ever produce a given APY.
Back-solving in Excel or Google Sheets
Spreadsheets reverse the conversion with the NOMINAL function, the mirror image of EFFECT. Type =NOMINAL(effective_rate, npery), where the effective rate is the APY as a decimal and npery is periods per year.
- Monthly:
=NOMINAL(0.043, 12)returns 0.042175, or 4.2175%. - Daily:
=NOMINAL(0.043, 365)returns 0.042104, or 4.2104%.
To recover the periodic rate directly, divide the result by n, or use the raw formula =(1+0.043)^(1/12)-1 for the monthly figure. For the continuous case, use =LN(1+0.043), which returns 0.042101.
APY-to-APR: the loan side of the conversion
Converting an APY to APR matters most for loans, where the same compounding math runs in the borrower's disfavor instead of the saver's favor. On a deposit, more frequent compounding is good for you; on a loan, it quietly raises your true cost above the stated APR.
Loan APR is a nominal-style rate. If a card states a 18.00% APR but compounds the balance daily, the effective annual rate you actually pay is (1 + 0.18/365)365 - 1 = 19.7164%. Compounded monthly instead, the same 18.00% APR works out to 19.5618% effective. Either way, the number that hits your balance is higher than the 18.00% on the offer.
To go the other direction - from a known effective rate (an APY-style figure) back to the stated APR a lender would quote - apply the same reverse formula. An effective annual cost of 19.7164% on a daily-compounding card back-solves to exactly the 18.00% nominal APR. This is why deposit accounts and loans should never be compared on their headline numbers alone: deposits advertise the higher APY, loans advertise the lower APR, and only converting both into the same language shows the real gap. The Consumer Financial Protection Bureau publishes the disclosure rules behind both.
Once you have a loan's true effective rate, you can weigh it against what your cash earns. If a card costs 19.72% effective and your savings yields 4.30% APY, paying down the card is worth more than four times the return of saving - a comparison only the rate conversion makes visible. To plan the payoff, use a credit card payoff calculator.
Try it yourself
Run your own numbers in the free APY Calculator — instant, private, no sign-up.
Open the APY Calculator →Frequently asked questions
- How do I find the nominal rate from an APY?
- Find the nominal rate from an APY with the reverse formula: nominal rate = n times ((1 + APY) to the power 1/n, minus 1), where n is compounding periods per year. For a 4.30% APY compounded monthly: 12 times ((1.043) to the 1/12 power minus 1) = 4.2175%. In Excel, the shortcut is =NOMINAL(0.043, 12).
- What is the nominal rate behind a 5% APY?
- The nominal rate behind a 5.00% APY depends on compounding frequency. Compounded monthly, it is 4.8889%; compounded daily, it is 4.8793%; compounded continuously, it is 4.8790%. If the account compounds only once a year, the nominal rate equals the APY at 5.00%. More frequent compounding means a lower nominal rate is needed to reach the same APY.
- Why is the nominal rate lower than the APY?
- The nominal rate is lower than the APY because APY already includes the extra interest earned from compounding within the year. The nominal rate is the simple annual rate before that effect. For a 4.30% APY compounded monthly, the nominal rate is only 4.2175% - the 0.0825-point gap is the compounding the APY captures and the nominal rate does not.
- How do I convert APY to APR for a loan?
- To convert a loan APY (effective annual rate) to APR, apply the reverse formula: APR = n times ((1 + effective rate) to the power 1/n, minus 1). A 19.7164% effective rate on a daily-compounding card back-solves to an 18.00% APR. Running it forward confirms it: (1 + 0.18/365) to the 365th power minus 1 equals 19.7164%.
- What is the periodic rate and how do I get it from APY?
- The periodic rate is the interest applied each compounding step, and you get it by taking the n-th root of (1 + APY) and subtracting 1 - not by dividing the APY by n. For a 4.30% APY compounded monthly, the monthly periodic rate is (1.043) to the 1/12 power minus 1 = 0.3515%, which annualizes to a 4.2175% nominal rate.
- How do I back-solve a nominal rate in Excel?
- Use the NOMINAL function: type =NOMINAL(effective_rate, npery), where the effective rate is the APY as a decimal and npery is periods per year. For a 4.30% APY compounded monthly, =NOMINAL(0.043, 12) returns 4.2175%. For continuous compounding, use =LN(1+0.043) instead, which returns 4.2101%.
- What is the lowest nominal rate that can produce a given APY?
- The lowest possible nominal rate for any APY comes from continuous compounding, and it equals the natural log of (1 + APY). For a 4.30% APY, that is ln(1.043) = 4.2101%. No real account can require a smaller nominal rate to reach that APY, so this figure is the theoretical floor for the conversion.
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