Equivalent Interest Rate Calculator
Two nominal rates are equivalent when they compound to the same effective annual rate. Enter a rate and its compounding frequency, choose a target frequency, and see the matching nominal rate.
Equivalent nominal rate (semi-annual)
12.3040%
- Input nominal rate (monthly)
- 12.0000%
- Effective annual rate (EAR)
- 12.6825%
- Equivalent nominal rate (semi-annual)
- 12.3040%
- Per-period rate (semi-annual, 2/year)
- 6.1520%
Two nominal rates are equivalent when they produce the same effective annual rate (EAR). The calculator first converts your input to its EAR, then re-expresses that EAR as a nominal rate at the target compounding frequency. Continuous compounding uses the natural exponential; discrete frequencies use the standard (1 + r/n)^n formula.
How to use this calculator
Enter the nominal (stated) rate — the headline number quoted before compounding is applied. Pick its compounding frequency (monthly, quarterly, and so on). Then pick the frequency you want to convert to. The calculator returns the equivalent nominal rate at the new frequency, the effective annual rate (EAR) the two share, and the per-period rate at the output frequency.
How the calculation works
Compounding frequency changes how interest accumulates within a year, so two loans quoted at the same nominal rate but different compounding schedules do not cost the same. Any nominal rate can be expressed as an effective annual rate (EAR): EAR = (1 + r_nom / n)^n − 1 for discrete compounding, or EAR = e^r_nom − 1 for continuous. The equivalent nominal rate at a new frequency inverts the same formula: r_out = n_out × ((1 + EAR)^(1/n_out) − 1), or r_out = ln(1 + EAR) for continuous. Two nominal rates that share an EAR are financially indistinguishable — a loan at either would produce identical interest cost.
Worked example
A credit card quotes 12.00% nominal compounded monthly. EAR = (1 + 0.12/12)^12 − 1 = 1.01^12 − 1 = 12.6825%. To express that as a semi-annual nominal rate: r_out = 2 × (1.126825^(1/2) − 1) = 2 × 0.061520 = 12.3040%. Both a 12.00% rate compounded monthly and a 12.3040% rate compounded twice a year cost the borrower exactly the same in interest per year. Convert the same 12.00% monthly to continuous: r_out = ln(1.126825) = 11.9403% — a slightly lower headline number with the same real cost.
Frequently asked questions
Why do two loans with the same nominal rate cost different amounts?
The nominal (or "stated") rate is the annual rate before compounding is applied. If interest is added to the balance more often than once a year — monthly on a credit card, quarterly on some bonds, daily on some deposit accounts — then interest starts earning interest within the year. The more frequent the compounding, the more the balance grows on the same nominal rate. A 12% credit card compounded monthly grows the balance by 12.68% per year in real terms; a 12% bond compounded annually grows it by exactly 12%. When you compare products, always compare their effective annual rates (EAR / APY), never the nominal rate alone.
What is the difference between nominal rate, effective annual rate, and APR?
The nominal rate is the raw annual rate before compounding — 12% "compounded monthly" is a nominal 12%. The effective annual rate (EAR, or APY in US retail banking) is the actual annual growth once compounding is folded in — 12.68% in that example. APR is a US-regulatory construct: it is broadly a nominal rate, but it also includes certain mandatory fees (origination, points) spread across the loan term. APR is a fee-adjusted nominal rate and does not fully reflect compounding, which is why credit card APRs and their EARs differ. When shopping for savings accounts, compare APY; when shopping for loans, be aware that a lower APR can still cost more if the compounding frequency is higher.
When does continuous compounding matter in practice?
Continuous compounding is the theoretical limit as the number of compounding periods per year goes to infinity — no real product actually compounds continuously. It matters heavily in derivatives pricing (the Black–Scholes model uses continuous compounding by construction), in fixed-income theory (spot and forward rate calculations, zero-coupon bond pricing), and in academic treatments of finance where the maths is cleaner. For consumer products it is mostly a rounding difference: 12% continuous ≈ 12.75% EAR versus 12% daily ≈ 12.7475% EAR. If you need the equivalent nominal rate at a discrete frequency to plug into another calculation, this calculator handles the conversion in either direction.
How do I compare a savings APY with a credit card APR?
Convert both to the same effective annual rate (EAR) first, then compare directly. Savings APYs are usually already effective rates — a 4.5% APY on a high-yield savings account grows a £10,000 balance to £10,450 in a year. Credit card APRs are nominal rates (usually compounded daily on the average balance): a 24% APR compounded daily is an EAR of (1 + 0.24/365)^365 − 1 = 27.11%. So carrying a £10,000 credit card balance costs about £2,711 per year in interest, while parking £10,000 in the savings account earns £450 — a real cost gap of £2,261 per year, meaningfully wider than the 24 − 4.5 = 19.5 point nominal spread suggests. Compounding always widens the gap.
Does this calculator handle simple interest?
No — simple interest is not "compounded" in any meaningful sense (interest is charged only on the original principal, never on accrued interest), so the concept of an equivalent rate at a different compounding frequency does not apply. On a simple-interest instrument, the nominal rate and the effective annual rate are the same number, and there is nothing to convert. Simple interest is uncommon in modern products: most mortgages, credit cards, deposits, and bonds compound. Payday loans, short-term auto loans in some US states, and some inter-company loans use simple interest — treat those separately.
Is the "equivalent rate" the same as the "converted rate" in the CFA curriculum?
Yes. The CFA Institute Level I Quantitative Methods reading defines the periodic rate as r_nom / m and the effective annual rate as (1 + r_nom / m)^m − 1, and covers exactly this conversion between nominal rates quoted at different compounding frequencies. Textbooks vary in terminology — Ross/Westerfield calls it the "effective annual yield", Brealey/Myers uses "effective annual rate" — but the underlying maths is identical: solve for the target-frequency nominal rate that produces the same EAR. This calculator implements the standard discrete and continuous forms of that identity.