What this calculator does
A nominal rate is a headline number; the effective rate is what you actually earn or pay once compounding is counted. An account advertising 6% compounded monthly does not return 6% over a year: it returns slightly more, because each month’s interest starts earning interest of its own.
The gap grows with both the rate and the compounding frequency. At 2% it is barely worth mentioning. At 20% on a credit card compounding monthly, the effective rate is nearly 22%, which is the figure that actually describes what the debt costs.
The formula
Dividing the nominal rate by the number of periods gives the rate per period. Compounding that across a full year and subtracting the original principal gives the effective annual yield. As the number of periods approaches infinity the expression converges on the continuous case, eʳ − 1.
| Term | Meaning |
|---|---|
| APR | The nominal annual rate, as quoted. |
| APY | The effective annual rate, after compounding. |
| m | The number of compounding periods per year. |
| r | The nominal rate expressed as a decimal. |
The inputs explained
| Field | What to enter |
|---|---|
| Nominal annual rate (%) | The nominal or quoted annual rate. |
| Compounds per year | How often interest is added to the balance. |
When to use it
Comparing savings accounts fairly
Two accounts quoting the same nominal rate at different compounding frequencies are not equivalent. Converting both to effective rates makes them directly comparable in a single number.
Understanding what a card really costs
Credit card rates are quoted nominally but charged monthly. The effective rate is the honest figure: typically one to two percentage points above the advertised one.
Checking a loan comparison rate
Comparison rates fold fees into an equivalent rate. This calculator handles the compounding half of that conversion, which is useful for checking whether a quoted comparison rate looks plausible.
Pricing short-term facilities
A facility charging 1.5% a month is not 18% a year. Compounded monthly, the effective rate is closer to 19.6%: a meaningful difference on a large balance.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Effective rate at different compounding frequencies
A 6% nominal rate, compounded at different intervals.
| Compounded | Effective annual rate | Periodic rate | On $10,000 for one year |
|---|---|---|---|
| Annually | 6.00% | 6.00% per period | $600.00 of interest |
| Half-yearly | 6.09% | 3.00% per period | $609.00 of interest |
| Quarterly | 6.14% | 1.50% per period | $613.64 of interest |
| Monthly | 6.17% | 0.500% per period | $616.78 of interest |
| Weekly | 6.18% | 0.115% per period | $618.00 of interest |
| Daily | 6.18% | 0.016% per period | $618.31 of interest |
How the gap widens as rates rise
With monthly compounding fixed, the difference between nominal and effective grows with the rate.
| Nominal rate | Effective annual rate | Continuous compounding equivalent | On $10,000 for one year |
|---|---|---|---|
| 2% | 2.02% | 2.02% | $201.84 of interest |
| 5% | 5.12% | 5.13% | $511.62 of interest |
| 10% | 10.5% | 10.5% | $1,047.13 of interest |
| 15% | 16.1% | 16.2% | $1,607.55 of interest |
| 20% | 21.9% | 22.1% | $2,193.91 of interest |
| 25% | 28.1% | 28.4% | $2,807.32 of interest |
Questions
What is the difference between APR and APY?
APR is the nominal annual rate, ignoring within-year compounding. APY, sometimes called the effective annual rate, includes it. For the same product, APY is always equal to or higher than APR.
Which one do lenders and banks quote?
It varies by product and jurisdiction. Deposit products often advertise the higher APY because it looks better; loans often quote the lower nominal rate for the same reason. Converting both to the same basis is the only fair comparison.
What is continuous compounding?
The theoretical limit of compounding infinitely often. It sets an upper bound on what any compounding frequency can produce and is used widely in options pricing, though few retail products use it.
Does APR include fees?
In some jurisdictions a regulated APR must include certain fees; the plain nominal rate does not. This calculator handles compounding only: for fee-inclusive comparisons, use a comparison rate.
To see the effect on a real balance over time, use the compound interest calculator.