How the result is calculated
An account that pays 5% a year, credited monthly, does not pay 5% in a year. It pays 5% ÷ 12 at the end of the first month, and in the second month that small credit earns interest along with your deposit. By the end of the year the balance has grown by slightly more than 5%. The quoted figure is the nominal rate; the growth over the year is the effective annual rate, called the annual percentage yield (APY) in the United States and the annual equivalent rate (AER) in Britain.
APY(1 + r ÷ m)m − 1
- r
- the nominal annual rate as a decimal, so 5% is 0.05
- m
- the number of times a year interest is credited: 1, 2, 4, 12, 52 or 365
Run backwards, the same relationship gives the nominal rate that produces a target yield: r = m × ((1 + APY)^(1 ÷ m) − 1), or r = ln(1 + APY) for continuous compounding. The calculator's APY to rate mode does that.
The third mode, From interest, works from what a deposit actually paid rather than from a quoted rate. US banks must calculate the APY they advertise this way, under the Truth in Savings Act's Regulation DD: take the interest the account would pay on a principal over its term and annualise it.
APY(1 + I ÷ P)365 ÷ d − 1
- I
- the interest paid over the term
- P
- the principal deposited
- d
- the term in days
Worked example
| Item | Amount | How it is obtained |
|---|---|---|
| Nominal rate, r | 5% | Nominal annual rate |
| Credits a year, m | 12 | Compounding: monthly |
| Rate per credit | 0.41667% | 5% ÷ 12 |
| Growth over a year | 1.0511619 | 1.004166712 |
| APY | 5.116% | 1.0511619 − 1 |
| On a 1,000 deposit | $51.16 | a year's interest, against $50.00 at 5% simple |
| Item | Amount | How it is obtained |
|---|---|---|
| Return for the term, I ÷ P | 0.03037 | 30.37 ÷ 1,000 |
| Terms in a year | 2.0055 | 365 ÷ 182 |
| Annualised growth | 1.0618369 | 1.030372.0055 |
| APY | 6.18% | 1.0618369 − 1, to two decimals as disclosed |
| Item | Amount | How it is obtained |
|---|---|---|
| One year's growth | 1.05 | 1 + 5% APY |
| Growth per month | 1.0040741 | 1.051 ÷ 12 |
| Nominal rate | 4.889% | 12 × 0.0040741 |
| Check | (1 + 0.04889 ÷ 12)^12 = 1.05 | the two figures describe one account |
Reading the result
The headline is the figure you asked for: the APY in the first and third modes, the nominal rate in the second. The first tile repeats the APY the way a US bank must disclose it, rounded to two decimals; the second gives the nominal rate it corresponds to at the compounding you chose; the third is the gap between them, which is what compounding is worth in a year.
The table lists the same rate at every compounding frequency. In the first mode the nominal rate is held and the APY rises down the table. In the second the APY is held and the nominal rate needed falls. Either way the steps shrink quickly: at 5%, half-yearly adds 0.06 of a point over yearly, monthly another 0.05, and daily only a hundredth more.
What changes the result
The nominal rate matters far more than the compounding. Moving from 5% to 5.25% adds a quarter of a point; moving from monthly to daily compounding at 5% adds about 0.01. When two accounts differ in both, compare APYs and ignore the schedule.
The gap between nominal rate and APY grows with the rate. At 1% monthly compounding adds less than half a hundredth of a point; at 20% it adds 1.94 points. That is why the difference matters little for savings and a great deal for expensive credit.
In the From interest mode, the term changes the answer through annualisation. A short term is scaled up as though it repeated for a full year, which is the convention, not a promise.
What is not included
The conversions assume a constant rate for a whole year with the interest left in the account. Fees, maintenance charges and minimum-balance penalties are not part of an APY and lower what you actually earn. Tax on interest is ignored. Tiered accounts, where a different rate applies to each part of the balance, need each tier worked on its own; Regulation DD's appendix shows how a bank must disclose them. And the APY of a loan is not what lenders quote: a loan's APR includes fees, which needs a different calculation.
Using the result
Compare savings accounts and deposits on their APY or AER, because that figure already contains the compounding schedule. If a provider quotes only a nominal rate and a frequency, convert it here first. If you hold a statement showing the interest an account paid over a period, the From interest mode tells you what it actually yielded, which you can set against what was advertised.