# APR and APY Calculator

> A nominal annual rate throws away the compounding; the effective rate puts it back. 12% compounded monthly is 12.6825% a year. This converts between the two in either direction, for any frequency including the continuous limit.

Use it: https://tessalor.com/en/money/apr-and-apy

This tool runs entirely in the browser. Nothing entered into it is uploaded.

## Formula

```
From a nominal rate to the effective one:

  effective = (1 + nominal/n)^n - 1

And back again:

  nominal = n x ((1 + effective)^(1/n) - 1)

Continuously compounded, the limit as n grows:

  effective = e^nominal - 1
  nominal   = ln(1 + effective)

n = periods a year: 1, 2, 4, 12, 52, 365

The two directions are exact inverses, which is
asserted rather than assumed.
```

## Assumptions

- The nominal rate is the periodic rate multiplied by the number of periods, which is what a nominal rate means and what almost every lender quotes.
- Nothing here includes fees. A regulatory APR in most jurisdictions folds compulsory charges into the figure as well as the compounding, so a legal APR and the effective rate computed here are not always the same number.
- Continuous compounding is a limit rather than a frequency, so it has no period and therefore no per-period rate. Reporting the instantaneous rate under that label would be a different quantity wearing the same name.
- The rate is constant for the year. A variable rate has no single effective figure.
- Nothing on this page is denominated in a currency, because a rate conversion is identical everywhere. That is why there are no money fields and no currency symbols.

## Inputs

| Name | Label | Type | Default | Range |
| --- | --- | --- | --- | --- |
| `rate` | Rate | percent | 12 | 0 to 1000 |
| `compounding` | Compounded | select | 12 |  |
| `direction` | What to work out | select | toEffective |  |

## Outputs

- `effectiveRate` — Effective annual rate (percent), primary
- `nominalRate` — Nominal annual rate (percent)
- `perPeriodRate` — Rate per period (percent)
- `periodsPerYear` — Periods a year (integer)

## Questions

### What is the difference between APR and APY?

APR is nominal — a periodic rate multiplied by the number of periods, with the compounding left out. APY, or AER on a savings account, folds the compounding back in. At 12% compounded monthly the periodic rate is 1% and the effective rate is 12.6825%, because each month's interest earns interest for the rest of the year.

### Why does my credit card cost more than its APR?

Because the APR is nominal and the interest compounds monthly. A 22.9% APR is 1.9083% a month, which over twelve months is an effective 25.4632%. On a balance you never clear, the effective figure is what you actually pay. The nominal one is what appears on the statement.

### Does daily compounding really beat monthly?

Barely. At 12% nominal, monthly gives 12.6825% and daily gives 12.7475% — six and a half hundredths of a percentage point. Almost all of the available gain arrives by monthly compounding, and everything after it is chasing a limit. Compounding frequency is rarely where a decision should turn.

### What is continuous compounding for?

It is the limit the others approach, not a product anybody sells. As the number of periods grows the effective rate converges on e to the power of the nominal rate, minus one — 12.7497% here, a fiftieth of a point above daily. It is used in mathematical finance because it is the closed form, and it is on this page because seeing how little it adds is the clearest argument that frequency is a small lever.

### Which figure should I compare two offers with?

The effective one, always, and only if both are computed the same way. Nominal rates are comparable only at the same compounding frequency; effective rates are comparable to each other by construction. Be aware that a regulatory APR usually includes fees as well as compounding, so it is not the same number as the effective rate this tool computes.

## Sources

- [Appendix J to Part 1026 — Annual Percentage Rate Computations for Closed-End Credit Transactions](https://www.consumerfinance.gov/rules-policy/regulations/1026/j/) — Consumer Financial Protection Bureau, applies to 2026. Retrieved 2026-07-31.
- [Appendix A to Part 1030 — Annual Percentage Yield Calculation (Regulation DD)](https://www.consumerfinance.gov/rules-policy/regulations/1030/A/) — Consumer Financial Protection Bureau, applies to 2026. Retrieved 2026-07-31.

## Variants

- [What is the difference between APR and APY?](https://tessalor.com/en/money/apr-and-apy/apr-against-apy)

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Estimate, not advice. See https://tessalor.com/en/disclaimer.
Machine-readable catalogue: https://tessalor.com/api/tools.json