# What APY will my rate earn?

Converts a stated interest rate to its annual percentage yield (APY) for daily, monthly, quarterly, twice-yearly, yearly or continuous compounding, and back.

- Page: https://www.acalculator.org/finance/apy-calculator
- JSON spec: https://www.acalculator.org/finance/apy-calculator.json
- Version: 18bc3811ea97

## Default answer

Example with the default inputs (Interest rate 4%, Interest is compounded daily): An interest rate of 4% compounded daily is an APY of 4.081%.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| rate | Interest rate | The stated yearly interest rate before compounding (the nominal rate). |
| apy | APY | The annual percentage yield: what the balance earns in a year with compounding, as a percent. |
| comp | Interest is compounded | How often interest is added to the balance: daily (365 times a year), monthly, quarterly, twice a year, yearly or continuously. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| rate | Interest rate | The stated yearly interest rate before compounding (the nominal rate). |
| apy | APY | The annual percentage yield: what the balance earns in a year with compounding, as a percent. |

## Method

APY = (1 + r ÷ n)^n − 1, where r is the stated yearly rate and n the times interest is compounded a year; APY = e^r − 1 for continuous compounding.

## Assumptions

- The rate stays the same for the whole year and all interest stays in the account.
- Daily compounding uses 365 days a year.
- No fees. Regulation DD’s APY for a set amount of interest earned over a term is APY = (1 + interest ÷ principal)^(365 ÷ days) − 1; for a full year of compounding this is the same as the formula above.

## Worked examples

1. rate = 6%, comp = 12 gives apy = 6.167781%. Source: Regulation DD (12 CFR 1030) appendix A: $61.68 of interest on $1,000 for a 365-day year is an APY of 6.17%; (1 + 0.06/12)^12 − 1 = 6.1678%.
2. rate = 5%, comp = 365 gives apy = 5.12675%. Source: hand calculation in content.mdx: (1 + 0.05/365)^365 − 1 = 5.1267%.
3. apy = 5%, comp = 12 gives rate = 4.888949%. Source: hand calculation in content.mdx: 12 × ((1.05)^(1/12) − 1) = 4.8889%.
4. rate = 5%, comp = continuous gives apy = 5.12711%. Source: hand calculation in content.mdx: e^0.05 − 1 = 5.1271%.
5. rate = 8%, comp = 4 gives apy = 8.243216%. Source: hand calculation in content.mdx: (1 + 0.08/4)^4 − 1 = 1.02^4 − 1 = 8.2432%.
6. apy = 5%, comp = continuous gives rate = 4.879016%. Source: hand calculation in content.mdx: ln(1.05) = 4.8790%.
7. rate = 3%, comp = 1 gives apy = 3%. Source: Regulation DD appendix A: for a 365-day term, APY = 100 × interest ÷ principal; yearly compounding at 3% pays 3%.

## FAQ

### How is APY calculated?

APY = (1 + r ÷ n)^n − 1, where r is the stated yearly rate as a decimal and n is how many times a year interest is compounded. A 6% rate compounded monthly is (1 + 0.06 ÷ 12)^12 − 1 = 6.168%. On $1,000 that is $61.68 of interest in a year, the example in Regulation DD.

### What is the difference between APY and the interest rate?

The interest rate is the yearly rate before compounding. The APY includes the interest earned on interest during the year, so it is higher whenever interest is compounded more than once a year. With yearly compounding they are equal.

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

APY is for savings: what a deposit earns in a year, with compounding. Banks must show it under the Truth in Savings Act (Regulation DD). APR is for loans: the yearly cost of credit, including some fees, without compounding. Lenders show it under the Truth in Lending Act (Regulation Z).

### Does daily compounding make a big difference?

A little. At 5%, yearly compounding gives an APY of 5%, monthly 5.116%, daily 5.127%, and continuous 5.127%. The gain from compounding more often gets smaller and smaller.

### How do I find the interest rate from an APY?

Reverse the formula: r = n × ((1 + APY)^(1/n) − 1). An APY of 5% with monthly compounding is a stated rate of 12 × (1.05^(1/12) − 1) = 4.889%.

### How do I use APY for a CD that is shorter than a year?

Regulation DD works out the APY from the interest paid over the term: APY = 100 × ((1 + interest ÷ principal)^(365 ÷ days in term) − 1). For a stated rate that compounds through the term, it gives the same APY as this calculator.

## Sources

- Regulation DD (Truth in Savings), 12 CFR part 1030, appendix A, Part I: APY = 100 [(1 + Interest/Principal)^(365/Days in term) − 1]; $61.68 of interest on $1,000 for a 365-day year is an APY of 6.17%. https://www.law.cornell.edu/cfr/text/12/appendix-A_to_part_1030
- Consumer Financial Protection Bureau, Regulation DD, 12 CFR 1030.2: definition of annual percentage yield. https://www.consumerfinance.gov/rules-policy/regulations/1030/2/