Enter the nominal interest rate and the compounding period into the calculator. The calculator will display the effective annual yield, also known as the true yield of your investment or loan.
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Effective Annual Yield Formula
The effective annual yield (EAY) converts a nominal rate into the actual yield you earn over one year once compounding is included.
EAY = (1 + r/m)^m - 1
For continuous compounding:
EAY = e^r - 1
To solve for the nominal rate that produces a given EAY:
r = m * [(1 + EAY)^(1/m) - 1]
- EAY = effective annual yield (decimal)
- r = nominal annual rate (decimal)
- m = number of compounding periods per year
- e = Euler's number, about 2.71828
The calculator has three modes. Nominal rate takes a stated annual rate and a compounding frequency and returns the EAY. Bond coupon divides the annual coupon by the face value to get the coupon rate, then applies the same compounding formula using the payment frequency. Target EAY runs the formula in reverse to tell you what nominal rate is needed to hit a desired yield at a chosen compounding frequency.
Reference Tables
The first table shows how a 6% nominal rate translates to EAY at common compounding frequencies. The second shows the reverse: the nominal rate needed to hit a 6% EAY.
| Compounding | Periods/year (m) | EAY from 6% nominal |
|---|---|---|
| Annual | 1 | 6.0000% |
| Semi-annual | 2 | 6.0900% |
| Quarterly | 4 | 6.1364% |
| Monthly | 12 | 6.1678% |
| Daily | 365 | 6.1831% |
| Continuous | ∞ | 6.1837% |
| Compounding | Nominal rate to hit 6% EAY |
|---|---|
| Annual | 6.0000% |
| Semi-annual | 5.9126% |
| Quarterly | 5.8695% |
| Monthly | 5.8411% |
| Daily | 5.8274% |
| Continuous | 5.8269% |
Worked Examples
Example 1: Monthly compounding. A savings account quotes 5% nominal interest compounded monthly. Plug in r = 0.05 and m = 12:
EAY = (1 + 0.05/12)^12 - 1 = (1.004167)^12 - 1 = 0.05116, or 5.1162%.
Example 2: Semi-annual bond. A bond has a $1,000 face value and pays $50 in coupons per year, split into two $25 payments. The coupon rate is 50/1000 = 5%. With m = 2:
EAY = (1 + 0.05/2)^2 - 1 = (1.025)^2 - 1 = 0.050625, or 5.0625%.
FAQ
How is EAY different from APR? APR is the nominal rate before compounding. EAY (also called APY or effective annual rate) reflects what you actually earn or pay once compounding is applied. EAY is always greater than or equal to the APR, with equality only when compounding is annual.
Why does the EAY barely change above daily compounding? The formula approaches its continuous limit, e^r - 1, very quickly. Going from daily to continuous compounding on a 6% rate adds less than one ten-thousandth of a percent.
Does EAY assume reinvestment? Yes. The bond mode assumes you reinvest each coupon at the same coupon rate. If you cannot reinvest at that rate, your realized yield will be lower.
Can I use EAY to compare two loans? Yes. Convert both stated rates to EAY using their respective compounding frequencies, then compare. The lower EAY is the cheaper loan.
