Effective Annual Rate (EAR) Calculator
Convert between nominal annual interest rate and effective annual rate (EAR) for any compounding frequency, including continuous compounding. Compare loans, savings accounts, and investments on a true apples-to-apples annual basis.
EAR Calculator
Typical EARs for 12% nominal rate
| Compounding | EAR |
|---|---|
| Annually (1×) | 12.00% |
| Quarterly (4×) | 12.55% |
| Monthly (12×) | 12.68% |
| Daily (365×) | 12.75% |
| Continuous | 12.75% |
What is the Effective Annual Rate (EAR)?
The effective annual rate (EAR) is the true annual interest rate you earn or pay once compounding is taken into account. It converts a nominal rate with a given compounding frequency (monthly, daily, etc.) into a single annual rate that produces the same growth over one year.
Because banks and lenders often quote different combinations of APR and compounding frequencies, EAR is the best way to compare products on an apples-to-apples basis.
EAR formulas
1. From nominal rate to EAR
Let:
- r = nominal annual interest rate (in decimal, e.g. 0.12 for 12%)
- m = number of compounding periods per year
Discrete compounding:
\[ \text{EAR} = \left(1 + \frac{r}{m}\right)^m - 1 \]
Continuous compounding:
\[ \text{EAR} = e^{r} - 1 \]
2. From EAR to nominal rate
If you know the effective annual rate and want the equivalent nominal rate with m compounding periods per year:
Discrete compounding:
\[ r = m \left[(1 + \text{EAR})^{1/m} - 1\right] \]
Continuous compounding:
\[ r = \ln(1 + \text{EAR}) \]
Worked example
Suppose a savings account advertises a 12% nominal annual rate compounded monthly. What is the EAR?
- Nominal rate: \( r = 0.12 \)
- Compounding periods: \( m = 12 \)
\[ \text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = (1 + 0.01)^{12} - 1 \approx 1.126825 - 1 = 0.126825 \]
So the effective annual rate is about 12.68%.
This means that \$1,000 deposited for one year would grow to:
\[ 1000 \times (1 + 0.126825) \approx \$1{,}126.83 \]
EAR vs APR vs nominal rate
- Nominal rate / APR: The stated annual rate, often used for marketing. It may or may not fully reflect compounding and fees.
- EAR (effective annual rate): The true annual rate after compounding is considered. It is always ≥ nominal rate for positive interest and at least annual compounding.
- Why it matters: Two loans can both advertise “12% APR” but have different EARs if one compounds monthly and the other daily. The daily-compounding loan is slightly more expensive.
When to use EAR
EAR is especially useful when you:
- Compare loans with different compounding frequencies (monthly vs. daily).
- Compare savings accounts or CDs from different banks.
- Evaluate credit cards that compound interest daily.
- Convert between nominal rates and effective yields in financial modeling.
Limitations and assumptions
- The formulas assume a constant interest rate over the year.
- They do not include fees, penalties, or changing balances (e.g., additional deposits or withdrawals).
- For negative rates, the same formulas apply, but interpretation changes (you are effectively losing value).
FAQ: Effective Annual Rate (EAR)
Is EAR always higher than the nominal rate?
For positive interest rates and at least one compounding period per year, yes. The more frequently interest is compounded, the higher the EAR. If there is no intra-year compounding (m = 1), then EAR equals the nominal rate.
What compounding frequency should I assume if it is not stated?
Common defaults are:
- Loans and mortgages: monthly compounding
- Credit cards: daily compounding
- Savings accounts: daily or monthly compounding
However, you should always check the product’s documentation to be sure.
Can I use EAR for non-annual periods?
Yes. Once you know the EAR, you can derive equivalent rates for other periods. For example, the equivalent monthly rate is:
\[ i_{\text{month}} = (1 + \text{EAR})^{1/12} - 1 \]
Similarly, you can compute daily or quarterly rates from EAR.