Expert in time value of money, compounding, and financial modeling with over 20 years in academia.
The **Effective Annual Rate (EAR) Calculator** determines the true rate of return earned on an investment or paid on a loan over a year, taking into account the effects of compounding. It is often referred to as Annual Equivalent Rate (AER) or Annual Percentage Yield (APY). Input any three variables (Nominal Rate, Compounding Frequency, Effective Rate, or Initial Principal) to solve for the missing one.
Effective Annual Rate Calculator
Effective Annual Rate Formula Variations
The core relationship between Nominal Rate (APR) and Effective Annual Rate (EAR):
Let $R_{nom}$ = Nominal Rate (decimal), $n$ = Compounding Periods.
Formula Source: Investopedia – Effective Annual Rate
Solving for each variable:
EAR (V) = (1 + R_{nom} / n)ᴺ - 1
APR (F) = n × [ (1 + EAR)^{(1/n)} - 1 ]
n (P): Requires numerical solver (complex)
Variables Explained
- F (Nominal Annual Rate – APR): The stated interest rate before compounding is considered. Also known as the Annual Percentage Rate.
- P (Compounding Periods – n): How many times per year the interest is calculated and added back to the principal. (e.g., Monthly = 12, Quarterly = 4).
- V (Effective Annual Rate – EAR): The actual annual rate earned or paid after accounting for compounding. Expressed as a percentage.
- Q (Initial Principal – P_init): The starting amount of the loan or investment. Used for context.
Related Calculators
- Annual Percentage Yield (APY) Calculator
- Compound Interest Growth Calculator
- Nominal vs. Effective Rate Comparison
- Loan Comparison Calculator
What is the Effective Annual Rate?
The **Effective Annual Rate (EAR)** is the standardized measure used to compare the profitability of different investments or the true cost of different loans. When interest is compounded more frequently than annually (e.g., monthly or daily), the total interest earned or paid over the year is actually higher than the stated Nominal Rate (APR).
The EAR is crucial for consumers because it strips away misleading compounding terms. For example, a loan with a 5% APR compounded daily has a higher effective cost than a loan with a 5% APR compounded semi-annually. By converting all options to their EAR, users can make an apples-to-apples comparison.
How to Calculate EAR (Example)
Scenario: A loan has a Nominal Rate (APR) of 6.0%, compounded monthly ($n=12$).
- Determine the Compounding Interest Rate (i):
$$i = \frac{6.0\%}{12} = 0.5\% \text{ or } 0.005$$
- Apply the EAR Formula:
$$EAR = (1 + 0.005)^{12} – 1$$
- Calculate the Result:
$$EAR \approx 1.061677 – 1 = 0.061677$$
- Final Result:
The Effective Annual Rate is approximately $6.168\%$.
Frequently Asked Questions (FAQ)
APR (Nominal Rate) is the stated rate, ignoring compounding frequency. EAR (Effective Rate) is the actual rate earned or paid after accounting for compounding frequency. EAR will always be equal to or higher than the APR (unless compounding is only annual).
Why is compounding frequency important?The more frequently interest is compounded (e.g., daily vs. annually), the faster the loan or investment grows, because interest starts earning interest sooner. This makes the EAR higher than the APR.
Does the initial principal (P_init) affect the EAR?No. The EAR is a rate, expressed as a percentage, and is independent of the initial principal amount. The principal is included in this calculator only to fulfill the 4-variable requirement and provide context for growth.
What happens if the compounding period is continuous?For continuous compounding, the formula changes to $EAR = e^{R_{nom}} – 1$. This calculator handles discrete compounding, which covers the vast majority of real-world loans and investments.