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Use the authoritative **Effective Annual Rate Calculator** to find the true, compounding rate of return on an investment or loan. Enter any three variables—Effective Annual Rate (EAR), Nominal Annual Rate (APR), Compounding Frequency, or Doubling Time—to solve for the remaining unknown value.
Effective Annual Rate Calculator
Effective Annual Rate Formula
Core Relationship: $\mathbf{EAR} = \left(1 + \frac{\mathbf{r}}{\mathbf{m}}\right)^{\mathbf{m}} – 1$
Where $\mathbf{EAR}$ and $\mathbf{r}$ are decimals, and $\mathbf{m}$ is the frequency.
The four solution formulas:
$\mathbf{EAR}$ (F) $= \left(\left(1 + \frac{P}{m}\right)^m – 1\right) \times 100$
$\mathbf{APR}$ (P) $= \left(\left((F/100) + 1\right)^{\frac{1}{m}} – 1\right) \times m \times 100$
$\mathbf{m}$ (Frequency, V) – *Requires iterative or graphical solution*
$\mathbf{n_d}$ (Time, Q) $= \frac{\ln(2)}{\ln(1 + EAR/100)}$
Formula Source: InvestopediaFormula Variables
- F ($\mathbf{EAR}$): Effective Annual Rate. The true, compounded rate of return or cost (%).
- P ($\mathbf{APR}$): Nominal Annual Rate (Annual Percentage Rate). The stated annual rate (%).
- V ($\mathbf{m}$): Compounding Frequency. The number of times interest is calculated per year (e.g., 12 for monthly).
- Q ($\mathbf{n_d}$): Doubling Time. The number of years required for an investment to double in value at the $\mathbf{EAR}$.
Related Calculators
- APY Calculator (Annual Percentage Yield)
- Compound Interest Calculator
- Present Value Calculator
- Loan Payoff Calculator
What is the Effective Annual Rate (EAR)?
The Effective Annual Rate (EAR), also known as the Effective Annual Yield (EAY) or Annual Percentage Yield (APY), is the true, comprehensive rate of return or cost on an investment or loan. It accounts for the effects of compounding interest. While the Nominal Annual Rate (APR) is the simple, stated rate, the EAR reflects the actual interest earned or paid over a year, given that interest is compounded more frequently than annually.
The EAR is always equal to or greater than the APR when compounding occurs more than once a year. For consumers, the EAR provides the necessary clarity to compare financial products accurately. A savings account with a 5% APR compounded daily is a better deal than a bond with a 5% APR compounded annually; the EAR reveals this difference.
How to Calculate Required Nominal Rate (Example)
Let’s find the Nominal Annual Rate ($\mathbf{r}$, P) required to achieve an Effective Annual Rate ($\mathbf{EAR}$, F) of 6.5\% when interest is compounded **quarterly** ($\mathbf{m}$, V = 4).
- Step 1: Convert Rates to Decimals
Target $EAR_{dec} = 6.5\% / 100 = 0.065$.
- Step 2: Apply the Inverse APR Formula
The formula for $r$ is: $r = \left(\left(EAR_{dec} + 1\right)^{\frac{1}{m}} – 1\right) \times m$.
- Step 3: Substitute and Solve for $\mathbf{r}$
$r = \left(\left(0.065 + 1\right)^{\frac{1}{4}} – 1\right) \times 4$. This yields $r \approx 0.06341$.
- Step 4: Determine the Required Nominal Rate
The calculation yields a required Nominal Annual Rate ($\mathbf{APR}$, P) of **6.341\%**.
Frequently Asked Questions (FAQ)
Compounding frequency ($\mathbf{m}$) is the number of times per year that interest is calculated and added back to the principal balance. Common frequencies are 1 (annually), 4 (quarterly), 12 (monthly), 365 (daily), or even continuous.
How does EAR differ from APR?APR (Nominal Rate) is the simple, stated annual rate, ignoring compounding. EAR is the *true* annual rate, incorporating the effect of compounding. If interest is compounded annually ($\mathbf{m}=1$), APR = EAR. If $\mathbf{m} > 1$, then EAR > APR.
What is the maximum possible EAR?As the compounding frequency ($\mathbf{m}$) approaches infinity (continuous compounding), the EAR approaches a limit defined by the natural exponential function, $e^r – 1$. For example, a 10% APR compounded continuously yields an EAR of $e^{0.10} – 1 \approx 10.517\%$.
Why is doubling time (Q) included?Doubling time ($\mathbf{n_d}$) is included as a common measure of investment performance that is directly calculated from the **Effective Annual Rate (EAR)**. It gives a practical, time-based context to the theoretical rate.