Calculate APR from EAR.

Steps to Solve

1. Identify the Effective Annual Rate (EAR)

Let's assume your effective annual rate (EAR) is given. For instance, let’s say the EAR is $0.05$ (or $5%$).

2. Determine the number of compounding periods per year

Decide how many times the interest compounds per year. Common values are:

• Annually: 1
• Semi-Annually: 2
• Quarterly: 4
• Monthly: 12

For this example, we will use monthly compounding, so $n = 12$.

3. Apply the formula to find APR

The relationship between EAR and APR, when compounding is considered, is given by the formula:

$$ EAR = (1 + \frac{APR}{n})^n - 1 $$

We need to rearrange this formula to solve for APR:

$$ APR = n \left( (1 + EAR)^{\frac{1}{n}} - 1 \right) $$

4. Plug in the values

Using EAR = $0.05$ and $n = 12$, we can substitute these values into the rearranged formula:

$$ APR = 12 \left( (1 + 0.05)^{\frac{1}{12}} - 1 \right) $$

5. Calculate APR

First, calculate ( (1 + 0.05)^{\frac{1}{12}} ):

$$ (1.05)^{\frac{1}{12}} \approx 1.004074123 $$

Now, calculate:

$$ APR \approx 12 \left( 1.004074123 - 1 \right) $$

This gives us:

$$ APR \approx 12 \times 0.004074123 \approx 0.048889476 $$

6. Convert APR to percentage

To convert APR to a percentage, multiply by 100:

$$ APR \approx 0.048889476 \times 100 \approx 4.89% $$