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Calculate APR from EAR.

Understand the Problem

The question is asking us to convert an Effective Annual Rate (EAR) into an Annual Percentage Rate (APR). This involves using the relationship between these two financial metrics, typically involving the number of compounding periods per year.

Answer

The APR is approximately $4.89\%$.
Answer for screen readers

The Annual Percentage Rate (APR) is approximately $4.89%$.

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 $5%$).

  1. 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 $n = 12$.

  1. 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:

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

  1. Plug in the values

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

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

  1. 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:

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

  1. Convert APR to percentage

To convert APR to a percentage, multiply by 100:

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

The Annual Percentage Rate (APR) is approximately $4.89%$.

More Information

Converting between EAR and APR is important for understanding the true cost of borrowing or the real return on investment over a period.

Tips

  • Ignoring the compounding frequency: A common mistake is to forget to account for how often interest is compounded, which can lead to incorrect results.
  • Using the wrong formula: Ensure you use the correct formula to convert between EAR and APR.
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