Practice Problem – Effective Rate

Practice Question

Intro to Finance
Time Value of Money
Effective Rate
Short Answer

An interest rate compounded twice a year yields an EAR of 5%. This rate is equivalent to what APR compounded daily?

Answer +
0.049
Explanation +

To find the equivalent APR compounded daily from an interest rate compounded twice a year that yields an EAR of 5%, we can use the following relationship between EAR and APR:

Step 1: Understand the Formulas

The formula for converting an effective annual rate (EAR) to an annual percentage rate (APR) is given by:

EAR = (1 + APR/m)^m - 1

Where:

  • m is the number of compounding periods per year.

Step 2: Set Up the Calculation

In this case:

  • EAR = 0.05 (5%)
  • For semi-annual compounding, m = 2

We need to find the APR that corresponds to this EAR. Rearranging the formula:

1 + EAR = (1 + APR/2)^2

Taking the square root:

√1.05 = 1 + APR/2

Calculating:

1.0247 ≈ 1 + APR/2
APR ≈ 0.0494 or 4.94%

Step 4: Convert to Daily Compounding

To convert this APR to an effective daily rate, assuming 365 days in a year:

Effective Daily Rate = (1 + APR/365)^365 - 1

However, since the question asks for the APR equivalent compounded daily, we can use the previously calculated APR.

Final Answer

The APR equivalent to an EAR of 5% compounded semi-annually is approximately 4.88% when compounded daily.