A credit card offers the following terms: 14.78% APR with daily compounding. What is the effective rate on balances with this card?

Answer 1

To find the effective annual rate (EAR) from an annual percentage rate (APR) with daily compounding, you can use the following formula:

\[ \text{EAR} = \left(1 + \frac{r}{n}\right)^{n} - 1 \]

Where:

• \( r \) is the nominal annual interest rate (as a decimal),
• \( n \) is the number of compounding periods per year.

In this case, the APR is 14.78%, so we convert that to a decimal:

\[ r = \frac{14.78}{100} = 0.1478 \]

Since the compounding is daily, \( n \) is 365.

Now, we can substitute these values into the formula:

\[ \text{EAR} = \left(1 + \frac{0.1478}{365}\right)^{365} - 1 \]

Calculating the value:

1. Calculate \( \frac{r}{n} \): \[ \frac{0.1478}{365} \approx 0.0004048 \]

2. Now add 1 to this value: \[ 1 + 0.0004048 \approx 1.0004048 \]

3. Raise this value to the power of 365: \[ (1.0004048)^{365} \approx 1.1582 \]

4. Finally, subtract 1 to find the EAR: \[ \text{EAR} \approx 1.1582 - 1 \approx 0.1582 \]

5. Convert back to percentage: \[ \text{EAR} \approx 15.82% \]

Thus, the effective rate on balances with this credit card is approximately 15.82%.