Question

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%**.