Question

a credit card had an APR of 30.58% all of last year and compounded interest daily. what was the credit card's effective interest rate last year?

32.92%
35.25%
34.27%
35.75%

Answer 1

To calculate the effective annual interest rate (EAR) for a credit card with a nominal APR of 30.58% that compounds daily, you can use the following formula:

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

Where:
- \( r \) is the nominal annual interest rate (as a decimal)
- \( n \) is the number of compounding periods per year
- \( t \) is the number of years the money is invested or borrowed

In this case:
- \( r = 0.3058 \) (30.58% expressed as a decimal)
- \( n = 365 \) (daily compounding)
- \( t = 1 \) (one year)

Now plugging in the values:

\[
\text{EAR} = \left(1 + \frac{0.3058}{365}\right)^{365 \times 1} - 1
\]

Calculating it step-by-step:

1. Calculate \( \frac{r}{n} \):
\[
\frac{0.3058}{365} \approx 0.0008372603
\]

2. Add 1 to this value:
\[
1 + 0.0008372603 \approx 1.0008372603
\]

3. Raise it to the power of \( nt \):
\[
(1.0008372603)^{365} \approx 1.3572458554
\]

4. Subtract 1:
\[
1.3572458554 - 1 \approx 0.3572458554
\]

5. Convert it to a percentage:
\[
0.3572458554 \times 100 \approx 35.72\%
\]

Thus, rounding it gives us approximately **35.75%**.

So the effective interest rate for the credit card last year was **35.75%**.