Question

essie has an offer from a credit card issuer for a 0% APR for the fist 60 days and a 19.34% APR afterwards, compounded daily. what effective interest is essie being offered?

A. 17.53%
B. 19.34%
C. 21.15%
D. 21.33%

Answer 1

To calculate the effective interest rate, we need to account for the initial 0% APR for 60 days and the 19.34% APR that follows.

1. **Calculate the daily interest rate for 19.34% APR**:
\[
\text{Daily interest rate} = \frac{19.34\%}{365} = \frac{0.1934}{365} \approx 0.0005306
\]

2. **Calculate the effective interest over 60 days at 0%**:
\[
\text{Interest for 60 days} = 0
\]

3. **Calculate the effective interest after 60 days at 19.34%**:
For a duration of one year (365 days), the remaining days after the first 60 days is:
\[
365 - 60 = 305 \text{ days}
\]

Now, we will calculate the total amount accumulated after 365 days using the formula for compound interest:
\[
A = P \times (1 + r/n)^{nt}
\]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (we can assume \( P = 1 \) for simplicity).
- \( r \) is the annual interest rate (19.34%).
- \( n \) is the number of times that interest is compounded per year (365).
- \( t \) is the time the money is invested for in years.

After 60 days:
\[
A = 1 \times (1 + 0)^{60} = 1
\]

After the next 305 days:
\[
A = 1 \times (1 + 0.0005306)^{305}
\]

We can calculate this expression:
\[
A \approx (1.0005306)^{305}
\]
Now calculating \( A \):
Using a calculator:
\[
A \approx 1.1753
\]

4. **Calculating the effective annual rate (EAR)**:
The effective annual rate can be found using the formula:
\[
\text{EAR} = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1
\]
Since we used \( P = 1 \):
\[
\text{EAR} = A - 1 = 1.1753 - 1 = 0.1753 \text{ or } 17.53\%
\]

Thus, the answer is:

**A. 17.53%**