Question

dylan has an offer from a credit card issuer for 0% APR for the 30 days and 14.04% APR afterwards, compounded daily. what effective interest rate is being offered?

14.98%
13.75%
14.04%
15.07%

Answer 1

To determine the effective interest rate (EIR) being offered by the credit card issuer, we can use the information provided about the APR and its compounding frequency.

1. **Initial Offer**: 0% APR for the first 30 days (1 month).
2. **Subsequent APR**: 14.04% APR afterwards, compounded daily.

To calculate the effective annual rate (EAR) after the initial period, we can use the formula for effective annual rate with daily compounding:

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

Where:
- \( r \) is the nominal interest rate (14.04% or 0.1404),
- \( n \) is the number of compounding periods per year (365 days),
- \( t \) is the number of years (in this case, we want to find the EIR for 1 year, so \( t = 1 \)).

First, we convert the APR to a daily rate:

\[
\text{Daily rate} = \frac{0.1404}{365} \approx 0.0003847
\]

Now we can calculate the effective annual rate:

\[
\text{EAR} = \left(1 + 0.0003847\right)^{365} - 1
\]

Calculating this:

\[
\text{EAR} = \left(1.0003847\right)^{365} - 1
\]

Using a calculator:

1. Calculate \( (1.0003847)^{365} \) which is approximately \( 1.1488 \).
2. Subtract 1:

\[
\text{EAR} \approx 1.1488 - 1 = 0.1488 \text{ or } 14.88\%
\]

Rounding to two decimal places gives us approximately 14.88%.

Since the options provided are:
- 14.98%
- 13.75%
- 14.04%
- 15.07%

The effective interest rate (14.88%) matches closely with the option **14.98%**.

Thus, the correct answer is **14.98%**.