Question

jay's credit card had an APR of 16.53% all of last year, and interest was compounded periodically throughout the year. which of these statements accurately describes the effective interest rate of jays credit card last year?

it was greater that 16.53% whether interest was compounded daily or monthly
it was less than 16.53% if interest was compounded daily, but not if interest was compounded monthly
it was less than 16.53% whether interest was compounded daily or monthly
it was greater than 16.53%if interest was compounded daily, but not if interest was compounded monthly

Answer 1

To determine the effective interest rate (EIR) based on the Annual Percentage Rate (APR) of 16.53%, we need to understand how compounding affects the overall rate.

The effective interest rate can be calculated using the formula:

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

where:
- \( r \) is the nominal interest rate (APR expressed as a decimal)
- \( n \) is the number of compounding periods per year (e.g., 365 for daily, 12 for monthly).

For an APR of 16.53%:
- As a decimal, \( r = 0.1653 \).

1. **Compounding Daily (n = 365)**:

\[
EIR_{daily} = \left(1 + \frac{0.1653}{365}\right)^{365} - 1 \approx 0.1802 \quad \text{or} \quad 18.02\%
\]

2. **Compounding Monthly (n = 12)**:

\[
EIR_{monthly} = \left(1 + \frac{0.1653}{12}\right)^{12} - 1 \approx 0.1795 \quad \text{or} \quad 17.95\%
\]

Both calculated effective interest rates are greater than 16.53%.

Given this analysis, the correct statement from the provided options would be:

**"It was greater than 16.53% whether interest was compounded daily or monthly."**