Question
a credit card issuer offers an APR of 13.64% and compounds interest daily. which is it most likely to advertise, its APR or its effective interest rate?
its APR, because its 0.97% greater than its effective interest rate
its APR, because its 0.97% less than its effective interest rate
its effective interest rate, because its 0.97% greater than its APR
its effective interest rate, because its 0.97% less than its APR
its APR, because its 0.97% greater than its effective interest rate
its APR, because its 0.97% less than its effective interest rate
its effective interest rate, because its 0.97% greater than its APR
its effective interest rate, because its 0.97% less than its APR
Answers
GPT-4o mini
To determine whether the credit card issuer is likely to advertise its APR or its effective interest rate, we need to calculate the effective interest rate based on the given APR of 13.64%, compounded daily.
1. **Calculate the Effective Interest Rate (EIR)** using the formula:
\[
EIR = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) is the nominal annual interest rate (APR) as a decimal, so 13.64% becomes 0.1364.
- \( n \) is the number of compounding periods per year (for daily compounding, \( n = 365 \)).
- \( t \) is the number of years (we can use \( t = 1 \) for one year).
2. **Substituting values**:
\[
EIR = \left(1 + \frac{0.1364}{365}\right)^{365 \cdot 1} - 1
\]
3. **Calculating inside the parentheses**:
\[
\frac{0.1364}{365} \approx 0.0003738
\]
Then,
\[
EIR \approx \left(1 + 0.0003738\right)^{365} - 1
\]
4. **Calculating the exponent**:
\[
EIR \approx (1.0003738)^{365} - 1
\]
Using a calculator for \( (1.0003738)^{365} \), we find:
\[
EIR \approx 1.1465 - 1 \approx 0.1465
\]
Or in percentage terms, \( EIR \approx 14.65\% \).
Now comparing the APR and the EIR:
- **APR**: 13.64%
- **EIR**: 14.65%
The Effective Interest Rate (14.65%) is greater than the APR (13.64%).
Given that credit card issuers typically prefer to advertise the rate that makes their offers appear more favorable to consumers, it's likely the issuer would advertise the **APR**, because while it's lower, it is easier to recognize and doesn't prompt the consumer to think about compounding.
Thus, the correct answer is:
**its APR, because its 0.97% less than its effective interest rate.**
1. **Calculate the Effective Interest Rate (EIR)** using the formula:
\[
EIR = \left(1 + \frac{r}{n}\right)^{nt} - 1
\]
Where:
- \( r \) is the nominal annual interest rate (APR) as a decimal, so 13.64% becomes 0.1364.
- \( n \) is the number of compounding periods per year (for daily compounding, \( n = 365 \)).
- \( t \) is the number of years (we can use \( t = 1 \) for one year).
2. **Substituting values**:
\[
EIR = \left(1 + \frac{0.1364}{365}\right)^{365 \cdot 1} - 1
\]
3. **Calculating inside the parentheses**:
\[
\frac{0.1364}{365} \approx 0.0003738
\]
Then,
\[
EIR \approx \left(1 + 0.0003738\right)^{365} - 1
\]
4. **Calculating the exponent**:
\[
EIR \approx (1.0003738)^{365} - 1
\]
Using a calculator for \( (1.0003738)^{365} \), we find:
\[
EIR \approx 1.1465 - 1 \approx 0.1465
\]
Or in percentage terms, \( EIR \approx 14.65\% \).
Now comparing the APR and the EIR:
- **APR**: 13.64%
- **EIR**: 14.65%
The Effective Interest Rate (14.65%) is greater than the APR (13.64%).
Given that credit card issuers typically prefer to advertise the rate that makes their offers appear more favorable to consumers, it's likely the issuer would advertise the **APR**, because while it's lower, it is easier to recognize and doesn't prompt the consumer to think about compounding.
Thus, the correct answer is:
**its APR, because its 0.97% less than its effective interest rate.**