Question
You have just purchased a home by borrowing
$
400
,
000
$400,000 for 30-years at a fixed APR of
3.87
%
3.87%. The loan payments are monthly and interest is compounded monthly.
What is the effective annual rate on the loan? (I.e., what is the interest rate once we take into account compounding?)
0.0394
0.0394
2.0394
2.0394
2.1239
2.1239
1.0394
1.0394
$
400
,
000
$400,000 for 30-years at a fixed APR of
3.87
%
3.87%. The loan payments are monthly and interest is compounded monthly.
What is the effective annual rate on the loan? (I.e., what is the interest rate once we take into account compounding?)
0.0394
0.0394
2.0394
2.0394
2.1239
2.1239
1.0394
1.0394
Answers
GPT-4o mini
To calculate the effective annual rate (EAR) from the nominal annual percentage rate (APR) given that interest is compounded monthly, you can use the following formula:
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) = nominal interest rate (as a decimal)
- \( n \) = number of compounding periods per year
In your case:
- \( r = 0.0387 \) (3.87% as a decimal)
- \( n = 12 \) (since the interest is compounded monthly)
Now, plug in the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.0387}{12}\right)^{12} - 1
\]
Calculating the values:
1. Calculate \( \frac{0.0387}{12} \):
\[
\frac{0.0387}{12} \approx 0.003225
\]
2. Calculate \( 1 + 0.003225 \):
\[
1 + 0.003225 \approx 1.003225
\]
3. Raise this result to the power of 12:
\[
(1.003225)^{12} \approx 1.039367
\]
4. Subtract 1 to find the EAR:
\[
1.039367 - 1 \approx 0.039367
\]
5. Convert to percentage:
\[
0.039367 \times 100 \approx 3.9367\%
\]
Thus, the effective annual rate (EAR) on the loan is approximately **3.94%**.
So none of the options you provided match the calculated EAR quite exactly, but the closest option available is:
- **0.0394**.
This is simply the decimal form of the approximate EAR calculated.
\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]
Where:
- \( r \) = nominal interest rate (as a decimal)
- \( n \) = number of compounding periods per year
In your case:
- \( r = 0.0387 \) (3.87% as a decimal)
- \( n = 12 \) (since the interest is compounded monthly)
Now, plug in the values into the formula:
\[
\text{EAR} = \left(1 + \frac{0.0387}{12}\right)^{12} - 1
\]
Calculating the values:
1. Calculate \( \frac{0.0387}{12} \):
\[
\frac{0.0387}{12} \approx 0.003225
\]
2. Calculate \( 1 + 0.003225 \):
\[
1 + 0.003225 \approx 1.003225
\]
3. Raise this result to the power of 12:
\[
(1.003225)^{12} \approx 1.039367
\]
4. Subtract 1 to find the EAR:
\[
1.039367 - 1 \approx 0.039367
\]
5. Convert to percentage:
\[
0.039367 \times 100 \approx 3.9367\%
\]
Thus, the effective annual rate (EAR) on the loan is approximately **3.94%**.
So none of the options you provided match the calculated EAR quite exactly, but the closest option available is:
- **0.0394**.
This is simply the decimal form of the approximate EAR calculated.