Question
What is the monthly payment of a loan where the loan is 20,000, a 4% interest rate, and a 4 year pay period? Use the future value of an annuity formula
Answers
GPT-4o mini
To find the monthly payment of a loan using the future value of an annuity formula can be a bit unconventional because we usually derive monthly payments from the present value of an annuity formula. However, let's clarify your situation and solve accordingly.
For loans, we typically want to see how much you would pay back monthly—not how much money you will have in the future. Thus, I will provide the formula for computing the monthly payment for the loan.
The formula for the monthly payment (PMT) of an annuity (which is the same as a loan payment) is calculated using the Present Value of Annuity formula as follows:
\[
PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- \( PMT \) = monthly payment
- \( P \) = loan amount (present value)
- \( r \) = monthly interest rate (annual rate / 12)
- \( n \) = total number of payments (loan term in months)
In your case:
- \( P = 20,000 \)
- Annual interest rate = 4%, so the monthly interest rate \( r = \frac{0.04}{12} = \frac{0.04}{12} \approx 0.0033333 \)
- Loan term = 4 years, so \( n = 4 \times 12 = 48 \)
Now, plug these numbers into the formula:
\[
PMT = 20000 \times \frac{0.0033333(1 + 0.0033333)^{48}}{(1 + 0.0033333)^{48} - 1}
\]
Calculating \( (1 + r)^n \):
Calculating \( (1 + 0.0033333)^{48} \):
\[
(1 + 0.0033333)^{48} \approx 1.17336
\]
Substituting back to find the monthly payment:
\[
PMT = 20000 \times \frac{0.0033333 \times 1.17336}{1.17336 - 1}
\]
Now calculating the denominator:
\[
1.17336 - 1 \approx 0.17336
\]
Now substituting everything back into the formula:
\[
PMT = 20000 \times \frac{0.0033333 \times 1.17336}{0.17336} \approx 20000 \times \frac{0.00391}{0.17336}
\]
Calculating the fraction:
\[
\frac{0.00391}{0.17336} \approx 0.02254
\]
Now multiplying by 20,000:
\[
PMT \approx 20000 \times 0.02254 \approx 450.76
\]
The monthly payment is approximately **$450.76**.
For loans, we typically want to see how much you would pay back monthly—not how much money you will have in the future. Thus, I will provide the formula for computing the monthly payment for the loan.
The formula for the monthly payment (PMT) of an annuity (which is the same as a loan payment) is calculated using the Present Value of Annuity formula as follows:
\[
PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- \( PMT \) = monthly payment
- \( P \) = loan amount (present value)
- \( r \) = monthly interest rate (annual rate / 12)
- \( n \) = total number of payments (loan term in months)
In your case:
- \( P = 20,000 \)
- Annual interest rate = 4%, so the monthly interest rate \( r = \frac{0.04}{12} = \frac{0.04}{12} \approx 0.0033333 \)
- Loan term = 4 years, so \( n = 4 \times 12 = 48 \)
Now, plug these numbers into the formula:
\[
PMT = 20000 \times \frac{0.0033333(1 + 0.0033333)^{48}}{(1 + 0.0033333)^{48} - 1}
\]
Calculating \( (1 + r)^n \):
Calculating \( (1 + 0.0033333)^{48} \):
\[
(1 + 0.0033333)^{48} \approx 1.17336
\]
Substituting back to find the monthly payment:
\[
PMT = 20000 \times \frac{0.0033333 \times 1.17336}{1.17336 - 1}
\]
Now calculating the denominator:
\[
1.17336 - 1 \approx 0.17336
\]
Now substituting everything back into the formula:
\[
PMT = 20000 \times \frac{0.0033333 \times 1.17336}{0.17336} \approx 20000 \times \frac{0.00391}{0.17336}
\]
Calculating the fraction:
\[
\frac{0.00391}{0.17336} \approx 0.02254
\]
Now multiplying by 20,000:
\[
PMT \approx 20000 \times 0.02254 \approx 450.76
\]
The monthly payment is approximately **$450.76**.