If you borrow $20,000 for 5 years at an annual rate of 8%, what would the monthly payment be?.
The Correct Answer and Explanation is :
To calculate the monthly payment for a loan of $20,000 borrowed for 5 years at an annual interest rate of 8%, we can use the formula for monthly payments on an amortizing loan, which is:
[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n – 1}
]
Where:
- ( M ) is the monthly payment
- ( P ) is the loan principal (in this case, $20,000)
- ( r ) is the monthly interest rate (annual rate divided by 12 months)
- ( n ) is the total number of payments (loan term in months)
Step-by-Step Calculation:
- Loan Principal (P): $20,000
- Annual Interest Rate: 8%, so the monthly interest rate is:
[
r = \frac{8\%}{12} = 0.08 / 12 = 0.006667
] - Number of Payments (n): 5 years × 12 months = 60 months
Now, substitute these values into the formula:
[
M = 20000 \times \frac{0.006667(1 + 0.006667)^{60}}{(1 + 0.006667)^{60} – 1}
]
First, calculate ( (1 + 0.006667)^{60} ):
[
(1 + 0.006667)^{60} = 1.006667^{60} \approx 1.48985
]
Now, plug this value into the equation:
[
M = 20000 \times \frac{0.006667 \times 1.48985}{1.48985 – 1} \approx 20000 \times \frac{0.00993}{0.48985} \approx 20000 \times 0.0203 \approx 406.12
]
So, the monthly payment is approximately $406.12.
Explanation:
This formula calculates the fixed monthly payment required to fully repay a loan over a set period with compound interest. The process involves determining the interest charged on the loan each month and spreading that over the total number of payments. Since interest compounds on the loan, the total amount paid each month covers both the principal (the original loan amount) and the interest charged on the remaining balance. Over time, the proportion of the payment going toward the principal increases, while the amount applied to interest decreases.