Using a spreadsheet program, create an amortization schedule for a 30-year mortgage of $500,000 at an annual interest rate of 4.25%.
(a) In which month does the amount of principal in a monthly payment first exceed the amount of interest?
Answer (2)
Calculate the monthly interest rate: 12 0.0425 = 0.003541666666666667 .
Calculate the total number of payments: 30 × 12 = 360 .
Calculate the monthly payment using the formula: M = 500000 × ( 1 + 0.003541666666666667 ) 360 − 1 0.003541666666666667 ( 1 + 0.003541666666666667 ) 360 ≈ 2459.70 .
Iterate through the amortization schedule to find the month where the principal paid exceeds the interest paid: 165 .
Explanation
Understanding the Problem We are given a 30-year mortgage of $500,000 at an annual interest rate of 4.25%. Our goal is to find the month in which the amount of principal in a monthly payment first exceeds the amount of interest.
Calculating Monthly Interest Rate First, we need to calculate the monthly interest rate. We divide the annual interest rate by 12: 12 4.25% = 12 0.0425 = 0.003541666666666667
Calculating Total Number of Payments Next, we calculate the total number of payments. Since it's a 30-year mortgage with monthly payments, we have: 30 years × 12 months/year = 360 payments
Calculating Monthly Mortgage Payment Now, we calculate the monthly mortgage payment using the formula: M = P ( 1 + i ) n − 1 i ( 1 + i ) n where:
M is the monthly payment
P is the principal loan amount ($500,000)
i is the monthly interest rate (0.003541666666666667)
n is the total number of payments (360) Plugging in the values, we get: M = 500000 × ( 1 + 0.003541666666666667 ) 360 − 1 0.003541666666666667 ( 1 + 0.003541666666666667 ) 360 ≈ 2459.70
Creating Amortization Schedule We create an amortization schedule. For each month, we calculate the interest paid, principal paid, and the remaining balance. The interest paid is the beginning balance multiplied by the monthly interest rate. The principal paid is the monthly payment minus the interest paid. The ending balance is the beginning balance minus the principal paid. We iterate through the months until the principal paid exceeds the interest paid.
Finding the Month After iterating through the amortization schedule, we find that the principal portion of the monthly payment first exceeds the interest portion in month 165.
Final Answer Therefore, the amount of principal in a monthly payment first exceeds the amount of interest in month 165.
Examples
Understanding amortization schedules is crucial when taking out loans, such as mortgages or car loans. For instance, if you're planning to sell your house after a certain number of years, knowing the amortization schedule helps you determine how much of the principal you've paid off and how much equity you have in your home. This knowledge is essential for making informed financial decisions about refinancing, selling, or making extra payments to reduce the loan term.
The principal portion of a mortgage payment first exceeds the interest portion in month 165 of a 30-year mortgage for $500,000 at an annual interest rate of 4.25%. This was determined by calculating the monthly payment, interest, and principal across the amortization schedule. By following these calculations, it can be concluded that this shift in payment dynamics occurs specifically at this point in time.
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