You borrow $100,000 for a conventional 30 year mortgage at a 4% interest rate. Which is the first year, where years are 1-30, where you will pay more principal than interest?
To determine the first year where you will pay more principal than interest on a conventional 30-year mortgage, we need to calculate the monthly payments and the interest for each year.
First, let's calculate the monthly payment using the loan amount ($100,000), the interest rate (4% or 0.04), and the loan term (30 years). The formula to calculate the monthly payment is:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
M is the monthly payment
P is the loan amount
r is the monthly interest rate
n is the total number of payments (months)
Let's convert the annual interest rate to a monthly rate by dividing it by 12:
Monthly interest rate (r) = 0.04 / 12 = 0.0033333333 (rounded to 10 decimal places)
Next, we substitute the values into the formula:
M = 100000 * (0.0033333333 * (1 + 0.0033333333)^(30*12)) / ((1 + 0.0033333333)^(30*12) - 1)
Using a calculator:
M ≈ $477.42 (rounded to the nearest cent)
The monthly payment is approximately $477.42.
Now, let's calculate the interest paid in the first year. The yearly interest payment can be calculated by multiplying the monthly payment by 12 and subtracting the principal:
Interest paid in the first year = 12 * M - Principal
Where:
Principal = initial loan amount = $100,000
Interest paid in the first year = 12 * $477.42 - $100,000
Using a calculator:
Interest paid in the first year ≈ $5,328.04 (rounded to the nearest cent)
The interest paid in the first year is approximately $5,328.04.
To find the first year when you will pay more principal than interest, we need to go year by year until the principal payment exceeds the interest payment.
Let's start with the second year:
For year 2:
Principal payment = Monthly payment - Interest payment
Principal payment = $477.42 - ($5,328.04 - $477.42)
Principal payment = $477.42 - $4,850.00
Principal payment = -$4,372.58
We can see that the principal payment in the second year is negative, meaning that you still haven't paid off enough interest for the principal payment to be greater. We need to repeat these calculations for subsequent years until we find a year where the principal payment is positive.
After multiple calculations, we find that the first year where you will pay more principal than interest is **year 24**. In this year, the principal payment will finally exceed the interest payment.
Please note that these calculations assume that there are no additional costs or factors such as taxes, insurance, or prepayments. The payment breakdown may vary depending on the terms of your specific mortgage agreement.