Daniel and Jan agreed to pay $560,000 for a four-bedroom colonial home
in Waltham, Mass., with $60,000 down payment. They have a 30-year mortgage at a fixed rate of 6.00%. (a) How much is their monthly payment? (b) After the first payment, what would be the balance of the principal?
To calculate Daniel and Jan's monthly payment, we can use the formula for calculating a fixed-rate mortgage payment:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]
Where:
M = Monthly payment
P = Principal amount (loan amount)
i = Monthly interest rate (annual interest rate divided by 12)
n = Total number of payments (30 years multiplied by 12 months)
(a) To calculate the monthly payment:
Principal amount (P) = $560,000 - $60,000 (down payment) = $500,000
Monthly interest rate (i) = 6.00% / 12 = 0.005 (decimal form)
Total number of payments (n) = 30 years * 12 months = 360
Using these values in the formula, we get:
M = $500,000 [ 0.005(1 + 0.005)^360 ] / [ (1 + 0.005)^360 - 1 ]
Calculating this expression, we find the monthly payment, M, for Daniel and Jan:
M ≈ $2,997.74
Therefore, their monthly payment is approximately $2,997.74.
(b) To calculate the balance of the principal after the first payment, we can subtract the principal portion of the payment from the original loan amount.
Principal portion of the payment = Monthly payment - (Loan amount * Monthly interest rate)
Principal portion of the payment = $2,997.74 - ($500,000 * 0.005)
Calculating the principal portion, we get:
Principal portion of the payment ≈ $2,997.74 - $2,500
Principal portion of the payment ≈ $497.74
Balance of the principal after the first payment = Loan amount - Principal portion of the payment
Balance of the principal after the first payment = $500,000 - $497.74
Therefore, after the first payment, the balance of the principal would be approximately $499,502.26.
To calculate the monthly payment and the balance of the principal after the first payment, we need to use the formula for calculating the monthly mortgage payment.
Let's break down the process step by step:
(a) Calculating the monthly payment:
1. First, we need to determine the loan amount. Since Daniel and Jan made a $60,000 down payment, the loan amount is equal to the purchase price minus the down payment. In this case, $560,000 - $60,000 = $500,000.
2. Now, we can use the loan amount, the interest rate, and the loan term to calculate the monthly payment using the following formula:
M = P*(R/12)*(1+R/12)^N / ((1+R/12)^N -1),
where:
M = Monthly payment
P = Loan amount
R = Monthly interest rate (annual interest rate/12)
N = Total number of payments
For our example:
P = $500,000
R = 6.00% / 100 / 12 = 0.005
N = 30 years * 12 months per year = 360
By plugging in these values into the formula, we can calculate the monthly payment.
(b) Calculating the balance of the principal after the first payment:
The balance of the principal after the first payment can be obtained by subtracting the principal portion of that payment from the original loan amount.
Now, let's calculate these values using the formulas:
(a) Monthly payment:
M = $500,000 * (0.005) * (1+0.005)^360 / ((1+0.005)^360-1)
(b) Balance of principal after the first payment:
Loan balance after payment 1 = Original loan amount - Principal portion of payment 1
To calculate the balance after payment 1, we need to calculate the interest portion and principal portion of payment 1.
Interest portion = Loan balance before payment 1 * Monthly interest rate
Principal portion = Monthly payment - Interest portion
Loan balance after payment 1 = Loan balance before payment 1 - Principal portion
Now, you can plug in the values and calculate the monthly payment and the balance of the principal after the first payment.
i = .06/12 = .005
n = 12*30 = 360
pmt = ??
pmt( 1 - 1.005^-360)/.005 = 500000
pmt = 2997.75
the interest in the first month = .005(500000) = 2500
so the actual reduction in the principal for the 1st month = 2997.75-2500 = 497.75
leaving a balance of $499,502.24