Your parents are buying a house for $187,500. They have a good credit rating, are making a 20% down payment, and expect to pay $1,575/month. The interest rate for the mortgage is 4.65%. What must their realized income be before each month and how much interest is paid at the end of the second month?
To find out what their realized income must be before each month, we need to consider the monthly mortgage payment, the down payment, and the interest rate.
First, let's calculate the loan amount. The down payment is 20% of the house price. Therefore, the loan amount is 80% of $187,500:
Loan amount = 0.8 * $187,500 = $150,000
Now, we need to calculate the monthly interest rate. Divide the annual interest rate by 12 to get the monthly interest rate:
Monthly interest rate = (4.65% / 100) / 12 = 0.03875
Next, we can calculate the monthly payment using the loan amount, interest rate, and the number of months:
Monthly payment = (Loan amount * Monthly interest rate) / (1 - (1 + Monthly interest rate)^(-n))
Where n represents the number of months in the term of the loan. In this case, we'll assume it's a 30-year mortgage, so n would be 360 (12 months * 30 years):
Monthly payment = ($150,000 * 0.03875) / (1 - (1.03875)^(-360)) ≈ $790.79
Since they expect to pay $1,575/month, we need to find out the difference between the monthly payment and their expected monthly payment:
Additional amount from realized income = $1,575 - $790.79 ≈ $784.21
Therefore, their realized income before each month would need to be at least $784.21.
To calculate the interest paid at the end of the second month, let's first calculate the interest portion of the monthly payment. Multiply the loan amount by the monthly interest rate:
Interest portion of the monthly payment = $150,000 * 0.03875 ≈ $5,812.50
Now, to find out the interest paid at the end of the second month, we need to consider that the first payment will cover both the principal and the interest for the first month. The second payment will cover the principal, plus the interest for the second month.
Principal portion of the monthly payment = Monthly payment - Interest portion of the monthly payment = $790.79 - $5,812.50 ≈ -$5,021.71
Since the principal portion is negative, it means the full payment goes towards interest in the second month.
Therefore, the interest paid at the end of the second month is approximately $5,812.50.
To determine the income required before each month, we need to calculate the monthly mortgage payment first.
The down payment is 20% of the house price of $187,500, which is $37,500 ($187,500 x 0.20).
The loan amount is the remaining $150,000 ($187,500 - $37,500).
To calculate the monthly mortgage payment, we can use the formula for the monthly payment on a fixed-rate mortgage:
M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1 ]
Where:
M is the monthly payment
P is the principal loan amount
i is the monthly interest rate (annual interest rate divided by 12)
n is the total number of monthly payments (loan term in years multiplied by 12)
In this case:
P = $150,000
i = 4.65% / 100 / 12 = 0.003875
n = loan term
To find the loan term, we need to know how many years the mortgage is for.
Now, since the monthly payment is given as $1,575, we can use this information to solve for the loan term.
Let's plug in the values and solve for n:
$1,575 = $150,000 [ 0.003875(1 + 0.003875)^n ] / [ (1 + 0.003875)^n - 1 ]
This equation is not solvable directly, but we can use iteration methods or an online calculator to find the approximate loan term.
Assuming the loan term is 30 years (360 months), we can calculate the monthly payment:
M = $150,000 [ 0.003875(1 + 0.003875)^360 ] / [ (1 + 0.003875)^360 - 1 ]
M ≈ $1,575
Now, to determine the required income before each month, we need to consider the monthly mortgage payment, property taxes, insurance, and other debts.
Let's assume property taxes are $200/month and insurance is $100/month (these are just estimated values, and it's best to consult with a mortgage professional to get accurate numbers).
The total monthly expenses would be:
$1,575 for the mortgage payment +
$200 for property taxes +
$100 for insurance
Total expenses = $1,575 + $200 + $100 = $1,875
To find out the required income before each month, it's generally recommended for housing expenses to not exceed 28% of the total monthly income.
So, we can use the formula:
Required income = Total expenses / Debt-to-income ratio
In this case, 28% is the debt-to-income ratio.
Required income = $1,875 / 0.28
Required income ≈ $6,696.43
Therefore, their realized income must be approximately $6,696.43 each month before expenses.
To calculate the interest paid at the end of the second month, we can use the amortization formula:
Interest for the first month = Loan amount x Monthly interest rate
Interest for the second month = (Loan amount - Principal paid in the first month) x Monthly interest rate
Principal paid in the first month = Monthly payment - Interest for the first month
Now let's calculate:
Interest for the first month = $150,000 x 0.003875 ≈ $581.25
Principal paid in the first month = $1,575 - $581.25 ≈ $993.75
Loan amount for the second month = $150,000 - $993.75 ≈ $149,006.25
Interest for the second month = $149,006.25 x 0.003875 ≈ $577.64
Therefore, the interest paid at the end of the second month is approximately $577.64.