A man buys a house for $350,000. He makes a $150,000 down payment and amortizes the rest of the purchase price with semiannual payments over the next 7 years. The interest rate on the debt is 13%, compounded semiannually.
(a) Find the size of each payment.
$
(b) Find the total amount paid for the purchase.
$
(c) Find the total interest paid over the life of the loan.
ljnl.
To find the size of each payment, you can use the amortization formula for a loan:
\[ P = I \times \left( \frac{1 - (1+r)^{-n}}{r} \right) \]
where:
P = Principal amount (the remaining loan balance)
I = Periodic payment amount
r = Interest rate per period
n = Total number of periods
In this case, the principal amount is $350,000 - $150,000 = $200,000 (since a $150,000 down payment was made). The loan will be amortized over 7 years, which corresponds to 14 semiannual periods (since payments are made semiannually). The interest rate per period is 13% / 2 = 6.5% (since it is compounded semiannually).
(a) To find the size of each payment (I), plug in the values into the formula:
\( I = 200,000 \times \left( \frac{1 - (1+0.065)^{-14}}{0.065} \right) \)
Calculating this will give you the size of each payment.
(b) To find the total amount paid for the purchase, you need to multiply the size of each payment by the total number of payments. In this case, the total number of payments is 14 (since it's amortized over 7 years with semiannual payments). Multiply the size of each payment by 14 to find the total amount paid.
(c) To find the total interest paid over the life of the loan, you can subtract the principal amount ($200,000) from the total amount paid calculated in part (b). This will give you the total interest paid.