A man buys a house for $310,000. He makes a $150,000 down payment and amortizes the rest of the debt with semiannual payments over the next 8 years. The interest rate on the debt is 11%, compounded semiannually.
(a) Find the size of each payment.
(b) Find the total amount paid over the life of the loan (including the down payment).
(c) Find the total interest paid over the life of the loan.
Update:
I've got parts a and c correct! But I'm still having difficulty with part b - I multiplied the answer from part a with the number of years and then added the down payment, but that isn't correct. I'm not really sure what I'm doing wrong.
Update #2:
I've figured it out! You're supposed to multiply the answer from part a with whatever the n value is (in this case it's 16 because 8x2), and then you add the down payment. Thanks to anyone who considered this question! Haha
I wonder
To find the size of each payment, we can use the formula for the present value of an ordinary annuity:
PV = PMT × (1 - (1 + r)^(-n)) / r
Where:
PV = Present value (the remaining debt)
PMT = Payment size
r = Interest rate per period
n = Total number of periods
In this case, the present value is the amount remaining after the down payment, which is $310,000 - $150,000 = $160,000. The interest rate is 11% compounded semiannually, so the interest rate per period is 11% / 2 = 0.055, and the total number of periods is 8 years * 2 = 16 half-years.
Using these values, we can substitute into the formula and solve for PMT:
$160,000 = PMT × (1 - (1 + 0.055)^(-16)) / 0.055
Now, let's calculate it step by step:
Step 1: Calculate the factor (1 - (1 + r)^(-n)) / r:
(1 - (1 + 0.055)^(-16)) / 0.055 ≈ 10.8825
Step 2: Solve for PMT:
$160,000 = PMT × 10.8825
PMT ≈ $160,000 / 10.8825 ≈ $14,699.06
So the size of each payment is approximately $14,699.06.
Now, let's move on to finding the total amount paid over the life of the loan, including the down payment.
To do that, we'll calculate the total number of payments made over 8 years, which is 8 * 2 = 16 half-year payments. We'll multiply this number by the payment size and add the down payment.
Total amount paid = Down payment + (Payment size × Number of payments)
Total amount paid = $150,000 + ($14,699.06 × 16)
Total amount paid ≈ $150,000 + $235,985.60 ≈ $385,985.60
So the total amount paid over the life of the loan, including the down payment, is approximately $385,985.60.
Finally, to find the total interest paid over the life of the loan, we subtract the initial loan amount (before the down payment) from the total amount paid.
Total interest paid = Total amount paid - Initial loan amount
Total interest paid = $385,985.60 - $310,000 ≈ $75,985.60
Therefore, the total interest paid over the life of the loan is approximately $75,985.60.