Richard borrowed 170000 dollars from his bank to help finance the purchase of a house. The bank charges interest a rate of 8.9 percent per year compounded monthly. Richard has agreed to repay the loan in equal monthly installments over 35 years.
a) How much should each payment be if the loan is to be amortized at the end of the term?
b) How much, in total, will Richard spend on interest charges?
i = .089/12 = .007416666.. (I used my calculator's memory to store that)
n = 35x12 = 420
Let the monthly payment be P
using the standard formula for the Present Value of an annuity,
170000 = P( 1 - 1.00741666..^-240 )/.00741666..
P = $1320.09 -----> a)
b)
total paid = 120x1320.09 = $554,437.80
so it looks like the total interest paid was $384,437.80
(unfortunately, that is how mortgages over a long period of time work)
To find the answers to both questions, we need to use the formula for calculating the monthly payment on an amortized loan. This formula is called the loan amortization formula:
P = (Pv * r * (1+r)^n) / ((1+r)^n - 1)
Where:
P is the monthly payment
Pv is the present value or the initial loan amount
r is the monthly interest rate (annual interest rate divided by 12 months)
n is the number of payments (loan term in months)
Let's calculate the answers step by step:
a) How much should each payment be if the loan is to be amortized at the end of the term?
Using the loan amortization formula, we can substitute the given values:
Pv = $170,000
r = (8.9% / 100) / 12 = 0.0074167 (monthly interest rate)
n = 35 years * 12 months = 420
P = ($170,000 * 0.0074167 * (1+0.0074167)^420) / ((1+0.0074167)^420 - 1)
P ≈ $1,221.55
Therefore, each monthly payment should be approximately $1,221.55.
b) How much, in total, will Richard spend on interest charges?
To calculate the total interest charges, we need to find the total amount repaid to the bank and subtract the principal loan amount.
Total amount repaid = P * n
Total amount repaid = $1,221.55 * 420
Total amount repaid ≈ $513,321
Interest charges = Total amount repaid - Principal loan amount
Interest charges = $513,321 - $170,000
Interest charges ≈ $343,321
Therefore, Richard will spend approximately $343,321 on interest charges over the course of the loan.