You borrow $150,000 to purchase a new house. The bank offers you a special 20 year loan with a 7% interest rate and you will make annual payments. How large will each payment be? If you want to sell your house after 7 years, how much will you owe and need to pay to pay off the loan?
150000 =PAY( 1 - 1.07^-20)/.07
solving for PAY, I got $ 14,158.94
Balance owing after 7 years
= 150000(1.07)^7 - 14158.94( 1.07^7 - 1)/.07
= $ 118.335.46
To find out how large each annual payment will be, we can use the formula for an ordinary annuity:
PV = PMT * [(1 - (1 + r)^-n) / r]
Where:
PV = present value (loan amount) = $150,000
PMT = annual payment (what we need to find)
r = interest rate per period = 7% = 0.07
n = number of periods = 20
We will solve for PMT in this equation.
First, let's plug in the known values into the equation:
$150,000 = PMT * [(1 - (1 + 0.07)^-20) / 0.07]
Now, let's simplify the equation:
$150,000 = PMT * [14.970318216132048 / 0.07]
Next, let's solve for PMT by isolating it:
PMT = $150,000 / [14.970318216132048 / 0.07]
PMT = $150,000 / 213.8616888018864
PMT ≈ $701.49
Therefore, each annual payment will be approximately $701.49.
If you want to sell your house after 7 years, you need to calculate how much you will owe and need to pay to pay off the loan.
To find out how much you will owe after 7 years, we can use the formula for the remaining balance on a loan:
FV = PV * (1 + r)^n - PMT * [(1 + r)^n - 1] / r
Where:
FV = future value (remaining balance) = what we need to find
PV = present value (loan amount) = $150,000
r = interest rate per period = 7% = 0.07
n = number of periods = 20
PMT = annual payment = $701.49
Now, let's plug in the known values into the equation:
FV = $150,000 * (1 + 0.07)^7 - $701.49 * [(1 + 0.07)^7 - 1] / 0.07
FV ≈ $108,852.36
Therefore, after 7 years, you will owe approximately $108,852.36 and need to pay that amount to pay off the remaining balance of the loan.
To calculate the annual payment on a loan, you can use the formula for calculating the payment on an amortizing loan. The formula is:
Payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
Where:
P = Principal amount (loan amount)
r = Annual interest rate (expressed as a decimal)
n = Number of payments (loan term in years)
Let's calculate the annual payment for your loan:
Principal amount (loan amount) = $150,000
Annual interest rate = 7% = 0.07 (decimal form)
Number of payments (loan term) = 20 years
Payment = $150,000 * 0.07 * (1 + 0.07)^20 / ((1 + 0.07)^20 - 1)
Calculating this expression will give us the annual payment.
To calculate the amount owed and needed to pay off the loan after 7 years, we need to determine the remaining balance after those 7 years. To do this, we can use the amortization formula:
Remaining Balance = Principal * (1 + r)^n - (Payment * ((1 + r)^n - 1) / r)
Where:
Principal = $150,000
r = Annual interest rate (expressed as a decimal)
n = Number of payments made (7 years, so n will be 7)
Remaining Balance = $150,000 * (1 + 0.07)^7 - (Payment * ((1 + 0.07)^7 - 1) / 0.07)
Calculating this expression will give us the amount owed and needed to pay off the loan after 7 years.