Final answer:
To calculate the monthly payment and amortization schedule, we can use the formula for the monthly payment of a mortgage. For a $500,000 mortgage with a 5% annual interest rate and a 40-year term, the monthly payment is $2,998.93. If Mr. X were to pay cash for the house instead, he would save $439,781.40 in interest. The payment at which the outstanding balance is less than 50% of the total loan is when the outstanding balance reaches $374,366.95.
Step-by-step explanation:
To calculate the monthly payment and amortization schedule, we can use the formula for the monthly payment of a mortgage:
P = (PV × r) / (1 - (1 + r)^(-n))
In this formula, P is the monthly payment, PV is the loan value, r is the monthly interest rate (annual interest rate divided by 12), and n is the number of months.
For the first question, let's calculate the monthly payment:
PV = $500,000
Annual interest rate = 5%
Monthly interest rate = 5% / 12 = 0.4167%
n = 40 years × 12 = 480 months
Plugging these values into the formula, we get:
P = ($500,000 × 0.004167) / (1 - (1 + 0.004167)^(-480))
P = $2,998.93
So, the monthly payment for a $500,000 mortgage with a 5% annual interest rate and a 40-year term is $2,998.93.
For the second question, let's calculate the amount of interest saved if Mr. X goes for a mortgage valued at $500,000:
Total interest paid = P × n - PV
Total interest paid = $2,998.93 × 480 - $500,000
Total interest paid = $439,781.40
If Mr. X were to pay cash for the house instead, he would save $439,781.40 in interest.
For the third question, let's find the payment at which the outstanding balance is less than 50% of the total loan:
Outstanding balance = Total loan value × (1 - (1 + r)^(-n))
Outstanding balance = $750,000 × (1 - (1 + 0.004167)^(-480))
Outstanding balance = $374,366.95
To find the payment, we need to check the amortization schedule for each monthly payment.