Jordan and Mike are both planning on attending university in Calgary. Jordan's parents rent him a one bedroom apartment for $750 per month. Mike's parents bought a 3 bedroom house for $285000 that required a down payment of 10% and offered a mortgage amortized over 15 years at an annual rate of 4.15% compounded semi-annually for a 5 year term. They rented the other two rooms out for $600 per month. The house depreciated in value by 1.5% a year and the cost of taxes and maintenance averaged $3000 a year.
a) How much was left to pay on the mortgage after 5 years?
Please help. Thank you.
To calculate how much is left to pay on the mortgage after 5 years, we first need to calculate the initial mortgage value and then determine the remaining balance after 5 years.
Step 1: Calculate the initial mortgage value
The down payment made on the house is 10% of $285,000, which is ($285,000 * 0.10) = $28,500.
Therefore, the initial mortgage value is $285,000 - $28,500 = $256,500.
Step 2: Calculate the semi-annual interest rate
The annual interest rate is 4.15%, compounded semi-annually. So, divide it by 2 (for semi-annual compounding) to get the semi-annual interest rate: 4.15% / 2 = 2.075%.
Step 3: Calculate the number of semi-annual periods
The mortgage is amortized over 15 years, which means there are 30 semi-annual periods (15 years * 2).
Step 4: Calculate the monthly mortgage payment
To calculate the monthly mortgage payment, we'll use the formula for calculating a mortgage payment using the present value of an annuity:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
M is the monthly payment
P is the principal amount (initial mortgage value)
r is the interest rate per period (semi-annual interest rate)
n is the number of periods (semi-annual periods)
Substituting in the values:
M = $256,500 * (0.02075 * (1 + 0.02075)^30) / ((1 + 0.02075)^30 - 1)
Using a calculator, you can find that the monthly mortgage payment is approximately $1,843.55.
Step 5: Calculate the remaining balance after 5 years
To calculate the remaining balance after 5 years, we need to determine the number of months in 5 years, which is 5 years * 12 months = 60 months.
Using the formula for calculating the remaining balance of a mortgage given the monthly payment:
B = P * ((1 + r)^n - (1 + r)^m) / ((1 + r)^n - 1)
Where:
B is the remaining balance after m periods
P is the principal amount (initial mortgage value)
r is the interest rate per period (semi-annual interest rate)
n is the number of periods (semi-annual periods)
m is the number of periods already paid
Substituting in the values:
B = $256,500 * ((1 + 0.02075)^30 - (1 + 0.02075)^60) / ((1 + 0.02075)^30 - 1)
Using a calculator, you can find that the remaining balance after 5 years is approximately $197,877.25.