Part III: Mathematics of Finance
You have just graduated from college and landed your first big job. You have always dreamed of being a homeowner, and after carefully shopping for your dream home, you find one that you would like to purchase at a cost of $250,000. After researching banks to find the best interest rate, you find that Banks for Homeowners offers the best rate of 6% interest that compounds monthly for 30 years.
•What is the monthly payment for this loan?
•What is the unpaid balance of the loan at the end of 5 years?
•What is the unpaid balance at the end of the 10th year?
what is your down payment? 20% ($50,000) or less
•What is the monthly payment for this loan? $1,498.88
•What is the unpaid balance of the loan at the end of 5 years? $229,200.00
•What is the unpaid balance at the end of the 10th year? $204,580.00
To calculate the monthly payment, unpaid balance at the end of 5 years, and unpaid balance at the end of the 10th year, we can use the formulas and concepts from the mathematics of finance. Specifically, we will use the formula for the monthly payment of a loan, as well as the formula for the remaining balance of a loan after a certain number of payments.
Before we begin, let's convert the interest rate from a percentage to a decimal by dividing it by 100. In this case, the interest rate is 6%, so it becomes 0.06.
To calculate the monthly payment, we can use the formula:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
M is the monthly payment
P is the principal amount (loan amount)
r is the monthly interest rate
n is the total number of payments
For the given loan, the principal amount (P) is $250,000, the monthly interest rate (r) is 0.06/12 (since the interest compounds monthly) and the total number of payments (n) is 30 * 12 = 360 (since the loan term is 30 years).
Plugging these values into the formula, we can calculate the monthly payment (M):
M = 250,000 * (0.005 * (1 + 0.005)^360) / ((1 + 0.005)^360 - 1)
Using a calculator, we find that the monthly payment is approximately $1,498.88.
To calculate the unpaid balance at the end of 5 years, we need to find the remaining balance of the loan after 5 years of regular monthly payments. We can use the formula for the remaining balance of a loan:
B = P * ((1 + r)^n - (1 + r)^p) / ((1 + r)^n - 1)
Where:
B is the remaining balance
P is the principal amount
r is the monthly interest rate
n is the total number of payments
p is the number of payments made
For 5 years of payments, the number of payments made (p) will be 5 * 12 = 60.
Plugging in the values into the formula, we can calculate the unpaid balance at the end of 5 years (B):
B = 250,000 * ((1 + 0.005)^360 - (1 + 0.005)^60) / ((1 + 0.005)^360 - 1)
Using a calculator, we find that the unpaid balance at the end of 5 years is approximately $203,364.84.
To calculate the unpaid balance at the end of the 10th year, we can use the same formula as above, but for 10 years of payments. The number of payments made (p) will be 10 * 12 = 120.
Plugging in the values into the formula, we can calculate the unpaid balance at the end of the 10th year (B):
B = 250,000 * ((1 + 0.005)^360 - (1 + 0.005)^120) / ((1 + 0.005)^360 - 1)
Using a calculator, we find that the unpaid balance at the end of the 10th year is approximately $148,467.61.