Question

Connor is a prospective student who has been accepted for a $50,000 student loan from two different loan providers. He wants to select the student loan with the lowest total cost. The first loan is a 10-year loan at 4% and the second loan is a 20-year loan at 3%. Both loans compound the interest monthly. Which student loan should Connor select?

Answer 1

Connor should select the 10-year loan because it has a lower total cost than the 20-year loan.

Step-by-step explanation: To calculate the total cost of each loan, first find the monthly payment of the loans. Use the formula below to find the monthly payments, where P is the monthly payment, L is the principal amount of the loan, i is the interest rate, and n is the duration of the loan.

P=L×i1−(1+i)−n

The loan amount needed, L, is $50,000 for both loans. Since the interest rate is compounded monthly, the interest rate offered is 4% per year or 0.0412 per month. The time period is 10 years or 120 months. Substituting the values into the formula and converting the interest rate to the monthly rate yields the following.

PP=50,000×0.04121−(1+0.0412)−120≈$506.23

Multiply the monthly payment by the number of months in the loan, 120, to determine the total cost of the loan.

$506.23×120=$60,747.60

Now calculate the monthly cost of the 20-year loan.

PP=50,000×0.03121−(1+0.0312)−240≈$277.30

$277.30×240=$66,552

Since the total cost of the 10-year loan, $60,747.60, is less than the total cost of the 20-year loan, $66,552, Connor should select the 10-year loan.

Your answer:

Connor should select the 20-year loan because it costs less than the 10-year loan.

The 20-year loan has a total cost of $66,552 while the 10-year loan has a total cost of $60,747.60.

Answer 2

To figure out which loan has the lower total cost, we need to calculate the total repayment amount for both loans.

The formula to calculate the monthly payment for a loan is:

P = [r*PV] / [1 - (1 + r)^-n]

where:

P is the monthly payment

r is the monthly interest rate (annual rate / 12)

PV is the present value, i.e., the loan amount

n is the number of payments (months)

For the first loan:

r = 4% / 12 = 0.00333 (approximately)

n = 10 years * 12 = 120 months

PV = $50,000

Plugging these numbers into the formula, we get the monthly payment for the first loan.

For the second loan, we do the same calculation, but with r = 3% / 12 and n = 20 * 12.

After we have the monthly payments, we multiply each by the total number of payments (n) to get the total repayment amount for each loan. The loan with the lower total repayment amount is the one Connor should select.

I hope that helps