Question

Consider the following two loans for P=$5,000. Loan A: 2.5-year loan, annual interest rate of 12%. Loan B: 5-year loan, annual interest rate of 6%. Both loans are paid monthly, and their interest is compounded monthly. Calculate the absolute difference between the total interest paid on both loans. Round your answer to the nearest cent. Do NOT round until you calculate the final answer.

Answer 1

For Loan A:

Number of months: 2.5 x 12 = 30

Monthly interest rate: 12% / 12 = 1%

Monthly payment: Pmt = PV x (r / (1 - (1 + r)^-n))

Pmt = 5000 x (0.01 / (1 - (1 + 0.01)^-30)) = $220.15

Total payment: 220.15 x 30 = $6,604.50

Total interest: 6604.50 - 5000 = $1,604.50

For Loan B:

Number of months: 5 x 12 = 60

Monthly interest rate: 6% / 12 = 0.5%

Monthly payment: Pmt = PV x (r / (1 - (1 + r)^-n))

Pmt = 5000 x (0.005 / (1 - (1 + 0.005)^-60)) = $95.12

Total payment: 95.12 x 60 = $5,707.20

Total interest: 5707.20 - 5000 = $707.20

Absolute difference in total interest: $1,604.50 - $707.20 = $897.30. Therefore, the absolute difference between the total interest paid on both loans is $897.30.

Answer 2

Answer: $12.60

Explanation:

First, we must calculate the monthly repayments on both loans, and calculate the interest from there: For Loan A: To calculate the monthly repayments we use

d=P(rn)(1−(1+rn)−n⋅t)

and substituting the relevant values gives

dA=5000(0.1212)(1−(1+0.1212)−12⋅2.5)

which yields dA=$193.74.

The total interest on Loan A is

IA=$193.74×12×2.5−$5,000=$812.20.

For Loan B: To calculate the monthly repayments we use

d=P(rn)(1−(1+rn)−n⋅t)

and substituting the relevant values gives

dB=5000(0.0612)(1−(1+0.0612)−12⋅5)

which yields dB=$96.66. The total interest on Loan B is

IB=$96.66×12×5−$5,000≈$799.60,

and therefore the absolute difference between the two is

|IA−IB|=$12.60.