Question

Sandy has taken out a $100,000 loan with Saint Jorge Bank to buy a new BMW 5 Series. The loan requires weekly repayments and the fixed interest rate on the loan is 4% p.a. compounded monthly. The duration of the loan is 5 years.

(a) Assuming that for the first 2 years, the loan repayments will be calculated based on the fixed interest rate. What is the weekly repayment for the first 2 years of the loan?

(b) For the remaining 3 years of the loan, the loan repayments will be calculated based on the new variable interest rate of 2.5% p.a. compounded weekly. What is the weekly repayment for the last 3 years of the loan?

Answer 1

Loan Amount = 100,000 | Rate = 4% compounded monthly, therefore, R = 4% / 12 | Time = 5 yrs or 5*12 = 60 months

To calculate weekly payment for first 2 years, we will assume the fixed interest rate for complete 60 months and we will be able to calculate the monthly payment. Converting the monthly payment (dividing by 4) into weekly payment, we will get our answer for part (a)

a) Since, the loan is an annuity, we will use Present Value of Annuity formula

PV of Annuity formula = (PMT / R) * (1 - (1+R)^-T)

The PV of loan is 100,000, R = 4% / 12 and T = 60 months

=> 100,000 = (PMT / (4%/12))*(1 - (1+4%/12)^-60)

=> PMT = (100,000 * (4%/12)) / (1-(1+4%/12)^-60)

Solving the above equation, we will get the monthly payment

Monthly Payment = $ 1,841.65

Weekly Payment = 1,841.65 / 4 = $ 460.41

For the first 2 years, weekly repayment is $ 460.41.

Before going to the part (b), we need principal remaining at the end of 2 years so that rate given in part(b) can be used on it for weekly payment calculation

Formula for Remaining Principal at 24 months = Future Value of Loan amount at 2 years - Future Value of monthly payments at 2 years

=> Remaining Prinicpal = 100,000 * (1+4%/12)²⁴ - (1841.65 / (4%/12))*((1+4%/12)²⁴ - 1)

=> Remaining Principal = 108,314.30 - 45,936.12

Remaining Principal at the end of 2 years = $ 62,378.17

b) New Interest rate = 2.5% weekly compounded, therefore, R = 2.5% / 52

Time remaining = 3 years or 3*52 = 156 weeks

Principal remaining at 2 years = 62,378.17 = PV of annuity

Since, all parameters for Annuity formula are weekly, therefore, we will get weekly payment from the calculation.

Using annuity formula similarly as part (a)

PV of Annuity formula = (PMT / R) * (1 - (1+R)^-T)

=> 62,378.17 = (PMT / (2.5% / 52))*(1-(1+2.5%/52)^-156)

=> PMT = (62,378.17 * 2.5% / 52) / (1 - (1+2.5%/52)^-156)

Solving the above equation, we will get the weekly payment

Weekly Payment = $ 415.14

Hence, weekly payment for remaining 3 years is $415.14.