Question

Tedros borrowed $2 million and planned to repay the loan by making equal month-end payments over

a period of 10 years. The interest rate on the loan is 6%, compounded monthly.

(a) Calculate the amount of monthly payment.

(b) Of the 60th payment, how much will be used to repay the interest and principal for the month?

(c) Tedros plans to pay off the loan immediately after making the 60th payment. What should the

size of the lump-sum (pre-)payment be?

(d) Immediately after the 60th repayment, the central bank increased the market interest rate

and the bank subsequently raised the loan's interest rate to 8% p.a., compounded

monthly. If Tedros decides to keep the number of remaining payments unchanged,

what is the size of the new monthly repayments?

(e) Calculate the total amount of principal repaid and interest paid in the first 60 monthly

payments. Assume that the relevant interest rate is still 6%, compounded monthly (that is,

not 8%).

[Hint: One make-sense way to think about this is by focusing on the relationships

between amount borrowed, total amount repaid to the bank and the amount still owed

to the bank after the 60th payment is made. You may think of the total amount of

principal repaid first].

(f) Suppose that if the bank allows Tedros to only pay the first five years of interests at

t=60 in a single payment (with principal repayment only to be made at the end of Year

10). How much interests will Tedros has to pay at t=60? What explains the difference

in the total amount of interests paid in part (e) and here [part (f)]? Assume that the

relevant interest rate is still 6%, compounded monthly (that is, not 8%).

Answer 1

a). PV = 2,000,000; N = 10*12 = 120; rate = 6%/12 = 0.5%, solve for PMT.

Monthly payment = 22,204.10

b). Principal at the end of 59 payments: N = 59; rate = 0.5%; PMT = 22,204.10; PV = -2,000,000, solve for FV.

Principal amount remaining to be paid = 1,164,899.15

Interest to be paid in the 60th installment = interest rate*pending principal amount = 0.5%*1,164,899.15 = 5,824.50

Principal component of the 60th installment = 22,204.10 - 5,824.50 = 16,379.60

c). Number of payments remaining after 60 payments have been made = 120 - 60 = 60 payments

Total amount to be paid (pre-payment lump sum) = 60*22,204.10 = 1,332,246.02

d). Principal pending after 60 payments: N = 60; rate = 0.5%; PMT = 22,204.10; PV = -2,000,000, solve for FV.

Principal pending = 1,148,519.54

Number of installments remaining = 60

New interest rate = 8%/12 = 0.667%

PV = 1,148,519.54; N = 60; rate = 0.667%, solve for PMT.

Monthly payment = 23,287.84

e). Total payment made in the first 60 installments = 22,204.10*60 = 1,332,246.02

Principal pending after 60 installments = 1,148,519.54 (calculated in part (d))

So, principal paid off in the first 60 installments = total principal - principal pending = 2,000,000 - 1,148,519.54 = 851,480.46

Interest paid off in the first 60 installments = total payment - principal paid = 1,332,246.02 - 851,480.46 = 480,765.57

f). If interest for first 5 years is paid at the end of 5 years then interest paid = principal*(1+rate)^n - principal = 2,000,000*(1+0.5%)^60 - 2,000,000 = 697,700.31

This total interest amount is different from the interest amount in part (e) because in this case, interest is calculated on the total principal whereas in case of monthly payments, interest is calculated on reducing principal amount.