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Bonus Problem 2 (25 marks) (a) A loan of amount L is repaid by n payments made at times t1,t2,.tn We let P be the amount of kth repayment (see the figure in P,2 of the lecture note). We let OLBx be the outstanding balance just after kth repayment. Show that for any non-negative integers k and m, we have Future value of OLBkm OLB at time tkm (Future value of repayments made between t and tk+m Hint: Think about the relation between OLB- and OLBx for any positive interger k and derive the equation using the method used in the lecture note) Note: This formula is very useful in solving some problems related to loan) (b) Ms. A borrows an amount L from a bank today and repay the loan by (4n)th payment made at the end of every quarter. The loan charges interest at a constant quarterly effect interest rate and the first repayment is made 3 months after today. You are given that The amount of each of first (3n) payments is X and the repayment amount of each of next n payments is X 462.9 The outstanding balance at the nth payment date is 48091.2093 . The outstanding balance at the 2nth payment date is 40331.3385 . The outstanding balance at the 3nth payment date is 28064.4485 Calculate the total amount of interest paid by the borrower over the loan Bonus Problem 2 (25 marks) (a) A loan of amount L is repaid by n payments made at times t1,t2,.tn We let P be the amount of kth repayment (see the figure in P,2 of the lecture note). We let OLBx be the outstanding balance just after kth repayment. Show that for any non-negative integers k and m, we have Future value of OLBkm OLB at time tkm (Future value of repayments made between t and tk+m Hint: Think about the relation between OLB- and OLBx for any positive interger k and derive the equation using the method used in the lecture note) Note: This formula is very useful in solving some problems related to loan) (b) Ms. A borrows an amount L from a bank today and repay the loan by (4n)th payment made at the end of every quarter. The loan charges interest at a constant quarterly effect interest rate and the first repayment is made 3 months after today. You are given that The amount of each of first (3n) payments is X and the repayment amount of each of next n payments is X 462.9 The outstanding balance at the nth payment date is 48091.2093 . The outstanding balance at the 2nth payment date is 40331.3385 . The outstanding balance at the 3nth payment date is 28064.4485 Calculate the total amount of interest paid by the borrower over the loan