Bonus Problem 2 (Optional, 25 marks) An-year loan of amount L is to be repaid by n annual payments. The first payment is made at the end of 1st year. The loan charges interest at an annual effective interest rate of i. We consider the following two repayment schemes: Scheme 1: The loan is repaid by n annual level payments of amount X. Scheme 2: Each of n payments consists of interest due plus an additional amount Question (a) We consider the scheme 1 and let PRK and Ik be the principal repaid and interest-due of kth payment respectively. Show that 11 > 12 > ... > In. (b) Show that for any k = 1,2, ..., 1 1, the outstanding balance under scheme 1 is larger than that under scheme 2 just after the kth repayment. Hence, show that the total interest paid under scheme 1 is larger than that paid under scheme 2. (Hint for (b): One way to argue this is to consider the fact that PRK = OLBk-1 - OLBk. Using the fact, one can express OLBk of scheme 1 in terms of PR1,PR2, ...) Bonus Problem 2 (Optional, 25 marks) An-year loan of amount L is to be repaid by n annual payments. The first payment is made at the end of 1st year. The loan charges interest at an annual effective interest rate of i. We consider the following two repayment schemes: Scheme 1: The loan is repaid by n annual level payments of amount X. Scheme 2: Each of n payments consists of interest due plus an additional amount Question (a) We consider the scheme 1 and let PRK and Ik be the principal repaid and interest-due of kth payment respectively. Show that 11 > 12 > ... > In. (b) Show that for any k = 1,2, ..., 1 1, the outstanding balance under scheme 1 is larger than that under scheme 2 just after the kth repayment. Hence, show that the total interest paid under scheme 1 is larger than that paid under scheme 2. (Hint for (b): One way to argue this is to consider the fact that PRK = OLBk-1 - OLBk. Using the fact, one can express OLBk of scheme 1 in terms of PR1,PR2, ...)