OLBk+m Bonus Problem 1 (Optional, 30 marks) (a) A loan of amount L is repaid by n payments made at times t, t2, ..., tn. We let Pk be the amount of kth repayment. We let OLBk be the outstanding balance just after kth repayment. Show that for any non-negative integers k and m, we have Future value (Future value of repayments TOLBk at time tk+m) made between tk and tk+m (Hint: Think about the relation between OLBk-1 and OLBk for any positive interger k) (b) Nancy borrows an amount L from the bank today. She repays the loan by m monthly repayments of amount X plus 1 balloon payment (smallest balloon payment) made 1 month after the mth repayment. The loan charges interest at a monthly effective interest rate 1.25%. You are given that The outstanding balance at kth repayment date is $57262.08157. The outstanding balance at (2k)th repayment date is $46864.35403 The outstanding balance at (4k)th repayment date is $19768.42662. Calculate the total amount of interest paid by the borrower over the loan. OLBk+m Bonus Problem 1 (Optional, 30 marks) (a) A loan of amount L is repaid by n payments made at times t, t2, ..., tn. We let Pk be the amount of kth repayment. We let OLBk be the outstanding balance just after kth repayment. Show that for any non-negative integers k and m, we have Future value (Future value of repayments TOLBk at time tk+m) made between tk and tk+m (Hint: Think about the relation between OLBk-1 and OLBk for any positive interger k) (b) Nancy borrows an amount L from the bank today. She repays the loan by m monthly repayments of amount X plus 1 balloon payment (smallest balloon payment) made 1 month after the mth repayment. The loan charges interest at a monthly effective interest rate 1.25%. You are given that The outstanding balance at kth repayment date is $57262.08157. The outstanding balance at (2k)th repayment date is $46864.35403 The outstanding balance at (4k)th repayment date is $19768.42662. Calculate the total amount of interest paid by the borrower over the loan