Bonus Problem 1 (Optional, 30 marks) (a) A loan of amount is repaid by payments made at times , , ... , . We let

be the

12 amount of h repayment. We let be the outstanding balance just after h

repayment. Show that for any non-negative integers and , we have

+ = { } { }. + +

(Hint: Think about the relation between 1 and for any positive interger

) (b) Nancy borrows an amount from the bank today. She repays the loan by monthly

repayments of amount plus 1 balloon payment (smallest balloon payment) made 1 month after the h repayment. The loan charges interest at a monthly effective interest rate 1.25%. You are given that

• The outstanding balance at h repayment date is $57262.08157.

• The outstanding balance at (2)h repayment date is $46864.35403

• The outstanding balance at (4)h repayment date is $19768.42662.

  Calculate the total amount of interest paid by the borrower over the loan.

Bonus Problem 1 (Optional, 30 marks) (a) A loan of amount L is repaid by n payments made at times ti, 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 of L_SFuture value of repayments) OLBk+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 m' 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. th Bonus Problem 1 (Optional, 30 marks) (a) A loan of amount L is repaid by n payments made at times ti, 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 of L_SFuture value of repayments) OLBk+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 m' 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. th