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Unformatted text preview: Problem Set 4: ACTSC 231 Mathematics of Finance, Fall 2010 Q1. (a) Noticing formulae s n e i = (1 + i ) n- 1 i and a n e i = 1- v n i , we immediately have i + 1 s n e i = i + i (1 + i ) n- 1 = i [(1 + i ) n- 1] + i (1 + i ) n- 1 = i (1 + i ) n (1 + i ) n- 1 = 1 (1+ i ) n- 1 i (1+ i ) n = 1 1- v n i = 1 a n e i . (b) Consider two ways to repay a loan of $1 in n years. One way is to repay level annual payments for n years at the end of each year. The level payment, say X , should satisfy the equation X a n e i = 1. In other words, the annual level payment is 1 a n e i . In the other way, at the end of each year, the borrower pays the interest due, which is $ i , and deposit $ Y into a savings account so that the savings account will accumulate to exactly $1 at the end of n years. This $1 will be used to payoff the loan amount. Therefore, the annual deposit Y is determined by Y s n e i = 1, i.e., the annual deposit Y = 1 s n e i . In these two ways, it costs the borrower either $ i + 1 s n e i or 1 a n e i at the end of each year for the same n years to repay the same loan of $. Thus, the annual total repayment must be equal, i.e.,loan of $....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
- Spring '10