Tutorial04_-_soln - Problem Set 4 ACTSC 231 Mathematics of...

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Problem Set 4: ACTSC 231 Mathematics of Finance, Fall 2010 Q1. (a) Noticing formulae s n i = (1 + i ) n - 1 i and a n i = 1 - v n i , we immediately have i + 1 s n 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 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 i = 1. In other words, the annual level payment is 1 a n 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 i = 1, i.e., the annual deposit Y = 1 s n i . In these two ways, it costs the borrower either $ i + 1 s n i or 1 a n 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., i + 1
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