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Chapter3 - Investment Science Chapter 3 Dr James A...

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Investment Science Chapter 3 Dr. James A. Tzitzouris <[email protected]> 3.1 Use A = rP 1 - 1 (1+ r ) n with r = 7 / 12 = 0 . 58%, P = $25 , 000, and n = 7 × 12 = 84, to obtain A = $377 . 32. 3.2 Observe that since the net present value of X is P , the cash flow stream arrived at by cycling X is equivalent to one obtained by receiving payment of P every n + 1 periods (since k = 0 , . . . , n ). Let d = 1 / (1 + r ). Then P = P k =0 ( d n +1 ) k . Solving explicitly for the geometric series, we have that P = P 1 - d n +1 . Denoting the annual worth by A , we must have A = rP 1 - d n , so that solving for P as a function of P and substituting the result into the equation for A , we arrive at A = r 1 - d n +1 1 - d n P . 1
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That is, A is directly proportional to P . 3.3 (a) To find the life expectancy, we multiply each age of death by its probability. Thus the life expectancy is L = 90 × 0 . 07 + 91 × 0 . 08 + · · · + 101 × 0 . 04 = 95 . 13 years. (b) To find the present value of an annuity that ends at age 95.13, we calculate the values for ages 95 and 96. From the standard formula P = A r 1 - 1 (1 + r ) n , with n = 5 and n = 6, we find that P 95 = $39 , 927 and P 96 = $46 , 228. Then, taking P = 0 . 87 × P 95 +0 . 13 × P 96
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