Chapter2 - Investment Science Chapter 2 Dr James A...

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Investment Science Chapter 2 Dr. James A. Tzitzouris <[email protected]> 2.1 (a) ($1)(1 . 033) 227 = $1 , 587 . 70 (b) ($1)(1 . 066) 227 = $1 , 999 , 300 . 00 2.2 We are given that (1+ r ) n = 2, so that taking the log of both sides, we have n ln (1 + r ) = ln 2 0 . 69. Using the first suggested approximation, we have that nr n ln (1 + r ) 0 . 69. Since i = 100 r , we must have that ni 69. Thus n 69 /i . If we use the more accurate approximation, we have that nr (1 - 0 . 5 r ) 0 . 69. Now, if r 0 . 08, then (1 - 0 . 5 r ) 0 . 96 and so we must have 0 . 96 n · (100 r ) = 0 . 96 ni 69 and we have n 72 /i . 2.3 Note that the rates calculated below are also commonly refered to as the “Annual Percentage Rates” (APRs), for example, on your monthly credit card statement. (a) (1 + 0 . 03 / 12) 12 - 1 = 3 . 04% (b) (1 + 0 . 18 / 12) 12 - 1 = 19 . 56% (c) (1 + 0 . 18 / 4) 4 - 1 = 19 . 25% 1
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2.4 Iteration λ f ( λ ) f ( λ ) 0 1 1 3 1 2/3 1/9 7/3 2 13/21 377/441 47/21 3 78/329 ... ... 2.5 First, denote the present value of the annual payment in the n th year (for n = 0 , . . . , 19) by PV n . Since the interest rate is 10%, we must have PV n (1 + 0 . 10) n = $500 , 000 , so that PV n = $500 , 000 / (1 . 1) n . Summing each yearly payment from n = 0 (since payment starts immediately) to n = 19, we arrive at the net present value of the lottery, denoted by PV and given by PV = 19 n =0 PV n = 19 n =0 $500 , 000 / (1 . 1) n . Recognizing the series on the right as a geometric series, we arrive at PV = ($500 , 000)(11)(1 - (1 / 1 . 1) 20 ) $4 , 682 , 460 . 2.6 First we consider the six month analysis. For simplicity, assume that “Plan A” is to remain in the first apartment and that “Plan B” is to switch to the second apartment. Under Plan A, the monthly cash flows are given as follows: ( - 1000 , - 1000 , - 1000 , - 1000 , - 1000 , - 1000) , and the present value of these cash flows are given by PV A = - 1000 5 n =0 1 (1 + 0 . 12 / 12) n .
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