Investment Scince

Investment Science Chapter 2

Dr. James A. Tzitzouris

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 = 100r, we must have that ni ≈ 69. Thus n ≈ 69/i. If we use the more accurate approximation, we have that nr(1 − 0.5r) ≈ 0.69. Now, if r ≈ 0.08, then (1 − 0.5r) ≈ 0.96 and so we must have 0.96n · (100r) = 0.96ni ≈ 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%

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2.4 Iteration 0 1 2 3 2.5 First, denote the present value of the annual payment in the nth year (for n = 0, . . . , 19) by P Vn . Since the interest rate is 10%, we must have P Vn (1 + 0.10)n = $500, 000, so that P Vn = $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 P V and given by

19 19

λ 1 2/3 13/21 78/329

f (λ) 1 1/9 377/441 ...

f (λ) 3 7/3 47/21 ...

PV =

n=0

P Vn =

n=0

$500, 000/(1.1)n .

Recognizing the series on the right as a geometric series, we arrive at P V = ($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

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P VA = −1000

n=0

1 . (1 + 0.12/12)n

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Note the fact that we divide the annual interest rate by 12 to arrive at the monthly interest rate. Then solving...