hw1q3b solution

# hw1q3b solution - g NPV @T = 4 5 : 2 R & 4 T ln R &...

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Here is another way to solve the infamous tokens question (3b) of our homework. E/ective daily rate is r = 1 : 10 1 = 365 1 = 0 : 00026116 Suppose we hoard tokens for T days in 2011. The amount of money we spend a day on hoarding is 2 : 5 ± 8 = 20 . Value of this spending as of January 1, 2012 is: FV ( Costs ) = 20 r (1 + r ) T 1 ± (1 + r ) We use two tokens a day, so our savings per day in 2012 will be 2 : 6 ± 2 = 5 : 2 4 T PV ( Benefits ) = 5 : 2 r 1 1 (1 + r ) 4 T ! (1 + r ) We want to maximize net present value with respect to T : NPV = PV ( Benefits ) FV ( Costs ) = 1 + r r " 5 : 2 5 : 2 (1 + r ) 4 T 20 (1 + r ) T + 20 # We can take the derivative with respect to T now, set it equal to zero, and we±ll get the right answer. Or we can simplify our life a little by realizing that 1 + r r is simply a number as are 5 : 2 and 20 , and they do not a/ect the maximization result. So we can rewrite our objective function that we are trying to maximize as g NPV = 5 : 2 (1 + r ) 4 T 20 (1 + r ) T = 5 : 2 R 4 T + 20 R T ; where R = 1 + r . Recalling some di/erentiation rules: @
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Unformatted text preview: g NPV @T = 4 5 : 2 R & 4 T ln R & 20 R T ln R = 0 Its just plain algebra after this: 5 R T = 5 : 2 R & 4 T R 5 T = 1 : 04 5 T = ln 1 : 04 ln R T = ln 1 : 04 5 ln (1 + 0 : 00026116) = 30 : So we will be hoarding tokens for 30 days, or from December 2 until December 31. In case you are wondering, hoarding for 30 days gives NPV = \$11 : 97 . You could check by hand to make sure that it doesnt make sense to be buying any tokens (not even one) on December 1. If we buy one token on December 1: FV ( Cost ) = 2 : 5 (1 + 0 : 00026116) 31 = 2 : 5203 The present value of the bene&t it gives you is less than this cost: PV ( Benefit ) = 2 : 6 (1 + 0 : 00026116) 30 4+1 = 2 : 5191 By contrast, you can verify that the &rst token we buy on December 2 has a positive NPV : 2 : 6 (1 + 0 : 00026116) 30 4 & 2 : 5 (1 + 0 : 00026116) 30 = 2 : 5198 & 2 : 5197 = 0 : 0001 > :...
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## This note was uploaded on 12/20/2011 for the course RSM 332 taught by Professor Raymondkan during the Fall '08 term at University of Toronto- Toronto.

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