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Unformatted text preview: Chapter 17 Real Options Question 17.1. The beneft oF waiting is to receive possibly higher proft, the cost is Forgoing interest. We should wait iF the increase in proft outweighs the interest lost on this (one time) higher proft. We use continuous growth rates to make the calculations easier. Assume the price oF a widget grows by g and the project cost grows at a rate h . IF we wait T years the NPV is NPV T = e − . 05 T ( e gT − . 9 e hT ) . (1) Not waiting gives an NPV oF NPV = . 1 (assuming we can immediately produce and sell the widget). IF g = h > 5% then NPV T = NPV exp ((g − . 05 )T ) and we would delay iF g > 5%. The net proft will be growing at a Faster rate than the risk Free bond (this would be unlikely in equilibrium). IF g = h < 5% we should invest now. IF h = 0 and g > 0 then our proft does not grow at a constant rate. Technically, NPV T = e (g − . 05 )T − e − . 05 T . 9 = e (g − . 05 )T − e − . 05 T . 9 (2) = e (g − . 05 )T ( 1 − . 9 e − gT ) . (3) IF g ≥ 5% we’d be better oFF waiting indefnitely (saving our .9 production cost). IF g < 5%, we’d wait (maximize NPV T ) T ∗ = ln ³ . 9 1 − 20 g ´ g (4) years. ±or example, iF g = 3%, we’d be better oFF waiting a year since the interest Forgone on our proft (in cents) 10 ( e . 05 − 1 ) = . 51271 is lower than our increased proft iF we wait (in cents) 100 ( e . 03 − 1 ) = 3 . 0455. In T ∗ = 27 . 031 years the two oFFset. Our proft is e . 03 ( 27 . 031 ) − . 9 = 1 . 35; at this point the interest Forgone From waiting, 135 ( e . 05 − 1 ) = 6 . 9216 cents, is greater than the increase in proft 225 ( e . 03 − 1 ) = 6 . 8523 cents. Question 17.2. IF invest at time T you receive the (at time T ) an “NPV” NPV T = . 8 ( 1 . 02 ) T + 1 1 . 05 + . 8 ( 1 . 02 ) T + 2 1 . 05 2 − 1 . 5 (5) = ( 1 . 02 ) T X − 1 . 5 (6) 232 Chapter 17 Real Options where X = . 8 ( 1 . 02 / 1 . 05 ) + . 8 ( 1 . 02 / 1 . 05 ) 2 . This is growing at a decreasing rate; we can show this by looking at the growth rate g t = NPV t + 1 − NPV t NPV t = . 02 1 − ³ 1 . 5 X( 1 . 02 ) t ´ . (7) Notice g = . 02 /( 1 − 1 . 5 /X) = . 955 and as t gets very large g t approaches .02. The key insight is that if we invest in the machine, you will be receiving the NPV and this cash grows at 5% (the risk free rate). Therefore, it is not optimal for you to invest if the NPV is growing at a higher rate; i.e. if g t > 5% then you should not invest. If g t < 5%, you should have invested. Therefore we need to solve g T = 5% for the optimal time T to invest: . 02 1 − ³ 1 . 5 X( 1 . 02 ) T ´ = . 05 =⇒ T = 24 . 727 years. (8) The NPV (today’s NPV) is X ( 1 . 02 ) 24 . 727 − 1 . 5 1 . 05 24 . 727 = . 29926 . (9) The harder way to do this problem is to maximize the log of the NPV directly. ln (NPV ) = ln Ã X ( 1 . 02 ) T − 1 . 5 1 . 05 T ! , (10) which is a calculus exercise (set the derivative equal to zero). The answer ( NPV = . 29931 and T = 25 . 224) will be slightly off due to using simple interest rates. Question 17.3. Using the same notation as in problem 16.2, g t = . 02 ( 1 . 5 ) 1 . 02 t X − ( 1 . 5 ) 1 . 02 (11) and the solution of g T = 5% is T = 16 . 639 with NPV equal to . 20122. If we directly maximized the NPV X − 1 . 5 1 . 02 T 1 . 05 T (12) we’d have T = 16 . 134 and NPV equal to . 20131....
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 Spring '06
 Adam

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