chapter12A - Additional Solutions Chapter Twelve...

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Additional Solutions: Chapter Twelve: Probability Analysis 12S.1 Suppose you deposit $100 with the bank. The six equally-likely possible outcomes are that you get back nothing, $110, $120, … , $150. So your expected return is: (110 + 120 + 130 + 140 + 150)/6 = 108.3 This is equivalent to an interest rate of 8.3%. *12S.2 Consider the fate of a single ship. To build and provision it required 100 000 gold pieces. The three pos- sible outcomes are that it is lost, that it returns empty – which is an outcome worth 50 000 gold pieces, since that is what it cost to build – or that it returns laded with merchandise, which outcome is worth 300 000 gold pieces – the value of the merchandise plus the ship itself. So the expected value at the end of the year is 0 ×0.25 + 50 000×0.25 + 300 000×0.5 = 162 500 This represents a 62.5% return on your original investment. As you increase the number of ships in your fleet, the rate of return remains the same, but the expected variance is reduced – there is less chance of finding yourself at the extreme ends of the distribution and either losing everything or getting a return much greater than 62.5%. 12S.3 You have a 50% chance of getting the first question right, so your expected income from the first round is ¥ 100 000 ×0.5 = ¥ 50 000 To get the ¥ 200 000 prize for the second round, you must both have passed the first round and got the second question right. The combined probability of this is 0.5 ×0.5 = 0.25, so your expected income in the second round is ¥ 200 000 ×0.25 = ¥ 50 000 By similar reasoning, your expected income in each subsequent round is ¥ 50 000, so your total expected income after ten rounds is ¥ 500 000, and this is what it would be rational to pay for a chance to play the game. In the variant where there is no upper limit to the series of questions, it would at first appear that you have an infinite series of payouts, each of expected value ¥ 50 000, and that it would therefore be worth paying an infinite amount of money for the privilege of playing. However, this reasoning is not correct. Even Japanese game shows have an upper limit to their budget; for example, the maximum feasible pay-
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out is not likely to exceed ¥4 50 000 000 000, since this was Japan’s Gross Domestic Product in 2006. But you would reach this payout after just 22 rounds of the game, so your maximum expected winnings would still only be 22× ¥ 50 000 = ¥1 100 000, and it would not be rational to pay more than this to play. *12S.4 The first thing to do is to calculate the value of a launch now, three months from now, and so on. The value of a launch now is an infinite series of payments of $100 000/month, where the interest rate is 2%. The capitalised cost of this series is $100 000/0.02 = $5 000 000. The capitalised cost of the same se- ries, starting 3 months later, is $5 000 000 (P/F,0.02,3) = 5 000 000 × 0.9423. A six-month delay gives a capitalised cost of 5 000 000 * 0.888; nine months, 5 000 000 × 0.8368; a year, 5 000 000 × 0.7885.
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