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Winter 2008 - Newhouse's Class - Problem Set 6

# Winter 2008 - Newhouse's Class - Problem Set 6 - Econ 171...

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Econ 171 Winter 2008 Problem Set 6 1. Nina is planning on making an investment for the next six months. She can invest in a bond fund or a stock fund. If she invests in the bond fund she’ll make \$100 if the market rises, she’ll make \$200 if the market stays even, and she’ll make \$400 if the market falls. If she invests in the stock fund she’ll make \$1600 if the market rises, she’ll break even if the market stays even and she’ll lose \$1200 if the market falls. Her prior places probability of 0.4 that the market will rise, probability 0.3 that it will stay even, and probability 0.3 that it will fall. a. Which decision maximizes her expected gain? E(Gain for bond) = 0.4(100) + 0.3(200) + 0.3(400) = \$220 E(Gain for bond) = 0.4(1600) + 0.3(0) + 0.3(–1200) = \$280 Investing in the stock fund maximizes her expected gain. b. What’s the value of perfect information? With perfect information she’ll choose the stock if the market rises (1600 > 100) and she’ll choose the bond if the market stays even (200 > 0) or if the market falls (400 > –1200). Her expected gain given perfect information is 0.4(1600) + 0.3(200) + 0.3(400) = \$820 EVPI = 820 – 280 = \$540 Maria consults people about market conditions. When the market rises, Maria predicts it will rise 80% of the time, she predicts it will stay even 10% of the time and she predicts it will fall 10% of the time. When the market stays even, Maria predicts it will rise 20% of the time, she predicts it will stay even 70% of the time and she predicts it will fall 10% of the time. When the market falls, she predicts it will rise 5% of the time, she predicts it will stay even 15% of the time and she predicts it will fall 80% of the time. c. Graph the decision tree for Nina’s problem. The tree is attached at the end. We want to update our probabilities for a specific change in the market conditional on the forecast. We’re given the probabilities for a specific forecast condition on the actual change in market conditions so we need to use Bayes’ rule.

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P(Forecast|Change in market) Change in market Rise (0.4) Even (0.3) Fall (0.3) Forecast Rise 0.8 0.2 0.05 Even 0.1 0.7 0.15 Fall 0.1 0.1 0.8 P(Forecast Rise Market Rises) = P(Forecast Rise|Market Rises)P(Market Rises) = 0.8(0.4) = 0.32 P(Forecast Rise Market Even) = P(Forecast Rise|Market Even)P(Market Even) = 0.2(0.3) = 0.06 P(Forecast Rise Market Falls) = P(Forecast Rise|Market Falls)P(Market Falls) = 0.05(0.3) = 0.015 P(Forecast Rise) = P(Forecast Rise Market Rises) + P(Forecast Rise Market Even) + P(Forecast Rise Market Fall) = 0.32 + 0.06 + 0.015 P(Forecast Change in market) Change in market
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Winter 2008 - Newhouse's Class - Problem Set 6 - Econ 171...

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