Question

1) Suppose the a priori probability of a bull market is 0.8, and a bear market is 0.2. In a bull market, there is a 0.7 probability of a rise in stock prices over a one-week period and 0.3 probability of a fall in stock prices over the same period. Alternatively, in a bear market there is a 0.4 probability of a rise in stock prices in a one-week period and a 0.6 probability of a decline in stock prices in a one-week period. Suppose, for simplicity, that stock price movements over a week are independent draws.

a) In the last 10 weeks, we have observed four weeks with rising prices and six weeks with declining prices. What is the probability that you are observing a bear market?

b) Suppose a cable news analyst behaves according to Grether’s generalized Bayes’ model of belief updating, with βP = 1.82 and βL = 2.25. What probability would the news analyst assign to a bear market?

c) Suppose a competing news analyst behaves according to Rabin’s mental urn model, refreshing after every two weeks of data. Suppose further that this analyst has 10 balls in each urn with distributions of balls labeled “rise” and “fall” corresponding to the true probabilities. What probability will he assign to a bear market? What if the analyst had 100 balls in each urn?

Answer 1