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Quantitative Methods Q183

183A researcher is studying the link between exchange rate movements and the discount rate set by the country’s bank. He uses historical data to determine that the probability of exchange rate rising or falling over the next month is 63% and 35% respectively, while the probability that the exchange rate stays the same is 2%. Some days later, he receives information that the central bank will increase the discount rate. The researcher estimates that given the new information regarding discount rates, the probabilities that the central bank will increase the discount rate given the scenarios that exchange rate rises, falls or stays the same are as follows:

P(increased discount rate| exchange rate increases) = 67% P(increased discount rate| exchange rate stays same) = 9% P(increased discount rate| exchange rate decreases) = 24%.

What is the probability that the exchange rate will fall given the new information that the central bank will increase the discount rate?
  • Option : C
  • Explanation : According to Bayes' Theorem: Updated probability of event given the new information = (Probability of new information given event / Unconditional probability of new information) * Prior probability of event In order to proceed with the given data, we need to calculate the unconditional probability of new information i.e. the probability of an increase in the discount rate. P (increased discount rate) = P (increased discount rate | exchange rate increases) * P (exchange rate increases) + P (increased discount rate | exchange rate stays same) * P (exchange rate stays same) + P (increased discount rate | exchange rate decreases) * P (exchange rate decreases) = (0.67 * 0.63) + (0.09 * 0.02) + (0.24 * 0.35) = 0.5079 = 50.79%. Using the unconditional probability and Bayes' Theorem, we can calculate updated probability of event given the new information about discount rates as: P (exchange rate decreases | increased discount rate) = [ P (increased discount rate | exchange rate decreases) ÷ P (increased discount rate) ] * P (exchange rate decreases) = ( 0.24 ÷ 0.5079) * 0.35 = 16.5%.
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