Quantitative Methods MCQ - Quantitative Methods Section 2

56 A 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?

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%.

57 An analyst has established the following prior probabilities regarding a company’s next quarter’s earnings per share (EPS) exceeding, equaling, or being below the consensus estimate.

	Prior Probabilities
EPS exceed consensus	15%
EPS equal consensus	40%
EPS less than consensus

Several days before releasing its earnings statement, the company announces an increase in its dividend. Given this information, the analyst revises his opinion regarding the likelihood that the company’s EPS will be below the consensus estimate. He estimates the likelihood of the company increasing the dividend given that EPS exceed/meet/fall below consensus as reported below:

	Probabilities the company increases dividends conditional on EPS exceeding/equaling/falling below consensus
P(increase div│EPS exceed)	75%
P(increase div │EPS equal)	20%
P(increase div │EPS below)	75%

Using Bayes’ formula, the updated (posterior) probability that the company’s EPS will be below the consensus given that the dividend has increased is closest to:

First, calculate the unconditional probability for an increase in dividends: P (Increase div) = P (Increase div | EPS exceed) * P (EPS exceed) + P (Increase div | EPS equal) * P (EPS equal) + P (Increase div | EPS below) * P (EPS below) = 0.75 * 0.15 + 0.20 * 0.40 + 0.05 * 0.45 = 0.215 Then update the probability of EPS falling below the consensus as: P (EPS below | Increase div) = [ P (Increase div | EPS below) / P (Increase div) ] * P (EPS below) = ( 0.05 / 0.215) * 0.45 = 0.1047

58 Finnish Mortgage Holding Company estimated that about 5 percent of its mortgage holders default. Out of those who default, 80 percent of them make payments a month late as compared to 60 percent of those who do not default. The probability that a mortgage with late payments will default is closest to:=

Based on the information presented, Bayes‟ formula can be applied. The first step is to note down the various probabilities given: P (Default) = 0.05 P (No default) = 0.95 P (Delayed payments | Default) = 0.80 P (Timely payments | Default) = 0.20 P (Delayed payments | No default) = 0.60 P (Timely payments | No default) = 0.40 P (Event | Information) = [ P (Information | Event) / P (Information) ] * P (Event) In this case, „delayed payments‟ is the information and „default‟ is the event. The formula can be written as. P (Default | Delayed payments) = [ P (Delayed payments | Default) * P (Default) ] / { [ P (Delayed payments | Default) * P (Default) + P (Delayed Payments | No default) ] } P (Default | Delayed payments) = [ 0.80 * 0.05 ] / [ (0.80 * 0.05) + (0.60 * 0.95) ] = 0.07

59 ABC Juices Limited has outlets in the city as well as the suburbs. 60% of the people live in the city, while the rest live in the suburbs. ABC’s juices are consumed by 50% of the people in the city and 25% of those in the suburbs. The probability that a person chosen at random lives in the city given that he consumes ABC Juices is closest to:

First, note down the various probabilities given in the problem: P (City) = 0.60 P (Suburbs) = 0.40 P (Consumers | City) = 0.50 P (Consumers | Suburbs) = 0.25 P (City | Consumer) = [ P (Consumer | City) * P (City) ] / { [ P (Consumer | City) * P (City) ] + [ P (Consumer | Suburb) * P (Suburb) ] } P (City | Consumer) = ( 0.50 * 0.60 ) / [ ( 0.50 * 0.60) + ( 0.25 * 0.40 ) ] = 0.75

60 The probability of boom is 60% and the probability of recession is 40% for the economy of Lorekia. If Lorekia’s economy is in a boom, the probability of Stock LMN outperforming is 85%, and the probability of the stock underperforming is 15%. On the other hand, during a recession, there is a 20% probability that Stock LMN will outperform and an 80% probability that it will underperform. The probability of the economy being in a recession, given that LMN is outperforming is closest to:

First, list the various probabilities given and determine the probability to be calculated:
P (Boom) = 0.60
P (Recession) = 0.40
P (Outperform | Boom) = 0.85
P (Underperform| Boom) = 0.15
P (Outperform | Recession) = 0.20
P (Underperform | Recession) = 0.80
P (Recession | Outperform)
= [ P (Outperform | Recession) * P (Recession) ] / { [ P (Outperform | Recession) * P (Recession) ] + [ P (Outperform | Boom) * P (Boom) ] } P (Recession | Outperform) = ( 0.20 * 0.40 ) / [ ( 0.20 * 0.40) + ( 0.85 * 0.60 ) ]