Quantitative Methods - Quantitative Methods Section 2
- 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%.
| 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:
- Option : A
- Explanation :
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
- Option : B
- Explanation : 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
- Option : C
- Explanation : 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
- Option : A
- Explanation : 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 ) ]
