Quant - Probability Trees and Conditional Expectations

Question 1

An analyst developed two scenarios with respect to the recovery of USD100,000 principal from defaulted loans:

The amount of the expected recovery is closest to which of the following?

A.USD36,400.
B.USD55,000.
C.USD63,600.

Answer 1

C is correct.

If Scenario 1 occurs, the expected recovery is 60% (USD50,000) + 40% (USD30,000) = USD42,000, and if Scenario 2 occurs, the expected recovery is 90% (USD80,000) + 10% (USD60,000) = USD78,000. Weighting by the probability of each scenario, the expected recovery is 40% (USD42,000) + 60% (USD78,000) = USD63,600. Alternatively, first calculating the probability of each amount occurring, the expected recovery is (40%)(60%)(USD50,000) + (40%)(40%)(USD30,000) + (60%)(90%)(USD80,000) + (60%)(10%)(USD60,000) = USD63,600.

Question 2

The probability distribution for a company’s sales is:

The standard deviation of sales is closest to which of the following?

A.USD9.81 million.
B.USD12.20 million.
C.USD32.40 million.

Answer 2

A is correct.

Question 3

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.

(see image)

Several days before releasing its earnings statement, the company announces a cut in its dividend. Given this new information, the analyst revises his opinion regarding the likelihood that the company will have EPS below the consensus estimate. He estimates the likelihood the company will cut the dividend, given that EPS exceeds/meets/falls below consensus, as reported below.

(see image)

The analyst thus determines that the unconditional probability for a cut in the dividend, P(Cut div), is equal to 23.75%. Using Bayes’ formula, the updated (posterior) probability that the company’s EPS are below the consensus is closest to:

A.
85%.
B.
72%.
C.
20%.

Answer 3

Answer is B.

Question 4

When working backward from the nodes on a binomial tree diagram, the analyst is most likely attempting to calculate:

A.
the number of potential outcomes.

B.
the probability of a given scenario.

C.
an expected value as of today.

Answer 4

C is Correct. In a tree diagram, a problem is worked backward to formulate an expected value as of today.

Question 5

The conditional expected value of a random variable is best described as the:

A.
expected value of a random variable given an event or scenario.

B.
probability-weighted average of the possible outcomes of the random variable.

C.
weighted average of the probabilities of an event given all possible scenarios.

Answer 5

A is Correct. The conditional expected value of a random variable is the expected value of a random variable given an event or scenario.

Question 6

A tree diagram is most likely used when dealing with investment problems that involve outcomes that are:

A.
mutually exclusive.
B.
independent at each node.
C.
unconditional at each node.

Answer 6

A. Mutually Exclusive

Question 7

An investor in Abco stock forecasts the probability that Abco exceeded, met, or fell short of consensus expectations for free cash flow (FCF) during the prior quarter:

P(FCF exceeded consensus) = 0.50
P(FCF met consensus) = 0.35
P(FCF fell short of consensus) = 0.15

While waiting for Abco to release last quarter’s FCF data, the investor learns that Abco will acquire a competitor. Believing that the upcoming acquisition makes it more likely that last quarter’s FCF will exceed the consensus, the investor generates a list of FCF events that may have influenced the acquisition:

P(Acquisition  |  FCF exceeded consensus) = 0.40
P(Acquisition  |  FCF met consensus) = 0.25
P(Acquisition  |  FCF fell short of consensus) = 0.35

Using Bayes’ Formula, calculate the probability that Abco is likely to exceed consensus FCF expectations for last quarter given the acquisition. P(FCF exceeded consensus  |  Acquisition) is closest to:

A.
34%.
B.
59%.
C.
27%.

Answer 7

B is Correct. The updated probability P(FCF exceeded consensus  |  Acquisition) is 59%.

Calculate the unconditional probability that Abco will acquire the competitor firm:P(Acquisition) = (0.50 × 0.40) + (0.35 × 0.25) + (0.15 × 0.35) = 0.34, or 34%.
Calculate the updated probability that Abco exceeded consensus expectations for FCF given that they acquire the competitor firm: P(FCF exceeded consensus  |  Acquisition) = [P(Acquisition  |  FCF exceeded consensus)/P(Acquisition)] × P(FCF exceeded consensus) = (0.40/0.34) × (0.50) = 0.59 or 59%.