Q1. An analyst expects that 20% of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 90% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 10% chance that the ratio will be negative. The analyst randomly selects a company with a negative ratio. Based on Bayes' theorem, the updated probability that the company will experience a decline is:

A)   26%.

B)   69%.

C)   18%.

Q2. John purchased 60% of the stocks in a portfolio, while Andrew purchased the other 40%. Half of John’s stock-picks are considered good, while a fourth of Andrew’s are considered to be good. If a randomly chosen stock is a good one, what is the probability John selected it?

A)   0.40.

B)   0.75.

C)   0.30.

Q3. Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?

A)    0.211.

B)    0.625.

C)    0.250.

Q4. The probability of A is 0.4. The probability of A^C is 0.6. The probability of (B | A) is 0.5, and the probability of (B | A^C) is 0.2. Using Bayes’ formula, what is the probability of (A | B)?

A)   0.625.

B)   0.125.

C)   0.375.

答案和详解如下：

Q1. An analyst expects that 20% of all publicly traded companies will experience a decline in earnings next year. The analyst has developed a ratio to help forecast this decline. If the company is headed for a decline, there is a 90% chance that this ratio will be negative. If the company is not headed for a decline, there is only a 10% chance that the ratio will be negative. The analyst randomly selects a company with a negative ratio. Based on Bayes' theorem, the updated probability that the company will experience a decline is:

A)   26%.

B)   69%.

C)   18%.

Correct answer is B)

Given a set of prior probabilities for an event of interest, Bayes’ formula is used to update the probability of the event, in this case that the company we have already selected will experience a decline in earnings next year. Bayes’ formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(company having a decline in earnings next year) = 0.20 is divided by 0.26 (which is the Unconditional Probability that a company having an earnings decline will have a negative ratio (90% have negative ratios of the 20% which have earnings declines) plus (10% have negative ratios of the 80% which do not have earnings declines) or ((0.90) × (0.20)) + ((0.10) × (0.80)) = 0.26.) This result is then multiplied by the Prior Probability of the ratio being negative, 0.90. The result is (0.20 / 0.26) × (0.90) = 0.69 or 69%.

Q2. John purchased 60% of the stocks in a portfolio, while Andrew purchased the other 40%. Half of John’s stock-picks are considered good, while a fourth of Andrew’s are considered to be good. If a randomly chosen stock is a good one, what is the probability John selected it?

A)   0.40.

B)   0.75.

C)   0.30.

Correct answer is B)

Using the information of the stock being good, the probability is updated to a conditional probability:

P(John | good) = P(good and John) / P(good).

P(good and John) = P(good | John) × P(John) = 0.5 × 0.6 = 0.3.

P(good and Andrew) = 0.25 × 0.40 = 0.10.

P(good) = P(good and John) + P (good and Andrew) =  0.40.

P(John | good) = P(good and John) / P(good) = 0.3 / 0.4 = 0.75.

Q3. Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?

A)    0.211.

B)    0.625.

C)    0.250.

Correct answer is A)

According to Bayes' formula: P(B / default) = P(default and B) / P(default).

P(default and B )= P(default / B) × P(B) = 0.250 × 0.300 = 0.075

P(default and CCC) = P(default / CCC) × P(CCC) = 0.400 × 0.700 = 0.280

P(default) = P(default and B) + P(default and CCC) = 0.355

P(B / default) = P(default and B) / P(default) = 0.075 / 0.355 = 0.211

Q4. The probability of A is 0.4. The probability of A^C is 0.6. The probability of (B | A) is 0.5, and the probability of (B | A^C) is 0.2. Using Bayes’ formula, what is the probability of (A | B)?

A)   0.625.

B)   0.125.

C)   0.375.

Correct answer is A)

Using the total probability rule, we can compute the P(B):
P(B) = [P(B | A) × P(A)] + [P(B | A^C) × P(A^C)]
P(B) = [0.5 × 0.4] + [0.2 × 0.6] = 0.32

Using Bayes’ formula, we can solve for P(A | B):
P(A | B) = [ P(B | A) ÷ P(B) ] × P(A) = [0.5 ÷ 0.32] × 0.4 = 0.625

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Q1. The covariance:

A)   must be between -1 and +1.

B)   can be positive or negative.

C)   must be positive.

Correct answer is B)         

Cov(a,b) = σ_aσ_bρ_a,b. Since ρ_a,b can be positive or negative, Cov(a,b) can be positive or negative.

Q2. With respect to the units each is measured in, which of the following is the most easily directly applicable measure of dispersion? The:

A)    covariance.

B)    variance.

C)    standard deviation.

Correct answer is C)

The standard deviation is in the units of the random variable itself and not squared units like the variance. The covariance would be measured in the product of two units of measure.

Q3. Personal Advisers, Inc., has determined four possible economic scenarios and has projected the portfolio returns for two portfolios for their client under each scenario. Personal’s economist has estimated the probability of each scenario as shown in the table below. Given this information, what is the covariance of the returns on Portfolio A and Portfolio B?

A)   0.890223.

B)   0.002019.

C)   0.001898.

Correct answer is C)

Q4. Given Cov(X,Y) = 1,000,000. What does this indicate about the relationship between X and Y?

A)  It is strong and positive.

B)  It is weak and positive.

C)  Only that it is positive.

Correct answer is C)

A positive covariance indicates a positive linear relationship but nothing else. The magnitude of the covariance by itself is not informative with respect to the strength of the relationship.

Q5. Which of the following statements is least accurate regarding covariance?

A)   Covariance can only apply to two variables at a time.

B)   Covariance can exceed one.

C)   A covariance of zero rules out any relationship.

Correct answer is C)

A covariance only measures the linear relationship. The covariance can be zero while a non-linear relationship exists. Both remaining statements are true.

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