LiquidAssets10

A two-sided but very thick coin is expected to land on its edge twice out of every 100 flips. And the probability of face up (heads) and the probability of face down (tails) are equal. When the coin is flipped, the prize is $1 for heads, $2 for tails, and $50 when the coin lands on its edge. What is the expected value of the prize on a single coin toss?

A) $1.50.

B) $2.47.

C) $17.67.

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Since the probability of the coin landing on its edge is 0.02, the probability of each of the other two events is 0.49. The expected payoff is: (0.02 × $50) + (0.49 × $1) + (0.49 × $2) = $2.47.

LiquidAssets10

The following information is available concerning expected return and standard deviation of Pluto and Neptune Corporations:

	Expected Return	Standard Deviation
Pluto Corporation	11%	0.22
Neptune Corporation	9%	0.13

If the correlation between Pluto and Neptune is 0.25, determine the expected return and standard deviation of a portfolio that consists of 65% Pluto Corporation stock and 35% Neptune Corporation stock.

A) 10.3% expected return and 2.58% standard deviation.

B) 10.0% expected return and 16.05% standard deviation.

C) 10.3% expected return and 16.05% standard deviation.

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ERPort	= (WPluto)(ERPluto) + (WNeptune)(ERNeptune)
	= (0.65)(0.11) + (0.35)(0.09) = 10.3%
σp	= [(w1)2(σ1)2 + (w2)2(σ2)2 + 2w1w2σ1σ2 r1,2]1/2
	= [(0.65)2(22)2 + (0.35)2(13)2 + 2(0.65)(0.35)(22)(13)(0.25)]1/2
	= [(0.4225)(484) + (0.1225)(169) + 2(0.65)(0.35)(22)(13)(0.25)]1/2
	= (257.725)1/2 = 16.0538%

LiquidAssets10

Assume two stocks are perfectly negatively correlated. Stock A has a standard deviation of 10.2% and stock B has a standard deviation of 13.9%. What is the standard deviation of the portfolio if 75% is invested in A and 25% in B?

A) 0.00%.

B) 4.18%.

C) 0.17%.

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The standard deviation of the portfolio is found by:
[W12 σ12 + W22 σ22 + 2W1W2σ1σ2r1,2]0.5, or [(0.75)2(0.102)2 + (0.25)2(0.139)2 + (2)(0.75)(0.25)(0.102)(0.139)(–1.0)]0.5 = 0.0418, or 4.18%.

LiquidAssets10

What is the standard deviation of a portfolio if you invest 30% in stock one (standard deviation of 4.6%) and 70% in stock two (standard deviation of 7.8%) if the correlation coefficient for the two stocks is 0.45?

A) 0.38%.

B) 6.83%.

C) 6.20%.

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The standard deviation of the portfolio is found by:
[W12 σ12 + W22 σ22 + 2W1W2σ1σ2r1,2]0.5, or [(0.30)2(0.046)2 + (0.70)2(0.078)2 + (2)(0.30)(0.70)(0.046)(0.078)(0.45)]0.5 = 0.0620, or 6.20%.

LiquidAssets10

An investor has two stocks, Stock R and Stock S in her portfolio. Given the following information on the two stocks, the portfolio's standard deviation is closest to:

• σR = 34%
• σS = 16%
• rR,S = 0.67
• WR = 80%
• WS = 20%

A) 7.8%.

B) 8.7%.

C) 29.4%.

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The formula for the standard deviation of a 2-stock portfolio is:
s = [WA2sA2 + WB2sB2 + 2WAWBsAsBrA,B]1/2
s = [(0.82 × 0.342) + (0.22 × 0.162) + (2 × 0.8 × 0.2 × 0.34 × 0.16 × 0.67)]1/2 = [0.073984 + 0.001024 + 0.0116634]1/2 = 0.08667141/2 = 0.2944, or approximately 29.4%.

LiquidAssets10

Given P(X = 20, Y = 0) = 0.4, and P(X = 30, Y = 50) = 0.6, then COV(XY) is:

A) 125.00.

B) 120.00.

C) 25.00.

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The expected values are: E(X) = (0.4 × 20) + (0.6 × 30) = 26, and E(Y) = (0.4 × 0) + (0.6 × 50) = 30. The covariance is COV(XY) = (0.4 × ((20 − 26) × (0 − 30))) + ((0.6 × (30 − 26) × (50 − 30))) = 120.

LiquidAssets10

Given P(X = 2, Y = 10) = 0.3, P(X = 6, Y = 2.5) = 0.4, and P(X = 10, Y = 0) = 0.3, then COV(XY) is:

A) -12.0.

B) 24.0.

C) 6.0.

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The expected values are: E(X) = (0.3 × 2) + (0.4 × 6) + (0.3 × 10) = 6 and E(Y) = (0.3 × 10.0) + (0.4 × 2.5) + (0.3 × 0.0) = 4. The covariance is COV(XY) = ((0.3 × ((2 − 6) × (10 − 4))) + ((0.4 × ((6 − 6) × (2.5 − 4))) + (0.3 × ((10 − 6) × (0 − 4))) = −12.

LiquidAssets10

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) 69%.

B) 26%.

C) 18%.

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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%.

LiquidAssets10

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.

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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.

cross-ied

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) 0.625.

B) 0.250.

C) 0.211.

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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