Reading 8: Probability Concepts-LOS l 习题精选 interpret, expected, standard, 2011

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Session 2: Quantitative Methods: Basic Concepts
Reading 8: Probability Concepts

LOS l: Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio.

 

 

There is a 30% chance that the economy will be good and a 70% chance that it will be bad. If the economy is good, your returns will be 20% and if the economy is bad, your returns will be 10%. What is your expected return?

A) 13%.

B) 17%.

C) 15%.

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Expected value is the probability weighted average of the possible outcomes of the random variable. The expected return is: ((0.3) × (0.2)) + ((0.7) × (0.1)) = (0.06) + (0.07) = 0.13.

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For assets A and B we know the following: E(R_A) = 0.10, E(R_B) = 0.10, Var(R_A) = 0.18, Var(R_B) = 0.36 and the correlation of the returns is 0.6. What is the variance of the return of a portfolio that is equally invested in the two assets?

A) 0.1500.

B) 0.2114.

C) 0.1102.

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You are not given the covariance in this problem but instead you are given the correlation coefficient and the variances of assets A and B from which you can determine the covariance by Covariance = (correlation of A, B) × Standard Deviation of A) × (Standard Deviation of B).

Since it is an equally weighted portfolio, the solution is:
[( 0.5² ) × 0.18 ] + [(0.5²) × 0.36 ] + [ 2 × 0.5 × 0.5 × 0.6 × ( 0.18^0.5 ) × ( 0.36^0.5 )] 
= 0.045 + 0.09 + 0.0764 = 0.2114

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Use the following data to calculate the standard deviation of the return:

• 50% chance of a 12% return
• 30% chance of a 10% return
• 20% chance of a 15% return

A) 3.0%.

B) 1.7%.

C) 2.5%.

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The standard deviation is the positive square root of the variance. The variance is the expected value of the squared deviations around the expected value, weighted by the probability of each observation. The expected value is: (0.5) × (0.12) + (0.3) × (0.1) + (0.2) × (0.15) = 0.12. The variance is: (0.5) × (0.12 ? 0.12)² + (0.3) × (0.1 ? 0.12)² + (0.2) × (0.15 ? 0.12)² = 0.0003. The standard deviation is the square root of 0.0003 = 0.017 or 1.7%.

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After repeated experiments, the average of the outcomes should converge to:

A) the expected value.

B) the variance.

C) one.

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This is the definition of the expected value. It is the long-run average of all outcomes.

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Given P(X = 2) = 0.3, P(X = 3) = 0.4, P(X = 4) = 0.3. What is the variance of X?

A) 3.0.

B) 0.6.

C) 0.3.

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The variance is the sum of the squared deviations from the expected value weighted by the probability of each outcome.
The expected value is E(X) = 0.3 × 2 + 0.4 × 3 + 0.3 × 4 = 3.
The variance is 0.3 × (2 ? 3)² + 0.4 × (3 ? 3)² + 0.3 × (4 ? 3)² = 0.6.

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Compute the standard deviation of a two-stock portfolio if stock A (40% weight) has a variance of 0.0015, stock B (60% weight) has a variance of 0.0021, and the correlation coefficient for the two stocks is –0.35?

A) 1.39%.

B) 2.64%.

C) 0.07%.

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The standard deviation of the portfolio is found by:

[W₁²σ₁² + W₂²σ₂ ²+ 2W₁W₂σ₁σ₂ρ_1,2]^0.5

= [(0.40)²(0.0015) + (0.60)² (0.0021) + (2)(0.40)(0.60)(0.0387)(0.0458)(–0.35)]^0.5

= 0.0264, or 2.64%.

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For assets A and B we know the following: E(R_A) = 0.10, E(R_B) = 0.20, Var(R_A) = 0.25, Var(R_B) = 0.36 and the correlation of the returns is 0.6. What is the expected return of a portfolio that is equally invested in the two assets?

A) 0.3050.

B) 0.2275.

C) 0.1500.

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The expected return of a portfolio composed of n-assets is the weighted average of the expected returns of the assets in the portfolio: ((w₁) × (E(R₁)) + ((w₂) × (E(R₂)) = (0.5 × 0.1) + (0.5 × 0.2) = 0.15.

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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.

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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 16.05% standard deviation.

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

C) 10.0% 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%

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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 + 2W₁W₂σ₁σ₂r_1,2]^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%.