Reading 8: Probability Concepts-LOS l习题精选 returns, standard, expected, interpret, 2010

bmaggie

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) 17%.

B) 15%.

C) 13%.

bmaggie

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) 17%.

B) 15%.

C) 13%.

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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.

bmaggie

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) 0.1102.

C) 0.2114.

bmaggie

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) 0.1102.

C) 0.2114.

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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

bmaggie

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) 2.5%.

C) 1.7%.

bmaggie

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) 2.5%.

C) 1.7%.

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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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bmaggie

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

A) the variance.

B) the expected value.

C) one.

bmaggie

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

A) the variance.

B) the expected value.

C) one.

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

bmaggie

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) 0.3.

C) 0.6.

bmaggie

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) 0.3.

C) 0.6.

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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.