Consider the following distribution of returns:
| Probability | Return on A | Return on B | Return on C |
|---|---|---|---|
| 30% | -20% | -5% | 5% |
| 40% | 5% | 10% | 3% |
| 30% | 40% | 15% | 2% |
Question 1
What is the expected return of security A? Round off your answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_A) = -20\% \cdot 0.30 + 5\% \cdot 0.40 + 40\% \cdot 0.30 = 8\%\)
Question 2
What is the expected return of security B? Round off your answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_B) = -5\% \cdot 0.30 + 10\% \cdot 0.40 + 15\% \cdot 0.30 = 7\%\)
Question 3
What is the expected return of security C? Round off your answer to two digit after the decimal point. State your answer as a percentage rate (such as 5.55).
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_C) = 5\% \cdot 0.30 + 3\% \cdot 0.40 + 2\% \cdot 0.30 = 3.3\%\)
Question 4
Using your answers for questions 1-3, now compute the expected return of a portfolio with 40% in A, 40% in B, and 20% in C. Round off your answer to two-digits after the decimal point. State your answer as a percentages rate (such as 5.55)
The expected return of a portfolio is calculated as:
\(E(R_p) = \sum w_i \cdot E(R_i)\), where \(w_i\) are the weights we choose for each investment
Note: \(\sum w_i = 1\)
In this case we form a portfolio with 40% in A, 40% in B and 20% in C, whereas \(w_A=40\%=0.40\), \(w_B=40\%=0.40\) and \(w_C=20\%=0.20\).
\(E(R_p) = 0.40 \cdot 8\% + 0.40 \cdot 7\% + 0.20 \cdot 3.3\% = 6.66\%\)
Question 5
Using your answers for questions 1-3, what is the expected return of an equally weighted portfolio? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)
The expected return of a portfolio is calculated as:
\(E(R_p) = \sum w_i \cdot E(R_i)\), where \(w_i\) are the weights we choose for each investment
Note: \(\sum w_i = 1\)
In an equally weighted portfolio: \(w_A=w_B=w_C={1\over3}=33.33\%\).
\(E(R_p) = {1\over3} \cdot 8\% + {1\over3} \cdot 7\% + {1\over3} \cdot 3.3\% = 6.1\%\)
Question 6
An investor has total wealth of $50,000 and wants to invest in a portfolio with 3 securities A, B, and C with expected returns \(E(R_A)=20%\), \(E(R_B)=15%\) and \(E(R_C)=17%\) respectively. If he chooses to invest $25,000 in security A, $12,500 in security B, and $12,500 in security C, what will be the expected return of this portfolio? State your answer as a percentage rate (such as 5.55)
The expected return of a portfolio is calculated as:
\(E(R_p) = \sum w_i \cdot E(R_i)\), where wi are the weights we choose for each investment
Note: \(\sum w_i = 1\)
We form a portfolio with:
\(w_A = {25,000 \over 50,000} = 0.50\), \(w_B = {12,500 \over 50,000} = 0.25\) and \(w_C = {12,500 \over 50,000} = 0.25\)
So the expected return of the portfolio would be:
\(E(R_p) = 0.50 \cdot 20\% + 0.25 \cdot 15\% + 0.25 \cdot 17\% = 18\%\)
Question 7
An investor has total wealth of $50,000 and wants to invest in a portfolio with 3 securities A, B, and C with expected returns \(E(R_A)=20%\), \(E(R_B)=15%\) and \(E(R_C)=17%\) respectively. If he chooses to invest $20,000 in security A, $10,000 in security B and $20,000 in security C, what will be the expected return of this portfolio? Round off your answer to digits after the decimal point. State your answer as a percentage rate (such as 5.55)
The expected return of a portfolio is calculated as:
\(E(R_p) = \sum w_i \cdot E(R_i)\), where wi are the weights we choose for each investment
Note: \(\sum w_i = 1\)
We form a portfolio with:
\(w_A = {20,000 \over 50,000} = 0.40\), \(w_B = {10,000 \over 50,000} = 0.20\) and \(w_C = {20,000 \over 50,000} = 0.4\)
So the expected return of the portfolio would be:
\(E(R_p) = 0.40 \cdot 20\% + 0.20 \cdot 15\% + 0.40 \cdot 17\% = 17.8\%\)
Question 8
Suppose your investment budget is $300,000. In addition, you borrow an additional $120,000 investing the total available funds in equities. If the expected rate of return in equities is 8%, and you borrow at 5%, what is your expected portfolio return?
☐ 18.2%
☐ 1.4%
☑ 9.2%
☐ 3%
Think about each asset’s weight in the total portfolio. Essentially you are creating a levered position in equities financed in part by borrowing.
The weight in equities = 420,000 / 300,000 = 140%
The weight in the short (borrowing position) = -40%
Expected portfolio return = 1.40 x 8% + (-0.40) x 5% = 9.2%
Consider the following distribution of returns:
| Probability | Return on A | Return on B | Return on C |
|---|---|---|---|
| 30% | -20% | -5% | 5% |
| 40% | 5% | 10% | 3% |
| 30% | 40% | 15% | 2% |
Question 1
What is the standard deviation of security A? Round off your answer to two digits after the decimal point (such as 5.55).
The standard deviation of the return is defined as the square root of variance, which is the expected value of the squared deviations from the expected return. The variance is calculated as:
\[\sigma^2_x = \sum \{[R_{x_i} - E(R_x)] ^ 2 \cdot P_i\}\]
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_A) = −20\% \cdot 0.30 + 5\% \cdot 0.40 + 40\% \cdot 0.30 = 8\%\)
The variance for A is calculated as:
\(\sigma^2_A = \sum \{[R_{A_i} - E(R_A)] ^ 2 \cdot P_i\}\)
\(\sigma^2_A = (-20 - 8) ^ 2 \cdot 0.30 + (5 - 8) ^ 2 \cdot 0.40 + (40 - 8) ^ 2 \cdot 0.30 = 546\)
Hence the standard deviation for A is equal to \(\sigma_A = \sqrt{546} = 23.37\%\)
Question 2
What is the standard deviation of security B? Round off your answer to two digits after the decimal point (such as 5.55).
The standard deviation of the return is defined as the square root of variance, which is the expected value of the squared deviations from the expected return. The variance is calculated as:
\[\sigma^2_x = \sum \{[R_{x_i} - E(R_x)] ^ 2 \cdot P_i\}\]
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_B) = −5\% \cdot 0.30 + 10\% \cdot 0.40 + 15\% \cdot 0.30 = 7\%\)
The variance for B is calculated as:
\(\sigma^2_B = \sum \{[R_{B_i} - E(R_B)] ^ 2 \cdot P_i\}\)
\(\sigma^2_B = (-5 - 7) ^ 2 \cdot 0.30 + (10 - 7) ^ 2 \cdot 0.40 + (15 - 7) ^ 2 \cdot 0.30 = 66\)
Hence the standard deviation for B is equal to \(\sigma_B = \sqrt{66} = 8.12\%\)
Question 3
What is the standard deviation of security C? Round off your answer to two digits after the decimal point (such as 5.55).
The standard deviation of the return is defined as the square root of variance, which is the expected value of the squared deviations from the expected return. The variance is calculated as:
\[\sigma^2_x = \sum \{[R_{x_i} - E(R_x)] ^ 2 \cdot P_i\}\]
The expected return is calculated as the probability-weighted average of the returns. \(P(R_i)\) is the probability of each scenario and Ri the return in each scenario, where scenarios are labeled by \(i\). So we write the expected return as:
\[E(R) = \sum R_i \cdot P(R_i)\]
\(E(R_C) = 5\% \cdot 0.30 + 3\% \cdot 0.40 + 2\% \cdot 0.30 = 3.3\%\)
The variance for C is calculated as:
\(\sigma^2_C = \sum \{[R_{C_i} - E(R_C)] ^ 2 \cdot P_i\}\)
\(\sigma^2_C = (5 - 3.3) ^ 2 \cdot 0.30 + (3 - 3.3) ^ 2 \cdot 0.40 + (2 - 3.3) ^ 2 \cdot 0.30 = 1.41\)
Hence the standard deviation for C is equal to \(\sigma_C = \sqrt{1.41} = 1.19\%\)
Question 4
What is the expected return of a portfolio with 40% in A, 20% in B and 40% in C? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55).
Recall that the expected portfolio return is the weighted average of the expected returns on individual securities.
We therefore need to first find the expected return for each security:
\(E(r_A) = 0.3 \times -20\% + 0.4 \times 5\% + 0.3 \times 40\% = 8\%\)
\(E(r_B) = 0.3 \times -5\% + 0.4 \times 10\% + 0.3 \times 15\% = 7\%\)
\(E(r_C) = 0.3 \times 5\% + 0.4 \times 3\% + 0.3 \times 2\% = 3.3\%\)
The weights of A, B, and C are respectively 40%, 20%, and 40%.
\(E(r_P) = 0.4 \times 8\% + 0.2 \times 7\% + 0.4 \times 3.3\% = 5.92\%\)
Question 5
Which of the following pairwise covariance measures are correct?
☑ \(σ_{AB} = 174\)
☑ \(σ_{AC} = - 26.4\)
☐ \(σ_{BC} = 26.4\)
☑ \(σ_{BC} = -9.60\)
Covariance between the returns of two assets is computed as the expected value of the product of the deviations from the mean:
\(\sigma_{AB} = \sum \{ [R_{A_i} – E(R_A)] \times [R_{B_i} – E(R_B)] \times p_i \}\)
You already computed the expected return on A and B in a previous quiz.
\(E(R_A) = 0.3 \times -20\% + 0.4 \times 5\% + 0.3 \times 40\% = 8\%\)
\(E(R_B) = 0.3 \times -5\% + 0.4 \times 10\% + 0.3 \times 15\% = 7\%\)
\(E(R_C) = 0.3 \times 5\% + 0.4 \times3\% + 0.3 \times 2\% = 3.3\%\)
Hence the covariance between A and B is calculated as:
\(\sigma_{AB} = (-20 - 8) \times (-5 - 7) \times 0.30 + (5-8) \times (10-7) \times 0.40 + (40-8) \times (15-7) \times 0.30 =174\)
Hence the covariance between A and C is calculated as:
\(\sigma_{AC} = (-20 - 8) \times (5 - 3.3) \times 0.30 + (5-8) \times (3-3.3) \times 0.40 + (40-8) \times (2-3.3) \times 0.30 = -26.4\)
Hence the covariance between B and C is calculated as:
\(\sigma_{BC} = (-5 - 7) \times (5 - 3.3) \times 0.30 + (10-7) \times (3-3.3) \times 0.40 + (15-7) \times (2-3.3) x 0.30 = -9.6\)
Question 6
Which of the following pairwise correlation coefficients are correct? (Hint: Your answers to the previous questions may be useful.)
☑ \(ρ_{AC} = −0.95\)
☐ \(ρ_{AC} = 0.95\)
☑ \(ρ_{CB} = −0.99\)
☐ \(ρ_{CB} = 0.99\)
☐ \(ρ_{AB} = -0.92\)
☑ \(ρ_{AB}= 0.92\)
Recall that the correlation coefficient is scaled covariance. That is, the correlation coefficient between A and C is calculated as:
\(ρ_{AC} = σ_{AC} / (σ_A \times σ_C)\)
We therefore need the covariance between A and C and the individual standard deviations for A and C. In previous questions, you computed these to be:
\(σ_A = 23.37\%\)
\(σ_C = 1.19\%\)
\(σ_{AC} = -26.4\)
The correlation coefficient between A and C is therefore computed as:
\(ρ_{AC} = -26.40 / (23.37 \times 1.19) = -0.95\)
The correlation coefficient between C and B is calculated as
\(ρ_{CB} = σ_{CB} / (σ_C \times σ_B)\)
We therefore need the covariance between C and B and the standard deviations for C and B. You have already computed these measures:
\(σ_B = 8.12\%\)
\(σ_C = 1.19\%\)
\(σ_{BC} = -9.60\)
The correlation coefficient between C and B is therefore computed as:
\(ρ_{CB} = -9.60 / (8.12 \times 1.19) = -0.99\)
The correlation coefficient between A and B is calculated as
\(ρ_{AB} = σ_{AB} / (σ_A \times σ_B)\)
We therefore need the covariance between A and B and the standard deviations for A and B. You already computed these measures:
\(σ_A = 23.37\%\)
\(σ_B = 8.12\%\)
\(σ_{AB} = 174\)
The correlation coefficient between A and B is therefore computed as:
\(ρ_{AB} = 174 / (23.37 \times 8.12) = 0.92\)
Question 7
Find the expected return of a portfolio that is equally weighted between securities A and C. Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55) (Hint: Your answers to previous questions may be useful)
Recall the expected return on A and C:
\(E(R_A) = 0.3 \times -20\% + 0.4 \times 5\% + 0.3 \times 40\% = 8\%\)
\(E(R_C) = 0.3 \times 5\% + 0.4 ×3\% + 0.3 \times 2\% = 3.3\%\)
The expected return of a portfolio is calculated as the weighted average of the individual expected returns:
\(E(r_P) = w_A \times E(R_A) + w_C \times E(R_C) = 0.5 \times 8\% + 0.5 \times 3.3\% = 5.65\%\)
Question 8
Compute the volatility of a portfolio that is equally invested in A and C. Round off your final answer to two digits after the decimal point (such as 5.55) (Hint: Your answers to previous questions may be useful.)
Portfolio volatility is measured by the standard deviation of the portfolio return. The variance of a two-asset portfolio is given by:
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_{12}\)
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_1 \times \sigma_2 \times \rho_{12}\)
Previously we computed:
\(σ_A = 23.37\%\)
\(σ_C = 1.19\%\)
\(σ_{AC} = -26.4\)
Hence:
\(σ^2_P = (0.5)^2 \times (23.37\%)^2 + (0.5)^2 \times (1.19\%)^2 + 2 \times 0.5 \times 0.5 \times -26.4 = 123.64\)
\(⇒ \sigma_P = 11.12\%\)
Question 9
Find the expected return of a portfolio with 60% in A and 40% in B. Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55) (Hint: Your answers to previous questions may be useful.)
Recall the expected return on A and B:
\(E(R_A) = 0.3 \times -20\% + 0.4 \times 5\% + 0.3 \times 40\% = 8\%\)
\(E(R_B) = 0.3 \times -5\% + 0.4 \times 10\% + 0.3 \times 15\% = 7\%\)
The expected return of a portfolio is calculated as the weighted average of the individual expected returns:
\(E(r_P) = w_A \times E(R_A) + w_B \times E(R_B) = 0.6 \times 8\% + 0.4 \times 7\% = 7.6\%\)
Question 10
Compute the volatility of a portfolio that is 60% invested in A and 40% invested in B. Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55) (Hint: Your answers to previous questions may be useful.)
Portfolio volatility is measured by the standard deviation of the portfolio return. The variance of a two-asset portfolio is given by:
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_{12}\)
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_1 \times \sigma_2 \times \rho_{12}\)
Previously we computed:
\(σ_A = 23.37\%\)
\(σ_B = 8.12\%\)
\(σ_{AB} = 174\)
Hence:
\(σ^2_P = (0.6)^2 \times (23.37\%)^2 + (0.4)^2 \times (8.12\%)^2 + 2 \times 0.6 \times 0.4 \times 174 = 290.64\)
\(⇒ \sigma_P = 17.05\%\)
Question 1
The volatility of a portfolio’s return is always equal to the weighted average of the standard deviations of the assets in the portfolio. True or false?
☐ True
☑ False.
As long as assets are less than perfectly correlated, portfolio volatility will be less than the weighted average of the standard deviations of the assets in the portfolio.
Question 2
Which of the following statements is correct?
☐ A well-diversified portfolio eliminates market risk.
☑ A well-diversified portfolio eliminates unique risk.
☐ Unique risk and market risk can both be eliminated through diversification.
☐ Increased insurance costs is an example of pure systematic risk for a corporation.
A well-diversified portfolio diversifies firm-specific (or unique) risk. It cannot reduce market risk (systematic risk).
Question 3
The measure of risk for an asset that is held in a diversified portfolio is:
☐ Specific risk
☐ Volatility
☐ Liquidity risk
☑ Covariance
Covariance with the rest of the portfolio ultimately determines how much risk it contributes to the portfolio.
Consider the following distribution of returns:
| Probability | Return on A | Return on B |
|---|---|---|
| 30% | -10% | 10% |
| 40% | 5% | 20% |
| 30% | 30% | 30% |
Question 4
Compute the covariance between A and B. Round off your final answer to two digits after the decimal point (such as 5.55).
The covariance between the returns of two assets is computed as the expected value of the product of the deviations from the mean:
\(σ_{AB} = \sum_i \{ [R_{A_i} – E(R_A)] × [R_{B_i} – E(R_B)] \} × p_i\)
We need to first therefore find the expected return of A and the expected return of B.
\(E(r_A) = 0.3 × -10\% + 0.4 × 5\% + 0.3 × 30\% = 8\%\)
\(E(r_B) = 0.3 × 10\% + 0.4 × 20\% + 0.3 × 30\% = 20\%\)
Hence:
\(σ_{AB} = (-10-8) \times (10-20) \times 0.30 + (5-8) \times (20-20) \times 0.40 + (30-8) \times (30-20) \times 0.30 =120\)
Question 5
Compute the correlation coefficient between A and B. Round off your final answer to two digits after the decimal point (such as 5.55)
Recall that the correlation coefficient is scaled covariance. That is, the correlation coefficient between A and B is calculated as:
\(ρ_{AB} = σ_{AB} / (σ_A × σ_B)\)
We therefore need to find the covariance and the standard deviations for A and B.
We found the covariance to be 120 in the previous question.
To find the standard deviation of each, recall that the variance is the probability weighted squared deviations from the mean:
\(σ_A^2 = \sum_i \{ [R_{A_i} – E(R_A)]\} × p_i\)
\(σ_B^2 = \sum_i \{ [R_{B_i} – E(R_B)]\} × p_i\)
\(σ_A^2 = (-10 – 8)^2 × 0.3 + (5-8)^2 × 0.4 + (30 – 8)^2 × 0.3 = 246 ⇒ σ_A = 15.68\)
\(σ_B^2 = (10 – 20)^2 × 0.3 + (20-20)^2 × 0.4 + (30 – 20)^2 × 0.3 = 60 ⇒ σ_B = 7.75\)
\(ρ_{AB} = σ_{AB} / (σ_A \times σ_B) = 120 / (15.68 * 7.75) = 0.99\)
Question 6
A portfolio consists of 120 shares of Jones stock, which sells for $50 per share, and 150 shares of Rice stock, which sells for $20 per share. What are the weights of the two stocks in this portfolio?
☑ For the Jones stock: \(w_J=66.67\%\) and for the Rice stock: \(w_R=33.33\%\)
☐ For the Jones stock: \(w_J=50\%\) and for the Rice stock: \(w_R=50\%\)
☐ For the Jones stock: \(w_J=70\%\) and for the Rice stock: \(w_R=30\%\)
☐ For the Jones stock: \(w_J=33.33\%\) and for the Rice stock: \(w_R=33.33\%\)
Let’s first find the dollar value of the Jones stocks would be:
120 shares × $50 / share = $6,000
Similarly, the dollar value of the Rice stocks would be: 150 shares × $20 / share = $3,000.
Hence the total value of the portfolio would be: $6,000 + $3,000 = $9,000.
The weights of the two stocks in this portfolio are therefore:
weight of Jones stock: \(w_J = 6000 / 9000 = 66.67\%\)
weight of Rice stock \(w_R = 3000 / 9000 = 33.33\%\)
Question 7
Jones stock has an expected return of 12 percent, and a standard deviation of 9 percent per year. Rice stock has an expected return of 18 percent, and a standard deviation of 25 percent per year. What is the expected return on a portfolio that consists of 30 percent Jones and 70 percent Rice? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)
Jones stock has an expected return of 12 percent and a standard deviation of 9 percent per year. Rice stock has an expected return of 18 percent, and a standard deviation of 25 percent per year.
The expected return on a portfolio that consists of 30 percent Jones and 70 percent Rice would be:
\(E(r_P) = w_J × E(R_J) + w_R × E(R_R) = 0.30 × 12\% + 0.70 × 18\% = 16.20\%\)
Question 8
If the correlation between the returns of Jones and Rice is 0.2, what is the return volatility of a portfolio described in the previous question? Round off your final answer to two digits after the decimal point (such as 5.55)
Portfolio volatility is measured by the standard deviation of the portfolio return. The variance of a two-asset portfolio is given by:
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_{12}\)
or
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \rho_{12} \times \sigma_1 \times \sigma_2\)
If the correlation between the returns of Jones and Rice is 0.2, then the variance of the portfolio is given by:
\(σ^2_P = 0.3^2×(9\%)^2+0.7^2×(25\%)^2+2×0.3×0.7×0.2×(9\%)×(25\%) = 332.44 ⇒ σ_P = 18.23\%\)
Question 9
Which of the following are examples of sources of market risk?
☐ Microsoft’s CEO is resigned.
☐ ExxonMobil decides to invest more heavily on R&D.
☑ The majority of British people votes at the referendum that Great Britain should leave the European Union That is correct, since this is likely to have market-wide effects so we can see it as a source of market risk.
☑ Oil prices rise. This is an example of market-wide risk.
Question 10
Assume there are N securities in the market. The expected return on every security is 10 percent. All securities have the same variance of 0.0025. The covariance between any pair of securities is 0.0064. What will happen to the variance of an equally weighted portfolio containing all N securities as N approaches infinity? Note: the weight of each security in the portfolio is 1/N.
☐ It will approach infinity
☐ It will approach zero
☑ It will approach the value 0.0064
☐ It will approach the value 0.0025
If \(N\) approaches infinity the first term of the portfolio variance will approach 0, while the second term will approach 0.0064 (since \((N-1) /N\) would approach 1). So the variance of the portfolio as \(N\) approaches infinity will approach the value 0.0064.
Question 11
Which of the following statements is false about the mean-variance frontier?
☑ The bottom half of the mean-variance frontier is efficient.
☐ For two assets, it consists simply of all possible portfolio combinations of the two assets.
☐ The mean variance frontier expands as we add more assets to the mix.
☐ Mean-variance frontier is the locus of the portfolios in expected return-standard deviation space that have the minimum variance for each expected return.
The bottom half of the mean-variance frontier is inefficient, because along the bottom half of the frontier, an investor obtains a lower expected return for a given level of risk than what she can earn along the top part of the frontier.
Question 12
Which of the following statements are true? Choose all that apply.
☑ The left-most point on the minimum variance frontier is called the minimum variance portfolio.
☑ Optimal portfolios are all the portfolios that lie on the minimum-variance frontier from the minimum-variance portfolio and upwards.
☐ Optimal portfolios can be obtained without diversification.
☐ Diversification removes idiosyncratic risk but does not influence the overall risk of the portfolio.
Question 13
Suppose you are considering to add real estate to your portfolio that currently includes only stocks, bonds, and cash. Which return characteristic of real estate would affect your portfolio risk?
☐ Standard deviation of real estate returns
☐ Expected return on real estate
☐ Age of the real estate properties
☑ Correlation with returns of the other asset classes
The greater an asset’s covariance with other assets in the portfolio, the more it contributes to portfolio variance. Therefore, it is the correlation that matters.
Question 14
All individual assets lie inside the efficient frontier in the expected return-volatility space. True or false?
☑ True.
☐ False.
Combining assets into portfolios reduce risk.
Question 15
Which of the following statements is correct?
☐ Portfolio theory is about elimination of systematic risk
☑ Portfolio theory is concerned with the effect of diversification on portfolio risk.
☐ Portfolio theory is about how active portfolio management can enhance returns.
☐ Portfolio theory is concerned with maximizing unsystematic risk to enhance returns.
Question 1
The term ‘efficient frontier’ refers to the portfolios that:
(Choose all that apply.)
☑ A. Yield the greatest return for a given level of risk
☑ B. Involve the least risk for a given level of expected return
☐ C. Yield the greatest return for maximum risk
☐ D. None of the above.
Question 2
An investor has total wealth of $100,000 and wants to invest in a portfolio with 3 securities A, B and C. If he chooses to invest $50,000 in security A, $25,000 in security B and $25,000 in security C, what will be the expected return of this portfolio? Round off your final answer to two digits after the decimal point. State your answer as a percentage rate (such as 5.55)
| E(r) | |
|---|---|
| Security A | 15.5% |
| Security B | 14.5% |
| Security C | 4.6% |
The expected return of the portfolio would be:
\(E(R_p) = 0.50 × 15.5\% + 0.25 × 14.5\% + 0.25 × 4.6\% = 12.53\%\)
Question 3
Consider the following distribution of returns:
| State of the economy | Prob. | Return on A | Return on B | Return on C |
|---|---|---|---|---|
| Depression | 30% | 10% | -5% | -3% |
| Normal | 50% | 15% | 20% | 5% |
| Expansion | 20% | 25% | 30% | 15% |
Which of the following statements are correct?
☑ \(E(R_A) = 15.5\%\) and \(σ_B = 13.31\%\)
☐ \(E(R_B) = 12.5\%\) and \(σ_A = 5.22\%\)
☑ \(E(R_C) = 4.6\%\) and \(E(R_B) = 14.5\%\)
☑ \(σ_A = 5.22\%\) and \(σ_C = 6.25\%\)
Question 4
Based on the distribution above compute the covariance between A and B. Round off your final answer to two digits after the decimal point (such as 5.55). (Hint: Your answers to previous questions may be useful.)
\(σ_{AB} = (10 − 15.5) × (−5 − 14.5) × 0.30 + (15 − 15.5) × (20 − 14.5) × 0.50+(25 − 15.5) × (30 − 14.5) × 0.20 = 60.25\)
Question 5
Calculate the correlation coefficient between A and B. Round off your final answer to two digits after the decimal point (such as 5.55). (Hint: Your answers to previous questions may be useful.)
The correlation coefficient between A and B is calculated as:
\(ρ_{AB} = σ_{AB} / (σ_A × σ_B)\)
Previously, we computed the following measures:
\(σ_{AB} = 60.25\)
\(σ_A = 5.22\%\)
\(σ_B = 13.31\%\)
The correlation coefficient between A and B is calculated as:
\(ρ_{AB} = σ_{AB} / (σ_A × σ_B) = 60.25 / (5.22 × 13.31) = 0.87\)
Question 6
What would be the expected return and standard deviation of a portfolio with 60% in A and 40% in B?
☑ \(E(R_P) = 15.10\%\) and \(σ_P = 8.19\%\)
☐ \(E(R_P) = 16.23\%\) and \(σ_P = 5.25\%\)
☐ \(E(R_P) = 10.13\%\) and \(σ_P = 7.50\%\)
☐ \(E(R_P) = 20.20\%\) and \(σ_P = 3.00\%\)
Question 7
What characteristics of a security are most important in the determination of the variance of a well-diversified portfolio?
☐ The expected return of a portfolio
☐ The selection of an equally weighted portfolio
☑ The correlation between the securities of a portfolio
☐ The number of the securities that form the portfolio
The most important characteristic in the determination of the variance of a well-diversified portfolio is the correlation between the securities. The lower the correlation between the securities of a portfolio, the greater the possibility of risk reduction.
Question 8
Which of the following options can be classified as a case of unique risk?
☐ A sudden change at the exchange rate between US dollar and euro.
☑ Kraft Foods buys Cadbury.
☐ Federal Reserve Bank tightens monetary policy.
☐ Oil prices fall.
While an acquisition may perhaps affect multiple companies, it is still industry specific.
Question 9
Which of the following statements are true about the mean-variance frontier? Choose all that apply.
☐ Mean-variance frontier is the locus of the portfolios in expected return-volatility space that have the maximum variance for a given level of expected return. The mean-variance frontier is the locus of the portfolios in expected return-standard deviation space that have the minimum variance for each expected return and not the maximum.
☑ When there are only two assets, mean-variance frontier consists simply of all possible portfolio combinations of these two assets.
☑ The left-most point on the minimum variance frontier is called the minimum variance portfolio.
☐ The mean variance frontier shifts to the right in mean-variance space as we add more assets to the mix. The mean variance frontier shifts to the left and expands as we add more assets to the mix. This is because the additional assets improve the diversification opportunities.
Question 10
Consider a portfolio of risky equities and Treasury bills. Suppose the expected return on equities is 12% per year with a volatility of 18%. Let’s also suppose that T-bills offer a risk-free 7% rate of return. What would be the volatility of your portfolio if you have 60% in equities and 40% in Treasuries?
☐ 10.0%
☐ 13.6%
☐ 1.94%
☑ 10.8%
The volatility of a two-asset portfolio is given by:
\(σ^2_P = w^2_1 \times \sigma^2_1 + w^2_2 \times \sigma^2_2 + 2 \times w_1 \times w_2 \times \sigma_{12}\)
Note, however, the risk-free asset has zero variance and zero covariance. Therefore,
\(σ^2_P = w^2_1 × σ^2_1 ⇒ σ_P = w_1 × σ_1 = 0.6 × 18\% = 10.8\%\)