FINA411 AA Fall2022 Assignment 2 solutions

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Concordia University *

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Apr 3, 2024

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John Molson School of Business FINA 411 AA - Page 1 Usual Copyright Disclaimer applies. ASSIGNMENT 2 FINA 411AA Solution sketch Selected Exercises 1. The following data is available on 3 stocks A, B, C: 𝐸(𝑅 ? ) = 𝐸(𝑅 ? ) = 𝐸(𝑅 ? ) 𝜎 ? = 𝜎 ? = 𝜎 ? 𝜌 (?,?) = +0.9 𝜌 (?,?) = −0.2 𝜌 (?,?) = +0.3 Portfolio of which two stocks will have the lowest standard deviation? Explain and include all steps necessary to arrive at the solution. The portfolio constructed containing stocks B and C would have the lowest standard deviation. Combining assets with equal risk and return but with low positive or negative correlations will reduce the risk level of the portfolio. 2. An analyst in the search for investment opportunities is evaluating the following five portfolios of risky assets, which are well-diversified (assume risk-free rate of 3.75 %): Portfolio Exp. Return St. Dev. A 8.00% 11.00% B 11.00% 14.50% C 5.00% 5.00% D 11.80% 20.00% E 6.40% 8.00%
John Molson School of Business FINA 411 AA - Page 2 Usual Copyright Disclaimer applies. a. Estimate the risk premium per unit of risk expected to be received per each portfolio. Estimates of (E(R)-Rf)/std: A 0.386 B 0.500 C 0.250 D 0.403 E 0.331 b. Which of the five portfolios is most likely to be the market portfolio? Draw the CML (define the numerical values of the intercept and slope next to the graph). The CML slope, [E(R MKT ) - RFR ]/ σ MKT , is the ratio of risk premium per unit of risk. Portfolio B has the highest ratio, 0.5000, of these five portfolios, so it is most likely the market portfolio. Thus, the slope of the CML is 0.5 and its intercept is 3.75%, which is the risk-free rate. c. Suppose the analyst ’s preference is only for making an investment with 𝜎 = 8% , and earning a 10% return. Would this be possible? The CML equation, based on the above analysis, is E(R portfolio ) = 3.75 % + (0.50) σ portfolio . If the desired standard deviation is 8.0%, then the expected portfolio return is 7.75%: E(R portfolio ) = 3.75% + (0.50) (8%) = 7.75% . The answer is no, it is not possible to earn an expected return of 10% with a portfolio with standard deviation of 8%. Risky portfolio R_f 0% 5% 10% 15% 20% 0% 5% 10% 15% 20% 25% 30% 35% Expected Return Standard Deviation Capital Allocation Line
John Molson School of Business FINA 411 AA - Page 3 Usual Copyright Disclaimer applies. d. What is the minimum level of risk necessary for an investment earning 10%? Estimate the portfolio composition (along CML) that can generate an expected return of 10%. Using the CML equation, we set the expected portfolio return equal to 10% and solve for the standard deviation: E(R portfolio ) = 10% = 3.75 % + (0.50) σ portfolio σ = 12.5%. Thus, 12.5% is the standard deviation consistent with an expected return of 10%. To find the portfolio weights which result in a risk of 10% and expected return of 10%, we use the fact that the covariance between the risk-free asset and the market portfolio is zero. Thus, the portfolio standard deviation calculation simplifies to σ portfolio = w MKT MKT ), and the weight of the risk-free asset is 1 - w MKT . Doing this, we have σ portfolio = 12.5% = w MKT (14.5% ), so w MKT = 0.8621 and w risk-free asset = 1 0.8621 = 0.1379. As a check, the weighted average expected return should equal 10%: 0.8621 (11%) + 0.1379 (3.75%) = 10%, which it does. e. Recently Bank of Canada has undertaken monetary policy tightening, which have led to a significant increase in the base interest rate in Canada. Given your answers to d), how would an increase in the risk- free interest rate affect the portfolio composition (i.e., portfolio weights)? Further increases in the risk-free rate will increase the portfolio weight for the risk-free asset as sensitivity analysis can show. At rf=9%, the weights of the risky and risk-free asset are equal to 0.5%. At rf=10%, it is optimal to invest in the risk-free asset only. 3. As a financial director working for a major financial institution, you are asked to evaluate two portfolio managers A and B hired externally. The following performance data based on the last three years is available for your review: Portfolio Actual return (Average) St. Dev. Beta Manager A 9.50% 11.50% 1.3 Manager B 9.00% 10.00% 0.7
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John Molson School of Business FINA 411 AA - Page 4 Usual Copyright Disclaimer applies. Based on your analysis, you find that the market portfolio has a risk-premium of 6 percent, and the risk- free rate is currently 3.75 percent. a) Define the Security Market Line (SML) - intercept and slope. With a risk premium of 6% and risk-free rate of 3.75%, the security market line is: E(return) = 3.75% + (6 %)β. Information about the level of diversification of the portfolios is not given, nor is information about the market portfolio. But a portfolio’s beta is the weighted average of the betas of the securities held in the portfolio so the SML can be used to evaluate the managers. b) Using CAPM, estimate the expected returns for manager A and manager B. Expected return (A) = 3.75% + (6 %)β = 3.75% + (6%)(1.3) = 0.1155 Expected return (B) = 3.75% + (6 %)β = 3.75% + (6%)(0.7) = 0.0795 c) Estimate each manager’s alpha over the last three years. Explain whether alpha will plot above the SML, or below the SML in either case. Alpha is the difference between the actual return and the expected return based on portfolio risk: Alpha of manager A = actual return expected return = 9.50% - 11.55% = -2.05% Alpha of manager B = actual return expected return = 9.00% - 7.95% = 1.05% A positive alpha means the portfolio outperformed the market on a risk-adjusted basis; it would plot above the SML. A negative alpha means the opposite, which is that the portfolio underperformed the market on a risk-adjusted basis; it would plot below the SML. d) Given the results in c), can you conclude whether 1) either manager outperformed the other, and 2) any of the managers outperformed market expectations in general. In this case, manager B outperformed the market portfolio on a risk-adjusted basis by 1.05%. Manager A underperformed, returning 2.05 % less than expected based upon the risk of A ’s portfolio.