FINA 6321: Portfolio Analysis and Management Professor Xixuji Lin Homework 4 Note: Answers must be justified. Correct answers without explanation will not be given credit. The total points of this homework is 25. The due data is November 30. Question (1) (Using a Factor Model to Obtain the Inputs for Mean Variance Analysis) (15 points) Consider two portfolio managers: the "HISTORICAL" portfolio manager and the "CSOM" portfolio manager (with these names, you know which one is the smart one....). Both managers follow a mean- variance analysis to construct their portfolios, but they follow different approaches on how to get the key inputs of the mean-variance analysis, namely the expected returns, the standard deviation and the correlations. In this question, I want to focus on the issue of obtaining good estimates of expected returns. So for simplicity, assume that the two portfolio managers know the true correlations and standard deviations of all the stocks in the market. Each portfolio manager cstimates the expected return of cach security as follows: The "HISTORICAL" portfolio manager estimates the expected return of each asset by ng for each assets i the historical average return of that asset. More specifically, she computes: ERL] = Historical average return of security i (i) The "CSOM" portfolio Manager does things differently. She uses a factor model to estimate the expected return of each asset. For each assets, she first runs the following regression: H = a; 8, X ACDI E (1) where AGDP is the growth rate of output and R = Rit-R is the excess return of asset i. By running this regression she obtains estimates of the intercepts ; and slopes 8: for each asset. Then she asks her research team to compute an estimate of the expected output growth E|AGDP for next year. Then she uses these estimates to produce an estimate of the ERA for each asset i using the regression in equation (1). To make things simple, let's assure that there are only two risky assets in the markel: stock A and stock B. Both managers know that the current risk-free rale in the market is Ri-5% and you also have the following information: E24 llistorical average return of aszel A - 20% E|NB Historical average return of asset B - 10.5% Corr(R 4, NB) Correlation between returns of assets A and B = 0.1 Sud(RA) Standard deviation of asset A Std(Ry) = Standard deviation of assct B = 10% 20% In addition, the "CSOM" portfolio manager told you that she obtained the following estimates for the factor model (NOTE: the superscrul e in the returns (RA) imeans that these are encers returns (ie. returns in excess of the risk free rale reluun), so if you need the actual reburns (R1), you need to add up the risk free rate. For example, for asset A, this means: RA= R +R;) 224, 15% + 2 x AGDP + est = 10% 0.2 x AGDP teit and that her researui team believes that the output next year will grow 10%. ie. E AGDP] = 10%. Answer the following questions. (a) Estimate the expected return of assets A and B according to the "CSOM" portfolio manager and according to the "HISTORICAL" portfolio manager. Compare the estimates. (b) Using the previous estimates for expected returns and the information provided in the introduction, find the Mean Variance Dllicient Portfolio (MVEP) of the "CSOM" portfolio manager. Also, find the MVEP of the "HISTORICAL" portfolio manager. Brielly comment on the differences between the two MVEP portfolios. (Note: you should use the Excel solver function to do this. It will be very easy if you use the file "Example Optimal CAL.xls" that is posted on the course webpage. Just plug in the new numbers there and maximize the Sharpe Ratio following the instructions there) (c) Find the expected return, the standard deviation and the Sharpe Ratio of the MVEP of each manager that you found in the previous question. (Note: il you used the Excel file "Example Optimal CAL.xls" to answer the previous question, you already have the answer to this question- feel free to use those number here) (BONUS: Do you think it makes sense to compare the two MVEP? If yes, why? If not, why not?) (a) Assume that the coefficient of risk aversion of each portfolio manager is A = 15. What is the optimal global portfolio of the "CSOM" portfolio manager and of the "HISTORICAL." portfolio manager? Find what proportion of their funds should each portfolio manager invest on the risk-free asset and what proportion should be invested in their MVEP. Bricfly comment on the differences in the proportion invested in cach asset by the two portfolio managers. Now suppose that the FED releases some new information stating that they are worried about the current state of the economy. Because of this, everybody now aggrees that the expected output growth for next year is no longer E|AGDP 10% but it is really E AGDP] 0%, i.e. there is a big drop in the expectations about how the economy will perform next year. The risk free rale stays the same (i.e. R: -5%). (e) Given this new information, will the "HISTORICAL" portfolio manager change her optimal global portfolio? What about the "CSOM" portfolio managers? If you answered yes, find the correspond- ing new global portfolio of each portfolio manager and comment on the difference between these portfolios and those found in part (d). In particular, find if they will invest more or less on the risk free asset than in part (d). Does this adjustment to the amount invested in the risk-free asset makes sense to you? 2 Question (2) (Minimum Variance Frontier, Efficient Portfolios and Optimal Portfolio Selec- Lion) (10 points) The Triad family of mutual funds allows investors to split their money between three portfolios managed by Triad. Portfolio C has an expected return of F[Rc) = 10% and a standard deviation of returns of (Re) - 15%. Portfolio B has an expected return of ER 1 19% and a standard deviation of return of (R) - 25%. The correlation coefficient between the returns of portfolios B and C is Pue - 0.2. Portfolio A consists entirely of riskfree securities, and has a certain return of 4%. Your client is leaning towards investing her money entirely in portfolio C, since she is unwilling to take the higher risk associated with portfolio B, but wants a higher return than offered by portfolio A. (a) As a Triad investment advisor; you suggest w her an alternative portfolio P (consisting of a com bination of only portfolios A and B) that has the same expected return as portfolio C but a lower standard deviation. If she has $200,000 to invest, how much should she invest in B and how much in A? What is the standard deviation of the return on her investment in this case? Sketch a mean standard deviation diagram that you would use to explain why the portfolio you suggest is better (b) However, after your convincing presentation of the alternative portfolio P, your client now says that she is really comfortable with the level of risk in portfolio C. So, you suggest to her another portfolio P1 (also consisting of a combination of only portfolios A and B) that has the same standard deviation as portfolio C, but higher expected return. If she has $200,000 to invest, how much should she invest in B and how much in A? What is the expected return on her investment in this case?? Sketch a mcan standard deviation diagram that you would use to explain why the portfolio you suggest is better. (c) Your client also recalls reading a while ago in the 1st edition of "Investments" by Sharpe and Alcxander (Jeff Bailey at the time she went to school was not a coauthor of this book), that people should hold cfficient portfolios, but she is unsure how to do it. She asks you for advice on how to optimally combine portfolios C and B in order to take advantage of diversification. You suggest her to invest in a mix of the portfolio A and a so called mean-variance ellicient portfolio (MVEP) which is made up of portfolios B and C. Find this MVEP portfolio (i.e. find the weiylits on portfolio B and C that generate the MVEP). Use Excel's solver function to do this (feel free to use the Example Optimal CAL.xls that is posted on the course webpage). What is the expected return, standard deviation and Sharpe ratio of the MVEP portfolio that you found? (d) Your client wants to keep the level of risk of her portfolio at the same level of portfolio C (i.e. standard deviation of 15%) hut take advantage of diversification. If she invest; $200,000 in A and MVEP, how much should she invest in portfolio A and how much in the MVEP ? How much in each of the portfolios B and C? What is the expected return on her portfolio in this case ? Compare it to the expected return on P1 obtained in part (B) and explain the difference, if any. (e) Suppose that now portfolio C has a standard deviation of 25% (the same as portfolio B, this is not a mistake), but the expected return on fund C is the same as before, only 10%. Would you still advise your client to hold portfolio C? If no, explain why. If yes, also explain why. Hint: find the new MVEP using Excel's solver and explain the result. 3 FINA 6321: Portfolio Analysis and Management Professor Xixuji Lin Homework 4 Note: Answers must be justified. Correct answers without explanation will not be given credit. The total points of this homework is 25. The due data is November 30. Question (1) (Using a Factor Model to Obtain the Inputs for Mean Variance Analysis) (15 points) Consider two portfolio managers: the "HISTORICAL" portfolio manager and the "CSOM" portfolio manager (with these names, you know which one is the smart one....). Both managers follow a mean- variance analysis to construct their portfolios, but they follow different approaches on how to get the key inputs of the mean-variance analysis, namely the expected returns, the standard deviation and the correlations. In this question, I want to focus on the issue of obtaining good estimates of expected returns. So for simplicity, assume that the two portfolio managers know the true correlations and standard deviations of all the stocks in the market. Each portfolio manager cstimates the expected return of cach security as follows: The "HISTORICAL" portfolio manager estimates the expected return of each asset by ng for each assets i the historical average return of that asset. More specifically, she computes: ERL] = Historical average return of security i (i) The "CSOM" portfolio Manager does things differently. She uses a factor model to estimate the expected return of each asset. For each assets, she first runs the following regression: H = a; 8, X ACDI E (1) where AGDP is the growth rate of output and R = Rit-R is the excess return of asset i. By running this regression she obtains estimates of the intercepts ; and slopes 8: for each asset. Then she asks her research team to compute an estimate of the expected output growth E|AGDP for next year. Then she uses these estimates to produce an estimate of the ERA for each asset i using the regression in equation (1). To make things simple, let's assure that there are only two risky assets in the markel: stock A and stock B. Both managers know that the current risk-free rale in the market is Ri-5% and you also have the following information: E24 llistorical average return of aszel A - 20% E|NB Historical average return of asset B - 10.5% Corr(R 4, NB) Correlation between returns of assets A and B = 0.1 Sud(RA) Standard deviation of asset A Std(Ry) = Standard deviation of assct B = 10% 20% In addition, the "CSOM" portfolio manager told you that she obtained the following estimates for the factor model (NOTE: the superscrul e in the returns (RA) imeans that these are encers returns (ie. returns in excess of the risk free rale reluun), so if you need the actual reburns (R1), you need to add up the risk free rate. For example, for asset A, this means: RA= R +R;) 224, 15% + 2 x AGDP + est = 10% 0.2 x AGDP teit and that her researui team believes that the output next year will grow 10%. ie. E AGDP] = 10%. Answer the following questions. (a) Estimate the expected return of assets A and B according to the "CSOM" portfolio manager and according to the "HISTORICAL" portfolio manager. Compare the estimates. (b) Using the previous estimates for expected returns and the information provided in the introduction, find the Mean Variance Dllicient Portfolio (MVEP) of the "CSOM" portfolio manager. Also, find the MVEP of the "HISTORICAL" portfolio manager. Brielly comment on the differences between the two MVEP portfolios. (Note: you should use the Excel solver function to do this. It will be very easy if you use the file "Example Optimal CAL.xls" that is posted on the course webpage. Just plug in the new numbers there and maximize the Sharpe Ratio following the instructions there) (c) Find the expected return, the standard deviation and the Sharpe Ratio of the MVEP of each manager that you found in the previous question. (Note: il you used the Excel file "Example Optimal CAL.xls" to answer the previous question, you already have the answer to this question- feel free to use those number here) (BONUS: Do you think it makes sense to compare the two MVEP? If yes, why? If not, why not?) (a) Assume that the coefficient of risk aversion of each portfolio manager is A = 15. What is the optimal global portfolio of the "CSOM" portfolio manager and of the "HISTORICAL." portfolio manager? Find what proportion of their funds should each portfolio manager invest on the risk-free asset and what proportion should be invested in their MVEP. Bricfly comment on the differences in the proportion invested in cach asset by the two portfolio managers. Now suppose that the FED releases some new information stating that they are worried about the current state of the economy. Because of this, everybody now aggrees that the expected output growth for next year is no longer E|AGDP 10% but it is really E AGDP] 0%, i.e. there is a big drop in the expectations about how the economy will perform next year. The risk free rale stays the same (i.e. R: -5%). (e) Given this new information, will the "HISTORICAL" portfolio manager change her optimal global portfolio? What about the "CSOM" portfolio managers? If you answered yes, find the correspond- ing new global portfolio of each portfolio manager and comment on the difference between these portfolios and those found in part (d). In particular, find if they will invest more or less on the risk free asset than in part (d). Does this adjustment to the amount invested in the risk-free asset makes sense to you? 2 Question (2) (Minimum Variance Frontier, Efficient Portfolios and Optimal Portfolio Selec- Lion) (10 points) The Triad family of mutual funds allows investors to split their money between three portfolios managed by Triad. Portfolio C has an expected return of F[Rc) = 10% and a standard deviation of returns of (Re) - 15%. Portfolio B has an expected return of ER 1 19% and a standard deviation of return of (R) - 25%. The correlation coefficient between the returns of portfolios B and C is Pue - 0.2. Portfolio A consists entirely of riskfree securities, and has a certain return of 4%. Your client is leaning towards investing her money entirely in portfolio C, since she is unwilling to take the higher risk associated with portfolio B, but wants a higher return than offered by portfolio A. (a) As a Triad investment advisor; you suggest w her an alternative portfolio P (consisting of a com bination of only portfolios A and B) that has the same expected return as portfolio C but a lower standard deviation. If she has $200,000 to invest, how much should she invest in B and how much in A? What is the standard deviation of the return on her investment in this case? Sketch a mean standard deviation diagram that you would use to explain why the portfolio you suggest is better (b) However, after your convincing presentation of the alternative portfolio P, your client now says that she is really comfortable with the level of risk in portfolio C. So, you suggest to her another portfolio P1 (also consisting of a combination of only portfolios A and B) that has the same standard deviation as portfolio C, but higher expected return. If she has $200,000 to invest, how much should she invest in B and how much in A? What is the expected return on her investment in this case?? Sketch a mcan standard deviation diagram that you would use to explain why the portfolio you suggest is better. (c) Your client also recalls reading a while ago in the 1st edition of "Investments" by Sharpe and Alcxander (Jeff Bailey at the time she went to school was not a coauthor of this book), that people should hold cfficient portfolios, but she is unsure how to do it. She asks you for advice on how to optimally combine portfolios C and B in order to take advantage of diversification. You suggest her to invest in a mix of the portfolio A and a so called mean-variance ellicient portfolio (MVEP) which is made up of portfolios B and C. Find this MVEP portfolio (i.e. find the weiylits on portfolio B and C that generate the MVEP). Use Excel's solver function to do this (feel free to use the Example Optimal CAL.xls that is posted on the course webpage). What is the expected return, standard deviation and Sharpe ratio of the MVEP portfolio that you found? (d) Your client wants to keep the level of risk of her portfolio at the same level of portfolio C (i.e. standard deviation of 15%) hut take advantage of diversification. If she invest; $200,000 in A and MVEP, how much should she invest in portfolio A and how much in the MVEP ? How much in each of the portfolios B and C? What is the expected return on her portfolio in this case ? Compare it to the expected return on P1 obtained in part (B) and explain the difference, if any. (e) Suppose that now portfolio C has a standard deviation of 25% (the same as portfolio B, this is not a mistake), but the expected return on fund C is the same as before, only 10%. Would you still advise your client to hold portfolio C? If no, explain why. If yes, also explain why. Hint: find the new MVEP using Excel's solver and explain the result. 3