Part 1: Preferences and the Equity Premium Puzzle Assume that you use a quadratic utility function to make your financial decisions. The average return for large US stocks is 11.26% with a standard deviation of 19.13%. Use these for your estimates of E() and o. 1. Suppose that in choosing a portfolio consisting of a risk-free asset (wherer 3%) and large US stocks, you invest 60% of your money in large US stocks (and the rest in the risk-free asset). What does this imply about your risk aversion coefficient, A? 2. If your preferences are consistent (i.e. you use the same utility function, including the same A as above), which would you prefer? (a) an asset which has E(r) = 5% and o=0 (b) an asset with E(r) = 10% and o = 20% Assume that you are only investing in (a) or (b) and not mixing the two assets into a portfolio 3. Suppose that we interview a group of investors who chose to invest 60% of their portfolio in large US stocks and 40% in the risk-free asset. We then ask them which asset from (2) that they prefer. Most answer that they prefer (b). If we believe that the investors in the group are consistent in their choices, what does this imply about the quadratic utility function? If we believe that the quadratic utility function is the correct utility function, what does this imply about the consistency of investors' preferences? Part 2: Markowitz Model and Optimal Portfolios For this part of the assignment, use the data in assignment2.data.xlsx 1. Generating some summary statistics: (a) Report the mean and standard deviation of monthly returns for each of the six stocks. (b) Report the covariance matrix of returns (6 by 6).' Are the covariances between returns on these stocks generally positive or negative? Are the signs of the co- variances surprising? 2. Solving the Markowitz Problem: Take the means above as the expected returns and the estimated covariance matrix above as your best estimate of the covariance between the returns of the six stocks. (a) Suppose that the target return is 0.8%. What are the portfolio weights for a portfolio with this return and the minimum possible variance? (b) Repeat for target returns of 0.9%, 1%, 1.1%, all the way to 1.8%. Report a table of portfolio weights, expected returns, and volatilities. Plot a graph of the expected return (y-axis) versus the volatility (x-axis) of the optimal portfolios. (c) Would an investor who likes higher expected returns and dislikes volatility ever invest in the portfolio constructed in (a)? Why or why not? 3. Now suppose that the risk-free rate is 0.4167%. (a) What is the optimal risky portfolio? (b) What is the expected return if we invest 50% in the optimal risky portfolio and 50% in the risk-free asset? What about 150% in the optimal risky portfolio and -50% in the risk-free asset? (c) For an investor with a utility function of E(r) - Ao and a coefficient of risk aversion of 4, what is the optimal asset allocation? Part 3: Diversification 1 Start with asset A which has an expected return of 10% and a volatility of 30%. 1. Suppose that we introduce asset B with an expected return of 10% and a volatility of 30%. The correlation between the two asset returns is 0.9. What is the optimal combination of A and B? What is the volatility of this portfolio? (Note: The choice of the risk-free rate is not important. You can use ry = 3%.] 2. Now suppose that instead of introducing B, we had introduced asset with an expected return of 10% and a volatility of 30%. The returns of asset C are uncorrelated with both the returns of asset A. What is the optimal combination of A and C? What is the volatility of this portfolio? 3. Did the introduction of B or C have a greater effect in decreasing the portfolio volatility? Why is this the case? Use the COVARIANCE.S function in Excel. B D E G KO GE 1 date BA JNJ MMM MRK 2 19460628 -0.0492 -0.0396 -0.08861 -0.02778 0.005128 -0.02593 3 19460731 -0.02247 -0.01579 0.018519 -0.00408 0.076531 -0.03053 4 19460830 -0.03448 -0.08021 -0.05 -0.03934 -0.0455 0.043307 5 19460930 -0.10863 -0.07791 -0.06699 -0.1453 -0.0275 -0.10566 6 19461031 -0.08725 -0.05733 -0.11282 -0.01 -0.01542 -0.00848 7 19461130 0.045037 -0.05405 -0.13295 0.062626 0.067363 0.064103 8 19461231 -0.00267 0.036429 0.173333 0.185714 0.185714 0.064257 9 19470131 0.1 0.090592 -0.09091 -0.09237 -0.03236 -0.075 10 19470228 0.025974 -0.03834 0.05 0.010619 -0.07918 -0.05198 11 19470331 -0.03006 -0.04585 0.059524 0.144737 -0.00943 -0.02895 12 19470430 -0.00984 -0.02113 -0.21177 -0.08046 0.038095 -0.07273 13 19470529 0.013245 -0.01079 -0.08955 -0.05417 0.033945 -0.09314 14 19470630 0.122549 0.044364 0.040984 0.118943 0.071429 0.107568 15 19470731 0.055556 0.073944 0.007874 0 0.008333 0.076167 16 19470829 0.00831 -0.03607 0.054688 -0.12284 -0.03554 -0.05023 17 19470930 -0.01099 0.004082 0.266667 -0.08108 0.137931 -0.05096 18 19471031 0.033124 0 -0.08772 0.107843 -0.01515 -0.03061 19 19471129 -0.00506 -0.0411 0.070513 -0.00531 -0.02308 0.042105 20 19471231 0.002755 0.032857 0.131737 -0.05357 0.111111 0.072727 21 19480131 -0.05357 -0.03846 0.031746 -0.07547 -0.12143 -0.03365 22 19480228 -0.09144 -0.05091 0.015385 -0.13367 -0.06504 -0.0597 23 19480331 0.041534 0.115709 0.186869 0.195266 0.078947 0.134392 24 19480430 0.049383 -0.03125 -0.04846 0.066832 0.097561 0.066038 25 19480528 -0.00147 0.204301 -0.03704 0.09884 0.051852 0.090708 26 19480630 0.001473 -0.01131 -0.00962 0.055085 0.006206 0.001623 27 19480730 -0.05917 -0.06991 -0.08738 -0.09237 -0.03436 -0.04704 28 19480831 -0.04088 0.03268 0.005319 -0.0031 -0.0146 0.053648 29 19480930 -0.03934 0 0.026455 0.084633 -0.08955 -0.05743 30 19481030 0.017182 0.083333 0.06701 0.164271 0.133197 0.065502 31 19481130 -0.04688 -0.11834 -0.23188 -0.0963 -0.13056 -0.04918 32 19481231 -0.03461 -0.057047 0.08805 0.091667 0.132353 0.031897 Sheet1 + HE > Part 1: Preferences and the Equity Premium Puzzle Assume that you use a quadratic utility function to make your financial decisions. The average return for large US stocks is 11.26% with a standard deviation of 19.13%. Use these for your estimates of E() and o. 1. Suppose that in choosing a portfolio consisting of a risk-free asset (wherer 3%) and large US stocks, you invest 60% of your money in large US stocks (and the rest in the risk-free asset). What does this imply about your risk aversion coefficient, A? 2. If your preferences are consistent (i.e. you use the same utility function, including the same A as above), which would you prefer? (a) an asset which has E(r) = 5% and o=0 (b) an asset with E(r) = 10% and o = 20% Assume that you are only investing in (a) or (b) and not mixing the two assets into a portfolio 3. Suppose that we interview a group of investors who chose to invest 60% of their portfolio in large US stocks and 40% in the risk-free asset. We then ask them which asset from (2) that they prefer. Most answer that they prefer (b). If we believe that the investors in the group are consistent in their choices, what does this imply about the quadratic utility function? If we believe that the quadratic utility function is the correct utility function, what does this imply about the consistency of investors' preferences? Part 2: Markowitz Model and Optimal Portfolios For this part of the assignment, use the data in assignment2.data.xlsx 1. Generating some summary statistics: (a) Report the mean and standard deviation of monthly returns for each of the six stocks. (b) Report the covariance matrix of returns (6 by 6).' Are the covariances between returns on these stocks generally positive or negative? Are the signs of the co- variances surprising? 2. Solving the Markowitz Problem: Take the means above as the expected returns and the estimated covariance matrix above as your best estimate of the covariance between the returns of the six stocks. (a) Suppose that the target return is 0.8%. What are the portfolio weights for a portfolio with this return and the minimum possible variance? (b) Repeat for target returns of 0.9%, 1%, 1.1%, all the way to 1.8%. Report a table of portfolio weights, expected returns, and volatilities. Plot a graph of the expected return (y-axis) versus the volatility (x-axis) of the optimal portfolios. (c) Would an investor who likes higher expected returns and dislikes volatility ever invest in the portfolio constructed in (a)? Why or why not? 3. Now suppose that the risk-free rate is 0.4167%. (a) What is the optimal risky portfolio? (b) What is the expected return if we invest 50% in the optimal risky portfolio and 50% in the risk-free asset? What about 150% in the optimal risky portfolio and -50% in the risk-free asset? (c) For an investor with a utility function of E(r) - Ao and a coefficient of risk aversion of 4, what is the optimal asset allocation? Part 3: Diversification 1 Start with asset A which has an expected return of 10% and a volatility of 30%. 1. Suppose that we introduce asset B with an expected return of 10% and a volatility of 30%. The correlation between the two asset returns is 0.9. What is the optimal combination of A and B? What is the volatility of this portfolio? (Note: The choice of the risk-free rate is not important. You can use ry = 3%.] 2. Now suppose that instead of introducing B, we had introduced asset with an expected return of 10% and a volatility of 30%. The returns of asset C are uncorrelated with both the returns of asset A. What is the optimal combination of A and C? What is the volatility of this portfolio? 3. Did the introduction of B or C have a greater effect in decreasing the portfolio volatility? Why is this the case? Use the COVARIANCE.S function in Excel. B D E G KO GE 1 date BA JNJ MMM MRK 2 19460628 -0.0492 -0.0396 -0.08861 -0.02778 0.005128 -0.02593 3 19460731 -0.02247 -0.01579 0.018519 -0.00408 0.076531 -0.03053 4 19460830 -0.03448 -0.08021 -0.05 -0.03934 -0.0455 0.043307 5 19460930 -0.10863 -0.07791 -0.06699 -0.1453 -0.0275 -0.10566 6 19461031 -0.08725 -0.05733 -0.11282 -0.01 -0.01542 -0.00848 7 19461130 0.045037 -0.05405 -0.13295 0.062626 0.067363 0.064103 8 19461231 -0.00267 0.036429 0.173333 0.185714 0.185714 0.064257 9 19470131 0.1 0.090592 -0.09091 -0.09237 -0.03236 -0.075 10 19470228 0.025974 -0.03834 0.05 0.010619 -0.07918 -0.05198 11 19470331 -0.03006 -0.04585 0.059524 0.144737 -0.00943 -0.02895 12 19470430 -0.00984 -0.02113 -0.21177 -0.08046 0.038095 -0.07273 13 19470529 0.013245 -0.01079 -0.08955 -0.05417 0.033945 -0.09314 14 19470630 0.122549 0.044364 0.040984 0.118943 0.071429 0.107568 15 19470731 0.055556 0.073944 0.007874 0 0.008333 0.076167 16 19470829 0.00831 -0.03607 0.054688 -0.12284 -0.03554 -0.05023 17 19470930 -0.01099 0.004082 0.266667 -0.08108 0.137931 -0.05096 18 19471031 0.033124 0 -0.08772 0.107843 -0.01515 -0.03061 19 19471129 -0.00506 -0.0411 0.070513 -0.00531 -0.02308 0.042105 20 19471231 0.002755 0.032857 0.131737 -0.05357 0.111111 0.072727 21 19480131 -0.05357 -0.03846 0.031746 -0.07547 -0.12143 -0.03365 22 19480228 -0.09144 -0.05091 0.015385 -0.13367 -0.06504 -0.0597 23 19480331 0.041534 0.115709 0.186869 0.195266 0.078947 0.134392 24 19480430 0.049383 -0.03125 -0.04846 0.066832 0.097561 0.066038 25 19480528 -0.00147 0.204301 -0.03704 0.09884 0.051852 0.090708 26 19480630 0.001473 -0.01131 -0.00962 0.055085 0.006206 0.001623 27 19480730 -0.05917 -0.06991 -0.08738 -0.09237 -0.03436 -0.04704 28 19480831 -0.04088 0.03268 0.005319 -0.0031 -0.0146 0.053648 29 19480930 -0.03934 0 0.026455 0.084633 -0.08955 -0.05743 30 19481030 0.017182 0.083333 0.06701 0.164271 0.133197 0.065502 31 19481130 -0.04688 -0.11834 -0.23188 -0.0963 -0.13056 -0.04918 32 19481231 -0.03461 -0.057047 0.08805 0.091667 0.132353 0.031897 Sheet1 + HE >