Olivia is an investor who has only two primary assets, asset A and asset B, in...
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Olivia is an investor who has only two primary assets, asset A and asset B, in which to hold her wealth, which is normalised to be 1. Asset A has mean mA = 4 and standard deviation A = 2, while asset B has mean mB = 2 and standard deviation OB = 1. Olivia can choose as her portfolio any mixture between the two assets defined by A, where A is the fraction of her wealth that is held in asset A (so a fraction 1-X is held in asset B). Portfolios are restricted to satisfy 0 ≤ ≤ 1. Olivia has constant absolute risk aversion preferences, with absolute risk aversion parameter equal to R, and her expected utility of a portfolio with mean m and standard deviation o is given by v(m, o) = m - R. Olivia makes her portfolio decisions maximising this function. 1. Plot the two assets in the MSD graph. 2. Assume, for this question only that the correlation between the two assets is PAB = 1. (a) Find the equation for the efficient portfolio curve. (b) Find Olivia's optimal investment strategy A*, and the coordinates of her optimal portfolio, (m, o). (c) If R=2, what is Olivia's optimal portfolio? 3. Now, assume for this question only that PAB = -1. (a) Find the equation for the efficient portfolio curve. (b) Find Olivia's optimal investment strategy A*, and the coordinates of her optimal portfolio, (m, o1). (c) If R=2, what is Olivia's optimal portfolio? 4. Finally, now assume for this question and the next one that PAB = 0. (a) Find the equation for the efficient portfol curve. (b) Find Olivia's optimal investment strategy X*, and the coordinates of her optimal portfolio, (m, o1). (c) If R = 2, what is Olivia's optimal portfolio? What is Olivia's level of expected utility at her optimal portfolio? 5. Assume now that a risk-free asset appears on the market, with return of mf = 3. Olivia can now divide her wealth between the two original assets and the risk-free asset. She does this by constructing the Capital Market Line corre- sponding to her new situation, and then choosing an optimal point along it. (a) Find the coordinates of the optimal risky portfolio, M, which is to be combined with the risk-free asset. (b) Find the mix of assets A and B that is needed to obtain the portfolio M. (c) Find Olivia's optimal point along the Capital Market Line as a function of R, including the fraction of wealth that she needs to dedicate to M in order to obtain her optimal position. (d) Assuming that R=2, what are the fractions of wealth that Olivia dedicates to (i) the risk-free asset, (ii) risky asset A, and (iii) risky asset B? (e) What is Olivia's level of expected utility at her optimal portfolio when R = 3? Compare this with your answer to question 4(c) above, and comment on how valuable it is to Olivia to have access to the risk-free asset. Given the existence of the risk-free asset, how valuable is it for Olivia to have access to the risky assets A and B? Olivia is an investor who has only two primary assets, asset A and asset B, in which to hold her wealth, which is normalised to be 1. Asset A has mean mA = 4 and standard deviation A = 2, while asset B has mean mB = 2 and standard deviation OB = 1. Olivia can choose as her portfolio any mixture between the two assets defined by A, where A is the fraction of her wealth that is held in asset A (so a fraction 1-X is held in asset B). Portfolios are restricted to satisfy 0 ≤ ≤ 1. Olivia has constant absolute risk aversion preferences, with absolute risk aversion parameter equal to R, and her expected utility of a portfolio with mean m and standard deviation o is given by v(m, o) = m - R. Olivia makes her portfolio decisions maximising this function. 1. Plot the two assets in the MSD graph. 2. Assume, for this question only that the correlation between the two assets is PAB = 1. (a) Find the equation for the efficient portfolio curve. (b) Find Olivia's optimal investment strategy A*, and the coordinates of her optimal portfolio, (m, o). (c) If R=2, what is Olivia's optimal portfolio? 3. Now, assume for this question only that PAB = -1. (a) Find the equation for the efficient portfolio curve. (b) Find Olivia's optimal investment strategy A*, and the coordinates of her optimal portfolio, (m, o1). (c) If R=2, what is Olivia's optimal portfolio? 4. Finally, now assume for this question and the next one that PAB = 0. (a) Find the equation for the efficient portfol curve. (b) Find Olivia's optimal investment strategy X*, and the coordinates of her optimal portfolio, (m, o1). (c) If R = 2, what is Olivia's optimal portfolio? What is Olivia's level of expected utility at her optimal portfolio? 5. Assume now that a risk-free asset appears on the market, with return of mf = 3. Olivia can now divide her wealth between the two original assets and the risk-free asset. She does this by constructing the Capital Market Line corre- sponding to her new situation, and then choosing an optimal point along it. (a) Find the coordinates of the optimal risky portfolio, M, which is to be combined with the risk-free asset. (b) Find the mix of assets A and B that is needed to obtain the portfolio M. (c) Find Olivia's optimal point along the Capital Market Line as a function of R, including the fraction of wealth that she needs to dedicate to M in order to obtain her optimal position. (d) Assuming that R=2, what are the fractions of wealth that Olivia dedicates to (i) the risk-free asset, (ii) risky asset A, and (iii) risky asset B? (e) What is Olivia's level of expected utility at her optimal portfolio when R = 3? Compare this with your answer to question 4(c) above, and comment on how valuable it is to Olivia to have access to the risk-free asset. Given the existence of the risk-free asset, how valuable is it for Olivia to have access to the risky assets A and B?
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Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
Posted Date:
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