Homework 5 Portfolio Optimization Question 1 (Bonus Question) The aim of this question is to review the fundamental logic behind the derivation of the First Order Condition (FOC) and its economic interpretation in the context of portfolio optimization. Following analogous steps to those in the slides leading to the FOC Eo((Wi) - 1.7 ) = 0 derive and interpret the FOC for a portfolio of all risky assets 2 Question 2 An investor preferences are summarized by the following utility function r(W) = (a +BW)+7 where a > 0, 0 and 1 1. Find restrictions on a, b and Wensuring that the above utility function displays non-satiation and risk aversion. Compute the absolute (ARA) and relative (RRA) risk aversion coefficients. hint 1: remember the association of non-satiation and risk aversion with the first two derivatives of (W)... hint 2: u(W) = (a + b )-and u(W)" --(a+b ))-1-1 2. The investors final wealth derives from investing in two assets, one risky and one risk free. The latter has return equal to ro. The return on the risky asset is equal to either rl or r2 with equal probabilities. The investor has initial wealth equal to 100$. Further a=0. Compute the optimal portfolio choice. 3. How does the optimal portfolio composition you found in part 2 change with wealth? hint: for this question 2 and 3 remember the exercise we did in class.... 3 Question 3 In a market are listed two risky assets whose returns are described by the following parameters HA=0.01. M = 0.07, OA 0.2 and ag = 0.12. The correlation among the securities is constant and equal to p=0.1. 1. Derive the equation for the frontier 2. Derive the minimum variance portfolio and the equation for the efficient frontier 3. Let's add a risk free asset among the possible investments with return the equation for the frontier 0.03 and derive (a) is this the Capital Allocation Line (CAL) or we should compute the efficient frontier? if this is the CAL explain why, if not derive the efficient frontier hint: think if there are inefficient portfolios on the frontier.... 4. Consider two investors, 1 and 2, with mean variance utility function, U(7) = Erol - 1X Var[F.). Investor 1 has risk aversion 1 = 5, while investor 2 has risk aversion of 10. Find the composition of their optimal portfolios. the formula for hint: a quick review of how to invert a 2 x 2 matrix A = the inverse is 4 = 4,102,2-0,3631|-22,1 21,1] 4 Question 4 The aim of this question is to make you better understand the characteristics of the CAL Stick to the market described in the previous question after we added the risk-free asset with return f = 0.03 1. Verify that the CAL is indeed generated by all possible portfolio (GCALT = [GALOSALT only containing the risky free asset (asset 0) and the tangency portfolio l' in the different proportion AGAL and OPAL respectively . hint 1: derive the CAL equation CAL (OCAL) and show that coincides with polo) - r.) + Hoe where is the Sharpe ratio of portfolio o hint 2: remember that the risk-free asset has zero risk 2. Stick to the market described in the Question 3 after we added the risk-free asset with return rf = 0.03 compute fee and op draw the CAL using the CAL equation CALOCAL locate the minimum variance portfolio, what are the weights in terms of *? locate the tangency portfolio t". what are the weights in terms of CAL? what happens to the weights CAL for (i) ACAL Mapa - hint 1: if OCAL )0 we borrow(lend) at the risk-free rate while GAL )0 we short(own) the tangency portfolio - hint 2: risk cannot be negative... Homework 5 Portfolio Optimization Question 1 (Bonus Question) The aim of this question is to review the fundamental logic behind the derivation of the First Order Condition (FOC) and its economic interpretation in the context of portfolio optimization. Following analogous steps to those in the slides leading to the FOC Eo((Wi) - 1.7 ) = 0 derive and interpret the FOC for a portfolio of all risky assets 2 Question 2 An investor preferences are summarized by the following utility function r(W) = (a +BW)+7 where a > 0, 0 and 1 1. Find restrictions on a, b and Wensuring that the above utility function displays non-satiation and risk aversion. Compute the absolute (ARA) and relative (RRA) risk aversion coefficients. hint 1: remember the association of non-satiation and risk aversion with the first two derivatives of (W)... hint 2: u(W) = (a + b )-and u(W)" --(a+b ))-1-1 2. The investors final wealth derives from investing in two assets, one risky and one risk free. The latter has return equal to ro. The return on the risky asset is equal to either rl or r2 with equal probabilities. The investor has initial wealth equal to 100$. Further a=0. Compute the optimal portfolio choice. 3. How does the optimal portfolio composition you found in part 2 change with wealth? hint: for this question 2 and 3 remember the exercise we did in class.... 3 Question 3 In a market are listed two risky assets whose returns are described by the following parameters HA=0.01. M = 0.07, OA 0.2 and ag = 0.12. The correlation among the securities is constant and equal to p=0.1. 1. Derive the equation for the frontier 2. Derive the minimum variance portfolio and the equation for the efficient frontier 3. Let's add a risk free asset among the possible investments with return the equation for the frontier 0.03 and derive (a) is this the Capital Allocation Line (CAL) or we should compute the efficient frontier? if this is the CAL explain why, if not derive the efficient frontier hint: think if there are inefficient portfolios on the frontier.... 4. Consider two investors, 1 and 2, with mean variance utility function, U(7) = Erol - 1X Var[F.). Investor 1 has risk aversion 1 = 5, while investor 2 has risk aversion of 10. Find the composition of their optimal portfolios. the formula for hint: a quick review of how to invert a 2 x 2 matrix A = the inverse is 4 = 4,102,2-0,3631|-22,1 21,1] 4 Question 4 The aim of this question is to make you better understand the characteristics of the CAL Stick to the market described in the previous question after we added the risk-free asset with return f = 0.03 1. Verify that the CAL is indeed generated by all possible portfolio (GCALT = [GALOSALT only containing the risky free asset (asset 0) and the tangency portfolio l' in the different proportion AGAL and OPAL respectively . hint 1: derive the CAL equation CAL (OCAL) and show that coincides with polo) - r.) + Hoe where is the Sharpe ratio of portfolio o hint 2: remember that the risk-free asset has zero risk 2. Stick to the market described in the Question 3 after we added the risk-free asset with return rf = 0.03 compute fee and op draw the CAL using the CAL equation CALOCAL locate the minimum variance portfolio, what are the weights in terms of *? locate the tangency portfolio t". what are the weights in terms of CAL? what happens to the weights CAL for (i) ACAL Mapa - hint 1: if OCAL )0 we borrow(lend) at the risk-free rate while GAL )0 we short(own) the tangency portfolio - hint 2: risk cannot be negative