An investor in financial markets has to consider both the average return (increase in price of an asset), represented by y, and its risk, represented by the standard deviation of the return, o. The target for the investor represents a combination of return and risk and takes the form (u-5)2 (62-7) + = 1.1 15 36 The utility function of the investor is U = 20u-3022 This question takes you through the steps to determine what combination of risk and return this investor will choose in order to maximize their utility. a) (1) Formulate this problem as one of constrained optimization, identifying the objective and constraint functions, and the choice variables. (2 marks) (ii) Hence specify the Largrangian, using 2 to represent the Lagrange multiplier. (4 marks) b) (1) Obtain the first order conditions of the problem. (4 marks) (ii) Show that there are two solutions for 2. (Hint: in solving the FOCs you get left with a quadratic equation in which then has to be solved.) (4 marks) (iii) Using the positive solution for 2, obtain solutions for u and o. (4 marks) c) Show that the solution based on x>0, is a maximum. (You need to check second order conditions, so the bordered hessian). (4 marks) d) By approximately how much will maximized utility increase if the tightness of the constraint, increases from 1 to 2? Explain your calculation. (4 marks) An investor in financial markets has to consider both the average return (increase in price of an asset), represented by y, and its risk, represented by the standard deviation of the return, o. The target for the investor represents a combination of return and risk and takes the form (u-5)2 (62-7) + = 1.1 15 36 The utility function of the investor is U = 20u-3022 This question takes you through the steps to determine what combination of risk and return this investor will choose in order to maximize their utility. a) (1) Formulate this problem as one of constrained optimization, identifying the objective and constraint functions, and the choice variables. (2 marks) (ii) Hence specify the Largrangian, using 2 to represent the Lagrange multiplier. (4 marks) b) (1) Obtain the first order conditions of the problem. (4 marks) (ii) Show that there are two solutions for 2. (Hint: in solving the FOCs you get left with a quadratic equation in which then has to be solved.) (4 marks) (iii) Using the positive solution for 2, obtain solutions for u and o. (4 marks) c) Show that the solution based on x>0, is a maximum. (You need to check second order conditions, so the bordered hessian). (4 marks) d) By approximately how much will maximized utility increase if the tightness of the constraint, increases from 1 to 2? Explain your calculation. (4 marks)
