A trading platform offers three assets. It prohibits selling any asset short, that is the investor can hold zero or a positive amount of either asset. There are two possible states of the world. The payoffs of the assets are given in the matrix below assets states [B/A : 1] Suppose that the prices of the assets are (91392393) = (1,1,1) and the probabilities of the two states are (T. 1 - ) = (1/2, 1/2). The initial wealth w = 6. We further adopt the mean-variance approach. a. (2 marks) If the investor likes higher mean, but dislikes higher variance, which assets will be bold in his portfolio? Explain your answer. (1 mark) For the assets you have chosen in a) derive the mean of the portfolio. Denote with the amount of asset i the investor hokis. The mean of the portfolio (21,22,23) should be the function of 21, 22, 23), of course, it can be that 21 = 0 for some i. (2 marks) For the assets you have chosen in a) derive the standard deviation of the portfolio o (31,32.-). This should be a function of (1, 22, 23) as well. d. (1 mark) Write the investor's budget constraint. 6. (3 marks) Derive the efficiency frontier. (The frontier should express in terms of o). Hint: you should use the expressions for 4 and a from 6. and c. and the investor's budget constraint ind. 4. (2 marks) Plot the frontier on the graph with the standard deviation o on the X axis and the expected payoff on the Y axis. Plot on the same graph all the three assets that are available. 9. (5 marks) Suppose the investor's preferences U (4,0) = x - 4o? What is his optimal portfolio? You need to show the optimal values of (a). Hint: approach this as an optimization problem, with the constraint derived in e. h. (2 marks) What asset holdings (1,2,3) form the optimal portfolio you found in g.? j. (3 marks) Explain qualitatively how would your answer to g. and h. change if assets can be also sold short. A trading platform offers three assets. It prohibits selling any asset short, that is the investor can hold zero or a positive amount of either asset. There are two possible states of the world. The payoffs of the assets are given in the matrix below assets states [B/A : 1] Suppose that the prices of the assets are (91392393) = (1,1,1) and the probabilities of the two states are (T. 1 - ) = (1/2, 1/2). The initial wealth w = 6. We further adopt the mean-variance approach. a. (2 marks) If the investor likes higher mean, but dislikes higher variance, which assets will be bold in his portfolio? Explain your answer. (1 mark) For the assets you have chosen in a) derive the mean of the portfolio. Denote with the amount of asset i the investor hokis. The mean of the portfolio (21,22,23) should be the function of 21, 22, 23), of course, it can be that 21 = 0 for some i. (2 marks) For the assets you have chosen in a) derive the standard deviation of the portfolio o (31,32.-). This should be a function of (1, 22, 23) as well. d. (1 mark) Write the investor's budget constraint. 6. (3 marks) Derive the efficiency frontier. (The frontier should express in terms of o). Hint: you should use the expressions for 4 and a from 6. and c. and the investor's budget constraint ind. 4. (2 marks) Plot the frontier on the graph with the standard deviation o on the X axis and the expected payoff on the Y axis. Plot on the same graph all the three assets that are available. 9. (5 marks) Suppose the investor's preferences U (4,0) = x - 4o? What is his optimal portfolio? You need to show the optimal values of (a). Hint: approach this as an optimization problem, with the constraint derived in e. h. (2 marks) What asset holdings (1,2,3) form the optimal portfolio you found in g.? j. (3 marks) Explain qualitatively how would your answer to g. and h. change if assets can be also sold short
