I need to get this problem set solved with all the results. It is investment and portfolio management.
1. Suppose that the economy is described by three growth states: strong, moderate, and weak. The probabilities of strong, moderate, and weak growth states are 0.3, 0.5 and 0.2, respectively. Assume that the risk-free rate is 2%. (a) Assume that you can buy or short three stocks: A, B, and C, with the following return distribution: Strong GrowthModerate GrowthWeak Growth A30%10%−10% B11.8%4.8%−2.2% C−5%0%5% Is there an arbitrage opportunity? Explain your answer, and if the answer is “yes”, describe in detail how you would take advantage of it. (b) Consider a different distribution of returns: Strong GrowthModerate GrowthWeak Growth A30%10%−10% B16.3%−3.9%5.3% C5.5%−2.7%18.5% Is there an arbitrage opportunity? Explain your answer, and if the answer is “yes”, describe in detail how you would take advantage of it. 2. Consider a two-factor model and assume that the factors, F1 and F2, are independent. The factors have the same risk premium of 5%, and volatilities of σF1 = 20% and σF2 = 10%. Consider two securities, A and B, that have the following dynamics: Security β1 β2 σєi A 1 1 5% B 0.5 2 10% where σєi denotes volatility of idiosyncratic component of asset i = {A, B}. (a) Assuming that both securities are fairly priced, what are their risk premia? (b) Compute volatility of each security. (c) What is the correlation between A and B returns? 1 3. Suppose that assets are priced according to the Fama-French 3-factor model. Consider three small firms: A, B and C. Assume that their risk characteristics are the same but otherwise the three firms are unrelated. In particular, Factor-betasABC βM1/21/21/2 βHML−1/2−1/2−1/2 βSMB111 Risk premia on the market, HML and SMB factors are 6%, 7% and 5% respectively. (a) Suppose that the three firms decide to merge and each will be operated as an independent unit of the merged company. What are risk characteristics of the merged company and its risk premium? (b) Are the dynamics of the merged company consistent with the prediction of the FF model? 4. Consider an equally weighted portfolio of 50 securities in the APT framework. Suppose that the average idiosyncratic variance across securities is 0.04. What is the non-systematic standard deviation of the portfolio?