Exercise 1. Consider an investor with log utility deciding how much to invest in one risky asset and one risk free asset. The risky asset will have an excess return of y > 0 with probability p, and an excess return of -y with probability (1-p). Compute the optimal fraction of wealth to hold in the risky asset. What is the probability of bankruptcy? Exercise 2. There are many assets in an economy. Among them are two stocks, stock A and stock B. Suppose that the expected returns are 4% and 7% respectively. The standard deviations are 12% and 9% respectively. The correlation between the two is-1 Suppose that there is a risk free asset with return 2 . How much would an investor buy of each asset? What would their return be? Risk? Sharpe ratio? Exercise 3. There exists two risky funds and a risk free asset. The first fund in mainly in stocks, and the second in bonds. The stock fund has an expected return of 10% and variance of 25%. The bond fund has an expected return of 6% and a variance of 15%. The correlation between the stock and bond funds is 0.1. The risk free rate is 4%. The agent is a mean-variance optimizer with coefficient A-15. What is the optimal fraction of their wealth to invest in each fund and the risk free asset? What is the Sharpe ratio of their portfolio? Exercise 4. There exists only two risky assets and a risk free rate r. Compute the portfolio with the highest Sharpe ratio. Exercise 1. Consider an investor with log utility deciding how much to invest in one risky asset and one risk free asset. The risky asset will have an excess return of y > 0 with probability p, and an excess return of -y with probability (1-p). Compute the optimal fraction of wealth to hold in the risky asset. What is the probability of bankruptcy? Exercise 2. There are many assets in an economy. Among them are two stocks, stock A and stock B. Suppose that the expected returns are 4% and 7% respectively. The standard deviations are 12% and 9% respectively. The correlation between the two is-1 Suppose that there is a risk free asset with return 2 . How much would an investor buy of each asset? What would their return be? Risk? Sharpe ratio? Exercise 3. There exists two risky funds and a risk free asset. The first fund in mainly in stocks, and the second in bonds. The stock fund has an expected return of 10% and variance of 25%. The bond fund has an expected return of 6% and a variance of 15%. The correlation between the stock and bond funds is 0.1. The risk free rate is 4%. The agent is a mean-variance optimizer with coefficient A-15. What is the optimal fraction of their wealth to invest in each fund and the risk free asset? What is the Sharpe ratio of their portfolio? Exercise 4. There exists only two risky assets and a risk free rate r. Compute the portfolio with the highest Sharpe ratio
