Consider three assets: two risky assets $($asset $1$ and asset $2)$ and the riskless asset. Asset $1$ has an expected return of $5\%$ and a volatility of $10\%.$ Asset $2$ has an expected return of $10\%$ and a volatility of $20\%.$ The riskless asset provides a return of $2\%.$

An investor has quadratic utility with a degree of relative risk aversion equal to $5$ and considers the following options:

a$)$ First, suppose that she uses the maximum utility strategy to form two portfolios. Portfolio $1$ invests in asset $1$ and the riskless asset. Portfolio $2$ invests in asset $2$ and the riskless asset. Solve for the optimal weights of the two portfolios. Compute the expected return, volatility and Sharpe ratio of the two portfolios. Which portfolio delivers the highest utility?

b$)$ Second, suppose that the investor uses the maximum expected return strategy to form two portfolios. Again, portfolio $1$ invests in asset $1$ and the riskless asset and portfolio $2$ invests in asset $2$ and the riskless asset. Using a target portfolio volatility of $5\%,$ solve for the optimal weights of the two portfolios. Compute the expected return, volatility and Sharpe ratio of the two portfolios. Which portfolio delivers the highest utility?

c$)$ Third, suppose that the investor uses the minimum volatility strategy to form two portfolios. Again, portfolio $1$ invests in asset $1$ and the riskless asset and portfolio $2$ invests in asset $2$ and the riskless asset. Using a target portfolio expected return of $4\%,$ solve for the optimal weights of the two portfolios. Compute the expected return, volatility and Sharpe ratio of the two portfolios. Which portfolio delivers the highest utility?

d$)$ Finally, fourth, suppose that the investor forms a portfolio that invests in both risky assets but not in the riskless asset. Given that you do not know the correlation between the two risky assets, compute the maximum possible volatility and the minimum possible volatility of the portfolio if the weights are $50\%-50\%.$ Repeat for weights of $25\%-75\%$ and $75\%-25\%.$