Question $1$

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 qudratic 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?