Suppose the average return of the S&P $500$ over the last $50$ years has been $8\mathrm{.}9\%$ with a standard

deviation of $21\%$ per year. Also assume the risk$-$free rate is $4\%.$

a$.$ Find the optimal weight of the risky asset in a complete portfolio for an individual with a

risk aversion coefficient of $2\mathrm{.}5.$ Additionally, calculate the return, standard deviation and

utility for the following portfolios.

Weight ${?}_{\text{I}\text{n}\text{d}\text{e}\text{x}}=1$

Weight ${?}_{\text{I}\text{n}\text{d}\text{e}\text{x}}=\mathrm{.}75$

Weight ${?}_{\text{I}\text{n}\text{d}\text{e}\text{x}}=\mathrm{.}35$

Weight ${?}_{\text{I}\text{n}\text{d}\text{e}\text{x}}=0$