$4.$ You are evaluating two investment alternatives. One is a passive market portfolio with an expected return of $10\%$ and a standard deviation of $16\%.$ The other is a fund that is actively managed by your broker. This fund has an expected return of $15\%$ and a standard deviation of $20\%.$ The risk$-$free rate is currently $7\%.$ Answer the questions below based on this information.

a$.$ Calculate the Sharpe ratio of the actively managed fund.

b$.$ What is the slope of the capital market line $($CML$)?$ Does the actively managed fund fall above or below the CML$?$

c$.$ If you have a quadratic utility function and you feel the two funds are equally good. What is your coefficient of risk aversion?

d$.$ If you decide to invest in the risk$-$free asset and the market portfolio, what is the optimal weight in the risk$-$free asset? Use the coefficient of risk aversion calculated in $($c$).$