You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P $500$ stock index, offers an expected rate of return of $13\%$ with a standard deviation of $22\%.$ You manage an active portfolio with expected return $18\%$ and standard deviation $31\%.$ The risk$-$free rate is $4\%.$ Your client's degree of risk aversion is A $=2\mathrm{.}4.$

Required:

If he chose to invest in the passive portfolio, what proportion, y$,$ would he select?

What is the fee $($percentage of the investment in your fund, deducted at the end of the year$)$ that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision $($i$.$e$.,$ his choice of y$)?$