For both problems $13$ and $16,$ assume that you manage a risky portfolio with an expected rate of return of $17\%$ and a standard deviation of $27\%.$ The risk$-$free tbill rate is $7\%.13.$ Suppose a client of yours decides to invest in your risky portfolio a proportion y of his total investment budget, so that his overall portfolio will have an expected rate of return of $15\%.$

a$)$ What is the proportion y$?$

b$)$ If your risky portfolio consists of $27\%$ stock A$,33\%$ stock B$,$ and $40\%$ stock C$,$ what are your client's investment proportions in A$,$ B$,$ C$,$ and tbills?

c$)$ What is the standard deviation of your client's portfolio?

$16.$ Your client wonders whether to shift his assets from your fund into an index fund with an expected return of $13\%$ and a standard deviation of $25\%.$

Find the maximum fee you could charge $($as a percentage of assets, deducted at the end of the year$)$ that would still leave your client at least as well off investing in your fund as in the passive one, by choosing proportions that give the two overall client portfolios the same risk level.