Suppose that the U$.$S$.$ market is your risky portfolio.

$\backslash$table$[[,\backslash$table$[[$Average Annual$],[$Returns$]],$U$.$S$.$ Equity Market,$],[,$U$.$S$.,1-$Month T$-,$Excess,Standard,Sharpe$],[$Period$,$equity,Bills,return,Deviation,Ratio$],[1927-2021,12\mathrm{.}17,3\mathrm{.}30,8\mathrm{.}87,20\mathrm{.}25,0\mathrm{.}44],[1927-1950,10\mathrm{.}26,0\mathrm{.}93,9\mathrm{.}33,26\mathrm{.}57,0\mathrm{.}35],[1951-1974,10\mathrm{.}21,3\mathrm{.}59,6\mathrm{.}62,20\mathrm{.}32,0\mathrm{.}33],[1975-1998,17\mathrm{.}97,6\mathrm{.}98,10\mathrm{.}99,14\mathrm{.}40,0\mathrm{.}76],[1999-2021,10\mathrm{.}16,1\mathrm{.}66,8\mathrm{.}50,18\mathrm{.}85,0\mathrm{.}45]]$

Required:

a$.$ If your risk$-$aversion coefficient is $A=5\mathrm{.}2$ and you believe that the entire $1927-2021$ period is representative of future expected performance, what fraction of your portfolio should be allocated to T$-$bills and what fraction to equity? Assume your utility function is $U=E(r)-0\mathrm{.}5A{}^2$

b$.$ What if you believe that the $1975-1998$ period is representative?