Consider historical data showing that the average annual return on the $S$&$P500$ portfolio over the past $80$ years have averaged roughly $8\mathrm{.}5\%$ more than the T $-$bill return and that the $S$&$P500$ standard deviation has been about $20\%$ per year. Assume these values are representative of investors' expectations for future performance and that the current T$-$bill rate is $5\%$

Calculate the expected return and variance of portfolios invested in T$-$bills and the S&P $500$ index with weights as follows and also calculate the utility levels for each of the portfolio for an investor with $A=3.$ Assume the utility function is $U=E(r)-0\mathrm{.}5A{}^2$

What do you conclude $($i$.$e$.,$ which portfolio gives the investor highest utility$)$

$\backslash$table$[[$W$($bills$),W($index$),{}^2$ rondolio $,U=E(r)-0\mathrm{.}5A{}^2,],[0,1\mathrm{.}0,],[0\mathrm{.}2,0\mathrm{.}8,],[0\mathrm{.}4,0\mathrm{.}6,],[0\mathrm{.}6,0\mathrm{.}4,],[0\mathrm{.}8,0\mathrm{.}2,],[1\mathrm{.}0,0,]]$