Assume that an investment in the S&P$500$ index gives an average return of $11\%.$ The short$-$term US government bond provides a risk$-$free return of $3\mathrm{.}5\%.$

a$.$ If you aim to earn an expected return of $8\mathrm{.}5\%,$ how would you construct a portfolio from these two assets?..

b$.$ If you aim to construct the $2-$asset portfolio with a beta of $0\mathrm{.}5,$ how much you would invest in the S&P $500$ index and in the risk$-$free bond?

c$.$ Calculate the risk$-$free premium in $($a$)$ and $($b$)$ and show that they are proportional to their betas.