Consider an efficient frontier that is described by the function

$E[R_P]=2\%+\sqrt[\phantom{2}]{{}_P-0\mathrm{.}05}$

The minimum variance portfolio is assumed to be given by ${}_P=5\%.$

Consider an investor that has mean$-$variance preferences given by

$E[R_P]-\frac{1}{2}\frac{{}_P^2}{T}$

where the level of risk tolerance is given by $T=5.$

$($a$)$ Find the expected return and variance of the portfolio that the investor will choose for the efficient frontier described above.

Suppose the investor can now lend and borrow at a risk$-$free rate$R_f=1\%.$

$($b$)$ Find the tangency portfolio of the efficient frontier that all investors will choose to invest in$.$

$($c$)$ Consider investors with different risk tolerance levels given by Tin$\{1,5,10\}.$ Find the optimal portfolios for these investors. How does the level of risk tolerance influence their choice?

Suppose that investors cannot borrow at all, but can lend at the risk$-$free rate of $R_f=1\%.$

$($d$)$ Does this alter the optimal portfolios for the investors that you have found in part

$($c$)$ of the question? If so$,$ find the new optimal portfolio$($s$).$