$2\mathrm{.}5$ Consider the mean$-$variance analysis covered in this chapter where there are $n$ risky assets whose returns are jointly normally distributed. Assume that investors differ with regard to their $($concave$)$ utility functions and their initial wealths. Also assume that investors can lend at the risk$-$free rate, $(\frac{?}{\text{b}\text{a}\text{r}}(R{)}_p,{}_p)R_f,$ but investors are restricted from risk$-$free borrowing; that $is,no$ risk$-$free borrowing $is$ permitted.

$a.$ Given this risk$-$free borrowing restriction, graphically show the efficient frontier for these investors $in$ expected portfolio return$-$standard deviation space $(\frac{?}{\text{b}\text{a}\text{r}}(R{)}_p,{}_p).$

$b.$ Explain why only three portfolios are needed $to$ construct this efficient frontier, and locate these three portfolios $on$ your graph. $(Note$ that these portfolios may not $be$ unique.$)$

$c.At$ least one $of$ these portfolios will sometimes need $tobe$ sold short $to$ generate the entire efficient frontier. Which portfolio$(s)isit(labeliton$ the graph$)$ and $in$ what range$(s)of$ the efficient frontier will $itbe$ sold short? Explain.