Consider a portfolio choice problem with a risk$-$free asset with return r$\_$f and two risky assets, the $$rst with mean return $\backslash$mu $\_1=0\mathrm{.}12$ and standard deviation $\backslash$sigma $\_1=0\mathrm{.}4$ and the second with mean $\backslash$mu $\_2=0\mathrm{.}08$ and standard deviation $\backslash$sigma $\_2=0\mathrm{.}3,$ with correlation $\backslash$rho $\_12=0.$ For any stock portfolio let $\backslash$lambda denote the proportion invested in stock $1.$

$($a$)$ Find the weight $\backslash$lambda $$ that minimizes portfolio standard deviation $\backslash$sigma $\_$p$.$

$($b$)$ Consider the tangency portfolio and let $\backslash$lambda $^*$ denote the weight it places on stock $1.$ Find the condition that de$$nes this value, but do not solve for it$,$ and explain how it would compare to $\backslash$lambda $.$

$($c$)$ Now consider varying the risk$-$free rate r$\_$f$.$ Again, without solving anything, explain how you would expect $\backslash$lambda $^*$ to vary as r$\_$f increases.

$($d$)$ Show how the slope of the tangent line changes with r$\_$f$.$ Recall a useful theorem that allows you to do this without ever actually solving for $\backslash$lambda $^*.$

$($e$)$ Suppose instead that $\backslash$rho $\_12=-1$ so that the stocks always move against each other. Find the weight $\backslash$lambda $\_$f that yields a risk$-$free portfolio and the expected return $\backslash$mu $\_$f to this portfolio.