Consider the following two assets A and B$,$ with normally distributed rates of return

$([tilde(r{)}_A],[tilde(r{)}_B])N[([0\mathrm{.}06],[0\mathrm{.}11]),([0\mathrm{.}002,0\mathrm{.}001],[0\mathrm{.}001,0\mathrm{.}004])]$

You seek to maximize the utility function: $u(widetilde(w))=-e^{-12\text{t}\text{i}\text{l}\text{d}\text{e}(w)}$

$($Hint: normalize initial wealth to one.$)$

$($a$)$ What is your optimal portfolio if A and B are the only assets available?

$($b$)$ There is now, in addition to A and B$,$ a risk free asset with a rate of return of $3\%.$ What is the new optimal portfolio?

$($c$)$ Assuming that the CAPM holds in part $($b$),$ derive the equations for the capital market line

and the security market line.

$($d$)$ There is no risk free asset in the economy anymore $-$ just assets A and B$.$ What is the global minimum variance? Which portfolio achieves this?

$($e$)$ As in part $($d$),$ there is no risk free asset. The market capitalization of asset A is $1,$ and that of asset B is $2.$ Is this compatible with the zero beta CAPM?