Mr Roman has a two$-$year investment project that either doubles or

halves with equal probability in each of its two years. Therefore there

are three possible outcomes for an investment I after two years: qua$-$

drupling to $4$I with probability $\frac{1}{4},$ remaining at I with probability $\frac{1}{2}$

or falling to $\frac{\text{I}}{4}$ with probability $\frac{1}{4}.$ Mr Roman's utility from his wealth

$W$ is given by $U(W)=\sqrt[\phantom{2}]{W}.$

$($a$)$ If Mr Roman invests his entire wealth in the project, $I=W,$

calculate each of the following associated with this investment:

$[20$ marks$]$

i$.$ Expected wealth Ew

ii$.$ Expected utility EU

iii. Certainty equivalent $CE$

iv$.$ Risk premium $RP.$

$($b$)$ Mr Roman wonders whether he should instead consider partial

investment $0IW.$ What is Mr Roman's optimal invest$-$

ment level $I^{**}$ in this case?

$[40$ marks$]$

$($c$)$ If Mr Roman is able to borrow at zero cost so that $IW,$ what

would be his optimal investment level?

$[10$ marks$]$

$($d$)$ Mr Thomas, another investor, would like to buy this project off

Mr Roman. If Mr Roman can choose any value of $I0,$ what is

the minimum price at which Mr Roman would sell this project?

$[10$ marks$]$

$($e$)$ How would your answers to $($b$),($c$)$ and $($d$)$ change if Mr Roman

was risk$-$neutral?

$[20$ marks$]$