$($b$)$ The utility that an investor derives from wealth is described by the function $U(W)=$

$\sqrt[\phantom{2}]{W}$ and the investor's current level of wealth is $600.$

$($i$)$ The investor is facing the gamble of winning $600$ with probability $\frac{4}{5}$ and losing $300$

with probability $\frac{1}{5}.$ Compute the risk premium using the Arrow$-$Pratt and Markowitz

definitions and comment on your findings

$(25\%)$

$($ii$)$ Considering the Markowitz premium, explain whether the investor will buy insurance

that completely removes the risk for a fee of $100$ or take the gamble.

$($iii$)$ Suppose the investor accepted the gamble outlined in $($ii$)$ and lost. If the investor is

now faced with the same gamble and has the same offer of insurance as before, explain

whether the investor will buy the insurance.