Question $3[5$ points$]$: Use the below information to answer the following question.

$\backslash$begin$\{$tabular$\}\{|$c$|$c$|$c$|$c$|\}$

$\backslash$hline Investment & Expected Return $\backslash ($ E$($r$)\backslash )$ & Standard Deviation $\backslash (($s$)\backslash )$ & Total Loss Probability $\backslash (($z$)\backslash )\backslash \backslash$

$\backslash$hline $1$ & $0\mathrm{.}12$ & $0\mathrm{.}19$ & $0\mathrm{.}02\backslash \backslash$

$\backslash$hline $2$ & $0\mathrm{.}15$ & $0\mathrm{.}20$ & $0\mathrm{.}02\backslash \backslash$

$\backslash$hline $3$ & $0\mathrm{.}21$ & $0\mathrm{.}25$ & $0\mathrm{.}02\backslash \backslash$

$\backslash$hline $4$ & $0\mathrm{.}24$ & $0\mathrm{.}30$ & $0\mathrm{.}00\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

An investor's utility is represented by $\backslash ($ U$=$E$($r$)-$A s$^\{2\}-$L z $\backslash ),$ where $\backslash ($ A $\backslash )$ and $\backslash ($ L $\backslash )$ capture the investor's aversion to symmetric standard deviation and aversion to the tail risk of total loss respectively.

$1.$ Suppose $\backslash ($ L$=0\backslash )$ and the investor's utility is governed by the standard mean$-$variance function. If $\backslash ($ A$=2\backslash )$ then based on the utility function above, which investment would the investor select? How about if $\backslash ($ A$=4\backslash )?$ What is the effect of increase in $\backslash ($ A $\backslash )$ on the investors choice? Discuss.

$2.$ Now suppose as you increase $\backslash ($ A$=2\backslash )$ to $\backslash ($ A$=4\backslash ),$ you also increase $\backslash ($ L$=0\backslash )$ to $\backslash ($ L$=10\backslash ).$ How does your answer to the previous question change? Discuss.