Problem $2$: The following table shows data on the returns over five $1-$year periods for six mutual funds. A firm's portfolio managers will assume that one of these scenarios will accurately reflect the investing climate over the next $12$ months. The probabilities of each of the scenarios occurring are $\backslash (0\mathrm{.}1,0\mathrm{.}3,0\mathrm{.}1,0\mathrm{.}1\backslash ),$ and $0\mathrm{.}4$ for years $1$ to $5,$ respectively. $\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|$c$|$c$|\}\backslash$hline Mutual Fund & Scenario $1$ & Scenario $2$ & Scenario $3$ & Scenario $4$ & Scenario $5\backslash \backslash \backslash$hline Large$-$Cap Stock & $35\mathrm{.}3$ & $20\mathrm{.}0$ & $28\mathrm{.}3$ & $10\mathrm{.}4$ & $-9\mathrm{.}3\backslash \backslash \backslash$hline Mid$-$Cap Stock & $32\mathrm{.}3$ & $23\mathrm{.}2$ & $-0\mathrm{.}9$ & $49\mathrm{.}3$ & $-22\mathrm{.}8\backslash \backslash \backslash$hline Small$-$Cap Stock & $20\mathrm{.}8$ & $22\mathrm{.}5$ & $6\mathrm{.}0$ & $33\mathrm{.}3$ & $6\mathrm{.}1\backslash \backslash \backslash$hline Energy$/$Resources Sector & $25\mathrm{.}3$ & $33\mathrm{.}9$ & $-20\mathrm{.}5$ & $20\mathrm{.}9$ & $-2\mathrm{.}5\backslash \backslash \backslash$hline Health Sector & $49\mathrm{.}1$ & $5\mathrm{.}5$ & $29\mathrm{.}7$ & $77\mathrm{.}7$ & $-24\mathrm{.}9\backslash \backslash \backslash$hline Technology Sector & $46\mathrm{.}2$ & $21\mathrm{.}7$ & $45\mathrm{.}7$ & $93\mathrm{.}1$ & $-20\mathrm{.}1\backslash \backslash \backslash$hline Real Estate Sector & $20\mathrm{.}5$ & $44\mathrm{.}0$ & $-21\mathrm{.}1$ & $2\mathrm{.}6$ & $5\mathrm{.}1\backslash \backslash \backslash$hline $\backslash$end$\{$tabular$\}$ a$.$ Develop a portfolio model for investors who are willing to risk a portfolio with a return no lower than $\backslash (2\backslash \%\backslash ).$ b$.$ Solve the model in part $($a$)$ and recommend a portfolio allocation for the investor with this risk tolerance. c$.$ Modify the portfolio model in part $($a$)$ and solve it to develop a portfolio for an investor with a risk tolerance of $0\backslash \%.$ d$.$ Is the expected return higher for investors following the portfolio recommendations in part $($c$)$ as compared to the returns for the portfolio in part $($b$)?$ If so$,$ do you believe the returns are enough higher to justify investing in that portfolio?