$(20$ points$)$ Congratulations! Today is your ${20}^{th}$ birthday, but you are starting with nothing in the bank. You just started working full$-$time, earning $$50,000$ per year. Your goal is to have $$2$ million by your ${65}^{th}$ birthday $($i$.$e$.,45$ years from today$).$ Your employer offers a $401(k)$ plan $($contributions by you are tax deductible, growth is tax deferred$),$ and within that plan you choose to invest in an extreme lowcost S&P $500$ index mutual fund $($like ones offered by Schwab, Fidelity, Vanguard, etc.$).$ The long$-$term expected return on the S&P $500$ index mutual fund is $10\%$ per year. Your employer pays you monthly.

$($a$)$ Ignoring taxes, if the employer offers no match on your contributions, how much would you need to save every month to reach your goal?

$($b$)$ Ignoring taxes, if the employer offers a match of $5\%$ on your contributions, how much would you need to save every month on top of your match to reach your goal?

$($c$)$ Assume your Federal marginal tax rate is $24\%$ and State marginal tax rate is $6\mathrm{.}9\%.$ What is the answer to question $($b$)$ on an after$-$tax basis $($i$.$e$.,$ how much do you have to contribute after the employer match and net of tax savings$)?$

$($d$)$ All of the above amounts are nominal. If your inflation expectation is $2\%$ per year, how much would you have to save after employer match, net of tax savings, and after accounting for inflation, to achieve a long$-$term goal of $$2$ million in today's $($real$)$ dollars by your ${65}^{th}$ birthday?

$($e$)$ Now you are retired and living off your more than $$2$ million investment, still invested in the S&P $500$ mutual fund. Will your return before and after retirement likely be higher or lower than that of the S&P $500,$ and why? $($Assume effectively zero management expense fees $($you can now do this with one of Fidelity's mutual funds$)$ with this extreme low$-$cost indexed mutual fund and ignore taxes$).$