Question

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 low-cost 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 6.9%. What is the answer to question (b) on an after-tax basis (i.e., how much do you have to contribute every month 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 monthly
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).

Answer #1

a. Interest rate = 10%, monthly rate = 10%/12 = 0.10/12

Number of years = 65-20 = 45 years = 4*12 = 540 months

Goal = FV = 2,000,000

The monthly savings needed if emp0loyers offers no match
=PMT(rate,nper,pv,fv) in excel =PMT(0.10/12,540,0,2000000) =
**$190.79**

b. If employer offeres a 10% match. Then monthly savings needed = 190.63/1.05 = 181.70

Monthly savings needed with 10% match by emplyer =
**$181.70**

c. Tax savings are 24+6.9 = 30.9%. So on the comntribution of 181.70, you save a 30.9% tax. So. tax savings = 181.70*0.309 = $ 56.15

So, monthly contribution taking into account tax savings and
employer match = 181.70-56.15 = **$125.55**

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