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All the arguments in favour putting money in the Roth 401K emphasise that the withdrawals are tax free. So if you compare putting the same amount of money during your working career in a trad 401K vs a Roth 401K, at retirement time, the latter enables you to withdraw more money. But I don't see them making any mention of the additional take home pay that you have with putting the same amount of money in a traditional 401K (pre tax). Theoretically if I could invest that addnl take home pay in a Roth IRA, say, at retirement time wouldn't the combination of trad 401K plus the Roth IRA exceed what I would get from just the Roth 401K?

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Only if your marginal tax rate ("tax bracket") at retirement is less than your tax rate when you contribute. They are equivalent if the tax rates are the same. Here's the math:

Tc = marginal tax rate at time of contribution
Tr = marginal tax rate at retirement
R  = annual return on investment (same in either scenario)
C  = after-tax contribution amount

* For both scenarios, contributions are made exactly 1 year before they are withdrawn (ignoring the 5-year Roth rule just to make the math easier)

Scenario 1: Invest in 401(k) and tax savings in Roth:

                     401(k)     + Roth
                     ----------   -------
Initial investment:  C          + Tc*C
After 1 year:        R*C*(1-Tr) + R*Tc*C        = R*C*(1-Tr+Tc) 

Scenario 2: Invest everything in Roth:

                     401(k)           + Roth
                     ----------------   -------
Initial investment:  0                + C
After 1 year:        0                + R*C     = R*C

So if Tc = Tr, the two are equivalent. If Tr > Tc, then the Roth is a better choice since you're paying a higher tax rate when withdrawing the 401(k) then you caved when contributing (and vice-versa).

If you able to max out your contribution, then the Roth becomes more attractive, because you are able to effectively contribute more after tax (20k in a 401(k) if worth less after-tax than 20k in a Roth 401(k)). But if you max out the 401(k) and invest the tax savings in a Roth (assuming you are eligible to do so), then they become equivalent again (modulo differences in tax rates).

There are other differences, like the required distribution on 401(k)s, that may influence one's decision as well.

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  • OK, the algebra definitely makes it easier to follow along- I would expect Tr to be < Tc under most circumstances? – ChuckLeviton Dec 3 '20 at 23:28
  • @ChuckLeviton probably, but remember that disbursements from a 401(k) count as income, and you never know what tax rates will do going forward. – D Stanley Dec 4 '20 at 13:50

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