Roth vs Traditional 401k – final take

Posted on November 10, 2019 by Siddharth Jain

This is my final take on Roth vs. Traditional 401k in which I extend the previous model to include two new variables:

  • a different tax rate t_2 for money withdrawn from 401k (the expectation is that you will be in a lower tax bracket during retirement) and
  • a variable e to denote the expenses in a year. These expenses need to be deducted from the amount that is invested in the ordinary non-tax sheltered account.

with these two variables, the new calculations for Roth vs. traditional are as follows:

Roth 401k

Total = Roth = (1+r)^N c + (p - c - e - f(p))(1 + r')^N

where:
f(x) is a function that computes tax on x, and
r' = r(1 - t_1) is the reduced rate of return that money in non-tax sheltered account earns
t_1 will be the marginal tax rate (bracket)

Traditional 401k

Total = Trad = (1+r)^N c (1 - t_2) + (p - c - e - f(p-c))(1 + r')^N

where:
t_2 = expected tax rate at time of withdrawal in retirement

Let’s see what we get if we subtract the two:

D = Roth - Trad = (1+r)^N c t_2 - (f(p) - f(p-c))(1 + r')^N
Now recall f is simply the function which computes tax, so f(p) - f(p-c) = t_1 c giving
D = (1+r)^N c t_2 - (1 + r')^N t_1 c
simplifying:
D = ((1+r)^N t_2 - (1 + r')^N t_1) c
For D > 0 (i.e., better to invest in Roth), we get a simple formula
(1+r)^N t_2 > (1 + r')^N t_1
This looks like a linear relationship in t_1 and t_2 but isn’t since r' is a function of t_1. The variables p,c,e have cancelled out and we see that the factors that influence whether you should choose Roth vs. Traditional 401k are:

  • your current tax bracket t_1
  • your expected tax bracket during retirement t_2
  • expected rate of return on investment r
  • time to retirement N

We can write a small function that gives us the breakeven point when both Roth and Traditional will give same return:

Let’s see what it gives:

Plot of breakeven point vs. N (# of years till retirement) with r=0.08 and two different values of t_1

Enough math. Can I get it in English please? Investing in Roth makes sense if you are young (i.e., N ↑) or in a low tax bracket (t_1 ↓). For most people, both these events happen at the same time. To use the graph above, calculate the # of years you have till retirement (x-axis) and your expected tax bracket in retirement (y-axis). That should give you a point on the graph. If that point is above the curve, you are better off investing in a Roth 401k.

Conclusion: Go Roth when you are young and switch to Traditional 401k once you cross 40-45 years of age. For most people that represents mid-point of their career. Thus, in other words, the conclusion is to split evenly between Roth and Traditional 401k. Choose Roth for first-half of your career and Traditional for the second half.

of course, again all this is just bookish exercise for fun. Code is here.

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