## Roth vs Traditional 401k – final take

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• a different tax rate $t_2$ for money withdrawn from 401k (the expectation is that you will be in a lower tax bracket during retirement) and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)

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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