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- a different tax rate for money withdrawn from 401k (the expectation is that you will be in a lower tax bracket during retirement) and
- a variable 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:
is a function that computes tax on , and
is the reduced rate of return that money in non-tax sheltered account earns
will be the marginal tax rate (bracket)
= expected tax rate at time of withdrawal in retirement
Let’s see what we get if we subtract the two:
Now recall is simply the function which computes tax, so giving
For (i.e., better to invest in Roth), we get a simple formula
This looks like a linear relationship in and but isn’t since is a function of . The variables have cancelled out and we see that the factors that influence whether you should choose Roth vs. Traditional 401k are:
- your current tax bracket
- your expected tax bracket during retirement
- expected rate of return on investment
- time to retirement
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 and two different values of
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 ( ↓). 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.