Hey all,

I recently got involved in my employer's Traditional 401(k) program, and was laughing at the idea of bothering with the Roth 401(k) that's also offered by the company. One of my coworkers challenged me on it though, saying how it may be more worth it since you can contribute $18k of post-tax money which is shielded from gains and dividends being taxed, rather than the Traditional which would have everything taxed at the end. I started running through the equations, and finally came out to an answer I was satisfied with. For a while I was confused, then slowly began to believe my results that the Roth was better a hefty majority of the time. In typing out this post, though, I realised I hadn't considered the difference in effective tax rates when you take the 401(k) contribution limit from your pre-tax income. Regardless, I've put so much work into writing the post now, I hope some of you enjoy it, and I'd love to improve my model, or be told about any errors that might sway me the other way!

I apologise for all the maths in this post, but it's how I came to my results, and I'd love if someone could correct me on any errors in my reasoning.

Let's start with some definitions:

**N** - Your gross annual income**L** - The annual 401(k) limit (currently $18k)**r**_{T} - Your current effective tax rate after the L has been subtracted from your gross annual income. Values like 0.33 expected**r**_{R} - Your current effective tax rate (For the Roth situation, as your tax rate is unaffected by contributions. Values like 0.33 expected**r**_{E} - Your expected tax rate at the end of the period of time we're considering. Values like 0.33 expected**n** - The number of years we're considering in our calculation**C** - Your annual spending, post-tax money NOT saved**i** - The annual return on investment from the funds you have invested in. This will have values like 0.08 for an 8% return**d** - The dividend income taxation rate that must be paid each year. Likely to be 15% for most tax bands, so likely value is 0.15

Whew, I know that was a lot, but each will be used slowly.

Let's set up a few of the variables here:

- N = 100,000. You make $100,000 a year
- L = 18,000. You contribute the $18k max
- r
_{R} = 0.1818. On federal income tax alone, $100k will have an effective tax rate of 18.18% - n = 10. We'll consider this over a 10 year time frame
- i = 0.1. It doesn't matter right now what value we make our investment income, so let's just go with it being 10%
- d = 0.15. At this income band, we'll be taxed 15% on dividends.

So let's assume that after tax and putting the $18k into our Roth 401(k), we spend everything else. Then the value of our investment grows at 10% a year for 10 years.

All fine and dandy. Now imagine we don't spend everything we make. What happens to the rest? A taxable savings account, of course! Let's say we invest in the same funds, and so it grows exactly the same, but now we have to pay taxes on the gains. Let's say the amount we save is X (just for now), and we let it grow for the 10 years too.

This gives us the amount we get after n years while paying d taxes on Xi amount of capital. Now, how much is X? Well, it's whatever we don't spend of course. That is our income N, which we're taxed on at rate r

_{R}, minus our Roth 401(k) contributions L, minus our cost of living C. So

So now we can put all this together and get the total value of our Roth 401(k) contribution after n years:

Next section: Traditional 401(k). A little more complicated now. Let's start on the value of the 401(k) contribution, then do the savings after.

We're gonna be taxed on our withdrawal at rate r

_{E}, so the value of the 401(k) contribution after n years will be:

Now, for our savings right now. We need to take the contribution from our pre-tax income (N - L), tax it at that rate (r

_{T}), and deduct our living costs. That's how much we save.

This grows at the taxable rate as before, so we can use that from the Roth section. Altogether, our Traditional 401(k) route would give the value:

Finding the DifferenceOkay, so we know the two values of each routes. But which is better? No idea, too many variables. So, let's take one away from the other. I wanted to prove that almost always, the Traditional was better than the Roth, so I'm gonna take the Roth away and hope it's positive most of the time.

Yes, that one looks pretty insane, but we can simplify it down pretty reasonably. I'm gonna skip the steps because LaTeX is annoying enough as it is, but please check these if you think I've simplified it down wrong.

Much simpler, right? Maybe just me... So everything actually looks very cute here, except the bunch in the middle that still cares about N and L. This is because you don't have the same effective tax rate with Roth and Traditional. Now, who cares what the actual dollar difference is? We want to know which is better! So if Traditional 401(k) is better, this value will be positive (Because we took the Roth value away from Traditional). So let's divide by L and check out how this inequality looks

If and only if

Alright, I know that was way too much, and I've been writing this so long I know I've gone too far, but let's plug some numbers in now we've come this far.

We make $100k, we live in San Francisco, we max out our 401(k) at $18k. We gain 10% per year for 10 years, and pay 15% taxes on our dividends. Our overall income tax burden is $32,209 (due to California's high state tax) with no Traditional 401(k) contributions, or 0.32209. If we make contributions, our income tax burden is $26,060, or 0.2606. Plugging all of this in, in order for it to be worth it to invest in the Traditional 401(k) instead of the Roth 401(k), we would need our tax bracket in 10 years to be...

**0.40386**, or a whopping 40.386% for it to be worth it!

I've made a

Google Spreadsheet if anyone is interested in plugging in their numbers.

As I said, when I started writing this I had come to the shocking conclusion that Roth was better than Traditional most of the time, but I managed to prove myself wrong again, and so I wanted to share the results with you not just because of my altruism, but because I spent so long typing this out that to delete it all because the results weren't as shocking didn't seem fair. I hope if you ever get into the argument of which is more valuable, this post can help out!