### Author Topic: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers  (Read 5030 times)

#### WorkingOnStubble

• Posts: 23
##### Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« on: October 27, 2016, 10:37:42 PM »
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.

• N - Your gross annual income
• L - The annual 401(k) limit (currently \$18k)
• rT - Your current effective tax rate after the L has been subtracted from your gross annual income. Values like 0.33 expected
• rR - Your current effective tax rate (For the Roth situation, as your tax rate is unaffected by contributions. Values like 0.33 expected
• rE - 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
• rR = 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.

$L \times (1+i)^n = 18,000 \times 1.1^n = 46,687$

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.

$X \times (1+i-id)^n$

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 rR, minus our Roth 401(k) contributions L, minus our cost of living C. So

$X = N(1-r_R) - L -C$

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

$[N(1-r_R) - L -C](1+i-id)^n + L(1+i)^n$

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 rE, so the value of the 401(k) contribution after n years will be:

$L(1-r_E)(1+i)^n$

Now, for our savings right now. We need to take the contribution from our pre-tax income (N - L), tax it at that rate (rT), and deduct our living costs. That's how much we save.

$(N-L)(1-r_T) - C$

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:

$[(N-L)(1-r_T) - C](1+i-id)^n + L(1-r_E)(1+i)^n$

Finding the Difference

Okay, 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.

$[(N-L)(1-r_T) - C](1+i-id)^n + L(1-r_E)(1+i)^n - \left \{ [N(1-r_R) - L - C](1+i-id)^n + L(1+i)^n \right \}$

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.

$L \left \{ (1+i-id)^n \Big[ r_T + \frac{N(r_R - r_T)}{L} \Big ] - (1+i)r_E \right \}$

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

$(1+i-id)^n \Big[ r_T + \frac{N(r_R - r_T)}{L} \Big ] - (1+i)r_E > 0$

If and only if

$r_E < \Big(\frac{1+i-id}{1+i} \Big)^n \Big[ r_T + \frac{N(r_R - r_T)}{L} \Big ]$

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!

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!
« Last Edit: October 27, 2016, 10:39:23 PM by WorkingOnStubble »

#### MDM

• Senior Mustachian
• Posts: 11529
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #1 on: October 27, 2016, 11:53:47 PM »
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.
....
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!
Math is good!

I didn't check yours in detail, but your conclusion differs from what others have found.  See Traditional versus Roth - Bogleheads, particularly the "Maxing out your retirement accounts" section.  In addition to The Finance Buff's spreadsheet referenced there, you can look at the '401k vs Taxable' tab in the case study spreadsheet.

In short, if your pre-tax amount is less than the tIRA maximum, then a simple "traditional is better if the withdrawal marginal tax rate is less than the contribution marginal rate and vice versa" strategy works.  Using "effective" rate is not correct.  The wiki article addresses that, but we can discuss more if it isn't clear.

If you have more pre-tax than the tIRA maximum, the tax drag occurring when you have to use "taxable plus traditional" gives a slight advantage to Roth.  In that case, you should find that Roth is preferable even with a slightly lower (vs. current marginal savings rate) withdrawal marginal tax rate.  See https://www.bogleheads.org/forum/viewtopic.php?f=10&t=140758 for a more detailed discussion.

#### MoonLiteNite

• Bristles
• Posts: 411
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #2 on: October 28, 2016, 05:13:21 AM »
I only make 40k a year. (i suck at math)
Can i get a TLDR? :D

nice post!

I made a copy of your spreed sheet and linked this forum  post, ill be sending some folks here.

Edit:
https://i.gyazo.com/ecbffb1650b25f0f4cbfaff2dc9cd60f.gif
« Last Edit: October 28, 2016, 05:22:55 AM by MoonLiteNite »

#### NoStacheOhio

• Handlebar Stache
• Posts: 2136
• Location: Cleveland
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #3 on: October 28, 2016, 06:38:22 AM »
I'm a little unclear on where you're taxing your dividends/cap gains, is that only in the taxable account?

401k distributions are taxed as ordinary income (higher than QD/LTCG).

#### ender

• Walrus Stache
• Posts: 7402
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #4 on: October 28, 2016, 07:18:30 AM »
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!

Don't you mean the opposite of this?

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.

Your effective tax rate while working has no impact on the "Roth vs traditional?" decision, except as a minor technicality that relates to AGI based phaseouts (because traditional 401k contributions are above the line deductions). Your marginal tax rate for the contributions is what matters when you contribute.

However, during withdrawal, your effective rate probably matters. But it's more the "what is the effective rate of all money you withdraw from your 401k contributions." For example, imagine  you have a \$100k pension too in retirement. Any money you take out of your 401k is now forced to be at a much higher marginal rate, such that the "effective tax rate of your IRA/401k money" becomes higher.

This is a fundamental misassumption in your analysis. The tax savings when contributing is the average marginal rate for the contributions, not anything related to your overall effective tax rate.

Likewise, because of how you are setting up your r_e at 33% it means if you withdraw \$20k, you pay \$6600 in taxes. This is clearly not correct but it is how your model is setup. This misassumption is made worse the lower your withdrawal is from your income during work (ie if you make \$100k during work but withdraw \$100k/year in retirement the problem is less bad than \$100k/working and \$40k/retirement).

I'm also not sure I think the 8% growth and 15% tax on that growth every year assumption is very valid either, if you have stock indexes that receive 2% dividends per year, you are saying that you will pay 60% of those dividends in taxes every year (0.02 - 0.08*0.15 = 0.02 - 0.012 = 0.8% actual dividend growth, which is effectively a 60% tax on your dividends). Most of that 8% return in stocks is non-realized capital gains growth while only a small percentage, which comes from dividends, is actually taxable every year. But you are taxing the entirety of the growth as if it is dividends. Or cashing out the stocks and rebuying every year, either of which are not good assumptions.

#### seattlecyclone

• Walrus Stache
• Posts: 7310
• Age: 39
• Location: Seattle, WA
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #5 on: October 28, 2016, 09:56:14 AM »
ender is right that you shouldn't consider your effective tax rate. Going in, your traditional 401(k) contributions are coming off the top of your income, and the tax savings are at your marginal rate.

Even coming out, the money is taxed at a marginal rate. This bit is harder to grasp sometimes. However you need to consider what your taxable income might be based on sources you've already earned.

Suppose you have \$250k in a traditional 401(k) and you plan for that money to double by the time you retire. You'll be taking 4% of \$500,000 each year, or \$20,000 per year. If you're single, that's already enough to put you right at the border between the 10% and 15% brackets, assuming you're taking the standard deduction and tax brackets stay where they are. You've already saved enough to fill up the 0% and 10% brackets for your whole retirement, so you should expect the next dollar you contribute to be taxed at 15% when you withdraw.

#### G. Thomas

• Posts: 78
• Age: 37
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #6 on: October 28, 2016, 10:14:10 AM »
Check out Madfientists post below.  His math shows traditional.  To maximize effectiveness convert to Roth over time.

#### MDM

• Senior Mustachian
• Posts: 11529
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #7 on: October 28, 2016, 10:28:13 AM »
Yes, if your marginal withdrawal rate is lower than your marginal contribution rate.  Which it will be for many (perhaps most) early retirees.  For those fortunate enough (expect a large pension; have a year with unusually low income; etc.) to have a marginal withdrawal rate higher than current marginal contribution rate, Roth is better.

#### Spork

• Walrus Stache
• Posts: 5742
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #8 on: October 28, 2016, 10:47:30 AM »
Yes, if your marginal withdrawal rate is lower than your marginal contribution rate.  Which it will be for many (perhaps most) early retirees.  For those fortunate enough (expect a large pension; have a year with unusually low income; etc.) to have a marginal withdrawal rate higher than current marginal contribution rate, Roth is better.

I sound like a broken record because I've mentioned this in a number of threads but:  If you expect to receive any sort of significant inheritance that is located inside an IRA... Roth is also probably better.  I don't think inheritance is something that should be "banked on"... but it's probably something worth considering for planning.

#### ender

• Walrus Stache
• Posts: 7402
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #9 on: October 28, 2016, 12:04:19 PM »
I sound like a broken record because I've mentioned this in a number of threads but:  If you expect to receive any sort of significant inheritance that is located inside an IRA... Roth is also probably better.  I don't think inheritance is something that should be "banked on"... but it's probably something worth considering for planning.

I'd say this depends primarily on your age you find out about MMM and whether you plan on an early retirement.

For someone who is 22 and planning on retiring after 15 years of work the result of this is different than someone who is 42 and plans on retiring 15 years later.

I plan on optimizing for the fastest retirement scenario - if I end up paying too much in taxes due to an inherited IRA, I guess that's a "good problem to have" (financially at least). But since most people do not know if/when they will receive these, my recommendation is to always plan on not having them. Particularly the younger you begin your MMM journey.

In other words, if I end up paying more in taxes because either my traditional IRA money grew way too fast or I got an inherited IRA with RMDs, I have "won" the early retirement game.

That is a situation I am much less worried about than choosing to contribute to Roth, trying to predictively optimize my total tax burden but adding a fairly significant risk to delay my ability to FIRE.

It takes looking at your personal situation to figure all this out. In our situation, it's seemingly likely that if we receive any meaningful inheritance it will happen post-FI (or very close to that happening).

Yes, if your marginal withdrawal rate is lower than your marginal contribution rate.  Which it will be for many (perhaps most) early retirees.  For those fortunate enough (expect a large pension; have a year with unusually low income; etc.) to have a marginal withdrawal rate higher than current marginal contribution rate, Roth is better.

The way I conceptualize this is that if it turns out that Roth would have been a better choice during my working years at 20-30% marginal rates, I probably won the ER game way and am ending up with way too much money.

This is far better than the opposite - choosing Roth over traditional and ending up working longer.

#### MDM

• Senior Mustachian
• Posts: 11529
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #10 on: October 28, 2016, 12:21:30 PM »
I plan on optimizing for the fastest retirement scenario - if I end up paying too much in taxes due to an inherited IRA, I guess that's a "good problem to have" (financially at least).
...
In other words, if I end up paying more in taxes because either my traditional IRA money grew way too fast or I got an inherited IRA with RMDs, I have "won" the early retirement game.
...
if it turns out that Roth would have been a better choice during my working years at 20-30% marginal rates, I probably won the ER game way and am ending up with way too much money.
This is far better than the opposite - choosing Roth over traditional and ending up working longer.

It takes looking at your personal situation to figure all this out.

#### TomTX

• Walrus Stache
• Posts: 5345
• Location: Texas
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #11 on: October 28, 2016, 04:58:00 PM »
Did OP remember that for an equal comparison* you don't compare \$18k Traditional versus \$18k Roth? I may have lost it in the number blur.

On the Traditional side, you've saved the marginal  Income Tax on the money put into the Traditional.

So, 25% bracket means you save \$4500 in income tax. In some states, you save more due to state income tax.

Therefore, you compare \$18k Traditional + \$4,500 taxable account (or TIRA)  VS \$18,000 in a Roth.

#### MidWestLove

• Bristles
• Posts: 316
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #12 on: October 30, 2016, 10:40:18 AM »
as others mentioned, if your model is off  the makes math irrelevant - standard garbage in garbage out situation. your savings on taxes are coming off the top and are last dollar tax rates while withdrawals are much more likely to be in lower tax brackets at the time of use which you can also control to certain amount.  the second bad assumption is that you assume Roth would be tax free which is unknown, even if government would not break the promise directly (which they may do), they can shown 100% of almost always breaking it indirectly by subjecting your roth withdrawals to many other 'means test' for purposes of subsidies or eligibility. Of cause this would all be sold to public under the guise of fairness, and paying their fair share.

so if you are not eligible for 401k, not getting a reduction for TIRA, by all means contribute to Roth. also, if you are in low enough tax bracket right now, there may be some consideration to Roth 401k. However, in all other situations, especially if you are in higher tax bracket, giving up a tool that reduces your AGI  (so you can play with and capture other tax benefits and credits) and gives you massive tangible benefit is likely less than optimal (aka stupid).

#### MidWestLove

• Bristles
• Posts: 316
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #13 on: October 30, 2016, 10:54:56 AM »
for taxes - I am consumed on how you arrived to these numbers

"
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!
"
putting two 25 year olds (age is irrelevant for this other than less than 50 at which point you get additional catch up contributions space)  at 100k wages your total tax in CA is 22k using standard deduction ( so this is worst case scenario assuming no other write offs, no dependents, no real estate taxes, nothing). 3.8k to CA and 19K to federal. How did you get the numbers above?

#### Indexer

• Handlebar Stache
• Posts: 1463
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #14 on: October 31, 2016, 09:29:27 PM »
Normally the rule is that if you are in a higher tax rate now than retirement the trad is better. Roth is the opposite rule.

This all changes if you are maxing out the account. If you compare 4k in a Roth to 4k+tax savings(lets say 5k total) in a Trad you get a level playing field. 18k in a trad is not the same thing as 18k in a Roth. You can't just put more in the Trad. That money goes into a taxable account which over the long term isn't going to be taxed as favorably as the Roth.

When I have run the numbers for max contributions the Roth wins out even when the tax rate drops, lets say from 33 to 25% at retirement. The longer the time horizon and the higher the rate of return the greater the impact.

I never created complex formulas like the ones above. I would just run a TVM calculation.

#### MDM

• Senior Mustachian
• Posts: 11529
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #15 on: October 31, 2016, 10:09:31 PM »
When I have run the numbers for max contributions the Roth wins out even when the tax rate drops....
Yes.  The answer to "by how much?" indeed depends on a variety of things.  See this post above and links therein.

#### shuffler

• Pencil Stache
• Posts: 579
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #16 on: October 31, 2016, 10:13:30 PM »
I was laughing at the idea of bothering with the Roth 401(k) that's also offered by the company.
I contribute pre-tax, but I've got a bunch of money in my Roth 401(k) because of the "mega" backdoor Roth.
It could be that your company is savvy to after-tax rollovers, and they offer the Roth account to receive the rollovers.
(That's what mine does.  I can't take the after-tax contributed money out, but I can do a "in-service" rollover to the Roth account.)

#### Indexer

• Handlebar Stache
• Posts: 1463
##### Re: Roth 401(k) vs Traditional 401(k), I'm getting some surprising numbers
« Reply #17 on: November 01, 2016, 05:50:17 AM »
Something else to consider if you want to retire before the age of 59 1/2 is the roth conversion ladder. If you retire with a lot of money in pre-tax accounts you can convert it to Roth over time. If it is all in Roth from the start you can only access the contributions, but not the earnings.