Wiki: Tax Effects of Maximizing Roth IRA Contributions

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
Post Reply
User avatar
Topic Author
FiveK
Posts: 8043
Joined: Sun Mar 16, 2014 2:43 pm

Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by FiveK » Sat Jun 07, 2014 2:03 am

[Moved into a stand-alone thread from Suggestions for the Wiki, see below --admin LadyGeek]

In the discussions of Traditional vs. Roth IRA, all the examples (that I found) assume a person starts with no more than the maximum contribution. E.g., start with $5000 and either invest it all in a tIRA and pay tax later, or pay taxes now and invest the remainder in a Roth tax free. Then, if the marginal tax rate doesn't change

G = Gross starting amount, $
i = Annual investment return, fraction
n = Number of years
T = Marginal tax rate, fraction
Roth = G * (1 - T) * (1 + i)^n
tIRA = G * (1 + i)^n * (1 - T)
...and the results are identical.

So far so good and that horse has been beaten to death. But...what if...the comparison is for a maximum Roth contribution? In that case (both options starting with equal amount "G"):

a = Fraction of marginal tax rate applied to investment returns in taxable accounts
Roth = G * (1 - T) * (1 + i)^n Same as above, with G * (1 - T) = IRS maximum
tIRA = (G - G*T) * (1 + i)^n * (1 - T) Same as above, except starting with G * (1 - T) to observe IRS rule
+ GT * (1 - T) * (1 + i*(1-a*T))^n Amount G*T is initially taxed, then annual returns are also taxed at rate a*T.

Dividing tIRA/Roth we get
tIRA/Roth = 1 - T * (1 - x), where x = (1 + i - i*a*T)^n / (1 + i)^n
If a = 0 (so x = 1) or T = 0 then tIRA = Roth. But if a > 0 and T > 0 then x < 1 and Roth > tIRA.

One gets a similar result if, instead of annual taxation of returns, a capital gains tax is applied at the end of the taxable account growth. In that case, tIRA/Roth = 1 - a*T^2*(1-1/(1+i)^n).
Again if a = 0 or T = 0 then tIRA = Roth. But if a > 0 and T > 0 then Roth > tIRA.

In other words, when the marginal tax rate does not change, and one has enough income to reach the maximum IRA contribution, the Roth is better than the traditional. Not what I expected but that is what the math seems to say.

Has this been covered elsewhere? Is the algebra wrong above? Something else wrong with the analysis?
Last edited by FiveK on Sun Jun 08, 2014 12:26 am, edited 1 time in total.

bpp
Posts: 2017
Joined: Mon Feb 26, 2007 12:35 pm
Location: Japan

Wiki Suggestion: Tax effects, maximizing Roth IRA contributions

Post by bpp » Sat Jun 07, 2014 7:03 am

Looks right to me, FiveK. You're pointing out that the proper comparison with a fully-funded Roth is a fully-funded TIRA plus some taxable investments so that the same gross pre-tax income is invested in either case. In this case, the taxable investments will undergo yearly dividend taxation, plus initial earned income taxation and final capital gains taxation, so will have a lower return than the singly-taxed Roth or TIRA contributions do.

Not sure if this has been covered before.

User avatar
LadyGeek
Site Admin
Posts: 58765
Joined: Sat Dec 20, 2008 5:34 pm
Location: Philadelphia
Contact:

Re: Suggestions for the Wiki

Post by LadyGeek » Sun Jun 08, 2014 11:13 am

FiveK is proposing an addition to the wiki - more discussion is needed. If the suggestion is appropriate for the wiki, where should it go?
Wiki To some, the glass is half full. To others, the glass is half empty. To an engineer, it's twice the size it needs to be.

stlutz
Posts: 5508
Joined: Fri Jan 02, 2009 1:08 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by stlutz » Sun Jun 08, 2014 11:24 am

Would probably go in the tax management area.

However, this type of discussion really all revolves around assumptions about future tax policy. I think the most helpful point is that the assumption that Roth IRAs are far and away better than deductible 401K/IRA contributions isn't really true, assuming tax policy in the future is at least somewhat similar to now.

User avatar
Doc
Posts: 9304
Joined: Sat Feb 24, 2007 1:10 pm
Location: Two left turns from Larry

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Doc » Sun Jun 08, 2014 11:31 am

FiveK wrote:In other words, when the marginal tax rate does not change, and one has enough income to reach the maximum IRA contribution, the Roth is better than the traditional. Not what I expected but that is what the math seems to say.
Not what I expected? Uh, if you invest more you often have more at the end. It even works in a taxable account. :wink:

We all get so involved with the nuances of tIRAs, ROTHs, tax changes etc that we often forget the basic premises in investing. But it's good to see a "newbie" that understands the math. :D
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

User avatar
Topic Author
FiveK
Posts: 8043
Joined: Sun Mar 16, 2014 2:43 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by FiveK » Sun Jun 08, 2014 12:47 pm

http://www.bogleheads.org/wiki/Traditional_versus_Roth seems the most appropriate place.

Under "Taxes" in that article is this (emphasis added): "For example, if your marginal tax rate is 25%, you can contribute $3000 to a Roth account, or $4000 to a traditional account for the same $3000 out of pocket. If the account doubles in value, then you could have $6000 in the Roth account, or $8000 in the traditional account. Thus, if you retire at a 25% marginal tax rate, you will pay $2000 in tax when you withdraw the $8000 and wind up with the same $6000."

That is essentially what popular press articles say also. E.g. from http://www.cbsnews.com/news/roth-vs-tra ... is-better/: "I should point out that if your current and future marginal tax brackets are equal, there's no difference mathematically between the two types of IRAs in the amount of after-tax income you'll receive in retirement."

It is somewhat of a fine point, but as "everyone" (have to believe contrary advice has been observed elsewhere, just didn't find it) says "for equal tax brackets it doesn't matter" that is indeed what I expected to find when inspecting the math. As bpp and Doc point out there is simple qualitative reasoning to make the math "obvious in hindsight" (at least to me - again, this may well have been obvious to others all along).

User avatar
Doc
Posts: 9304
Joined: Sat Feb 24, 2007 1:10 pm
Location: Two left turns from Larry

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Doc » Sun Jun 08, 2014 3:55 pm

FiveK wrote:at least to me - again, this may well have been obvious to others all along
Nope, we all our guilty of getting lost in the words at times. "Forget the words, do the math."
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

bpp
Posts: 2017
Joined: Mon Feb 26, 2007 12:35 pm
Location: Japan

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by bpp » Sun Jun 08, 2014 6:55 pm

FiveK wrote:http://www.bogleheads.org/wiki/Traditional_versus_Roth seems the most appropriate place.
Actually, it looks like this may have been covered on that page:

http://www.bogleheads.org/wiki/Traditio ... an_max_out
If you can max out

The IRS sets a maximum contribution to retirement accounts; if you have reached this maximum, anything else you contribute must be in a taxable account, which will lose much more to taxes than either a traditional or Roth account. The IRS limits do not distinguish between traditional and Roth accounts. If you contribute the maximum to a traditional account, you will pay less in taxes, but you will have to invest any tax savings in a taxable account. Therefore, if you can hit the IRS limit, a Roth account has an advantage because it effectively lets you tax-defer more money.
(Followed by qualifications about the effect of differences in tax rates at time of contribution and at time of withdrawal.)

Does that cover what you want to say, FiveK?

User avatar
Topic Author
FiveK
Posts: 8043
Joined: Sun Mar 16, 2014 2:43 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by FiveK » Sun Jun 08, 2014 7:08 pm

Thanks bpp, that's it exactly. Maybe someone could add something like "but see below if you can maximize your Roth contribution" to the "Taxes" section?

Or just leave things alone - it is "in there" already, so now we're talking style points and I defer to the wiki editors.

bpp
Posts: 2017
Joined: Mon Feb 26, 2007 12:35 pm
Location: Japan

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by bpp » Sun Jun 08, 2014 7:24 pm

FiveK wrote:Thanks bpp, that's it exactly. Maybe someone could add something like "but see below if you can maximize your Roth contribution" to the "Taxes" section?
Done: http://www.bogleheads.org/wiki/Traditio ... Roth#Taxes
Last edited by bpp on Sun Jun 08, 2014 7:50 pm, edited 1 time in total.

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Sun Jun 08, 2014 7:38 pm

If you like numbers, use different ones. Make the max contribution $6,000 (it will get there one day). Be in the 25% bracket now and forever. Make $8,000.

1) Pay $2,000 tax. Put $6,000 in Roth.
2) Put $6,000 in traditional, of which only $4,500 is yours (the rest belongs to the IRS). Pay $500 in tax. Keep $1,500 in taxable.

With (1), you never pay tax again. You keep whatever $6,000 grows to.

With (2), you keep what $4,500 grows to plus whatever $1,500 grows to against the annual tax drag.

$6,000 with no taxes or $4,500 with no taxes plus $1,500 with taxes. It doesn't seem like a hard choice, given the assumption of 25% bracket now and forever.

Or is my math wrong?

Bob's not my name
Posts: 7407
Joined: Sun Nov 15, 2009 9:24 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Bob's not my name » Sun Jun 08, 2014 7:45 pm

I believe the opposite is true: that the saturated case gets too much emphasis. Note that the saturated case argument says "if you can max out" whereas it should say "if you will max out every year for the rest of your career". Many people have so much tax-advantaged space available to them that they can save about a third of their gross income*, but few people can actually save that much, or want to. There are the house down payment years, the expensive family years, and the super-expensive college years.

Here I am using the terms "many" and "few" in their strict scientific senses, meaning "I don't actually know what I'm talking about, but I'll extrapolate from my personal experience."

Anyway, just because you can max out this year doesn't mean the saturated case arguments apply to you. Obviously if you leave tax-advantaged space unused next year your case is not saturated -- you can just take this year's surplus and put it in next year's IRA.

*I can put $23,000 in my 403b and my employer matches with about $21,000. My wife puts 100% of her net profit in her SIMPLE, which is around $6,000. And we can put $13,000 in Roth IRAs, which consumes about $22,000 of gross income. So when we max our tax-advantaged space we save about $72,000 of gross income, which is more than a third. In thirty years I have almost never maxed out my tax-advantaged space.

Another example: a working couple under 50 with $120,000 of gross income. They can contribute $35,000 to their 401k's and have employer matching of, say, 3% = $3,600. They can save over a third of their gross income before they even touch Roth IRAs.

Take the same couple with access to only one 401k. That's $17,500 and, say, $2,500 of employer matching = $20,000. Two Roth IRAs maxed consume about $14,000 of gross income. Their tax-advantaged space allows them to save about 28% of their gross income, not including 529s or HSA.

freddie
Posts: 920
Joined: Sat Feb 08, 2014 11:06 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by freddie » Sun Jun 08, 2014 8:21 pm

You assumption of 25% is questionable. Maybe the investor should think about buying stocks in the taxable account:) Everyone needs to run their own math. You start taking the IRA out using deductions and exemptions and the 10-15% brackets, you can end up way ahead with the traditional. Have your income change so that you in the 40% bracket when you retire and the ROTH looks mightly fine:)
sscritic wrote:If you like numbers, use different ones. Make the max contribution $6,000 (it will get there one day). Be in the 25% bracket now and forever. Make $8,000.

1) Pay $2,000 tax. Put $6,000 in Roth.
2) Put $6,000 in traditional, of which only $4,500 is yours (the rest belongs to the IRS). Pay $500 in tax. Keep $1,500 in taxable.

With (1), you never pay tax again. You keep whatever $6,000 grows to.

With (2), you keep what $4,500 grows to plus whatever $1,500 grows to against the annual tax drag.

$6,000 with no taxes or $4,500 with no taxes plus $1,500 with taxes. It doesn't seem like a hard choice, given the assumption of 25% bracket now and forever.

Or is my math wrong?

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Sun Jun 08, 2014 8:33 pm

freddie wrote:You assumption of 25% is questionable.
The OP started with T, not T1, T2, T3, ... , T43. If the argument is being made about one T, then you have to use one T. If you want to tell the OP not to use one T, go right ahead. I haven't had one T for my whole life, but I have been pretty close. There were two years when my income dropped way off, which were the only ones in which I could make a deductible IRA contribution ($2000 back then). T variability is a key to taxation, I agree, but only if you can pull it off.

* Of course I am lying. Changes in the tax law have changed T from time to time. But we can't argue about laws that don't exist.

User avatar
Topic Author
FiveK
Posts: 8043
Joined: Sun Mar 16, 2014 2:43 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by FiveK » Sun Jun 08, 2014 10:06 pm

Short version
Wiki change looks good to me - thanks again bpp.

Long version
The context of the question is this: someone (outside this forum) asked the "should I use tIRA or Roth?" question. I was going to suggest tIRA, based on "if your retirement tax rate is lower you made the right choice, if your tax rate doesn't change it doesn't matter, and if your retirement tax rate is higher - you will have otherwise done well enough to forgive yourself for the tIRA choice." But then, as noted above in various ways (sscritic, that seems a fine "for example"), it became clear that the statement "if your tax rate doesn't change it doesn't matter" is plain wrong (when maximizing the IRA contribution).

So now (unless someone has already done so...?) I'll have to hedge that original thought, and determine when the crossover occurs as after-retirement taxes drop below pre-retirement....

User avatar
Doc
Posts: 9304
Joined: Sat Feb 24, 2007 1:10 pm
Location: Two left turns from Larry

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Doc » Mon Jun 09, 2014 9:42 am

FiveK wrote:"if your retirement tax rate is lower you made the right choice, if your tax rate doesn't change it doesn't matter, and if your retirement tax rate is higher - you will have otherwise done well enough to forgive yourself for the tIRA choice."
The rule is for 1) equal pre-tax contributions and 2) no change in tax rate a ROTH and a tIRA/401k are equal.

All the nuances that apply to us as individuals are what happens if one or both of those two conditions are not met.


FWIW I postulated the maxed out ROTH 401k situation a while back and got jumped on not because my math was wrong but because hardly anyone can max out their 401k's of any stripe. :|
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

manwithnoname
Posts: 1584
Joined: Mon Jul 22, 2013 7:52 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by manwithnoname » Mon Jun 09, 2014 4:23 pm

sscritic wrote:If you like numbers, use different ones. Make the max contribution $6,000 (it will get there one day). Be in the 25% bracket now and forever. Make $8,000.

1) Pay $2,000 tax. Put $6,000 in Roth.
2) Put $6,000 in traditional, of which only $4,500 is yours (the rest belongs to the IRS). Pay $500 in tax. Keep $1,500 in taxable.

With (1), you never pay tax again. You keep whatever $6,000 grows to.

With (2), you keep what $4,500 grows to plus whatever $1,500 grows to against the annual tax drag.

$6,000 with no taxes or $4,500 with no taxes plus $1,500 with taxes. It doesn't seem like a hard choice, given the assumption of 25% bracket now and forever.

Or is my math wrong?
Your math is wrong because you have excluded the opportunity cost of the taxes paid on the roth, but included them for the TIRA. Opportunity cost is the future investment value forfeited on the taxes paid in the year of contribution to the roth. In a TIRA the opportunity costs are the taxes paid upon distribution.

Contribute $6000 to a Roth invested at 7% for 20 years. Future value will be 23,218.

However the future value of the roth will be reduced by the opportunity cost of the 2000 in taxes paid on the 8000 in income in year of contribution which could have been invested at 7% for 20 years. Future value of opportunity cost is 7339. 23,218 -7739 = net value 15,879 before reduction for any taxes due on the 5339 gain.

TIRA contribution of 6000 invested at 7% for 20 years is 23,218 less opportunity cost of 5804 (25% x 23,218) results in a net value of 17,413. You get the same result if 4500 is invested at 7% for 20 years.

Its up to each taxpayer to calculate tax efficiency of investments.

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Mon Jun 09, 2014 4:35 pm

manwithnoname wrote: Your math is wrong because you have excluded the opportunity cost of the taxes paid on the roth, but included them for the TIRA. Opportunity cost is the future investment value forfeited on the taxes paid in the year of contribution to the roth. In a TIRA the opportunity costs are the taxes paid upon distribution.

Contribute $6000 to a Roth invested at 7% for 20 years. Future value will be 23,218.

However the future value of the roth will be reduced by the opportunity cost of the 2000 in taxes paid on the 8000 in income in year of contribution which could have been invested at 7% for 20 years. Future value of opportunity cost is 7339. 23,218 -7739 = net value 15,879.
Opportunity costs are not costs you pay. They are the difference between what you could have made if you had followed a different strategy than the one you did follow.
[Edit for Correction: this is the opportunity loss - see game theory, but the idea is the same, what you gave up by the choice you made]

Let's try your numbers.

Contribute $6000 to a Roth invested at 7% for 20 years. Future value will be 23,218.

When you take it out, how much do you have? Is it $23,218 or $15,879? Who takes the extra $7,739 away from you? Is it Vanguard? Does the IRS charge you $7,339 as tax on your $23,218 withdrawal from your Roth? Where do you find that in the tax code?

Try again.

Perhaps you can tell us your investment strategy that yields $30,957. If I use a strategy that yields $23,218, then indeed there is an opportunity cost of $7,739 by choosing my strategy instead of yours since your $30,957 minus my $23,218 is $7,739. Please tell us how you made your $30,957.

P.S. I will give you a hint or two. First hint: $30,957 is 33 1/3 % more than $23,218. Second hint: $8,000 is 33 1/3% more than $6,000. If you need more hints, I will be glad to provide them.
Last edited by sscritic on Mon Jun 09, 2014 6:14 pm, edited 1 time in total.

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Mon Jun 09, 2014 5:14 pm

Actually, by the formal definition of opportunity cost, if I go to a movie instead of reading a book, the opportunity cost of the movie is the book I didn't get to read.

Plugging in the numbers, if I have $23,218 in retirement (the movie) instead of $7,739 in retirement (the book), the opportunity cost of my $23,218 is the $7,739 I could have had instead of my $23,218.

Note that you don't get both the book and the movie, you get one or the other. You don't get both the $23,218 and the $7,739, you get one or the other. I choose the $23,218, but you can choose the $7,739 if you wish.

In a fantasy world, you could have both the movie and the book, the $23,218 and the $7,739. Then you would have a mythical 30,957 fantasy dollars you can use to buy E tickets for rides in Fantasyland.

freddie
Posts: 920
Joined: Sat Feb 08, 2014 11:06 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by freddie » Mon Jun 09, 2014 7:11 pm

The math
Roth: 6k @ 7% = 23.2k

Traditional
6k*7%*.75 = 17.34
2k*6.25%= 2000+(6724-2000)*.85 = 6015
23.34k

As expected given the way the tax code works the TIRA wins.

You can tweak the 6.25% number to match you estimate for the yearly tax drag but it is a pretty reasonable one. As expected contributing to the ROTH is tax inefficient and in the end it costs you money. The ROTH always looks good if you ignore the tax cost at the start. Once you include it, it comes down to the changes in the tax rates.



sscritic wrote:
manwithnoname wrote: Your math is wrong because you have excluded the opportunity cost of the taxes paid on the roth, but included them for the TIRA. Opportunity cost is the future investment value forfeited on the taxes paid in the year of contribution to the roth. In a TIRA the opportunity costs are the taxes paid upon distribution.

Contribute $6000 to a Roth invested at 7% for 20 years. Future value will be 23,218.

However the future value of the roth will be reduced by the opportunity cost of the 2000 in taxes paid on the 8000 in income in year of contribution which could have been invested at 7% for 20 years. Future value of opportunity cost is 7339. 23,218 -7739 = net value 15,879.
Opportunity costs are not costs you pay. They are the difference between what you could have made if you had followed a different strategy than the one you did follow.
[Edit for Correction: this is the opportunity loss - see game theory, but the idea is the same, what you gave up by the choice you made]

Let's try your numbers.

Contribute $6000 to a Roth invested at 7% for 20 years. Future value will be 23,218.

When you take it out, how much do you have? Is it $23,218 or $15,879? Who takes the extra $7,739 away from you? Is it Vanguard? Does the IRS charge you $7,339 as tax on your $23,218 withdrawal from your Roth? Where do you find that in the tax code?

Try again.

Perhaps you can tell us your investment strategy that yields $30,957. If I use a strategy that yields $23,218, then indeed there is an opportunity cost of $7,739 by choosing my strategy instead of yours since your $30,957 minus my $23,218 is $7,739. Please tell us how you made your $30,957.

P.S. I will give you a hint or two. First hint: $30,957 is 33 1/3 % more than $23,218. Second hint: $8,000 is 33 1/3% more than $6,000. If you need more hints, I will be glad to provide them.

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Mon Jun 09, 2014 8:41 pm

freddie wrote:The math
Roth: 6k @ 7% = 23.2k

Traditional
6k*7%*.75 = 17.34
2k*6.25%= 2000+(6724-2000)*.85 = 6015
23.34k
I thought you had to pay tax on your income. In the first year, you make $8,000. Your choices are

1) pay $2000 tax and put $6000 into a Roth
2) put $6000 into a tIRA and deduct $6,000 from your $8,000 income. You still have $2,000 of income you have to pay tax on. That would be $500 of tax (using the same rate as in 1), leaving you with $1500 in your taxable account to invest, not $2000.

Am I misinterpreting the 2k in your tIRA calculation? (I know your * doesn't mean multiply, at least not without some compounding at the stated rates first). The 0.85 is what's left after a 15% tax on your long-term cap gain, but the basis should be $1500, not $2000. I still am not following. Could you explain the $2,000?

kaneohe
Posts: 6023
Joined: Mon Sep 22, 2008 12:38 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by kaneohe » Mon Jun 09, 2014 10:37 pm

freddie wrote:The math
Roth: 6k @ 7% = 23.2k

Traditional
6k*7%*.75 = 17.34
2k*6.25%= 2000+(6724-2000)*.85 = 6015
23.34k

As expected given the way the tax code works the TIRA wins.

You can tweak the 6.25% number to match you estimate for the yearly tax drag but it is a pretty reasonable one. As expected contributing to the ROTH is tax inefficient and in the end it costs you money. The ROTH always looks good if you ignore the tax cost at the start. Once you include it, it comes down to the changes in the tax rates.

freddie......your conclusion is opposite from what is normally given. Another tool that I like is not to use a certain % return/yr for N yrs but to just assume things double(or go up by K x) in
N yrs so no calculator is necessary and you can usually get round numbers. I saw this used by someone some yrs ago but don't recall who it was. For this issue assume 6K income and 1.5K in a taxable side fund.
1) for the Roth, 6K goes into the Roth, and the 1.5K taxable side fund is used to pay the 25% tax on the 6K income , leaving just the Roth.
2) for the TIRA, 6K goes into the TIRA but is deducted so no tax is due and the 1.5K taxable side fund remains.
1A) for the Roth, N yrs later, the 6K has quadrupled to 24K. No tax is due so the after tax value is 24K.
2A) for the TIRA, it similarly has quadrupled to 24K but after 25% tax, the after tax value is 18K. The taxable side fund has also quadrupled to 6K but after taxes along the
way or at the end, the after tax value is somewhat less.....say 6K - so the after tax value of TIRA and side fund is 24K - whatever you calculate

In this case, the Roth is ahead by the tax drag on the taxable side fund. This applies assuming starting/final tax rates are the same.

User avatar
Topic Author
FiveK
Posts: 8043
Joined: Sun Mar 16, 2014 2:43 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by FiveK » Tue Jun 10, 2014 12:14 am

Comparing tIRA and Roth, we know that if
- the tax rate doesn't change between contributing to and withdrawing from the IRA, and
- the comparison is based on a pre-tax amount that allows maximum Roth contribution
- (which also allows maximum tIRA, plus some extra that is taxed up front and then on investment results), then...
- ...Roth is better.

Given that, what drop in after-retirement tax rate (with all other above conditions holding) makes Roth and tIRA equivalent? In other words, below what after-retirement tax rate does tIRA becomes a better option?

Symbols
a = Fraction of tax rate applied to investment returns in taxable accounts (e.g., for a 15% dividend tax instead of 25% ordinary tax, a = 15/25 = 0.6)
i = Annual investment return, fraction
n = Number of years
R = Marginal post-retirement tax rate, fraction
T = Marginal pre-retirement tax rate, fraction

To solve, take the formulas in the OP and change the after-retirement tax from T to R. Set tIRA/Roth = 1 and solve for R.

If the taxable account pays no annual taxes (e.g., all capital gain at the end of the comparison period, no dividends along the way)
(1) R = T/(1+a*(1-1/(1+i)^n)*T)

If the taxable account investment return is taxed every year (e.g., all annual dividends and no capital gain occurs)
(2) R = T*((1+i-a*i*R)^n/(1+i)^n) (note: formula is implicit in R. Fortunately, direct substitution iteration in Excel works.)

Equations #1 and #2 should bracket most actual investments.

In both cases, if a=0 then R=T. In other words, if there is no tax on the taxable account investment return then Roth and tIRA are equivalent at any initial tax rate. Both formulas also calculate R=0 when T=0.

Calculations below assume i=5% and n=25 years. Just to give a few examples:
T=28%, a=15/28=0.5357, the "capital gain" (formula #1) R=25.3% and the "annual dividend" (formula #2) R=24.0%.
T=28%, a=1, #1 R = 23.4% and #2 R = 21.6%
T=25%, a=0.6, #1 R = 22.6% and #2 R = 21.4%
T=39.6%, a=15/39.6, #1 R = 35.8%, #2 R = 34.0%

In practical terms, dropping from the 25% to 15% bracket makes tIRA better. In the higher tax brackets, it may take more than a 1 bracket drop to make tIRA better than Roth.


For anyone interested, copy the table below, paste into spreadsheet cell A1 (with iterative calculations enabled), and try any combination....

Symbol Capital_Gain Annual_Dividends Placeholder
R =B3/(1+B4*(1-1/(1+B5)^B6)*B3) =B3*((1+B5-B4*B5*D2)^B6/(1+B5)^B6) =C2
T 0.396
a 0
i 0.05
n 25

Bmac
Posts: 337
Joined: Sun Mar 03, 2013 8:58 am
Location: Seattle

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Bmac » Tue Jun 10, 2014 1:59 am

Of course, the "free lunch" is the back-door Roth, converting maximum non-deductible TIRA contributions immediately into a Roth annually. Regardless of current or future tax rates, this is a win-win. No fancy math required.

manwithnoname
Posts: 1584
Joined: Mon Jul 22, 2013 7:52 pm

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by manwithnoname » Tue Jun 10, 2014 9:52 am

From the above posts it is clear that there are many different ways to calculate whether a Roth or TIRA option will yield the maximum benefit based on the criteria used to determine the metrics. My own calculation allows me to determine how achieve maximize tax arbitrage of a specific investment. Tax arbitrage is applying two or more tax provisions together to achieve the maximum tax benefit of a particular investment such as placing retirement contributions in pre tax401k/ TIRAs when the marginal tax rate is high and then converting to a roth IRA when the tax rate declines by 15% or more and state income tax is eliminated.

Investors can use any calculation methodology that they believe analyzes their investment/tax strategy. There is no one correct answer for all possible investment scenarios.

User avatar
Doc
Posts: 9304
Joined: Sat Feb 24, 2007 1:10 pm
Location: Two left turns from Larry

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by Doc » Tue Jun 10, 2014 10:03 am

manwithnoname wrote:Investors can use any calculation methodology that they believe analyzes their investment/tax strategy. There is no one correct answer for all possible investment scenarios.
There is no one answer as you say but tax code and the arithmetic remain constant. Opinions vary but facts are facts.
A scientist looks for THE answer to a problem, an engineer looks for AN answer and lawyers ONLY have opinions. Investing is not a science.

sscritic
Posts: 21858
Joined: Thu Sep 06, 2007 8:36 am

Re: Wiki: Tax Effects of Maximizing Roth IRA Contributions

Post by sscritic » Tue Jun 10, 2014 10:04 am

I agree with the man with no name. However, make sure that when you are doing your calculations that your four function calculator works correctly and that you know how to use it. If you hit the x key when you meant to hit the + key and don't notice that your result looks a little screwy, your answers won't be worth squat.

Post Reply