Ben Mathew Economics
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Tripping Up On IRA Math

10/4/2013

11 Comments

 
If you read an article about retirement accounts (IRAs, 401(ks), etc.) in a respectable news outlet like the New York Times or the Wall Street Journal, it's often only a matter of time before someone says something blatantly wrong. I don't mean wrong as in maybe-probably-kinda wrong. I mean wrong as in mathematically-algebraically-demonstrably-provably wrong. I don't know what it is about the math of retirement accounts that trips people up. It's simple eighth grade algebra, but it's gotten a lot of people confused--even professionals managing hundreds of millions of dollars.

So when I was reading an article about IRAs in today's New York Times, I was expecting an error to pop up at some point. And sure enough, it did. And I'm going to use it as a jumping point to clarify how retirement accounts work.

Here's the offending paragraph:
Michael E. Goodman, a certified public accountant and the president of Wealthstream Advisors, says he has a client who is a 45-year-old widow with children who looks each year at how much she can convert from a traditional retirement account to a Roth. Since she makes large charitable donations, she is often able to use the tax deduction to offset some of the tax bill from the Roth conversion.

“It’s much more powerful for my client at 45 than at 65,” he said, pointing out that it takes years for investment gains to surpass the tax losses. The exception is someone who is doing the conversion for heirs, who will have a longer time horizon.
(emphasis mine)
The bolded part is wrong. It does not matter at all when the tax is paid. The only thing that matters is the tax rate. The time horizon left after the payment of the tax is irrelevant. That may sound a little counterintuitive. But the algebra behind it is ironclad. Let me explain.

The U.S. government allows us to save money in special accounts called Individual Retirement Accounts or IRAs. There are other special accounts called 401(k) and 403(b)s and 457 that work in much the same way. So understanding how IRAs work will help you understand these other accounts as well.

What's special about an IRA is that you don't have to pay capital gains tax on the growth of the money you put in that account. [Edit: More accurately, you don't have to pay tax on the growth of that money, whether it takes the form of capital gains, interest, or dividends.] That's a huge advantage. And that's why you are strongly encouraged to save your money in an IRA rather than in a regular ol' account.

But IRA accounts do not save you from the income tax. [Edit: More accurately, they do not save you from paying income tax on the original amount you place into the account. In other words, the original amount is taxable; the growth is tax free.] One way or the other, you have to pay your income tax in full. There are two types of IRA accounts, and the key difference between them is the timing of when you have to pay your income tax. With a Roth IRA, you pay the income tax when you put the money into the account. With a traditional IRA, you pay the income tax when you pull the money out of the account. Whichever you choose, there's no escaping the income tax.

So which is better? Paying your income tax at the beginning for the Roth or paying your income tax at the end for the Traditional account? Let's work it out. Suppose you want to put $100 in an IRA account today. You'll take the money out some time in the future--like thirty years from now. Over that time, your investment will grow by x percent.

Now suppose you go with the Roth IRA. Because it's a Roth account, you'll have to pay tax on that $100 today. If your income tax rate this year is T1 percent, after paying the tax, you'll have only $ 100*(1-T1)  to put in your Roth IRA account. It grows by x percent in the account, so you'll end up with $ 100*(1-T1)*(1+x) in thirty years.

Now suppose you had gone with the Traditional IRA. Because it's a Traditional account, you don't have to pay tax on that $100 today. So the full $100 goes into the account. It grows x percent in thirty years, so you'll end up with $ 100*(1+x) in thirty years. But now you have to pay the income tax. Say your income tax rate when you're pulling out the money is T2 percent. After paying the tax, you'll have only $ 100*(1+x)*(1-T2) left.

Now, compare the two amounts. If your tax rate when you put the money in (T1) is the same as the tax rate when you take the money out (T2), then both Roth and Traditional accounts will net you exactly the same amount in thirty years. It comes down to the commuatitive property of multiplication: A x B = B x A. Whether the money is taxed first and then grows, or the money grows and then is taxed, if the growth rate and the tax rate are the same, you'll end up with the same amount of money at the end of it all. The timing of the tax does not matter. Only the tax rate does.

Converting from a Traditional account to a Roth account in a particular year is basically a decision to pay the income tax in that year as opposed to paying it when the money is pulled out. The above algebra shows that the timing of the tax is irrelevant. The only thing that matters is the tax rate. So to maximize the money you will walk away with, you should pay your taxes in the years where your tax rate is the lowest. For most people, tax rates are likely to be lowest in retirement, because they're not earning much in retirement. So, for most people, a traditional IRA account makes sense.

(Now there are other reasons why you might still opt for the Roth account. If you've maxed out your Traditional IRA and want to shield more of your money, or you want to leave your IRA to your kids, it might make sense to switch to Roth. But that's another post for another day.)

The reason a lot of people, including professionals, often get this math wrong is that usually time horizons matter. But in this case, it simply doesn't.

FYI, one person in the personal finance field who I trust to be on top of these kinds of things is Kaye Thomas, founder of Fairmark.com. He's a real pro. And he has written a great primer on investing, That Thing Rich People Do, which I 'm always recommending to everyone.
11 Comments
dcdoc
10/9/2013 02:17:54 pm

Kinda, sorta, but not quite.

There is no reason why the value of assets within a traditional IRA should grow any faster or slower than those within a Roth. So, it is correct to assume, as you have, that both will grow by exactly the same "x" percent going along. Can one expect that same "x" rate of return, no more or less, on assets outside a tax-advantaged account. No, most definitely not.

Assets held outside a tax-deferred retirement account will be taxed all along the way up until the day they must be marshalled to pay the taxes previously deferred. And that day ALWAYS comes. The money to pay the taxes will been earning an after-tax rate of return that nets to <"x." That less then "x" difference will always be there, and can be huge depending on a number of factors (number of years; tax rates in the beginning and in the end, as well as along the way, etc.)

Unless you or the person who will eventually be paying the taxes (surviving spouse or heirs) will be in a substantially lower bracket when the taxes must be paid, a Roth will almost always come out ahead of a TIRA, and may come out very much ahead.

So, yes on the law of commuativity, but there is more than that to this story, namely that the <x rate of growth there will be on the funds that will be need to pay the taxes on any distributions from a TIRA. And there will ALWAYS be those taxes, the only question(s) being who will pay them, when, and at what rate.

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Ben Mathew
10/10/2013 10:08:39 am

You're right that money saved outside of tax-shielded accounts will grow at a slower rate (because of the capital gains tax). But that does not affect the equivalence conclusion above. That's because the saver would (and should) be paying the income tax on the traditional IRA with money they're pulling out of the traditional IRA, so it's shielded money.

Let's look at how it works in the example. Let T1=T2=30%. You have a $100 today pre-tax which you can put into a traditional or Roth IRA. If you went traditional, the account would get the full $100. If you went Roth, it would get only $100*(1-.30)=$70 after the tax. So $100 pre-tax today will become $100 in the traditional account but only $70 in the Roth account. (The extra money in the traditional will be just enough to cover the deferred income tax.)

Whatever account you choose, you invest in IBM stock that doubles by the time you withdraw the money. The $100 in the traditional account doubles to $200. But then you have to pay the 30% income tax, so you're left with $200*(1-.30)=$140. With the Roth, the $70 doubles to $140, and you don't have to pay any income tax. In both scenarios, you end up with $140 after tax.

The big exception is if you've maxed out your traditional IRA. At that point you will be forced to save the eventual tax money outside of the IRA in an unshielded account which, as you said, won't grow fast enough. If that's the case, you might be better off putting the money in the Roth as long as you're not in too high of a tax bracket now relative to retirement. So if you haven't maxed out, traditional is usually the best bet. If you have maxed out, it's more complicated and you may want to consider switching to the Roth.

I think a simple rule that should work for most people is traditional until maxed out, then switch to Roth to save more.

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dcdoc
10/10/2013 12:28:04 pm

"because of the capital gains tax" No, it is simply because that money outside a tax-advantaged account will be subject to ordinary income taxes as well as capital gains if there are any. (Interest income is generally treated as ordinary income.) The effective rate of return on that "outside" money is the after-tax one, and will always be less than "x %".

"That's because the saver would (and should) be paying the income tax on the traditional IRA with money they're pulling out of the traditional IRA, so it's shielded money." Huh?! If it's sitting outside a tax-advantaged account for no matter how short a period of time, there will be taxes to pay on any earnings or gains, and that will mean <x% rate of return. And one can't withdraw money from a traditional IRA whenever they chose, since for one thing they must be at least 59 1/2.

I don't have the time to go through it with you, but you are fundamentally wrong that everything comes out the same in the wash because of the law of commuativity. If one wants to max the advantage of tax-free (Roth) for years to come (no RMDs with Roth IRAs, and the miraculous possibility of stretch Roths for heirs), they won't pay the tax due on tax-deferred accounts (TIRAs) by reducing the corpus of their IRA funds either upfront or in the end; instead, they will use non-IRA funds to pay the taxes, thus getting a great deal more of the advantage. Indeed, many will advise individuals not to convert at a time when they don't have the money outside an IRA to pay the taxes on a conversion. You're approach would take very much sub-maximal advantage of the Roth option.

(Someone with a PhD in economics from the University of Chicago has a huge amount of cred on the subject of economics, but I think you have this personal finance thing wrong. If you were right, the Roth option wouldn't be all that attractive. Unless there is the prospect of 'tax arbitrage" through a substantially lesser rate of taxation in the future, there is no downside to a Roth and a huge amount of upside, especially is one is going to maximally fund it by paying the taxes upfront with "outside" money, not reducing the corpus of what they can stuff away in a Roth. There is more to this than the rule of commuativity.)

Ben Mathew
10/10/2013 02:50:46 pm

I should have said that growth in an unshielded account is slower because of the tax on the returns to capital, whether it's through capital gains, dividends, interest income, or anything else. Sorry about the sloppy language.

<"If it's sitting outside a tax-advantaged account for no matter how short a period of time, there will be taxes to pay on any earnings or gains, and that will mean <x% rate of return.">

It's not sitting outside the tax-advantaged account for any period of time. On day one, $100 went into the traditional and only $70 went into the Roth. The extra $30 in the traditional is what ends up paying the tax. If you work through the algebra in the above example carefully before dismissing it out of hand, I think you will be convinced. If it helps, I'm not the only one saying this. Kaye Thomas's Go Roth, Chapters 1 to 6, goes through the mechanics of all of this in some detail. He calls it the parity principle. I just googled around and here's a random brochure I found that says the same thing (page 11, Why Bother - the parity principle): http://www.raymondjames.com/melvillewealthmanagement/pdfs/RothConversionAnalysis.pdf. Of course, that doesn't prove I'm right, because I'm sure there are brochures and booklets and books that say the opposite. I just want to lay it out there that I'm not alone in making this claim.

The Roth does have some advantages, as I've acknowledged in both posts. I didn't want to get into the details. But here's the quick version. $100 in the Roth is more money after-tax than $100 in the traditional account, because it's post-tax money. Since we can only contribute $5,500/yr in our IRA accounts, the Roth account effectively has a higher limit. So, if we are maxed out on the traditional and want to go higher, we can use the Roth instead. Also, with a traditional account, you are forced to withdraw starting at the age of 70.5. If you want more flexibility about the timing of your withdrawals, the Roth might be the way to go. For these reasons, I actually have most of my retirement money in Roth accounts. So I'm not against the Roth at all.

A Ph.D. in economics does not automatically qualify a person to be a personal finance expert. But it helps. Economics underpins most of the models used in personal finance--saving decisions, discounting, insurance, risk preferences, asset pricing models, etc. But it's also true that one can have a Ph.D. and not know much about the institutional details of how things work. But in this instance, you and I don't disagree about any of the institutional rules or any of the economics--only about the algebra of how the rules play out. And I doubt either of us have the advantage in eighth grade algebra. :)

dcdoc
10/9/2013 02:19:44 pm

BTW, does anyone know how "tax losses" figure in a TIRA to Roth conversion? I don't.

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Ben Mathew
10/10/2013 10:20:17 am

If you're referring to using losses in your IRA accounts to offset capital gains taxes on unshielded accounts, the traditional to roth conversion won't help or hurt. But losses in your unshielded account might bump you into a lower tax bracket which would make it more attractive to convert to roth.

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dcdoc
10/10/2013 11:57:25 am

I'm referring to what was said in that Sullivan column in the NYT last Saturday about "tax losses." I think there are no "tax losses," at least as that term is generally understood, in connection with conversions from tax-deferred (TIRAs) to tax-free (Roth) or in connection with "recharacterizations."

The speaker was very confused and confusing when he used the term, and flat out wrong with the part about it taking years of investment gains to make up for the "tax loss."

dcdoc
10/11/2013 01:06:22 am

" Since we can only contribute $5,500/yr in our IRA accounts, the Roth account effectively has a higher limit."

Have a look at what the annual limits are on SEP-IRA contributions. And note that much of what goes into TIRAs and Roth IRAs comes out of qualified retirements plans (e.g., 401(k)s) that allow much greater contributions and internal build up.

Maybe we can do this as "bullets." For instance, why don't you in summary fashion cite distinctions that you think favor conversion and those you think disfavor conversion. Afterwards, I will come back and give me list. Then we can see what we agree and disagree on here.

(And the way this blogs comment thread operates, it is difficult or impossible for me to elaborate fully because the administrator rejects follow-ons.)

Ben Mathew
10/11/2013 01:58:33 am

Most people are not self-employed, so they can only contribute $5,500 to their IRAs. So I said $5,500 is the max. It's impossible to list all the caveats--there are just too many and they're not central to the story. The important thing is that there is some limit to how much you can contribute to your IRAs, and if you hit the limit on traditional, the only way to shield more money is to switch to the Roth because the after-tax dollar in the Roth is effectively more money than the pre-tax dollar in the traditional. The precise limit is not important to the mechanism.

Your original point of disagreement was that you felt that money had to be saved in an unshielded account to pay for the traditional IRA tax. I'd like to reiterate that the example did not assume this in any way. All taxes come out of the $100 pre-tax money that you're going to save--either at the start with the Roth, or after it grows with the traditional. There is no separate unshielded account in the picture. Much of what you say about IRAs is right, but it does not change the parity conclusion.

I'm afraid I don't see us headed towards a resolution on this issue. We'll just have to agree to disagree on this one. Thanks for taking the time to share your views on this topic.

Avinash
11/4/2013 06:57:05 am

Simple, and clean. I like the basic math explanation and it makes sense.

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Ben Mathew
11/4/2013 11:43:22 am

Glad you found it useful.

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    Ben Mathew

    Author of Economics: The Remarkable Story of How the Economy Works

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