### Roth IRA vs Traditional IRAs

Which is better for you: investing in a Traditional IRA, or a Roth IRA?
• Traditional IRA, ``tax deferred'': you take a chunk of money, don't pay taxes on it now, and let it grow until you retire. When you withdraw the money, then you pay taxes on it. You pay more taxes (because the money has grown), but you are likely in a lower tax bracket (because your taxable income is much less -- perhaps only your IRA distributions).
• Roth IRA, ``tax advantaged savings'': You pay taxes on some money now, put it in the Roth IRA, and let the money grow. When you withdraw the money, you pay no taxes. (This is different than a regular investment, where you would pay tax on the investment's growth.)

I hear it said that if you're still a decade or more away from retiring, you should prefer a Roth IRA is preferred over a Traditional IRA, since the former frees you from taxes on all the accumulated growth.

I'm puzzled though, because when I try to work the math myself, I get a different answer (one that depends only on the tax rates when you put the money in and take it out, and nothing to do with how many years you keep it, or the rate of return).

### Doing the math

#### Setup

So at the end of the year, you're sitting there with a pile of money that you want to set aside for retirement. You've paid all your other taxes except the taxes on this pile of money. Just for illustration, let's suppose it's \$4000, and suppose that your marginal tax rate is 25%, and the maximum allowed IRA contribution is \$3000. You can either
1. Put \$3000 in a Roth IRA. You still owe taxes on 25% of \$4000, so all the remainder goes immediately to taxes.
2. Put \$3000 in a Traditional IRA, and since you can deduct that from your income you only owe taxes on the remaining \$1000; you cough up that \$250 of taxes, and you're left with \$750 in addition to the \$3000 in the Traditionoal IRA. You invest this \$750 parallel to the IRA, and will choose not to touch it until you cash it for living expenses during retirement.

#### Pertinent(?) Quantities

• n, the number of years until you take out your money to live on.
• r, the average annualized rate of growth of your investement (e.g., 1.05).
• t0, your marginal tax rate this year (e.g. 25%).
• t1, your overall tax rate in the year you'll withdraw the money (e.g. 15%). We use ``overall'' on the assumption that most of your taxable income during retirement will be IRA distributions. (If not, then use the marginal rate here.)
• I, the amount you'll put into the IRA this year (e.g. \$3000).
• I0, the amount you have around to invest. This is the amount you'll put into the IRA
(Okay, actually we'll be interested only in (1-t0) and (1-t1), and they would be the better amounts to abstract over, oh well, we'll just keep carrying around a bunch of ``(1 - …)''.)

#### The goal

We'll ultimately be interested in the sum of ``amount withdrawn IRA after tax'' and ``amount withdrawn bank after tax'', for each of the two types of IRA. (By ``bank'', I really mean an investment that grows over time, but you only realize taxes when you ``withdraw'' it; a more accurate term would be ``capital investment''.) We'll fill in the following table step by step.
Amount in IRA initially Amount in IRA later Amount withdrawn IRA after tax Amount in bank initially Amount in bank later Amount withdrawn bank after tax
No IRA 0 0 0
Roth IRA 0 0 0

The very first thing we'll note is that in the case of the Roth IRA, all the money is either put into the IRA or is used immediately to pay taxes, so there is 0 money initially in the bank, and alas, nothing will come of nothing. So we can fill in three 0s. And if we eschew IRAs all together, we'll always have 0 in those entries.

#### No IRA

This case is pretty easy. Of our initial I0, we will owe taxes right away and be left with only I0 ⋅ (1-t0) = I, all in the bank initially. After n years of growth we have I ⋅ rn, and when we cash it out we'll owe taxes on the whole thing (at the overall rate t1), leaving us with I rn ⋅ (1-t1).

Amount in IRA initially Amount in IRA later Amount withdrawn IRA after tax Amount in bank initially Amount in bank later Amount withdrawn bank after tax
No IRA 0 0 0 I I rn I rn ⋅ (1-t1)
Roth IRA 0 0 0

#### The Roth case

This case is similar to the No-IRA case, except that we won't pay taxes upon cashing out our investment:
Of our initial I0, we will owe taxes right away and be left with only I0 ⋅ (1-t0) = I, all in the Roth IRA initially. After n years of growth we have I ⋅ rn, and when we cash it out we pay no taxes, leaving us with the full I rn.

Amount in IRA initially Amount in IRA later Amount withdrawn IRA after tax Amount in bank initially Amount in bank later Amount withdrawn bank after tax
No IRA 0 0 0 I I rn I rn ⋅ (1-t1)
Roth IRA I I rn I rn 0 0 0

Here we have I0, and we immediately squirrel I away into the original IRA. This leaves us I0 - I which we need to pay taxes on, so
Amount in IRA initially Amount in IRA later Amount withdrawn IRA after tax Amount in bank initially Amount in bank later Amount withdrawn bank after tax
No IRA 0 0 0 I I rn I rn ⋅ (1-t1)
Trad. IRA I (I0-I) ⋅ (1-t0)
Roth IRA I I rn I rn 0 0 0
With a bit of arithmetic, we see
(I0-I) ⋅ (1-t0)
= (I/(1-t0) - I) ⋅ (1-t0)
= I - I (1-t0)
= I - (I - t0 I)
= t0 I.
From here, the remaining columns are easy: we have rn growth (both in the Traditional IRA and the bank account), and upon withdrawl we'll have to pay taxes in both cases, leaving us with (1-t1) times the pre-tax amount.
Amount in IRA initially Amount in IRA later Amount withdrawn IRA after tax Amount in bank initially Amount in bank later Amount withdrawn bank after tax
No IRA 0 0 0 I I rn I rn ⋅ (1-t1)
Trad. IRA I I rn I rn ⋅ (1-t1) t0 I t0 I rn t0 I rn ⋅ (1-t1)
Roth IRA I I rn I rn 0 0 0

#### Reaching Conclusions

I am not a tax attorney. I'm not any type of attorney. I don't even know any attorneys. I glean my guesses to the tax code mostly by second-guessing why turbotax is asking me certain things. Consult with somebody who knows what they're doing (and read the "shortcomings" section below) before deciding your investments! (I am fairly confident about the math, though.)

After summing the ``amount withdrawn'' columns, which option is best? Clearly the ``No IRA'' option loses -- as we'd hope -- since both IRA options are at least as big as I rn ⋅ (1-t0).

So do we prefer a Roth IRA or a Traditional IRA? We should prefer a Roth IRA exactly when:
I rn > I rn ⋅ (1-t1) + t0 I rn ⋅ (1-t1) = (I rn (1-t1)) ⋅ (1+t0).
But the I rn can be canceled on both sides, and we prefer a Roth exactly when

1 > (1-t1) (1+t0).
• If t0t1 (you will be in a higher tax bracket upon retirement), then (1-t1) (1+t0) = 1 - t1 + t0 - t0 t1 = 1 - (t1 - t0) - t0 t1 ≤ 1 - t1 ⋅ t0 ≤ 1, and a Roth IRA is always preferable in this case.
(This result is backwards from my original impression.)
• If your current marginal tax rate t0 = 28%, then you should prefer a Roth iff t1 > 21.875%.

Non-shortcomings: This analysis uses several simplifying assumptions. Most of them don't change the outcome; the above formula holds even if

• The return rate r changes over time (including, if you change your investement strategies to be more conservative as you near and enter retirement).
• you combine this single-year's-contribution with the contributions made in other years.
• upon retirement, you don't withdraw an entire year's contribution all at once
These issues should all come out in the wash.

Shortcomings of the analysis: The biggest issues which this ignores are other legal differences between Traditional and Roth IRAs:

• If we have Traditional plus some non-IRA bank investments, that other money is (a) more flexibly accessible before retirement, and (b) in danger of not actually being there at retirement (due to (a), and the desire to have that jacuzzi).
• My Dad pointed out to me that in a traditional IRA, you must make withdrawls starting at age 70.5. That's could be a huge difference, depending on your have other sources of post-retirement income.
• You can't withdraw Roth contributions for 5yrs after depositing them, unlike Traditional?
• There are earnings-cap differences on what you can contribute to each type of IRA.
• If you already have one type of IRA, and now prefer the second,
Plus, in the math itself, there are some points that have been glossed over:
• I suspect it's fine even if the ``bank'' investments need to pay their taxes incrementally, and not only upon withdrawl. I haven't actually worked this one through though.
• It's a "all-other-things-held-equal" analysis. That is, using t0 as the marginal tax rate is saying that your contribution is being made after paying all other income taxes, and your choice of IRA contribution isn't simultaneously affecting other tax choices. This seems fair (esp. because all the investment assumptions seem monotonic, if not linear), but I'm not sure, there could be optima found not at extremes.