If you're the impatient type, here are the math formulas that, but for some subtle differences between traditional and Roth IRAs, will usually result in the most profitable account for a given contribution:
Quick Links
1. If you intend to invest any tax savings in a standard taxable account, click here.
2. If you intend to invest any tax savings in a traditional IRA or 401(k), click here.
Otherwise, let's dig in.
The IRS allows a traditional IRA to be converted, all or in part, to a Roth IRA. Transitively, a 401(k) can be converted to a Roth by first rolling to an IRA.
One would want to do this in order to take advantage of the features of the Roth. In addition to some comparatively trivial benefits, the Roth IRA has the advantage of taxfree withdrawals after age 59 1/2. However, contributions occur with posttax dollars, and conversions incur income tax, as though the converted amount had been earned on top of one's existing income. Therefore, it only makes sense to convert when one's annual income falls into a low bracket, which may occur due to unemployment or flowthrough losses from real estate or businesses. Generally, one does not want to convert an entire traditional IRA all at once, unless it's sufficiently small, or your other losses provide room. Furthermore, I suggest that Roth conversions should occur in December, when your tax bracket is more predictable. (After much discussion with the IRS, it seems to me that April 15 is the due date for Roth conversions for the previous year, regardless of whether or not you file an extension. However, this is a gray area, so I advise December.) On the other hand, if your stocks have suffered immensely early in the year, it might be wise to perform a conversion now, before they snap back up and make tax-minimal conversion more difficult.
Does all this monkey business really help?
Let's look at the math. A Roth conversion is just a like a Roth contribution, except that it is subject only to income limits (including, I should add, the converted amount itself), and not contribution limits. We can look at this amount -- call it P1 for "principal" -- as either being contributed to (or remaining within) a traditional IRA, or being contributed to (or converted into) a Roth IRA.
We assume that you will invest in the same securities in either account, since, after all, growth is taxfree in both cases, and investment expenses are likely to be identical. Let G1 be the growth rate, e.g. G1 = 0.08 for a growth rate of 8%.
But before we consider conversion, let's look at the difference in the investment yield of contribution.
The investment will grow for Y years in either case. In either account, P1 will grow at rate G1 within a traditional IRA. In the traditional IRA case, due to the tax deduction we receive up front, we will also have P2 -- the tax we didn't spend -- to invest. P2 will grow at rate G2 in a taxable account, with G2 <= G1 due to taxation. (In principle, you could throw P2 into a tax-deferred account if you have not yet exhausted all contribution limits. At the moment, we're assuming that you throw P2 into a taxable account, but we'll look at this alternative later.)
After Y years, we will have T, the traditional IRA account balance before taxes:
T = P1(1+G1)^Y+P2(1+G2)^Y
But we know that P2 is simply the tax deduction received as a reward for contributing P1:
P2 = CtP1
where Ct is the traditional IRA contribution tax deduction rate.
For example, suppose you contribute $1000. This reduces your taxable income by $1000. The first $400 of deduction occur in the 28% bracket. The next $600 occur in the 25% bracket. Thus:
Ct = (400/1000)(0.28)+(600/1000)(0.25)
Ct = 0.262 (or 26.2%)
So rewriting P2 in terms of Ct, we have:
T = P1(1+G1)^Y+CtP1(1+G2)^Y
Upon liquidation of the investment and subsequent withdrawal of the proceeds, the traditional IRA will be taxed at rate W, which is the ordinary income rate. The taxable account, which held P2, will have only its growth taxed, namely, at rate Q, known as the "capital gains rate". Currently, Q= 0.15 for 15%, although may return to 0.2 in the next few years. The principal (CtP1) will not be taxed. Thus, we will be left with the following net proceeds, Nt, from the traditional IRA and its invested tax deduction:
Nt = (1-W)P1(1+G1)^Y+CtP1+[1-Q][CtP1(1+G2)^Y-CtP1]
Nt = (1-W)P1(1+G1)^Y+CtP1{1+[1-Q][(1+G2)^Y-1]}
In the case of a Roth contribution, there is no P2 because we do not receive an upfront tax deduction, which we might then invest. Therefore, Ct, Q, and G2 are also irrelevant. There is also no tax upon withdrawal, so W is zero. So then, the net proceeds, Nr, from the Roth IRA would be:
Nr = R = P1(1+G1)^Y
We want to end up with as much money as possible. Put another way, we should only contribute to a Roth instead of a traditional when:
Nr > Nt
or
P1(1+G1)^Y > (1-W)P1(1+G1)^Y+CtP1{1+[1-Q][(1+G2)^Y-1]}
which simplifies to the following contribution-size-independent relation:
How to Choose Traditional or Roth Contribution, Where Tax Savings in the Former Case Are Invested in a Taxable Account
If the following relation is true, then contribute to a Roth. Otherwise,
contribute to a traditional:
W(1+G1)^Y > Ct{1+[1-Q][(1+G2)^Y-1]} (The "Roth Test Relation".)
where:
Ct = Rate of tax deduction for the traditional IRA contribution, e.g. 0.28 for 28%. This is not necessarily equal to the bracket at which the deduction occurs, as the deduction may span multiple brackets. Instead, it's the weighted average of all spanned brackets, as illustrated in the above example with 25% and 28% brackets.
G1 = Growth rate inside either IRA, presumed to be equal in both IRAs, e.g. 0.10 for 10%.
G2 = Aftertax growth rate outside the IRA, e.g. 0.09 for 9%.
Q = Capital gains tax rate on the growth of the invested tax deduction, in the traditional IRA case, which is imposed simultaneously with the IRA withdrawal, e.g. 0.2 for 20%.
W = Tax rate upon withdrawal from a traditional IRA, on top of all other retirement income in the same year, e.g. 0.35 for 35%.
Y = Number of years until withdrawal of investment proceeds, presumed to be equal in both IRAs.
Exmaple:
Ct = 28%
G1 = 10%
G2 = 9%
Q= 20%
W = 35%
Y = 40
W(1+G1)^Y >? Ct{1+[1-Q][(1+G2)^Y-1]}
0.35(1+0.1)^40 >? 0.28{1+[1-0.2][(1+0.09)^40-1]}
15.84 > 7.09
The statment is true. Therefore, we should contribute to a Roth.
It's theoretically possible, by the way, that the optimal solution is to contribute to both accounts, as the value of Ct varies stepwise with the size of the contribution. However, this is unlikely to make a material difference to the value of your retirement savings, and is more likely to just increase transaction costs.
Now assume that we have made our contribution to one or the other account. Now, we want to know whether we should convert some of our traditional holdings to Roth holdings. (In theory, we could move the other way, but the ramifications would including filing amended tax returns for many prior years, and would probably not be profitable anyway. Don't go there.)
In most respects, this is exactly the same exercise as determining to which account to contribute:
In the case where we do the conversion, we must first pay some tax on the converted amount, counting it as ordinary income on top of all our other income the same year. After that, the amount will be moved from the traditional to the Roth, where it grows tax free. (I suggest using the same broker for both accounts. Otherwise, you may incur substantial transaction costs.) Upon subsequent withdrawal from the Roth, neither principal nor growth is taxed.
In the case where you leave the money in the traditional, i.e. do nothing, it will grow tax free, then get taxed as ordinary income upon withdrawal. Additionally, you can take the money that you would have spent on conversion taxes, and invest it in a taxable account. (Of course, you might still have room to dump it into your 401(k). But this is unlikely, as the very fact that you've decided against conversion implies that your income is high, and therefore you have already maxed out your tax deferrals for the year. So we won't investigate this possibility.)
Guess what? This is exactly the same as the contribution analysis, except for one small and critical difference: the conversion tax rate may not equal the contribution tax deduction rate. Think of it this way: In one case, you contribute $5000 at the (hypothetical) 30% tax bracket. Congratulations, you saved 30%, or $1500. In the other case, you convert the $5000 from traditional to Roth. This act pushes you from the 30% to the 40% tax bracket. The tax rate associated with this act is 40%, or $2000. Thus adding income does not necessarily have the opposite tax impact as subtracting the same amount.
From above, we have the Roth Test Relation:
W(1+G1)^Y > Ct{1+[1-Q][(1+G2)^Y-1]}
Now, however, there is no Ct, because we are not contributing to a traditional IRA; we are instead converting to a Roth IRA. The tax rate on this latter transaction is Cr:
W(1+G1)^Y > Cr{1+[1-Q][(1+G2)^Y-1]}
Thus we have the same Roth Test Relation, as applied to conversion. Remember, generally, Ct and Cr are different, even if all other values are the same. Thus the decision as to whether or not to contribute $1000 is not the same decision as to whether or not to convert the same amount.
Let's get back to contributions. As I said above, there is some possibility that the tax deduction due to traditional contribution could itself be invested in a tax-deferred account, if one has a roomy 401(k). We assume that this additional investment will always target a traditional IRA or 401(k); otherwise, it is likely too small to justify investing independently. (We know that the major investment is occurring in a traditional IRA or 401(k), as otherwise, there would be no tax savings to invest.) In this case, we need to change the Roth Test Relation:
W(1+G1)^Y > Ct{1+[1-Q][(1+G2)^Y-1]}
to account for the facts that:
1. G2 = G1, because all principal values are invested tax-deferred.
2. Q = W, because all proceeds are taxed as orindary income.
3. All proceeds are taxed -- not just growth.
So in this case,
Nt = (1-W)(P1+P2)(1+G1)^Y
Nt = (1-W)P1(1+Ct)(1+G1)^Y
and Nr is unchanged:
Nr = P1(1+G1)^Y
As before, we only want to contribute or convert to a Roth if Nr > Nt:
P1(1+G1)^Y > (1-W)P1(1+Ct)(1+G1)^Y
So, we have...
How to Choose Traditional or Roth Contribution, Where Tax Savings in the Former Case Are Invested in a Tax-Deferred Account
(1-W)(1+Ct) < 1 (The "Reinvested Roth Test Relation".)
If the Reinvested Roth Test Relation is true, then contribute to a Roth. Otherwise, contribute to a traditional, and dump the proceeds of the tax savings into a traditional IRA or 401(k).
Finally, in all of the above contribution analysis, you can interchange traditional IRAs and 401(k)'s. You cannot, however, directly convert a 401(k) to a Roth IRA; as far as I know, this must be done via a one-year detour through a traditional IRA.
Granted, there are some subtle advantages to the Roth IRA that I haven't discussed. You may wish to review these to help you make a final decision.
Happy investing!
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