To Roth or not to Roth...


"Now the general who wins a battle makes many calculations in his temple ere the battle is fought. The
general who loses a battle makes but few calculations beforehand. Thus do many calculations lead to victory, and
few calculations to defeat: how much more no calculation at all! It is by attention to this point that I can
foresee who is likely to win or lose." The Art of War
A financial planning textbook makes the following remarks regarding when to use Roth vs.. Traditional
IRAs: "Even though individual analysis is best, here are some general considerations...when the individual expects
to be in a higher tax bracket in retirement than at the time of the contribution, the Roth IRA is the more
appropriate vehicle...if an individual expects his or her tax rate to go down in retirement, the opposite is
true...However, for many taxpayers this is a relatively unlikely scenario...if the tax rates remain the same, there
are still advantages to the Roth IRA...one of the Roth's powerful features is that distributions are not required
during the participant's lifetime." The financial planning industry is rife with "rules of thumb." But upon
inspection we find the following evidence: if the tax rate goes up in retirement, they are correct  the Roth is
better. If the tax rate goes down, the Traditional IRA may or may not be better, but this is
actually quite common for those in their 30's or 40's. If tax rates stay the same there is absolutely no reason to
use the Traditional IRA, but this has nothing to do with the distribution requirements as they suggest.
Adhering to rules of thumb often applies a level of rigor to a question that is not likely to produce
the proper course of action. In the Art of War, they explain that the general who does many calculations
before battle wins, whereas the general who is lax typically loses. Here you want to review your circumstances and
apply sufficient rigor to make a sound determination for the proper course of action. If you hire an advisor, you
want to be sure they also apply such a level of rigor. Let's look at this situation specifically as the formulas
are not difficult. We will focus on an investment in bonds and determine the break even tax rate at withdrawal
that tells you whether you are better off in Roth vs. a Traditional IRA.
You can read this article in two ways: you can review the algebra so you understand why the formula works. Or
you can cut to the chase by going to the area where the needed formulas are highlighted. Below the formulas is a description of how to use the formulas.
We will assume the investment is in bonds which are taxed based upon the income the produce every year in
a taxable account. So we have:
R = return
N = number of periods where we receive R
T = tax rate (we will split this by period
later)
The formula of the future value of $1 invested in a Roth IRA at R for N periods is simply
Future value Roth = (1 + R)^N
or 1 + return raised to the power of the number of periods invested. This approach takes into account the
compounding of returns. For example, if I earn 10% on $1 for 2 years, I will not have $1.2 at the end (or 10% on
the original dollar added for each year invested). The reason is that I will earn 10% on the first period of
interest of $0.10, which is $0.01. So I will actually have $1.21. This seems small, but if you increase the amount
of money and extend the period it becomes quite large. So if I use the formula above as 1.1^2, I get the $1.21, so
that is the correct way to consider compounding.
For a taxable account, the formula compounds the after tax returns in the same way:
Future value taxable = (1 + R  T * R)^N
So we can see that as long as T > 0 and R > 0 the future value of the Roth will always be more than
the taxable account.
For the Traditional IRA, we pay tax at the end on the full amount, but we also receive a tax benefit in the
beginning and invest it in a taxable account. The formula is
Future value Traditional IRA = (1 + R)^N  T * (1 + R)^N + T * (1 + R  T * R)^N
So the first group is the return of a Roth, the second group subtracts the taxes on that return and the third
group adds back the returns on the taxes saved when you made the contribution. It is instructive to note that is
you invest the tax savings in a Roth, the third term would be T * (1 + R)^N. This would be wiped out by the
second term and the Roth would be precisely equal to the Traditional IRA. The value in the Roth is that for a
Traditional IRA you take away the tax free earnings on the taxes you pay up front and add back taxable earnings on
it. As long as the tax rate T > 0, the Roth is better.
Now let's see what happens when you subtract the Roth from the Traditional IRA to see when they are
the same.
(1 + R)^N  T * (1 + R)^N + T * (1 + R  T * R)^N  (1 + R)^N = 0
Some simple algebra leads to the following results:
((1 + R)^N) / ( (1 + R  T * R)^N) = 1
Since we already know the top term is the future value of a Roth and the bottom is the future value of a taxable
account, as long as T > 0 and the returns are positive, this fraction will always be greater than 1. The only
time it equals one practically speaking is when the tax rate = 0.
Now we need to do the same algebra, but consider that the tax rates change over time.
T1 = tax rate when the contribution is made
T2 = tax rate when a withdrawal is made
K is the ration of tax rates: K = T1 / T2
For simplicity we assume T1 is the rate up until retirement or some sort of period of unemployment and then T2
starts for bonds. The reason for this assumption is that as soon as the tax rate drops, you would be motivated to
convert the Traditional IRA to the Roth. So we now do the same operation and substitute the proper tax
rates:
(1 + R)^N  T2 * (1 + R)^N + T1 * (1 + R  T1 * R)^N  (1 + R)^N = 0
This reduces to
(T2 * (1 + R)^N) / (T1 * (1 + R  T1 * R)^N) = 1
Now the tax rate to use should be the marginal tax rate considering both state and federal taxes where
applicable. The marginal rate is the tax rate if you changed only those dollars that were switched from Traditional
IRA to Roth IRA. It is not your average tax rate, but would be the highest tax rate to which you are subject on the
tax scale published by the IRS. If you transfer so much that it pushes you down the scale for a portion of the
taxes, you would need to consider the specific tax dollars saved by the change as the marginal amount and divide by
the amount you contributed. To include state taxes, your total tax rate T for TF (federal tax rate) and TS (state
tax rate) is
T = 1(1  TF) * (1  TS)
The substitution for T2 = T1 / K
Bond Formula: ( (1 + R)^N) / ( (1 + R  T1 * R)^N) = K
So here we have a formula that will allow us to calculate the break even tax rate we need at withdrawal in
order to find if the Traditional IRA is better than a Roth. It is interesting to note what it is saying. The
numerator is the future value of a Roth and the denominator is the future value of a taxable account. When you
save taxes up front using a Traditional IRA these funds are invested in a taxable account. As you saw above,
Traditional IRA future value formula takes away a return on these tax savings that
is invest like the Roth and substitutes a return for a taxable account. So you have exchanged tax free
income for taxable income for the life of the IRA. The taxes on these earnings must be recovered. So you save
taxes up front at a high tax rate when you contribute to the Traditional IRA and then need to pay taxes at
withdrawal at a lower tax rate. The proportion of these two tax rates is the same as the proportion of
the future value you would have in the two account types (Roth or taxable) at the end of the
period. If you do not expect tax savings from a lower tax rate at a later point in time, you have no reason to
use a traditional IRA at all irrespective of how long you live. But if your tax rate does fall, you need to be sure
it falls sufficiently to recoup the taxes paid.
To apply it say we know our current tax rate T1 is 25%, we expect a return of 10% for 40 years before our tax
rate drops. There we have K = 2.51. So T2, or the break even withdrawal rate is 9.97%. If we think our tax rate in
retirement will be below 10%, the traditional IRA makes sense. This works in precisely the same manner if you
convert from a Traditional IRA to a Roth, except you use today's tax rate for T1. So if I pay tax at 25% now and
convert to a Roth, I would need to save taxes when I would have withdrawn the Traditional IRA in 40 years at a rate
of 10% or more to justify the move. So either way I use the Traditional IRA if I expect my tax rate at withdrawal
to be lower than 10%, otherwise the Roth is better. Let's look at a chart to see how different scenarios
work:
25% Initial tax rate 
15 Years 
40 Years 
10% Bonds 
17.17% 
9.97% 
5% Bonds 
20.89% 
15.48% 
Here we can see the typical rule of thumb suggesting that the traditional IRA be used if tax rates fall may not
be useful. In the case of a 10% return for 40 years, the taxable income would have to drop literally to $0 from
$83,600 for there to be any benefit. Using 5% for 15 years, a period of unemployment with $34,500 taxable income
would produce a nice gain. To see the impact, consider that you are in the 28% tax bracket and drop to the 25%
bracket upon retirement. You can use the future value formulas above to determine the future value of a Roth
and Traditional IRA invested at 10% for 40 years. The rule of thumb suggests you use a Traditional IRA, which
would have a value that is roughly 85% of the value of the Roth. This is a substantial loss. Using 5% for returns,
that figure is still about 91%. (By discounting back to present value the loss is the same
percentage).
The formulas above you are free to make whatever assumptions you wish. If you are using an advisor, you can
check to see if you are comfortable with their assumptions and check their decision process to see if it
works. This understanding may increase your confidence in your advisor.
This analysis only considers bonds. The formulas for stocks are based upon the ones above with some minor
adjustments for tax considerations. All the formulas can be used to determine whether stocks or bonds are best
placed in an IRA, when conversions should be made, which tax deferred or exempt vehicles make the most sense, etc.
For more information, please inquire.
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