Investment Performance Comparison Between Roth and Traditional Individual Retirement Accounts

With the recent introduction of the Roth Individual Retirement Account (IRA) along with a significantly improved Traditional IRA, there has been considerable interest in comparing the performance of these investment vehicles. Some confusion regarding rhese comparisons has evolved. In this paper we show that lhis confusion may be attributed to scale and tax differences between [he two investment vehicles. We adjust for these differences by focusing on the after-tax rate-of-relUm on investment for each IRA vehicle. We find that peiformance depends crudaUy on the relationship between an individual's tax rates at the time of investment and at the time o.fwithdrawal.


W
ith the introduction of the Roth Individua.I Retirement Account (JR.A.) in 1997 along with a significantly improved Traditional IRA, the long-standing btent i.nterest in these investment vehicles has reappeared.Coupled with the fact that the U.S. population is aging and the increased emphasis on self-directed retirement planning, the IRA continues to enjoy increasing popularity and importance.
With this success; however, new questions have emerged.Many of these questions have focused generally on a comparison between the Roth and Traditional IR..A.s as retirement vehicles.These questions have been widely discussed in the literature and popular press.For example, Johnson (1999) and Stevens (1999)

focused on
Readers with comments or questions are encouraged to contact the authors via email.whether one should.convert an.eXIstmg Traditional IRA to a Roth IRA.Which IRA is better for estate planning purposes has been addressed by Amhony (1999) and Kilpatrick (1999).Fi-naHy, Shanney-Saborsky (1999) touched on which IRA is a beuer investment vehicle.The fundamental reason these questions have emerged is that the Roth and Traditional IRAs provide significantly different tax incentives.
The Traditional IRA is a taxable investment that provides an immediate tax deduction in the amount of the sum invested subject to limitations.However, any withdrawals including the initial amounts invested are taxable as they occur.In effect, there are two tax savings benefits from investing in a Traditional IRA.There is an immediate up-front deduction; and second, any future earnings are not taxed until the funds are withdrawn.For exa.,nple, if one invests $2000 in a Traditional IRA and is in a 40 percent marginal tax bracket (i.e. from federal, state, and The Journal of Applied Business Research local taxes), the immediate tax savings would be $800.Moreover, taxes on any future earnings as well as the $2000 invested are postponed until withdrawals occur.The Roth IRA, on the oLlIer hand, is a nontaxable investment.That is to say that, unlike the Traditional IRA, there is no immediate tax savings benefit; however, the total account including the investment amounts and any future earnings are never subject to taxes, even if withdrawn.
The purpose of this paper is to compare the Traditional IRA with ti'1e Roth IRA from an after-tax investment point of view.The most appropriate investment performance measure is the after-tax rate-of-rerurn on investi"Ilent.The analysis reveals that which IRA is a better investment depends crucially on an individual's tax rates at the time funds are invested as well as at the time funds are withdrawn.Specifically, this analysis finds that the Traditional IRA will have a higher after-tax return-on-investment if the individual's tax rate at the time of withdrawal is less than the individual's tax rate at the time of investment.On the other hand, if the individual's tax rate at the time of withdrawal is greater than the tax rate at the time of investment, the Roth IRA will perform better.If the individual's tax rates at the times of investment and wit.lIdrawal are the same, both lRAs 'NiH perform equally as well.
This paper is organized as follows.The next section of this paper discusses dollar accumulations in the two IRA vehicles.This is followed with a discussion of IRA after-tax returns.Sensitivity of IRA returns with respect to the various input parameters is then investigated.A discussion of IRA choice follows.Finally, conclusions and suggestions for future research are presented.

IRA Accumulations
Much of the confusion regarding the performance of both the Roth and Traditional IRAs is a consequence of focusing on the accumulation amounts for each investment vehicle rather than 56 Volume 17, Number 1 on the return-on-investment.
The difficulty stems from the fact that the Roth and Traditional IRAs are taxed differently as described above.
To see this, consider a $2000 investment in a Roth IRA.This IRA provides no immediate tax benefit, but rather, the initial investment amount and any future earnings are never taxed, even if withdrawn.Suppose then, one invests the $2000 for ten years with an annual rate-ofreturn of 10 percent.The accumulated amount would be $5187, which is not taxable.Alternatively, if one invests $2000 in a Traditional IRA, the accumulated amount would also be $5187.However, this amount is taxable and would be less than the Roth IRA accumulation on an aftertax basis.Assuming the individual's tax rate is 40 percent, the after-tax accumulation would only be $3112.However, the up-front tax savings on the Traditional IRA could also be invested in a taxable account and, assuming a 40 percent tax rate, the $800 would grow to $1433 after taxes in ten years.Therefore, the total Traditional IRA after-tax accumulation would be $4545, which is still less than the Roth IRA accumulation.Consequently, the Roth IRA would be perceived as a better investment on the basis of after-tax accumulations.
We can formalize these arguments as follows.Letting Thus, the total Traditional IRA and initial tax-savings would accumulate to:

The Journal of Applied Business Research
The net amount, Roth accumulation less total Traditional accumulation, would be: Roth and Traditional accumulations are presented in Tables 1 and 2 using different input parameters.The tables reveal that if an individual's tax rate remains constant, the after-tax accumula.tion in a Roth IRA will always be greater L~an the after-tax accumulation for a Traditional IRA investment.Thus, the perception is that the Roth IRA is a better investment vehicle,

IRA Investment Returns
Although the previous section presents a convincing argument indicating L"1at the Roth IRA shouid be the better investment vehicle, this section investigates this proposition further.The previous discussion found that the Roth IRA accumulations wiU always be larger than the Traditional IRA accumulations.However, the results rest on two important assumptions.First, the Traditional IRA accumulation includes the upfront tax savings as an investment even though it is not apart of the IRA.Second, the individual investor tax rate is assumed to remain the same at both the time of investment and at the time of withdrawaL These two assumptions are most certainly inappropriate for comparing the two IRA investment vehicles.The first assumption includes the tax savings accumulation as part of the IRA accumulation.However, the tax savings is not part of the IRA and it is highly unlikely, from a practical manner, that any investor would include it as such.The correct interpretation is to recognize that, with the differential tax treatment, the two IRA investments will necessarily differ in scale.The Traditional IRA initial net investment amount will be less than the Roth IRA investment amount by the size of the t3-X savings.With differences in the scale of the investments, it is crucial to compare them on the 57 Volume 17, Number 1 basis of their after-tax rates-of-return.Second, the assumption that the individual tax rates at the times of investment and withdrawal are the same is clear;y inappropriate.Although reasons for this are obvious, this reality win be discussed in a later section of this paper.In any event, a proper comparison should certainly allow for L;e possibility of differential tax rates at the times of investment and withdrawal.
We now compare the Roth and Traditional IR.A.s on the basis of their after-tax rates-ofreturn.Letting R = annuaL rate-of-retum on IRA funds N "' e number of years until funds are withdrawn A '" amount invested in IRA T! '" individual tax rate at the time of investment Tw"" individual tax rate at the time of withdrawal Then, sInce the Roth IRA is not taxable, the after-tax rate-of-rerurn is defined as: Thus, the Roth IRA return is simply the rate-ofreturn earned on the fund.This makes sense since thl~ Roth IRA is never taxable.
For the Traditional IRA, the after-tax rateof-rerum is defined as: Comparing equations (1) and ( 2), we can see that: Therefofe, a comparison of a Roth IRA with a Traditional IRA on the basis of after-tax rates-ofreturn shows that performance is independent of the amount invested, but depends crucially on the relationship between the individual's tax rates at the times of investment and withdrawal.The Traditional IRA performs better (worse) than the   Roth IRA if the tax rate at withdrawal is less (greater) than the tax rate at the time of investment.If the tax rates are the same then the Roth and Traditional IRAs perform equally as well.
Table 3 provides a sample of acmaI calculated returns for both a Roth and Traditional IRA for different representative individual tax rates and different times of withdrawal.The average annual ret . . . .1Tn (before taxes) is assumed to be 10 percent.
We now investigate how L.1.eafter-tax rateof-return on L.1.eTraditional and Roth IRAs vary with respect to the different input parameters.Letting X == (l-Tw)/(l-Tl), we can write: RTrold = (l+R) X IIN -1 X represents the relative tax rate ratio.We first compute the sensitivity of RTmd with respect to X: We next compute the sensitivity of RTrad with respect to the time of withdrawal N. U sing the property d/dx (aU(x) = aU In(a) (du/dx), see Selby (1969) for example, we can write: The sign of this expression depends upon the value of X as follows: Note again that dRRoth/dN is zero.Therefore, equation ( 4) can be interpreted as follows.When the Roth IRA performs better than the Traditional IRA, that is X < 1 (see (3)), then the Traditional IRA return performance approaches that of the Roth.In the limit that N ~ "", notice L.1.atRTrad ~ RRotll and the two lRAs are equivalent.However, in the case that the Tradi- tional IRA performs better that the Roth (Le.X > 1), then the performance of the Traditional IRA declines relative to the Roth as N gets large.
In the limit as N ~ co, the RTrad will approach RRoth (from above).Finally, if the two IRt\s are equivalent (Le, X = 1), then like the Roth, the Traditional IRA performance does not change with N.An in all, as L'Ie withdrawal time is delayed, the two IRAs become closer in return performance.
Which IRA is Better?
The analysis presented above indicates that, on the basis of after-tax return on investment, the Traditional IRA performs better (worse) than the Roth IR.t>.For most individuals, one can argue persuasively ::hat the tax rate at the time of investment will be greater than the tax rate at the time of withdrawal.There are a number of reasons why this statement is likely to be true.First, most individLRals will have lower income during retirement than during their working years.The design of many retirement plans favors this situation.Second, many individuals move in their retirement years to states which tend to have lower individual tax rates.Table 4 shows individual income tax rates for selected states.There is a tendency for people to work in states with higher tax rates.Third, a person can and win most probably "time" the IRA investment decision to coincide with periods of high income.Fourth, Co~ri nt © 2-001.All ri tits reserved.within limits one can "time" IRA withdrawals with periods of lower income when the need is highest.Finally, for a variety of reasons, the general expectation is that tax rates will probably fall in the foreseeable future.For all of these reasons, most individual tax rates will be lower at the time of withdrawal than at the time of investment.As such, the Traditional IRA will most likely be the preferred investment choice.

Conclusions
In this paper we show that which IRA, Roth or Traditional, is better from an after-tax return on investment viewpoint depends cruciaHy on the relationship between an individual's tax rates at the time of invesffi1ent and at the time of withdrawal.We also emphasize the importance of focusing on the return on investment.Since the two IRAs are treated differently with respect to taxes, a proper comparison must not ignore the differences in scale of the two investments.Finally, we argue that individual tax rates at the time of withdrawal will generally be less than individual tax rates at the time of investment; and as a consequence, the Traditional IRA will be a better investment vehicle for most individuals.

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Volume 17, Number 1 Sugge!>'tions for Future Research There are at least two potentially fruitful areas of future research rdated to this topic that may provide additional insight.One would be to focus on theoretical aspects not addressed in this analysis.Specifically, although allowance for differential tax rates is incorporated ill this study, this analysis assumes one investment and one withdrawal.
However, multiple withdrawals (with a series of different tax rates) of a single investment may be more realistic and may provide further insight.Observation of how the IRAs differ in withdrawal restrictions and how each can be used as a tool in estate planning could be incorporated into the analysis.These subtle differences are virtually ignored in this present study.
A second area of focus could he an empirical investigation of current investor behavior regarding IRA choice.IRA choice could be investigated demographically.Data collection could employ the careful use of either surveys or isolated samples.Trends in individual tax rates both before and during retirement could also be investigated.UJ

N
= the number of years until funds are withdrawn R = the annual rate-of-return on IRA (before taxes) A = dollar amount invested T = individual tax rate Then, an investment of $A in a Roth IRA would accumulate to A (1 + Rt in N years.Alternatively, an investment of $A in a Traditional IRA would accumulate to A(1-T)(l + R)N on an aftertax basis.However, the Traditional IRA upfront tax savings would accumulate to: AT[l+R(l-T)]N.

Table 3
After-tax Percentage Returns on TmditionallRA * (Time of Withdrawal Varies) Nominal Retm:n Per Year: 10%, Amount lnv('stcd (Befm:e-fux) $2,000 * Note: Roth IRA After-tax Return is 10% in all Cases when the i.ndividual tax rate at the time of investment is less (greater) than the individual's tax rate at LfJ.e time of withdrawaL So_