# Simple math undercuts case for Roth IRA's reputed higher payoff

August 27, 2006|By Humberto Cruz | Humberto Cruz,Tribune Media Services

An astute reader uncovered a basic mathematical truth that seems to elude many die-hard Roth individual retirement account supporters:

"I've been reading people who sing the praises of Roth IRAs and how they can earn more than traditional IRAs," the reader wrote. "I didn't see how this could be true, so I sat down with a computer spreadsheet program. I put a hypothetical \$2,000 into a traditional IRA and figured a 5 percent return per year for 15 years.

"After 15 years, I deducted the ordinary income tax (at my 15 percent bracket) on withdrawal. I then put a hypothetical \$1,700 into a Roth IRA (the amount left after starting with \$2,000 and paying income tax at 15 percent). I applied the same 5 percent rate of return and let it grow for 15 years. Lo and behold, both accounts returned exactly the same dollars!"

You didn't need a spreadsheet program to figure this out. All you had to do was remember an elementary school math rule: When multiplying a series of numbers, order does not matter. For example, 3 times 4 times 8 is the same as 8 times 4 times 3, or 4 times 3 times 8 (the product is always 96).

As basic as this rule is, it is overlooked by many financial commentators who make the blanket statement that a Roth IRA is sure to give you a better return, or who, without caveats or qualifications, advocate converting a traditional IRA to a Roth.

With a traditional IRA, you may be entitled to a tax deduction on your contribution, but then withdrawals are fully taxed. With a Roth IRA, you cannot claim an upfront deduction, but withdrawals could be tax-free after age 59. This reader saves \$300 in taxes the year he contributes \$2,000 to the traditional IRA and saves nothing right away by contributing to the Roth.

But when he withdraws the money from the Roth IRA in 15 years, there's no tax to pay, while everything in the traditional IRA would be taxed.

Assuming he stays in the 15 percent bracket, the impact of the tax can be expressed mathematically by multiplying the IRA account value by 0.85 (he is left with 85 percent of it). The impact of the assumed investment return of 5 percent a year can be expressed by multiplying the account value by 1.05 15 times in a row, once for each year.

Now for the basic math rule: In this series of multiplications (\$2,000 times 1.05 15 times, times 0.85), it does not matter whether the "times 0.85" comes at the beginning (applied to the \$2,000 Roth contribution) or at the end (applied to the ending traditional IRA balance).

Admittedly, you could contribute \$2,000 rather than \$1,700 to the Roth IRA, paying the additional \$300 in taxes from other funds and leaving more money to grow potentially tax-free.

Also, making periodic partial withdrawals, which is the common course of action rather than taking everything out at once, would affect the calculations. And it could be that, when you take the money out, the value of the IRA itself could push you beyond the 15 percent tax bracket, accentuating the advantage of tax-free Roth withdrawals.

Two other Roth advantages: You can keep contributing after age 70, and you never have to start taking required distributions as long as you live. But many statements about their certain higher tax-free returns are erroneous, ignoring the basic multiplication rule.

yourmoney@tribune.com

Humberto Cruz writes for Tribune Media Services.

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