Traditional vs Roth IRA - Island of Sanity

# Island of Sanity

Personal Finance

Should you put your retirement money in a "traditional" IRA or a Roth IRA? I've seen a lot of discussions about the relevant merits. And frankly, many of them clearly didn't do the arithmetic.

The key difference between the two is this: WIth a traditional IRA, when you deposit money to your account, you can deduct it from your taxes in the year that you deposit, and you then have to pay taxes on it when you withdraw it. With a Roth IRA, there is no deduction when you deposit, and therre is no tax when you withdraw. So effectively, with a traditional IRA, you don't pay taxes on the money the year you deposit, but you do pay taxes the year you withdraw. You "defer" your taxes. With a Roth IRA, you pay taxes the year you deposit, but you don't pay taxes the year you withdraw.

So which is better? Let's calculate for a simple example. Let's make the following assumptions for an example: You have \$1000 to save. Between when you depoit the money and when you withdraw it, interest or profits on your investment will result in the money doubling. And you are in a 20% marginal tax bracket both when you put the money in and when you take it out. I'll discuss the validity of these assumptions in a moment, but please just go with them for now.

So with a traditinoal IRA: You have \$1000. This is tax deductible, so you deposit the full \$1000 into your IRA. The money doubles to \$2000. Then you withdraw it. You now have to pay 20% taxes, or \$400. You end up with \$1600.

With a Roth IRA: You have \$1000. You have to pay 20% tax on this, so that leaves you with \$800 to deposit into your account. The money doubles. You now have \$1600. There is no tax when you withdraw it, so you end up with \$1600.

Note that the two numbers are exactly the same. Instead of using example numbers we could use a little algebra and prove that the two numbers will always be exactly the same no matter what numbers you use, as long as you assume that the amount invested, growth rate, and tax rate are the same in both cases.

If you don't want to get into the algebra, skip the next paragraph.

Let p be the amount to invest, t be the tax rate, and r be the growth rate. So in this example, p=1000, t=20%, and r=2. But they could be anything. With a traditional IRA, you're ending amount is p x r x (100% - t). With a Roth IRA, the ending amount is (p x (100% - t)) x r. The only difference is the order of the factors. Multiplication is commutative, that is, when you multiply two or more numbers together, you get the same result regardless of the order. 2 x 3 x 4 is the same as 4 x 2 x 3, etc.

I said we'd get back to the question of whether these assumptions are valid. The amount to invest is just an example. Plug in any number and, as long as you invest the same amount regardless of whether it's traditional or Roth, the result is the same. There's no reason why you'd have more money to invest with one type of IRA versus the other, so this is valid.

At this point some say, "Wait, if I have \$1000 to invest in a Roth IRA, who says I have to subtract the taxes from that amount before saving? I could pay the taxes with other money." Well, sure. But then you're not allocating \$1000 to your investment, you're allocating \$1200. If you have an extra \$200 available when you make your deposit, then if you were depositing to a traditional IRA, why not add in that \$200? Comparing putting \$1000 into one investment versus putting \$1200 into a different investment is not a fair comparison.

Likewise, assuming that the money doubles is just an example, and you could substitute any growth rate you believe is plausible in there. Note that I didn't say how long it takes for the money to double because it doesn't matter to the calculation. And it's irrelevant, as long as you assume the same growth for both types of account. Roth IRA money and traditional IRA money are typically invested in exactly the same funds, so both should get the same growth rates.

Finally, what about the tax rates? Here the assumption is NOT valid. The tax rate you pay changes over the course of your life. We have a progressive tax system: if you make more money, not only do you pay more, but you pay at a higher rate. If you make \$20,000 per year you pay 12%, but if you make \$600,000 per year you pay 37%. (As of 2022. The rates and brackets change all the time.)

Most people start out making very little money. Then as they grow in experience and their career advances, their income goes up, and their tax rate also goes up. When they retire their income usually falls rather sharply, so their tax rate goes down.

Let's run the above example again but with one difference: You are in a 20% tax bracket when you deposit, but 10% when you withdraw.

With the traditional IRA: You have \$1000. You pay no tax when you deposit. The money doubles to \$2000. When you withdraw, you pay 10%, or \$200. You keep \$1800.

With the Roth IRA: You have \$1000. You pay 20% tax, leaving you \$800 to deposit. The money doubles to \$1600. There is no tax when you withdraw so you keep \$1600.

Note that in this example, the traditional IRA is better. If we switched the tax rates, if we assumed 10% tax when you deposit and 20% when you withdraw, the Roth IRA would be better.

The difference is that with a traditional IRA you pay the tax rate that applies to you when you WITHDRAW the money, but with a Roth IRA you pay the tax rate that applies when you DEPOSIT the money. So if you are paying a lower tax rate when you deposit than when you withdraw, than a Roth IRA is better. Conversely, if you are paying a higher tax rate when you deposit than when you withdraw, a traditional IRA is better.

There is, of course, the catch that you can't know the future. You don't know exactly what your income will be 5 years or 10 years or 50 years from now. And you don't know what will happen with tax laws. Will rates go up or down? What deductions will be available? Etc.

But still, in general, you know that your income is likely to go up over the course of your working life and then fall sharply the day you retire.

So the simple answer to the question of which is better is: When you are young, not making much money and thus paying a low tax rate, your taxes now are probably lower than they will be when you retire, so Roth is better. At some point your income will rise to the point that your taxes are higher than what you will be paying in retirement, and at that point you should switch to a traditional IRa. Exactly when you hit that point is hard to say. You can only guess. But you should be able to guess in the ballpark.