# Why a Roth IRA is Better - With Math

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Which is better? A Roth or a Traditional?

That’s a hard question to answer. Better in what way? And a Roth or Traditional *What*? Since IRAs aren’t the only thing that can be “Roth” now. Hello, Roth 401k.

I’m going to attempt to answer this question with math. And I’m going to assume we’re talking about IRAs, and that “better” means more money in retirement - or whenever you cash out.

I’m also going to make some maybe-not-so-good assumptions:

- The tax rate now and in retirement (later) is the same.

Before I go into details, I’d like to point out what I used to think: there really was no difference between the two, given the same tax rate now and later. This is because of the associative property of multiplication:

```
Roth: ((1 - tax) * invested) * growth
Traditional: (1 - tax) * (invested * growth)
```

So, if you invested $10,000, the tax rate was 10%, and it doubled by the time you withdrew, you’d end up with $18,000 either way:

```
Roth:
((1 - 0.1) * $10,000) * 2
(0.9 * $10,000) * 2
$9,000 * 2
$18,000
Traditional:
(1 - 0.1) * ($10,000 * 2)
0.9 * $20,000
$18,000
```

So, yeah, it doesn’t matter *if all you had was that ten grand*. But I forgot about the rest of your paycheck.

For the following math, I’m going to make some assumptions. Maybe they’re a bit off the mark, maybe they won’t apply to you. But here they are:

- I’m only going to look at the growth of one year of investment. That is, what is invested in a year, and how much from that year is available years later.
- Taxable income is $100,000.
- IRA max, Roth or otherwise, is $10,000. Which is more than it currently is, but it’s to make the math easier.
- Spend is $50,000. The rest is available to invest.
- The tax rate, both now and later, is 10%.

First, let’s look at how much someone would have after their investment doubles *without* investing in either a Roth or Traditional IRA:

```
Taxable Income = $100,000
After-Tax Income = 0.9 * $100,000 = $90,000
Spend = $50,000
Investment = $90,000 - $50,000 = $40,000
Investment Return = $40,000 * 2 = $80,000
Investment Gain = $40,000
Tax on Gain = 0.1 * $40,000 = $4,000
After-Tax Return = $80,000 - $4,000 = $76,000
```

Now, let’s look at how much someone would have after their investment doubles with investing in a Traditional IRA:

```
Taxable Income = $100,000
Traditional IRA Investment = $10,000
Taxable Income after IRA = $100,000 - $10,000 = $90,000
After-Tax Income = 0.9 * $90,000 = $81,000
Spend = $50,000
Investment = $81,000 - $50,000 = $31,000
Investment Return = $31,000 * 2 = $62,000
Investment Gain = $31,000
Tax on Gain = 0.1 * $31,000 = $3,100
Traditional IRA Return = $20,000
Tax on Traditional IRA Return = 0.1 * $20,000 = $2,000
Total Return = $62,000 + $20,000 = $82,000
After-Tax Return = $82,000 - $3,100 - $2,000 = $76,900
```

Finally, let’s look at how much someone would have after their investment doubles with investing in a Roth IRA:

```
Taxable Income = $100,000
After-Tax Income = 0.9 * $100,000 = $90,000
Roth IRA Investment = $10,000
Spend = $50,000
Investment = $90,000 - $50,000 - $10,000 = $30,000
Investment Return = $30,000 * 2 = $60,000
Investment Gain = $30,000
Tax on Gain = 0.1 * $30,000 = $3,000
Roth IRA Return = $20,000
Tax on Traditional IRA Return = $0
Total Return = $60,000 + $20,000 = $80,000
After-Tax Return = $80,000 - $3,000 = $77,000
```

So the clear winner of these three is the Roth IRA, with the investor ending up with $77,000.

The traditional IRA comes in second with the investor ending up with $76,900.

And doing neither comes in last, as expected, with the investor ending up with $76,000.

In case you want to run the numbers yourself, I made a handy-dandy calculator below. Perhaps you can find a situation where a Roth IRA is not better?