And Roth IRAs are funded post-tax. That means he paid taxes on the income already, before he put the money into the IRA. That's why it's "tax-free".
0.8 * (1.07 ^ 20) == (1.07 ^ 20) * 0.8
(0.8 * 1.07) ^ 20 and (1.07 ^ 20) * 0.8, which are not equal. I'm assuming that 0.8 is the percentage after tax, 1.07 is the deposit, and 20 is the growth.
1. Roth IRA: Pay taxes on the value of the principal up front, but not pay taxes on appreciation. Let’s say a 20% tax rate (made up for simplicity), so 20% x .01 x 1M = $2,000. Thiel writes a check for this relatively small number.
2. Traditional IRA: pay no (immediate) taxes on the principal up front, but pay taxes on the appreciated value. So assuming the same tax rate: 20% x 1M x $1000 = $200M.
Your math is correct, of course. The mistake is assuming that Theil would pay taxes by selling 20% of his inordinately valuable $.01 PayPal shares, even though technically (after appreciation) those would be worth $200M. That would be stupid —- obviously he wouldn’t do that, he would just write a check from his bank account — selling that principal would be absurd since such shares can’t be purchased or sold on any liquid market. The entire purpose of using a Roth IRA here was that Theil had access to a unique, non-liquid asset with a potential for extremely high appreciation, and he wanted to insure himself against paying taxes in the likely event that happened.
IMHO a part of the problem here is that the initial “price” of PayPal wasn’t determined by the market. So allowing Theil to pay taxes at that rate made it absurd for Theil to sell any of his principal to pay taxes. Even if you grant that PayPal was basically a lottery ticket, we shouldn’t use a middle class tax vehicle to shield taxes on assets that the middle class (broadly) doesn’t have access to in public markets.
If $1,667 is a correct valuation for the stock -- meaning, the stock is valued by a liquid and well-functioning market -- then maybe there is no difference! From that perspective, there's no difference between writing a cash check for $333 and keeping the principal, vs. selling $333 of your principal and having a lower return in the long run. The long-term expected value is basically the same, you're just investing your money in a lottery ticket. But the lottery ticket is available to anyone else, and it's (in theory) fairly valued by the market.
But let's suppose that $1,667 isn't a fair value determined by a liquid and well-functioning market. Let's suppose you have strong reason to believe that this stock will be worth quite a bit of money, and moreover the market valuation is low because it's an asset that only you have access to purchase (e.g., because you're a founder with pre-IPO shares that can't be purchased for any price on any liquid market, and so the valuation is something absurdly, comically low that your lawyer scribbled on an Operating Agreement in order to minimize your taxes.) You're certainly not going to sell $333 of your massively undervalued assets to pay the taxes.
In that case the whole logic of the Roth IRA falls apart. The Roth IRA is supposed to be a savings vehicle for the middle class, where investors pay a tax on income received, then get a tax deduction on appreciation (i.e., one that is subsidized by the US tax payer.) But what is a "fair" way to calculate your income when it includes illiquid stocks? For certain assets, the "fair price" is literally whatever my lawyer says it is. So allowing highly illiquid (and not-well-priced) assets into this system seems like an invitation for abuse.
There is a separate question about whether the total Roth IRA tax benefit should be capped to something like $10m in your lifetime even when restricted to fairly valued liquid assets. After all: the US taxpayer is subsidizing this as a middle-class retirement program. But disallowing the inclusion of weird, illiquid assets might be a good idea as well.
It's totally possible that I missed modeling some part of the situation that somehow makes more money from the Roth way of doing things. But the models above produce the same results.
That being said, I tend to agree. Its to prevent double tax in which you're trying to promote a better investment. Your long term health and retirement.