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So, I suspect I am missing something huge about this. But I notice there is a lot of talk about retirement accounts recently such as 401-Ks and IRAs. I understand these come in two flavors, traditional and Roth.

Roth

I understand Roth accounts entirely and the advantages, one deposits money, paying the taxes on it, it then grows tax free and later in life one has a larger amount of money than they deposited with and no tax liability, they don't pay any tax on the gains, this means they get capital gains income at a 0% rate. That is a very obvious advantage.

Traditional

Traditional accounts however I fail to understand. It seems one deposits money without paying taxes, this money then grows to a larger sum over time from capital gains. Finally one takes this money out later in life, paying taxes on both the gains and the original deposits. It seems one ultimately pays taxes on all the money they saved, albeit later on.

Why is this advantageous to do as opposed to just paying taxes as one goes on the income and capital gains, particularly when taking taxable income in retirement can jeopardize social security payments and Medicare?

Example

To give a concrete example, say between the ages 20 and 30 one places $5000 a year into either a normal brokerage account, a traditional retirement account or a Roth account, and keeps this money in the account for 30 years. (The numbers are contrived but are mostly to illustrate my confusion with only simple math.

Roth

Pays a total of $50,000 into account, pays taxes on $50,000.
Makes... Say $400,000 in capital gains over 30 years.
Withdraws $450,000 and has paid tax on $50,000.

Traditional

Pays a total of $50,000 into account, pays no taxes on the $50,000.
Makes... Say $400,000 in capital gains over 30 years.
Withdraws $450,000 and pays taxes on all $450,000.

Brokerage

Pays a total of $50,000 into account, pays taxes on $50,000.
Makes... Say $400,000 in capital gains over 30 years, pays taxes on the $400,000, but at an advantaged capital gains rate
Withdraws $450,000 and has paid taxes on all $450,000, but less on the $400,000 of capital gains.

The core of my question, is why then does the traditional retirement account save money, it still seems one has paid taxes on the entire sum, and possibly losing the advantageous capital gains rate on it, that would otherwise reduce the taxes on a large proportion. What is the advantage?

Please note for this question I am explicitly avoiding the potential different rates of fees and returns for the various accounts, and assuming the same investment at the same expense ratio, as well as ignoring any potential employer match.

The piece you're missing about a traditional IRA is the up-front tax savings that may apply. If you were taxed at 25% and able to deduct the full contribution then you could contribute 33% more to a traditional IRA (assuming room under the annual contribution limit) than you could to a Roth IRA for the same net cost, or have 25% more cash in hand after contributing. Note that 25% was just an easy number to use, there's a deduction phase out that varies based on whether you have a 401k.

If you don't qualify for deductible traditional IRA contributions, then the only benefit is deferred taxation on the growth, you'll pay tax in retirement at a likely lower rate than you pay now. If you can't get a deduction but can still contribute to a Roth IRA, then contributing to the Roth IRA is most likely the better option. If you can't get the deduction for traditional IRA contributions and can't contribute to a Roth IRA due to income, then the deferred tax is still most likely a benefit over a brokerage account. Though many would advocate a backdoor Roth contribution in that case.

The point you are missing in the examples is what you are doing with the money saved on taxes with the Traditional IRA.

Assuming that that money is also invested, if you have a completely flat tax with no deductions on both contributions and distributions, a Roth IRA and a Traditional IRA are equivalent; because taxes are multiplicative, rate of return is multiplicative, and multiplication is commutative, it doesn't matter whether you pay taxes now or later.

As an example, assume you have $10,000 of pretax money and a flat tax rate with no deductions of 20%. If you left it for 10 years at a rate of return of 7% per year, you'll find that both 10000 * (1 - 0.2) * 1.07^10 and 10000 * 1.07^10 * (1 - 0.2) give you the same answer: $15,744.

Let's go to your examples:

Roth

Pays a total of $50,000 into account, pays taxes on $50,000. Makes... Say $400,000 in capital gains over 30 years. Withdraws $450,000 and has paid tax on $50,000.

Let's assume the same 20% flat tax rate with no deductions, so $10,000 is being paid in taxes. That means that $40,000 is being invested. Assume a x10 rate of return to get $400,000 in gains. End result: $440,000.

Traditional

Pays a total of $50,000 into account, pays no taxes on the $50,000. Makes... Say $400,000 in capital gains over 30 years. Withdraws $450,000 and pays taxes on all $450,000.

No taxes are paid on the $50,000, so the extra $10,000 that would be paid can be invested. Because more is invested, more will be earned; instead of getting $400,000 of capital gains, with the x10 rate of return you'd get $500,000, for a grand total of $550,000. At a 20% tax rate, that's $110,000 in taxes, leaving you with $440,000, the same as the Roth.

The core of my question, is why then does the traditional retirement account save money, it still seems one has paid taxes on the entire sum, and possibly losing the advantageous capital gains rate on it, that would otherwise reduce the taxes on a large proportion. What is the advantage?

If your effective tax rate in retirement is lower than your effective tax rate while working, you will obviously wind up in a better position: you may pay more absolute dollars in tax, but that's because you have more money; the effective tax rate is lower.

However, because the US tax structure is not flat, and has deductions, even if the tax brackets and your income stay exactly the same in retirement as when working, you will pay less in taxes. This is because money that you contribute to a Roth account is generally taxed at your marginal (highest) tax rate, whereas money contributed to a Traditional IRA generally deducts from the marginal tax rate. Additionally, when you withdraw from your Traditional IRA, at least some portion of the withdrawal will fall under deductions and lower-than-marginal tax brackets, resulting in less overall tax.

I agree with you about Roth accounts. Pay the tax now and you only pay tax on your contribution not any gains. But the advantage of Traditional arrangements is as you put it "albeit later on."

Generally speaking people begin earning at the lowest tax rates. Through their lives they increase in earning power and thanks to progressive tax schemes your rate increases as you earn. The theory is, you avoid paying the tax now at today's rates because when you retire you'll be earning less and subject to a lower rate.

When you retire, you're not distributing $450,000 and paying tax at once. You're taking, maybe $30,000 for the year as income, which is subject to income related deductions. A single retired person who took $30,000 in distributions from their traditional account in 2018 will pay tax on $18,000 of income thanks to the $12,000 standard deduction. So you're only paying a tax on 60% of the distribution, at presumably a much lower bracket than you were in when you contributed the money. Granted, using your numbers, this is still a disadvantage because only 11% of your funds are contributed money ($50k/$450k). So 11% of the money is subjected to a 35% tax versus 60% of your money being subjected to a 10% tax.

Obviously, the assumptions related to future income taxation could be completely wrong. It's possible that we live in a higher rate and lower deduction future that totally invalidates the value of traditional retirement accounts. Additionally, Traditional arrangements are subjected to "Required Minimum Distributions" that Roth arrangements are not. People are generally working, living and earning longer and it's possible that RMDs would require you to take a taxable distribution before you would otherwise want to when you are still earning at your peak earnings and the distributed funds would be subjected to the then current top marginal rate.

But its also possible that we go crazy in the future and subject Roth distributions to income taxes as well.

The reality is most people want a deduction this year over any other consideration, and traditional arrangements offer that.