Ignore all other factors, I only want to compare the final total return on my IRA account. I only want to compare between Roth IRA and traditional IRA in terms of final total return. Assume the current Tax rate is T1 and the tax rate when withdraw is Tn, after n number of years, we want to find the final return:

both start with M1 amount of funding:

- Traditional IRA final Mt =?
- Roth IRA Mr=?

Assume

M1: the start funding

T1: current tax rate

Tn: withdraw tax rate

G: gain per year

assume both contribute the same amount just for this year and they both have the same annual return rate

or a Traditional IRA: The initial contribution is made with pre-tax dollars, so the initial investment is M1.

The annual gains are not taxed, so after n years the total balance is M1*(1+G)^n. When withdrawing the money, the entire balance is subject to income tax at the rate Tn. So the final value after taxes is Mt = M1*(1+G)^n*(1-Tn).

For a Roth IRA: The initial contribution is made with after-tax dollars, so the initial investment is M1*(1-T1). The annual gains are not taxed, so after n years the total balance is M1*(1-T1)*(1+G)^n. When withdrawing the money, there is no income tax to pay. So the final value after taxes is Mr = M1*(1-T1)*(1+G)^n. To compare the two options, we need to calculate which one will provide a higher final return. We can simplify the two equations by dividing both sides by M1*(1+G)^n:

Traditional IRA: Mt/M1 = (1-Tn)

Roth IRA: Mr/M1 = (1-T1)

Since both IRAs have the same annual return rate, the comparison between the two comes down to tax rates. If Tn < T1, then the Traditional IRA will have a higher final value after taxes. If T1 < Tn, then the Roth IRA will have a higher final value after taxes. If Tn = T1, then both options will have the same final value after taxes.

Therefore, the choice between Traditional IRA and Roth IRA depends on your tax situation. If you expect to be in a higher tax bracket when you withdraw the money, then a Roth IRA may be the better option. If you expect to be in a lower tax bracket when you withdraw the money, then a Traditional IRA may be the better option.

Conclusion: as long as T1>Tn, traditional IRA is always better in terms of total return. If the current tax rate T1 is higher than the tax rate at withdrawal Tn, then the Traditional IRA will have a higher final value after taxes and therefore be the better option in terms of total return. This is because with a Traditional IRA, you defer paying taxes until you withdraw the money, so if your tax rate at withdrawal is lower than your current tax rate, you will pay less in taxes overall.