Roth IRA's are post-tax investments, and tax-free withdrawals, while Traditional IRA's (and 401k's) are pre-tax investments, and taxable withdrawals. We want to know how much money will be withdrawn (denoted as Withdrawal) with each type of investment vehicle.
Roth:
We are able to invest
Investment * (1 - Tax_Now).
Limit goes into a Roth IRA, and
Investment * (1 - Tax_Now) - Limit
goes into a taxable account. So we have a tax-free withdrawal of
Limit * (1 + Return)^Years.
We also have a taxable withdrawal of
(Investment * (1 - Tax_Now) - Limit) * (1 + Return)^Years at a rate of Tax_Withdraw,
which results in a withdrawal of
(Investment * (1 - Tax_Now) - Limit) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1).
The -1 and +1 are because you are only taxed on your earnings, not on the principal, when you withdraw.
Add this all together we have:
Withdrawal_Roth = Limit * (1 + Return)^Years + (Investment * (1 - Tax_Now) - Limit) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1)
Traditional IRA:
The traditional IRA is only taxed on withdrawal. So we are able to invest Limit into an IRA, and
(Investment - Limit) * (1 - Tax_Now)
into a taxable account. The IRA results in
Limit * (1 + Return)^Years,
which, after taxes, equals
Limit * (1 + Return)^Years * (1 - Tax_Withdraw).
The taxable account contains
(Investment - Limit) * (1 - Tax_Now)
dollars invested, resulting in
(Investment - Limit) * (1 - Tax_Now) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1)
So our total withdrawal from a traditional IRA is:
Withdrawal_Traditional = Limit * (1 + Return)^Years * (1 - Tax_Withdraw) + (Investment - Limit) * (1 - Tax_Now) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1).
Example:
We'll go with $10,000 gross to be invested with a $5,000 limit for 8 years, at 9% return (so that our investments roughly double). The tax rates will both be 30%.
Withdrawal_Roth = Limit * (1 + Return)^Years + (Investment * (1 - Tax_Now) - Limit) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1)
= $5,000 * 1.09 ^ 8 + ($10,000 * 0.70 - $5,000) * ((1.09^8 - 1) * 0.70 + 1)
= $13,352.41
Withdrawal_Traditional = Limit * (1 + Return)^Years * (1 - Tax_Withdraw) + (Investment - Limit) * (1 - Tax_Now) * (((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1)
= $5,000 * 1.09 ^ 8 * 0.70 + ($10,000 - $5,000) * 0.70 * ((1.09^8 - 1) * 0.70 + 1))
= $12,905.77
Combination:
We can combine some like terms if we define several common variables:
IRA_Return = Limit * (1 + Return)^Years
Other_Fraction = ((1 + Return)^Years - 1) * (1 - Tax_Withdraw) + 1
Now, we can write them as:
Withdrawal_Roth = IRA_Return + (Investment * (1 - Tax_Now) - Limit) * Other_Fraction
Withdrawal_Traditional = IRA_Return * (1 - Tax_Withdraw) + (Investment - Limit) * (1 - Tax_Now) * Other_Fraction
subtracting, we get
Difference = Withdrawal_Roth - Withdrawal_Traditional
= IRA_Return * (1 - 1 + Tax_Withdraw) + Other_Fraction * (Investment * (1 - Tax_Now) - Limit - (Investment - Limit) * (1 - Tax_Now))
= IRA_Return * Tax_Withdraw + Other_Fraction * (-Limit * Tax_Now)
= IRA_Return * Tax_Withdraw - Other_Fraction * Limit * Tax_Now
So in our example, we have
IRA_Return = $5,000 * 1.09^8 = $9,962.81
Other_Fraction = (1.09^8 - 1) * (1 - 0.30) + 1 = 1.6948
Difference = $9,962.81 * 0.30 - 1.6948* $5,000 * 0.30
= $446.64
Just to check, we have:
$13,352.41 - $12,905.77 = $445.64. Good.
So Difference is how much you gain by using a Roth IRA over a traditional IRA.
