How to Calculate the Future Value of a Tax-Free Account

Recall, that the formula utilized to calculate the future value of a lump sum is as follows:

Where:
FV = Future Value
PV = Present Value
r = rate
n = periods

Calculating the future value of a tax-free account incorporates the tax paid on the money prior to investing it in the tax-free account. We can account for the taxes paid by adjusting the present value for taxes:

In essence, the present value is reduced by the tax owed today and becomes the net amount invested. This net amount is then grown tax-free through all periods and no tax liability is owed when the money is withdrawn.

Assume you have a present value of $1,000, that will grow at a 7% rate for 10 years, and the initial tax owed is 30%. We can calculate the future value of the tax-free account by plugging those variables into the formula as follows:

Using Excel, we can model what occurs during each of the ten periods:

Year	PV	rate	FV
1	$    700.00	7.00%	$    749.00
2	$    749.00	7.00%	$    801.43
3	$    801.43	7.00%	$    857.53
4	$    857.53	7.00%	$    917.56
5	$    917.56	7.00%	$    981.79
6	$    981.79	7.00%	$ 1,050.51
7	$ 1,050.51	7.00%	$ 1,124.05
8	$ 1,124.05	7.00%	$ 1,202.73
9	$ 1,202.73	7.00%	$ 1,286.92
10	$ 1,286.92	7.00%	$ 1,377.01

Notice how the initial present value is reduced by the current tax rate. In the United States, this is how the future value of a Roth IRA would be calculated. We can illustrate the table above visually with the following chart:

Using an HP12C calculator, you can calculate the future value of a tax-free account using the following keystrokes:

[1000][ENTER]
[.][7][*][PV]
[7][i]
[10][n]
[FV]

The formula can be rearranged as follows to find the present value of a tax-free account:

The present value version of the tax-free account formula is usually only seen on tests which require you to calculate the initial investment an investor made in the past, given some current value in the future.

A copy of the Excel model can be found here.