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 This net amount is then grown tax-free through all periods and no tax liability is owed when the money is withdrawn.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.