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:

future value formula

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:

future value of a tax-free account

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 This net amount is then grown tax-free through all periods and no tax liability is owed when the money is withdrawn.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

where; PV = $1,000, rate = 0.07, n = 10, t = 30%

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

Year PV rate FV 
1 $    700.007.00% $    749.00
2 $    749.007.00% $    801.43
3 $    801.437.00% $    857.53
4 $    857.537.00% $    917.56
5 $    917.567.00% $    981.79
6 $    981.797.00% $ 1,050.51
7 $ 1,050.517.00% $ 1,124.05
8 $ 1,124.057.00% $ 1,202.73
9 $ 1,202.737.00% $ 1,286.92
10 $ 1,286.927.00% $ 1,377.01
future value of a tax-free account table

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:

future value of a tax-free account chart

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

hp12c

[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:

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.

Traditional IRAs vs Roth IRAs

Many people are familiar with the different tax treatment between Traditional IRAs and Roth IRAs; unfortunately, often times most individuals are not sure which contribution type is right for them.

Typically, advocates for either type of IRA tend to elect 100% of their annual contribution limits towards their IRA flavor of choice.

Ultimately, if taxes never changed then mathematically it makes zero difference if you elected Traditional IRA or Roth IRA contributions, the ending balances would be the same.

Recall, that the future value of a present value today can be solved for with the following formula:

future value formula

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

In order to determine the after-tax value of a Traditional IRA or a Roth IRA, a new quantity for taxes is added. The key difference is the order that the taxes are taken out, which becomes clearer when comparing the two formulas. Here is the formula for the after-tax value of a Traditional IRA:

after-tax value of traditional IRA

Where:
t = taxes

Comparatively, here is the formula for the after-tax value of a Roth IRA (which can also be referred to as the future value of a tax-free account):

after-tax value of roth IRA

Notice how the quantity (1 – t) is moved to the front of the equation, which makes sense given that Roth IRA contributions are taxed up front, and the remaining amount is invested. Since multiplication is cumulative, then the ending balances after paying taxes should be identical between both accounts.

Let’s assume that you invest $1,000 in a Traditional IRA growing at a rate of 7.00% for 10 years, when you retire your marginal tax bracket is 30%. Plugging those values into the formula for a Traditional IRA would yield the following:

where; PV = $1,000, rate = 0.07, n = 10, t = 30%

Your initial $1,000 contribution grew tax free for 10 years, and after paying a 30% tax in the tenth year you are left with $1,377.01. The table below illustrates the tax free growth for 10 years, the tax paid, and the net after-tax future value:

Traditional IRA
Year PV RateFVTax (30%)
1 $  1,000.007.00% $            1,070.00
2 $  1,070.007.00% $            1,144.90
3 $  1,144.907.00% $            1,225.04
4 $  1,225.047.00% $            1,310.80
5 $  1,310.807.00% $            1,402.55
6 $  1,402.557.00% $            1,500.73
7 $  1,500.737.00% $            1,605.78
8 $  1,605.787.00% $            1,718.19
9 $  1,718.197.00% $            1,838.46
10 $  1,838.467.00% $            1,967.15
TAX $  590.15
Net ATFV $            1,377.01
traditional IRA table

We can represent the data above in the following chart:

traditional IRA ATFV

Let’s plug the same values into the Roth IRA formula to prove that they are identical:

where; PV = $1,000, rate = 0.07, n = 10, t = 30%

Despite starting with a lower initial investment since the tax with Roth IRA contributions are paid up front, the ending value is still identical to the Traditional IRA account:

Roth IRA
Year PV RateFVTax (30%)
1 $     700.007.00% $     749.00
2 $     749.007.00% $     801.43
3 $     801.437.00% $     857.53
4 $     857.537.00% $     917.56
5 $     917.567.00% $     981.79
6 $     981.797.00% $  1,050.51
7 $  1,050.517.00% $  1,124.05
8 $  1,124.057.00% $  1,202.73
9 $  1,202.737.00% $  1,286.92
10 $  1,286.927.00% $  1,377.01
TAX $           –  
Net ATFV $  1,377.01
roth IRA table
roth IRA ATFV

To reiterate, if your marginal tax bracket in the future is identical to your marginal tax bracket today, there is zero difference between a Traditional IRA and a Roth IRA (there may be a perceived difference due to emotional or cognitive biases).

Obviously, the future state of taxation is a huge assumption to make and the world is not that simple. There are a multitude of additional variables and factors that need to be taken into account in order to determine what blend between Traditional IRA or Roth IRA contributions makes the most financial sense for you.

Since there are exactly zero people on the planet who can accurately predict what the marginal tax brackets will look like when you retire (I’d be skeptical of anyone claiming that they can), having money in both account types will provide you with the maximum flexibility in determining your actual tax bracket when you retire.

Remember that the next time anyone says, “Roth IRAs are better than Traditional IRAs so I put 100% of my annual contribution limits into my Roth IRA”, or vice versa. Generalized advice, generally has good intentions, but produces generally bad outcomes, generally speaking.

Taxes aren’t the only variable that should be taken into account when determining the optimal Traditional IRA and Roth IRA blend. One should also consider the following:

  • what will your burn rate be when you retire
  • what will your projected federal tax bracket be
  • do you pay state income taxes
  • will you move to a state that has no income taxes
  • will you move to a state that does have an income tax
  • what is your current blend of qualified vs non-qualified assets
  • what proportion of your employer matching contributions into any qualified plans is made on your behalf

The list above, while somewhat lengthy, is nowhere near exhaustive despite the potential to get exhausted while thinking about it.

All of the concepts above are also applicable to the world of 401(k)s. From a taxation standpoint, Traditional IRAs (which are a type of qualified accounts) are taxed identically to Traditional 401(k) contributions. Similarly, Roth IRAs are taxed identically to Roth 401(k) contributions.

Other important subtleties that you should be aware of for Roth IRAs and Roth 401(k)s are the lack of required minimum distributions. RMDs do not apply to Roth IRAs or Roth 401(k)s which make them an extremely potent estate planning wealth transfer tool.

In my opinion, the best kept secret about Roth 401(k) contributions are the fact that the income limitations that apply to Roth IRAs, do not apply to Roth 401(k) contributions. You could make $1mm dollars a year and still contribute the IRS maximum into a Roth 401(k).

If you don’t have a Traditional IRA or a Roth IRA you can open one at Charles Schwab for commission free trading with an initial funding bonus of up to $500 by clicking here.