The future value interest factor formula that accounts for an annual tax on wealth is as follows:

Where:

Assume that an individuals portfolio which comprises their entire net worth will grow at a required rate of 7% for 10 years and the tax rate on wealth is 1% annually, plugging those values into the formula will yield the following:

An individual with an initial net worth of $1,000,000 would be worth $1,779,100 in ten years.

On the surface, a 1% tax on wealth doesn’t seem like much when this individual’s portfolio is now worth $1.7mm dollars; however, let’s quantify the tax-drag that wealth based taxation has on net worth.

Recall that the formula used to calculate tax-drag is as follows:

where:

In order to calculate the tax-drag we need to compute the value of the portfolio without taxes as follows:

We can now plug the ending values of the tax-free and taxable accounts into the tax-drag formula:

The tax-drag in this particular scenario is 19.44%. In other words, a 1% tax on wealth every ten years eroded 19.44% of the appreciation relative to the tax-free account.

Constructing a tax-drag table that quantifies the effect wealth based taxation has over time will illustrate how destructive wealth taxes are even if wealth taxes are in the low single digits per annum.

First let’s construct the future value table for the tax-free account:

We’ll do the same for the taxable account assuming an annual wealth tax of 1%:

Now we can calculate the tax-drag table using the two tables above:

Notice how the tax-drag is much higher if your net worth grows at a lower assumed rate of return. Higher rates can somewhat offset the tax-drag; however, a 2% rate of return over ten years has a tax-drag of over 53.22%, and that’s if the annual wealth tax is “only” 1%. Imagine having a tax-drag of 53.22% over ten years before adjusting for inflation. Can you say, “Capital flight”?

The Excel model used to calculate the tax-drag of annual wealth taxes can be found here.

Where:

We can define the ending value of a tax-free account as follows:

The formula above is essentially the future value formula with a different set of variables. Recall, that the future value of a taxable account adjusts the rate of return by a quantity that takes into account annual accrual based taxation:

Assume you made a $1,000 investment into a security that was projected to pay a 4% dividend for ten years, with an annual dividend tax of 30%. What is the tax-drag on the taxable account relative to the tax-free account?

In order to calculate the tax-drag we need to plug the variables into both future value formulas:

Plugging those same values to solve for the ending value of the taxable account yields the following:

Now that we have calculated the ending values for both the tax-free and taxable accounts, we can plug those values into the tax-drag formula:

Notice that the tax-drag of 33.7% is higher than the annual tax rate of 30%. The additional tax erosion above and beyond the 30% annual rate of taxation is due to paying 30% every year in the taxable account, relative to not paying any taxes in the tax-free account.

We can illustrate the longer term detrimental effects of tax-drag by constructing a tax-drag table. Using Excel, let’s compute a future value table for a tax-free account, assuming an annual tax rate of 30%:

Let’s do the same for a taxable account:

With these two tables, we can now compute a tax-drag table using the values above:

As you can see, the importance of sheltering gains from taxation (in a legal manner of course) becomes extremely important the longer the time frame, the higher the tax rate, and the higher the expected level of return. Based on the table above, the tax-drag on an investment with a 7% rate of return over 10 years is 36.57%, which is much higher than 30%.

Understanding the impact of tax-drag illustrates the importance of having a tax-diversification strategy in addition to an asset diversification strategy. For instance, hold longer term capital appreciating securities which pay no dividends or interest in a taxable account, while investing in interest bearing and dividend paying securities inside of tax-sheltered accounts such as Traditional IRAs and Roth IRAs or employer sponsored retirement plans.

Generally speaking, if taxes are paid on an annual basis the tax-drag will be greater than the tax rate. Conversely, if taxes are deferred until the end of the period the tax-drag will be equal to the tax rate.

A copy of the Excel model used to construct the tax-drag table can be found here.

The formula utilized to calculate the FVIF which takes into account this method of taxation is as follows:

The annual taxation on dividends is reflected in the formula above by adjusting the rate of return by the quantity (1 – tax on dividends). The formula for accrual based taxes on interest is identical:

The only difference between the two formulas is the use of subscript “d” versus subscript “i”.

Let’s assume that an investment paid an annualized rate of interest of 4% for ten years, and the annual tax on interest was 15%. Plugging those values into the formula above would yield the following:

If your initial investment was $1,000, you could calculate the future value in the tenth year by multiplying $1,000 by the FVIF of 1.3970:

Using Excel we can construct a FVIF table that takes into account different rates of taxation on interest or dividends, the FVIF table below is constructed with a discount rate of 4%:

Obviously, the higher the rate of taxation the more it will reduce the FVIF.

The Excel model for FVIFs based on annual accrual taxation can be found here.

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

Calculating the future value of a tax-deferred account incorporates the tax paid on the money when it is withdrawn during the final period. We can account for the taxes paid by adjusting the present value after it has been compounded by the specified rate and number of periods:

The addition of the quantity (1 – t) adjusts the future value in the final period by the tax that is owed.

Assume you have a present value of $1,000, the will grow at a rate of 7% for ten years, with an assumed tax rate of 30%. Plugging those values into the formula will yield the following:

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

Notice how the tax is paid during the final period. In the United States, this is how the future value of a Traditional IRA would be calculated. We can represent the table above visually with the following chart:

Using an HP12C calculator, we can calculate the future value of a tax-deferred account with the following keystrokes:

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

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

The present value of a tax-deferred account formula is usually only seen on tests which require you to calculate the present value of a tax-deferred account based on an 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.

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:

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.

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:

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:

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):

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:

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:

We can represent the data above in the following chart:

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

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:

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.