# Compounding benefits of tax protected accounts

Here, we highlight one of the most important benefits of tax protected accounts (eg Traditional and Roth IRAs and 401ks). Specifically, we review the fact that not having to pay taxes on any investment growth that occurs while the money is held in the account results in compounding / exponential growth with a larger exponent than would be obtained in a traditional account.

## The growth equations

Here, we consider three types of investment account: A standard bank account without tax protection, a traditional tax protected account, and a Roth tax protected account. We’ll consider an idealized situation where we earn regular income of $D_0^{\prime}$ at time $0$ and then place this wealth (taxed, as appropriate for each case) into an investment that always returns a growth factor of $g$. For simplicity, we’ll assume that our tax rate never changes and is given by $t$. In the next three sections, we calculate expressions for the final wealth at time $T$ that results from each account. Following that, we compare the results.

### Standard account

In the standard account, the initial income must be taxed before it can be invested. Again, we define $t$ as the tax rate per year, so that the money left after tax at the start is
\begin{eqnarray} \tag{1} \label{1}
D_0 = D_0^{\prime} (1 – t).
\end{eqnarray}
We place this money into an idealized investment that always returns a growth of $g$. Therefore, after one year, the net wealth before tax is
\begin{eqnarray}\tag{2} \label{2}
D_1^{\prime} = D_0 (1 + g).
\end{eqnarray}
The portion $D_0 g$ is new income that must be taxed, so after tax we have
\begin{eqnarray}\tag{3} \label{3}
D_1 = D_0 + D_0 g (1 – t) = D_0[1 + g(1-t)].
\end{eqnarray}
If we iterate this expression up to time $T$, we obtain
\begin{eqnarray}\nonumber
D_T &=& D_0[1 + g(1-t)]^T \\
&\equiv & D_0^{\prime} (1 – t)[1 + g(1-t)]^T \tag{4} \label{4}
\end{eqnarray}
This is our equation for the final, post-tax wealth obtained from the standard account.

In the traditional account, we do not need to pay tax at time $0$ on our initial $D_0^{\prime}$ dollars. Instead, this wealth is immediately put into our growth investment for $T$ years. This gives a pretax wealth at time $T$ of
\begin{eqnarray}\tag{5} \label{5}
D_T^{\prime} = D_0 [1 + g]^T.
\end{eqnarray}
However, when this money is taken out at time $T$ it must be taxed. This gives
\begin{eqnarray}\tag{6} \label{6}
D_T = D_0^{\prime} (1-t) [1 + g]^T.
\end{eqnarray}
This is the equation that describes the net wealth generated by the traditional tax protected account.

### Roth tax protected account

In the Roth account, we do pay taxes on the initial $D_0^{\prime}$ at time $0$. However, once this is done, we never need to pay taxes again, even when taking the money out at expiration. Therefore, the net wealth at time $T$ is
\begin{eqnarray}\tag{7} \label{7}
D_T = D_0^{\prime} (1-t) (1 + g)^T
\end{eqnarray}
Notice that this expression is identical to that for the traditional tax protected account.

## Comparison

Now that we have derived expressions for the final wealth in the three types of account, we can easily compare them. First, note that (\ref{4}), (\ref{6}), and (\ref{7}) all share the common factor of $D_0^{\prime} (1-t) \equiv D_0$, which can be considered the initial post-tax wealth. This means that the only difference between the standard and tax protected accounts is the effective growth rate: The growth rate term for the standard account is
\begin{eqnarray}
\text{growth factor (standard account)} = [1 + g(1-t)]^T \tag{8} \label{8}
\end{eqnarray}
while that for the two tax protected accounts is
\begin{eqnarray}
\text{growth factor (tax protected)}=[1 + g]^T \tag{9}\label{9}
\end{eqnarray}
These two factors may look similar, but they represent exponential growth with different exponents. Consequently, for large $T$, the growth from (\ref{9}) can be much larger than that from (\ref{8}). To illustrate this point, we tabulate the two functions assuming $7$ percent growth for $30$ years at a few representative tax rates below. Notice that the growth rates are similar when that tax rates are lower — which makes sense because taxation does not have much of an effect in this limit. However, the tax protected account has a much larger value in the opposite limit — 7.61 vs 2.81 for the standard account!

def standard(T, g, t):
return (1 + g * (1-t))** T

def tax_protected(T, g, t):
return (1 + g) ** T

taxes = [0.1, 0.2, 0.3, 0.4, 0.5]
standard_values = [standard(30, 0.07, t) for t in taxes]
protected_values =  [tax_protected(30, 0.07, t) for t in taxes]

# output:
TAX RATE:   0.1,   0.2,   0.3,   0.4,   0.5
STANDARD:  6.25,  5.13,  4.20,  3.44,  2.81
PROTECTED: 7.61,  7.61,  7.61,  7.61,  7.61