# Determining the effective tax rate in a tax on tax situation

There are taxation situations where the taxable amount includes the tax calculated on the taxable amount (e.g. this is a recursive calculation, as follows)...

Iteration  Taxable Amount   Tax per iteration
0          $100,000,000.00$5,000,000.00
1          $105,000,000.00$250,000.00
2          $105,250,000.00$12,500.00
3          $105,262,500.00$625.00
4          $105,263,125.00$31.25
5          $105,263,156.25$1.56
6          $105,263,157.81$0.08
7          $105,263,157.89$0.00
8          $105,263,157.89$0.00
9          $105,263,157.89$0.00
10         $105,263,157.89$0.00

Tax Rate                    5.00%
Effective Tax Rate          5.26%


I would like to determine the Effective Tax Rate without the need to apply the calculation recursively - because no matter what the starting taxable amount the Effective Tax Rate is always the same.

What you are looking at is a geometric series. Let's say the tax rate is $x$ and the principal $p$. Then what you are doing is computing

$$p+ p x + p x^2 + \cdots = p \sum_{k=0}^{\infty} x^k$$

You may recognize the sum of the geometric series, which may be written simply as

$$p' = \text{net taxable amount} = p \sum_{k=0}^{\infty} x^k = \frac{p}{1-x}$$

The effective tax rate is then

$$\frac{p'-p}{p} = \frac{1}{1-x} - 1 = \frac{x}{1-x}$$

With $x=0.05$, the effective tax rate is about $0.0526$, which agrees with your spreadsheet.

• So simple... thanks! Commented Jul 5, 2013 at 18:33