Estimating tax payable#

In this example we are given the following scenario:

Personal tax (before deductions) in Australia is based on the table below. The tax payable at the end of the financial year depends on the individual’s income. The higher the income, the higher the tax rate, as defined by tax brackets (or tiers). Given a list of incomes, calculate the corresponding tax payable for each income.


Income Thresholds

Rate

Tax payable

$0 - $18,200

0%

Nil

$18,200 - $45,000

19%

19c for each $1 over $18,200

$45,000 - $120,000

32.5%

$5,092 plus 32.5c for each $1 over $45,000

$120,000 - $180,000

37%

$29,467 plus 37c for each $1 over $120,000

$180,000 and over

45%

$51,667 plus 45c for each $1 over $180,000

We start by importing pandas, numpy and piso, and creating an interval index for the tax brackets.

In [1]: import pandas as pd

In [2]: import numpy as np

In [3]: import piso

In [4]: tax_brackets = pd.IntervalIndex.from_breaks(
   ...:     [0,18200,45000,120000,180000,np.inf],
   ...:     closed="left",
   ...: )
   ...: 

In [5]: tax_brackets
Out[5]: 
IntervalIndex([      [0.0, 18200.0),   [18200.0, 45000.0),
                [45000.0, 120000.0), [120000.0, 180000.0),
                    [180000.0, inf)],
              dtype='interval[float64, left]')

With each interval in the tax bracket, we’ll associate three values:

  1. the lower threshold for the tax bracket

  2. the fixed amount payable

  3. the tax rate for each dollar above the threshold (as a fraction)

We describe this data as a pandas.DataFrame indexed by tax_brackets.

In [6]: tax_rates = pd.DataFrame(
   ...:     {
   ...:         "threshold":tax_brackets.left,
   ...:         "fixed":[0, 0, 5092, 29467, 51667],
   ...:         "rate":[0, 0.19, 0.325, 0.37, 0.45],
   ...:     },
   ...:     index = tax_brackets,
   ...: )
   ...: 

In [7]: tax_rates
Out[7]: 
                      threshold  fixed   rate
[0.0, 18200.0)              0.0      0  0.000
[18200.0, 45000.0)      18200.0      0  0.190
[45000.0, 120000.0)     45000.0   5092  0.325
[120000.0, 180000.0)   120000.0  29467  0.370
[180000.0, inf)        180000.0  51667  0.450

For the income, we’ll generate some random integers (and plot the distribution) corresponding to 100,000 individuals.

In [8]: income = pd.Series(np.random.beta(5,50, size=100000)*1e6).astype(int)

In [9]: income.plot.hist(bins=20);
../../_images/case_study_tax_income_dist.png

We are now in a position to use piso.lookup(), which take two parameters:

  1. a pandas.DataFrame or pandas.Series which is indexed by a pandas.IntervalIndex

  2. the values which are will be compared to the interval index

In [10]: tax_params = piso.lookup(tax_rates, income)

In [11]: tax_params
Out[11]: 
        threshold  fixed   rate
54151     45000.0   5092  0.325
60904     45000.0   5092  0.325
160305   120000.0  29467  0.370
65786     45000.0   5092  0.325
84227     45000.0   5092  0.325
...           ...    ...    ...
43453     18200.0      0  0.190
94646     45000.0   5092  0.325
150276   120000.0  29467  0.370
66109     45000.0   5092  0.325
79964     45000.0   5092  0.325

[100000 rows x 3 columns]

The result is a dataframe, indexed by the values of income, sharing the same columns as tax_rates.

We can then use a vectorised calculation for the tax payable:

In [12]: tax_params["fixed"] + (tax_params.index-tax_params["threshold"])*tax_params["rate"]
Out[12]: 
54151      8066.075
60904     10260.800
160305    44379.850
65786     11847.450
84227     17840.775
            ...    
43453      4798.070
94646     21226.950
150276    40669.120
66109     11952.425
79964     16455.300
Length: 100000, dtype: float64

Alternative approaches#

There are a couple of alternative, straightforward solutions which do not require piso which we detail below.

Alternative 1: pandas.cut

The tax_params dataframe that was produced above by piso.lookup() can be reproduced using pandas.cut() which can be used to assign bins to data with an interval index.

In [13]: tax_params = tax_rates.loc[pd.cut(income, tax_brackets)].set_index(income)

In [14]: tax_params
Out[14]: 
        threshold  fixed   rate
54151     45000.0   5092  0.325
60904     45000.0   5092  0.325
160305   120000.0  29467  0.370
65786     45000.0   5092  0.325
84227     45000.0   5092  0.325
...           ...    ...    ...
43453     18200.0      0  0.190
94646     45000.0   5092  0.325
150276   120000.0  29467  0.370
66109     45000.0   5092  0.325
79964     45000.0   5092  0.325

[100000 rows x 3 columns]

This approach however runs approximately 20 times slower than piso.lookup().

Alternative 2: applying function

The second approach involves writing a function which takes a single value (an income for an individual) and returns the tax payable. The function can then used with pandas.Series.apply

In [15]: def calc_tax(value):
   ....:     if value <= 18200:
   ....:         tax = 0
   ....:     elif value <= 45000:
   ....:         tax = (value-18200)*0.19
   ....:     elif value <= 120000:
   ....:         tax = 5092 + (value-45000)*0.325
   ....:     elif value <= 180000:
   ....:         tax = 29467 + (value-120000)*0.37
   ....:     else:
   ....:         tax = 51667 + (value-180000)*0.45
   ....:     return tax
   ....: 

In [16]: income.apply(calc_tax)
Out[16]: 
0         8066.075
1        10260.800
2        44379.850
3        11847.450
4        17840.775
           ...    
99995     4798.070
99996    21226.950
99997    40669.120
99998    11952.425
99999    16455.300
Length: 100000, dtype: float64

This approach runs approximately 3 times slower than piso.lookup(). It also requires a function to be defined which is relatively cumbersome to implement. This approach becomes increasingly unattractive, and error prone, as the number of tax brackets increases.

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