.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_value/compounding/plot_01_pacioli_rule_of_72.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_time_value_compounding_plot_01_pacioli_rule_of_72.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_time_value_compounding_plot_01_pacioli_rule_of_72.py:


Pacioli's rule of 72
====================

Luca Pacioli's *Summa de arithmetica* (1494) gives, without proof, a rule
for how long money takes to double: divide 72 by the annual interest
rate in percent. The exact doubling time with annual compounding is
:math:`\ln 2 / \ln(1 + r)`, and since :math:`\ln(1 + r) \approx r - r^2/2`
the best constant drifts upward with the rate; 72 is close to right near
8% and has many convenient divisors.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_value import doubling_time, rule_of_72








.. GENERATED FROM PYTHON SOURCE LINES 20-22

The rule against the exact answer
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-26

.. code-block:: Python


    for r in (0.02, 0.06, 0.08, 0.12, 0.24):
        print(f"{r:5.0%}: rule {rule_of_72(r):6.2f} years, exact {doubling_time(r, compounding=1):6.2f} years")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

       2%: rule  36.00 years, exact  35.00 years
       6%: rule  12.00 years, exact  11.90 years
       8%: rule   9.00 years, exact   9.01 years
      12%: rule   6.00 years, exact   6.12 years
      24%: rule   3.00 years, exact   3.22 years




.. GENERATED FROM PYTHON SOURCE LINES 27-31

Relative error of the rules of 69.3, 70 and 72
-----------------------------------------------
69.3 is exact for continuous compounding; with annual compounding a
larger constant is better.

.. GENERATED FROM PYTHON SOURCE LINES 31-44

.. code-block:: Python


    r = np.linspace(0.01, 0.30, 300)
    exact = doubling_time(r, compounding=1)
    fig, ax = plt.subplots(figsize=(7, 4.5))
    for c in (69.3, 70.0, 72.0):
        ax.plot(100 * r, 100 * (rule_of_72(r, numerator=c) / exact - 1), label=f"rule of {c:g}")
    ax.axhline(0.0, color="k", lw=0.8)
    ax.set_xlabel("annual rate (%)")
    ax.set_ylabel("error vs exact doubling time (%)")
    ax.set_title("Pacioli's rule of 72")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/time_value/compounding/images/sphx_glr_plot_01_pacioli_rule_of_72_001.png
   :alt: Pacioli's rule of 72
   :srcset: /api/gallery/time_value/compounding/images/sphx_glr_plot_01_pacioli_rule_of_72_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.031 seconds)


.. _sphx_glr_download_api_gallery_time_value_compounding_plot_01_pacioli_rule_of_72.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/time_value/compounding/plot_01_pacioli_rule_of_72.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_pacioli_rule_of_72.ipynb <plot_01_pacioli_rule_of_72.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_pacioli_rule_of_72.py <plot_01_pacioli_rule_of_72.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_pacioli_rule_of_72.zip <plot_01_pacioli_rule_of_72.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_