.. 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_03_bernoulli_continuous_compounding.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_03_bernoulli_continuous_compounding.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_03_bernoulli_continuous_compounding.py:


Bernoulli and continuous compounding
====================================

In 1683 Jacob Bernoulli asked what a sum lent at interest becomes if the
interest is added not yearly but half-yearly, monthly, daily, at every
instant. At 100% a year, 1 grows to :math:`(1 + 1/m)^m` with ``m``
compounding periods, which increases with ``m`` but stays bounded; he
showed the limit lies between 2.5 and 3. The limit is :math:`e`, and
continuous compounding, :math:`e^{rt}`, is the convention financialkit
uses by default.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_value import equivalent_rate, future_value, plot_compounding_growth








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Approaching e
-------------

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: Python


    for m in (1, 2, 4, 12, 52, 365, 8760):
        print(f"m = {m:5d}: {future_value(1.0, 1.0, 1.0, m):.6f}")
    print(f"continuous: {future_value(1.0, 1.0, 1.0):.6f}  (e = {np.e:.6f})")





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

 .. code-block:: none

    m =     1: 2.000000
    m =     2: 2.250000
    m =     4: 2.441406
    m =    12: 2.613035
    m =    52: 2.692597
    m =   365: 2.714567
    m =  8760: 2.718127
    continuous: 2.718282  (e = 2.718282)




.. GENERATED FROM PYTHON SOURCE LINES 29-33

Equivalent rates
----------------
The continuously compounded rate giving the same growth as 100% annual
interest is ln 2.

.. GENERATED FROM PYTHON SOURCE LINES 33-36

.. code-block:: Python


    print(f"100% annual = {equivalent_rate(1.0, 1, None):.4%} continuous")





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

 .. code-block:: none

    100% annual = 69.3147% continuous




.. GENERATED FROM PYTHON SOURCE LINES 37-39

Growth paths and the limit
--------------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-51

.. code-block:: Python


    m = np.unique(np.logspace(0, 4, 60).astype(int))
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    plot_compounding_growth(1.0, 1.0, compoundings=(1, 2, 4, 12, None), ax=axes[0])
    axes[1].semilogx(m, [future_value(1.0, 1.0, 1.0, int(k)) for k in m], "o", ms=3, label="(1 + 1/m)^m")
    axes[1].axhline(np.e, color="k", ls="--", label="e")
    axes[1].set_xlabel("compounding periods per year, m")
    axes[1].set_ylabel("value after one year")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/time_value/compounding/images/sphx_glr_plot_03_bernoulli_continuous_compounding_001.png
   :alt: Compounding at 100%
   :srcset: /api/gallery/time_value/compounding/images/sphx_glr_plot_03_bernoulli_continuous_compounding_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.192 seconds)


.. _sphx_glr_download_api_gallery_time_value_compounding_plot_03_bernoulli_continuous_compounding.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_03_bernoulli_continuous_compounding.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_bernoulli_continuous_compounding.ipynb <plot_03_bernoulli_continuous_compounding.ipynb>`

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

      :download:`Download Python source code: plot_03_bernoulli_continuous_compounding.py <plot_03_bernoulli_continuous_compounding.py>`

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

      :download:`Download zipped: plot_03_bernoulli_continuous_compounding.zip <plot_03_bernoulli_continuous_compounding.zip>`


.. only:: html

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

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