.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_value/present_value/plot_02_fisher_real_rate.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_present_value_plot_02_fisher_real_rate.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_present_value_plot_02_fisher_real_rate.py:


Fisher's real and nominal rates of interest
===========================================

In *Appreciation and Interest* (1896) Irving Fisher argued that lenders
who foresee inflation demand compensation for it, so that a nominal rate
:math:`i` and a real rate :math:`r` satisfy
:math:`1 + i = (1 + r)(1 + \pi)`. The familiar shortcut
:math:`r \approx i - \pi` is accurate only when inflation is low; this
script compares the two across inflation rates, up to hyperinflation.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_value import fisher_nominal_rate, fisher_real_rate








.. GENERATED FROM PYTHON SOURCE LINES 20-22

A loan at 12% with 8% inflation
-------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-27

.. code-block:: Python


    print(f"exact real rate:   {fisher_real_rate(0.12, 0.08):.4%}")
    print(f"approximate (i-pi): {fisher_real_rate(0.12, 0.08, exact=False):.4%}")
    print(f"nominal rate needed for a 3% real return at 8% inflation: {fisher_nominal_rate(0.03, 0.08):.4%}")





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

 .. code-block:: none

    exact real rate:   3.7037%
    approximate (i-pi): 4.0000%
    nominal rate needed for a 3% real return at 8% inflation: 11.2400%




.. GENERATED FROM PYTHON SOURCE LINES 28-31

The approximation breaks down as inflation rises
------------------------------------------------
Hold the real rate at 3% and ask what nominal rate a lender needs.

.. GENERATED FROM PYTHON SOURCE LINES 31-49

.. code-block:: Python


    infl = np.linspace(0.0, 1.0, 201)
    exact = fisher_nominal_rate(0.03, infl)
    approx = 0.03 + infl

    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(100 * infl, 100 * exact, label="exact: (1+r)(1+pi) - 1")
    axes[0].plot(100 * infl, 100 * approx, "--", label="approximate: r + pi")
    axes[0].set_xlabel("inflation (%)")
    axes[0].set_ylabel("required nominal rate (%)")
    axes[0].legend()
    axes[1].plot(100 * infl, 100 * (exact - approx))
    axes[1].set_xlabel("inflation (%)")
    axes[1].set_ylabel("gap = r * pi (percentage points)")
    axes[1].set_title("Error of the first-order Fisher equation")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/time_value/present_value/images/sphx_glr_plot_02_fisher_real_rate_001.png
   :alt: Error of the first-order Fisher equation
   :srcset: /api/gallery/time_value/present_value/images/sphx_glr_plot_02_fisher_real_rate_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.058 seconds)


.. _sphx_glr_download_api_gallery_time_value_present_value_plot_02_fisher_real_rate.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/present_value/plot_02_fisher_real_rate.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_fisher_real_rate.ipynb <plot_02_fisher_real_rate.ipynb>`

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

      :download:`Download Python source code: plot_02_fisher_real_rate.py <plot_02_fisher_real_rate.py>`

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

      :download:`Download zipped: plot_02_fisher_real_rate.zip <plot_02_fisher_real_rate.zip>`


.. only:: html

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

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