.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fixed_income/duration/plot_02_hicks_modified_duration.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_fixed_income_duration_plot_02_hicks_modified_duration.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_fixed_income_duration_plot_02_hicks_modified_duration.py:


Hicks: duration as an elasticity (modified duration)
=====================================================

John Hicks (1939) found Macaulay's average period again as the
elasticity of a capital value with respect to the discount factor. Measured
per unit of yield, that elasticity is *modified duration*,

.. math:: D_{mod} = -\frac{1}{P}\frac{dP}{dy} = \frac{D_{mac}}{1 + y/m},

for a yield compounded :math:`m` times a year. The script checks the
relation across compounding frequencies and shows that :math:`-D_{mod}`
is the slope of the log-price curve.

.. GENERATED FROM PYTHON SOURCE LINES 17-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import bond_cash_flows, macaulay_duration, modified_duration, price_from_yield

    cf = bond_cash_flows(100.0, 0.05, 15.0, freq=2)
    y0 = 0.05








.. GENERATED FROM PYTHON SOURCE LINES 26-28

Macaulay and modified duration by compounding frequency
-------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-35

.. code-block:: Python


    for m in (1, 2, 4, 12, None):
        d_mac = macaulay_duration(cf, y0, m)
        d_mod = modified_duration(cf, y0, m)
        ratio = 1.0 if m is None else 1 + y0 / m
        print(f"m={str(m):>4}: D_mac={d_mac:.4f}  D_mod={d_mod:.4f}  D_mac/(1+y/m)={d_mac / ratio:.4f}")





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

 .. code-block:: none

    m=   1: D_mac=10.7423  D_mod=10.2308  D_mac/(1+y/m)=10.2308
    m=   2: D_mac=10.7268  D_mod=10.4651  D_mac/(1+y/m)=10.4651
    m=   4: D_mac=10.7188  D_mod=10.5865  D_mac/(1+y/m)=10.5865
    m=  12: D_mac=10.7134  D_mod=10.6689  D_mac/(1+y/m)=10.6689
    m=None: D_mac=10.7107  D_mod=10.7107  D_mac/(1+y/m)=10.7107




.. GENERATED FROM PYTHON SOURCE LINES 36-38

Modified duration is the slope of log-price against yield
---------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-50

.. code-block:: Python


    y = np.linspace(0.0, 0.10, 200)
    log_p = np.log(price_from_yield(cf, y, 2))
    d_mod = modified_duration(cf, y0, 2)
    p0 = np.log(price_from_yield(cf, y0, 2))
    fig, ax = plt.subplots()
    ax.plot(100 * y, log_p, label="log price")
    ax.plot(100 * y, p0 - d_mod * (y - y0), ls="--", label=f"slope $-D_{{mod}}$ = {-d_mod:.2f}")
    ax.set_xlabel("yield, semiannual (%)")
    ax.set_ylabel("log price")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/fixed_income/duration/images/sphx_glr_plot_02_hicks_modified_duration_001.png
   :alt: plot 02 hicks modified duration
   :srcset: /api/gallery/fixed_income/duration/images/sphx_glr_plot_02_hicks_modified_duration_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.024 seconds)


.. _sphx_glr_download_api_gallery_fixed_income_duration_plot_02_hicks_modified_duration.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/fixed_income/duration/plot_02_hicks_modified_duration.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_hicks_modified_duration.ipynb <plot_02_hicks_modified_duration.ipynb>`

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

      :download:`Download Python source code: plot_02_hicks_modified_duration.py <plot_02_hicks_modified_duration.py>`

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

      :download:`Download zipped: plot_02_hicks_modified_duration.zip <plot_02_hicks_modified_duration.zip>`


.. only:: html

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

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