.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_value/annuities/plot_02_consol_perpetuity.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_annuities_plot_02_consol_perpetuity.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_annuities_plot_02_consol_perpetuity.py:


Consols and the perpetuity
==========================

In 1751 Henry Pelham consolidated several British government debts into
a single issue of "3 per cent consolidated annuities", which were never
to be repaid: the holder received 3 a year per 100 of face value
forever. A perpetuity paying :math:`P` is worth :math:`P/i` at yield
:math:`i`, so a consol's price moves inversely with long-term rates and
at a 3% yield it trades at par. This script prices consols across yields
and compares them with long but finite annuities.

.. GENERATED FROM PYTHON SOURCE LINES 15-20

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.time_value import annuity_present_value, perpetuity_present_value








.. GENERATED FROM PYTHON SOURCE LINES 21-23

Consol prices
-------------

.. GENERATED FROM PYTHON SOURCE LINES 23-27

.. code-block:: Python


    for y in (0.025, 0.03, 0.035, 0.05):
        print(f"yield {y:.1%}: consol price {perpetuity_present_value(3.0, y):6.2f}")





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

 .. code-block:: none

    yield 2.5%: consol price 120.00
    yield 3.0%: consol price 100.00
    yield 3.5%: consol price  85.71
    yield 5.0%: consol price  60.00




.. GENERATED FROM PYTHON SOURCE LINES 28-30

A long annuity is nearly a perpetuity
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-49

.. code-block:: Python


    yields = np.linspace(0.02, 0.08, 121)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(100 * yields, perpetuity_present_value(3.0, yields), "k", lw=2, label="consol (perpetuity)")
    for n in (20, 50, 100):
        axes[0].plot(100 * yields, annuity_present_value(3.0, yields, n), label=f"{n}-year annuity")
    axes[0].axhline(100.0, color="gray", ls=":")
    axes[0].set_xlabel("yield (%)")
    axes[0].set_ylabel("price per 100 face")
    axes[0].legend()

    n = np.arange(1, 301)
    axes[1].plot(n, annuity_present_value(3.0, 0.03, n) / perpetuity_present_value(3.0, 0.03))
    axes[1].set_xlabel("years of payments")
    axes[1].set_ylabel("fraction of consol value")
    axes[1].set_title("Annuity / perpetuity at 3%")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/time_value/annuities/images/sphx_glr_plot_02_consol_perpetuity_001.png
   :alt: Annuity / perpetuity at 3%
   :srcset: /api/gallery/time_value/annuities/images/sphx_glr_plot_02_consol_perpetuity_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.064 seconds)


.. _sphx_glr_download_api_gallery_time_value_annuities_plot_02_consol_perpetuity.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/annuities/plot_02_consol_perpetuity.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_consol_perpetuity.ipynb <plot_02_consol_perpetuity.ipynb>`

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

      :download:`Download Python source code: plot_02_consol_perpetuity.py <plot_02_consol_perpetuity.py>`

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

      :download:`Download zipped: plot_02_consol_perpetuity.zip <plot_02_consol_perpetuity.zip>`


.. only:: html

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

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