.. 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_01_de_witt_life_annuity.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_01_de_witt_life_annuity.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_01_de_witt_life_annuity.py:


De Witt's life annuities
========================

In 1671 Johan de Witt, Grand Pensionary of Holland, sent the States
General his *Waerdye van Lyf-renten*, arguing that the life annuities the
province sold at 14 years' purchase were too cheap. He valued an annuity
as the sum of each payment, discounted at 4% and weighted by the chance
that the buyer is still alive to receive it, under a simple mortality
hypothesis: an equal chance of death in every half-year up to age 53,
rising to 1.5, 2 and 3 times that in the bands 53-63, 63-73 and 73-80.

.. 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_factor, de_witt_survival, life_annuity_value








.. GENERATED FROM PYTHON SOURCE LINES 21-24

Survival and the value of a life annuity
----------------------------------------
Annual payments of 1 florin, discounted at 4%.

.. GENERATED FROM PYTHON SOURCE LINES 24-32

.. code-block:: Python


    ages = [3, 20, 40, 60, 70]
    k = np.arange(1, 78)
    for age in ages:
        p = de_witt_survival(age, k, base_hazard=0.015)
        print(f"age {age:2d}: life annuity worth {life_annuity_value(1.0, 0.04, p):5.2f} years' purchase")
    print(f"perpetual annuity-certain at 4%: {annuity_factor(0.04, 10_000):.2f}")





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

 .. code-block:: none

    age  3: life annuity worth 17.57 years' purchase
    age 20: life annuity worth 16.99 years' purchase
    age 40: life annuity worth 15.11 years' purchase
    age 60: life annuity worth 10.32 years' purchase
    age 70: life annuity worth  6.28 years' purchase
    perpetual annuity-certain at 4%: 25.00




.. GENERATED FROM PYTHON SOURCE LINES 33-35

The ingredients
---------------

.. GENERATED FROM PYTHON SOURCE LINES 35-56

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    t = np.linspace(0.0, 77.0, 400)
    for age in ages:
        axes[0].plot(age + t, de_witt_survival(age, t, base_hazard=0.015), label=f"bought at {age}")
    axes[0].set_xlabel("age")
    axes[0].set_ylabel("probability alive")
    axes[0].set_title("De Witt's survival curves")
    axes[0].legend()

    buy_ages = np.arange(3, 79)
    values = [life_annuity_value(1.0, 0.04, de_witt_survival(a, k, base_hazard=0.015)) for a in buy_ages]
    axes[1].plot(buy_ages, values)
    axes[1].axhline(14.0, color="gray", ls="--", label="price charged: 14 years' purchase")
    axes[1].set_xlabel("age of annuitant")
    axes[1].set_ylabel("value of 1 a year for life")
    axes[1].set_title("Value of a life annuity at 4%")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/time_value/annuities/images/sphx_glr_plot_01_de_witt_life_annuity_001.png
   :alt: De Witt's survival curves, Value of a life annuity at 4%
   :srcset: /api/gallery/time_value/annuities/images/sphx_glr_plot_01_de_witt_life_annuity_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.083 seconds)


.. _sphx_glr_download_api_gallery_time_value_annuities_plot_01_de_witt_life_annuity.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_01_de_witt_life_annuity.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_de_witt_life_annuity.ipynb <plot_01_de_witt_life_annuity.ipynb>`

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

      :download:`Download Python source code: plot_01_de_witt_life_annuity.py <plot_01_de_witt_life_annuity.py>`

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

      :download:`Download zipped: plot_01_de_witt_life_annuity.zip <plot_01_de_witt_life_annuity.zip>`


.. only:: html

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

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