.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fixed_income/immunization/plot_02_fong_vasicek_m_squared.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_immunization_plot_02_fong_vasicek_m_squared.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_immunization_plot_02_fong_vasicek_m_squared.py:


Fong-Vasicek M-squared
======================

Redington's conditions protect against parallel shifts only. Fong and
Vasicek (1984) showed that when the forward curve twists with slope at
most :math:`K`, a duration-matched portfolio loses at most

.. math:: \frac{\Delta V_H}{V_H} \ge -\tfrac12 K M^2, \qquad M^2 = \sum_i w_i (t_i - H)^2,

where :math:`M^2` measures how far the cash flows are spread around the
horizon :math:`H`. Among immunized portfolios the bullet, with the
smallest :math:`M^2`, is the least exposed to twists.

.. GENERATED FROM PYTHON SOURCE LINES 17-39

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import (
        CashFlows,
        FlatCurve,
        bond_cash_flows,
        fong_vasicek_bound,
        horizon_value,
        linear_forward_shock,
        m_squared,
        redington_immunize,
    )

    curve, H = FlatCurve(0.04), 8.0
    target = CashFlows([H], [1000.0])
    portfolios = {
        "bullet 7y/9y": redington_immunize(target, [bond_cash_flows(1.0, 0.0, 7.0), bond_cash_flows(1.0, 0.0, 9.0)], curve).portfolio,
        "ladder 4y/14y": redington_immunize(target, [bond_cash_flows(1.0, 0.0, 4.0), bond_cash_flows(1.0, 0.0, 14.0)], curve).portfolio,
        "barbell 1y/30y": redington_immunize(target, [bond_cash_flows(1.0, 0.0, 1.0), bond_cash_flows(1.0, 0.0, 30.0)], curve).portfolio,
    }








.. GENERATED FROM PYTHON SOURCE LINES 40-42

Horizon value under twists of the forward curve
-----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 42-57

.. code-block:: Python


    slopes = np.linspace(-0.004, 0.004, 41)
    fig, ax = plt.subplots()
    for name, port in portfolios.items():
        m2 = m_squared(port, H, curve)
        base = horizon_value(port, curve, H)
        change = np.array([horizon_value(port, curve.shifted(linear_forward_shock(0.0, b, H)), H) / base - 1 for b in slopes])
        (line,) = ax.plot(1e4 * slopes, 100 * change, label=f"{name}, $M^2$ = {m2:.1f}")
        ax.plot(1e4 * slopes[slopes > 0], 100 * fong_vasicek_bound(m2, slopes[slopes > 0]), ls=":", color=line.get_color())
    ax.axhline(0.0, color="0.6", lw=0.8)
    ax.set_xlabel("forward-curve slope change $b$ (bp per year)")
    ax.set_ylabel("change in horizon value (%)")
    ax.set_title("Dotted: Fong-Vasicek bound $-K M^2/2$")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/fixed_income/immunization/images/sphx_glr_plot_02_fong_vasicek_m_squared_001.png
   :alt: Dotted: Fong-Vasicek bound $-K M^2/2$
   :srcset: /api/gallery/fixed_income/immunization/images/sphx_glr_plot_02_fong_vasicek_m_squared_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.038 seconds)


.. _sphx_glr_download_api_gallery_fixed_income_immunization_plot_02_fong_vasicek_m_squared.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/immunization/plot_02_fong_vasicek_m_squared.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_fong_vasicek_m_squared.ipynb <plot_02_fong_vasicek_m_squared.ipynb>`

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

      :download:`Download Python source code: plot_02_fong_vasicek_m_squared.py <plot_02_fong_vasicek_m_squared.py>`

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

      :download:`Download zipped: plot_02_fong_vasicek_m_squared.zip <plot_02_fong_vasicek_m_squared.zip>`


.. only:: html

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

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