.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/expected_shortfall/plot_01_coherence.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_risk_expected_shortfall_plot_01_coherence.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_risk_expected_shortfall_plot_01_coherence.py:


Artzner-Delbaen-Eber-Heath: coherent risk measures
==================================================

Artzner, Delbaen, Eber and Heath (1999) listed four properties a
capital requirement should have: monotonicity, subadditivity, positive
homogeneity and translation invariance. VaR fails subadditivity: two
desks can each look safe while their merger needs capital. The classic
counterexample is a pair of independent bonds that each default with
4% probability. At 95% neither default is in the tail, so each bond's
VaR is negative; together, a default happens with probability 7.8%, and
the portfolio's VaR is a loss.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import ExpectedShortfall, ValueAtRisk, check_coherence








.. GENERATED FROM PYTHON SOURCE LINES 22-27

Two defaultable bonds
---------------------

Each bond pays 2 if it survives and loses 100 on default. The 10,000
equally likely states reproduce the joint default probabilities exactly.

.. GENERATED FROM PYTHON SOURCE LINES 27-42

.. code-block:: Python


    p = 0.04
    n = 10_000
    counts = np.round(np.array([(1 - p) ** 2, p * (1 - p), p * (1 - p), p**2]) * n).astype(int)
    A = np.repeat([2.0, -100.0, 2.0, -100.0], counts)
    B = np.repeat([2.0, 2.0, -100.0, -100.0], counts)

    for measure in (ValueAtRisk(0.95), ExpectedShortfall(0.95)):
        res = check_coherence(measure, A, B)
        print(f"{measure.name}: rho(A) = {res.rho_x:7.2f}, rho(B) = {res.rho_y:7.2f}, rho(A + B) = {res.rho_sum:7.2f}")
        print(
            f"    subadditive {res.subadditivity}, monotone {res.monotonicity}, homogeneous {res.positive_homogeneity}, "
            f"translation invariant {res.translation_invariance}: coherent = {res.coherent}"
        )





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

 .. code-block:: none

    VaR: rho(A) =   -2.00, rho(B) =   -2.00, rho(A + B) =   98.00
        subadditive False, monotone True, homogeneous True, translation invariant True: coherent = False
    ES: rho(A) =   79.60, rho(B) =   79.60, rho(A + B) =  101.26
        subadditive True, monotone True, homogeneous True, translation invariant True: coherent = True




.. GENERATED FROM PYTHON SOURCE LINES 43-49

Merging gets punished, at some levels only
------------------------------------------

The diversification benefit, :math:`\rho(A) + \rho(B) - \rho(A + B)`,
is negative for VaR at confidence levels between 92.2% and 96%, and
never for ES.

.. GENERATED FROM PYTHON SOURCE LINES 49-65

.. code-block:: Python


    levels = np.linspace(0.90, 0.995, 200)
    benefit = {
        m: [(lambda r: r.rho_x + r.rho_y - r.rho_sum)(check_coherence(cls(a), A, B)) for a in levels]
        for m, cls in [("VaR", ValueAtRisk), ("ES", ExpectedShortfall)]
    }
    fig, ax = plt.subplots(figsize=(7, 3.8))
    for m, b in benefit.items():
        ax.plot(levels, b, label=m)
    ax.axhline(0, color="k", lw=0.8)
    ax.set_xlabel("confidence level")
    ax.set_ylabel("diversification benefit")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/expected_shortfall/images/sphx_glr_plot_01_coherence_001.png
   :alt: plot 01 coherence
   :srcset: /api/gallery/risk/expected_shortfall/images/sphx_glr_plot_01_coherence_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.167 seconds)


.. _sphx_glr_download_api_gallery_risk_expected_shortfall_plot_01_coherence.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/risk/expected_shortfall/plot_01_coherence.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_coherence.ipynb <plot_01_coherence.ipynb>`

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

      :download:`Download Python source code: plot_01_coherence.py <plot_01_coherence.py>`

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

      :download:`Download zipped: plot_01_coherence.zip <plot_01_coherence.zip>`


.. only:: html

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

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