.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/var/plot_01_cornish_fisher.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_var_plot_01_cornish_fisher.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_var_plot_01_cornish_fisher.py:


Cornish-Fisher: VaR beyond the normal
=====================================

Cornish and Fisher (1937) expanded the quantile of a nearly normal
distribution in its cumulants. To second order, with skewness :math:`S`
and excess kurtosis :math:`K`,

.. math::

   z_{cf} = z + \frac{(z^2 - 1) S}{6} + \frac{(z^3 - 3z) K}{24} - \frac{(2z^3 - 5z) S^2}{36},

so a VaR needs only four sample moments to account for crash-prone
skew and fat tails. The expansion is accurate for moderate departures
from normality and breaks down when the tails are very heavy.

.. GENERATED FROM PYTHON SOURCE LINES 19-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from financialkit.risk import cornish_fisher_var, historical_var, parametric_var








.. GENERATED FROM PYTHON SOURCE LINES 26-32

A negatively skewed, fat-tailed return
--------------------------------------

Minus a gamma variable: losses are rarer than gains but larger. We
compare the exact 99% VaR with the normal and Cornish-Fisher figures as
the shape parameter, and so the skew, varies.

.. GENERATED FROM PYTHON SOURCE LINES 32-45

.. code-block:: Python


    shapes = np.array([2.0, 4.0, 8.0, 16.0, 64.0])
    alpha = 0.99
    rows = []
    for a in shapes:
        d = stats.gamma(a)
        mu, var, skew, kurt = (float(m) for m in d.stats(moments="mvsk"))
        exact = d.ppf(alpha) - mu  # loss quantile of mu - G
        rows.append((-skew, kurt, exact, parametric_var(0.0, np.sqrt(var), alpha), cornish_fisher_var(0.0, np.sqrt(var), -skew, kurt, alpha)))
    print(" skew   kurt   exact  normal      CF")
    for r in rows:
        print(" ".join(f"{v:6.3f}" for v in r))





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

 .. code-block:: none

     skew   kurt   exact  normal      CF
    -1.414  3.000  4.638  3.290  4.688
    -1.000  1.500  6.045  4.653  6.072
    -0.707  0.750  8.000  6.580  8.014
    -0.500  0.375 10.743  9.305 10.750
    -0.250  0.094 20.067 18.611 20.069




.. GENERATED FROM PYTHON SOURCE LINES 46-52

From sample moments
-------------------

In practice the moments come from data. On 2,000 skewed returns the
Cornish-Fisher VaR tracks the empirical quantile across confidence
levels, where the normal VaR understates every one.

.. GENERATED FROM PYTHON SOURCE LINES 52-68

.. code-block:: Python


    rng = np.random.default_rng(1)
    x = 0.01 * (4.0 - stats.gamma(4.0).rvs(2000, random_state=rng)) / 2.0
    m, s = x.mean(), x.std()
    S, K = stats.skew(x), stats.kurtosis(x)
    levels = np.linspace(0.9, 0.995, 30)
    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(levels, [historical_var(x, a).var for a in levels], "k.", label="historical")
    ax.plot(levels, parametric_var(m, s, levels), label="normal")
    ax.plot(levels, cornish_fisher_var(m, s, S, K, levels), label=f"Cornish-Fisher (S = {S:.2f}, K = {K:.2f})")
    ax.set_xlabel("confidence level")
    ax.set_ylabel("VaR")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_01_cornish_fisher_001.png
   :alt: plot 01 cornish fisher
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_01_cornish_fisher_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.040 seconds)


.. _sphx_glr_download_api_gallery_risk_var_plot_01_cornish_fisher.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/var/plot_01_cornish_fisher.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_cornish_fisher.ipynb <plot_01_cornish_fisher.ipynb>`

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

      :download:`Download Python source code: plot_01_cornish_fisher.py <plot_01_cornish_fisher.py>`

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

      :download:`Download zipped: plot_01_cornish_fisher.zip <plot_01_cornish_fisher.zip>`


.. only:: html

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

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