.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/backtesting/plot_01_kupiec.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_backtesting_plot_01_kupiec.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_backtesting_plot_01_kupiec.py:


Kupiec's proportion-of-failures test
====================================

A 99% VaR should be exceeded on about 1% of days. Kupiec (1995) turned
"about" into a likelihood-ratio test: with :math:`x` exceptions in
:math:`n` days and :math:`\hat p = x/n`,

.. math::

   LR_{uc} = -2 \ln \frac{(1 - p)^{n - x} p^x}{(1 - \hat p)^{n - x} \hat p^x} \sim \chi^2_1.

He also showed how weak the test is on a year of data: a model that
is badly wrong is often not rejected.

.. GENERATED FROM PYTHON SOURCE LINES 18-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import kupiec_test, parametric_var, var_exceptions








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Non-rejection regions
---------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-31

.. code-block:: Python


    for n in (250, 500, 1000):
        accepted = [x for x in range(60) if not kupiec_test(np.arange(n) < x, 0.99).reject()]
        print(f"n = {n:4d}: a 99% VaR passes with {accepted[0]} to {accepted[-1]} exceptions (expected {n // 100})")





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

 .. code-block:: none

    n =  250: a 99% VaR passes with 1 to 6 exceptions (expected 2)
    n =  500: a 99% VaR passes with 2 to 9 exceptions (expected 5)
    n = 1000: a 99% VaR passes with 5 to 16 exceptions (expected 10)




.. GENERATED FROM PYTHON SOURCE LINES 32-38

Testing a model that underestimates volatility
----------------------------------------------

Returns have a 1.2% daily volatility; the model assumes 1%. With 250
days the test catches the error only about half the time; with four
years, almost always.

.. GENERATED FROM PYTHON SOURCE LINES 38-60

.. code-block:: Python


    rng = np.random.default_rng(0)
    var = parametric_var(0.0, 0.01, 0.99)
    power = {}
    for n in (250, 1000):
        pnl = 0.012 * rng.standard_normal((1000, n))
        power[n] = np.mean([kupiec_test(var_exceptions(row, var)).reject() for row in pnl])
        print(f"n = {n:4d}: rejection rate {power[n]:.0%}")

    pnl = 0.012 * rng.standard_normal(1000)
    res = kupiec_test(var_exceptions(pnl, var))
    print(f"one sample: {res.n_exceptions} exceptions vs {res.expected:.0f} expected, LR = {res.statistic:.2f}, p = {res.p_value:.4f}")

    x = np.arange(0, 25)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(x, [kupiec_test(np.arange(1000) < k).statistic for k in x], ".-", label="LR statistic, n = 1000")
    ax.axhline(3.841, color="C3", ls="--", label="5% critical value")
    ax.set_xlabel("number of exceptions")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/backtesting/images/sphx_glr_plot_01_kupiec_001.png
   :alt: plot 01 kupiec
   :srcset: /api/gallery/risk/backtesting/images/sphx_glr_plot_01_kupiec_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    n =  250: rejection rate 48%
    n = 1000: rejection rate 97%
    one sample: 28 exceptions vs 10 expected, LR = 21.99, p = 0.0000





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

   **Total running time of the script:** (0 minutes 0.553 seconds)


.. _sphx_glr_download_api_gallery_risk_backtesting_plot_01_kupiec.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/backtesting/plot_01_kupiec.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_kupiec.ipynb <plot_01_kupiec.ipynb>`

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

      :download:`Download Python source code: plot_01_kupiec.py <plot_01_kupiec.py>`

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

      :download:`Download zipped: plot_01_kupiec.zip <plot_01_kupiec.zip>`


.. only:: html

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

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