.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/backtesting/plot_03_christoffersen.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_03_christoffersen.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_03_christoffersen.py:


Christoffersen's independence test
==================================

Kupiec's test counts exceptions but ignores when they happen. A VaR
that is right on average but slow to react produces exceptions in
bunches, exactly when a bank can least afford them. Christoffersen
(1998) fitted a two-state Markov chain to the hit sequence and tested
whether a hit is more likely the day after a hit,
:math:`\pi_{11} > \pi_{01}`; adding Kupiec's statistic gives the
conditional-coverage test, :math:`LR_{cc} = LR_{uc} + LR_{ind} \sim \chi^2_2`.

.. GENERATED FROM PYTHON SOURCE LINES 15-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import christoffersen_independence, christoffersen_test, kupiec_test, parametric_var, var_exceptions
    from financialkit.risk.visualizers import plot_backtest
    from financialkit.time_series import GARCH








.. GENERATED FROM PYTHON SOURCE LINES 23-32

A constant VaR on a GARCH market
--------------------------------

Returns follow a GARCH(1,1) process, simulated with
:class:`financialkit.time_series.GARCH`. One VaR uses the unconditional
volatility, the other the GARCH conditional volatility. Both are
exceeded about as often as they should be, so Kupiec's test is
satisfied, but the constant VaR's exceptions arrive in the volatile
spells.

.. GENERATED FROM PYTHON SOURCE LINES 32-51

.. code-block:: Python


    n = 2000
    model = GARCH(omega=2e-6, alpha=0.20, beta=0.78)
    path = model.simulate(n, rng=0)
    r, sig2 = path.returns, path.variance
    static = parametric_var(0.0, np.sqrt(model.unconditional_variance), 0.99)
    dynamic = parametric_var(0.0, np.sqrt(sig2), 0.99)
    for name, var in [("constant VaR", static), ("GARCH VaR", dynamic)]:
        hits = var_exceptions(r, var)
        uc, ind, cc = kupiec_test(hits), christoffersen_independence(hits), christoffersen_test(hits)
        print(f"{name:12s}: {uc.n_exceptions} hits; p(uc) = {uc.p_value:.3f}, p(ind) = {ind.p_value:.4f}, p(cc) = {cc.p_value:.4f}")

    fig, axes = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
    plot_backtest(r, np.full(n, static), ax=axes[0])
    axes[0].set_title("constant VaR: exceptions cluster")
    plot_backtest(r, dynamic, ax=axes[1])
    axes[1].set_title("conditional VaR: exceptions scattered")
    fig.tight_layout()




.. image-sg:: /api/gallery/risk/backtesting/images/sphx_glr_plot_03_christoffersen_001.png
   :alt: constant VaR: exceptions cluster, conditional VaR: exceptions scattered
   :srcset: /api/gallery/risk/backtesting/images/sphx_glr_plot_03_christoffersen_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    constant VaR: 18 hits; p(uc) = 0.648, p(ind) = 0.0093, p(cc) = 0.0305
    GARCH VaR   : 18 hits; p(uc) = 0.648, p(ind) = 0.5674, p(cc) = 0.7648




.. GENERATED FROM PYTHON SOURCE LINES 52-60

How often each test notices
---------------------------

On one path the evidence can go either way, so repeat the experiment.
The independence test rejects the constant VaR on most paths, and the
conditional VaR at about its 5% size. Kupiec also rejects the constant
VaR fairly often: GARCH returns are fat-tailed, so a normal VaR with
the right variance is exceeded too often on average.

.. GENERATED FROM PYTHON SOURCE LINES 60-71

.. code-block:: Python


    paths = [model.simulate(n, rng=seed) for seed in range(1, 301)]
    R = np.array([p.returns for p in paths])
    S2 = np.array([p.variance for p in paths])
    for name, V in [("constant VaR", np.full_like(R, static)), ("GARCH VaR", parametric_var(0.0, np.sqrt(S2), 0.99))]:
        H = var_exceptions(R, V)
        uc = np.mean([kupiec_test(h).reject() for h in H])
        ind = np.mean([christoffersen_independence(h).reject() for h in H])
        print(f"{name:12s}: Kupiec rejects {uc:.0%}, independence rejects {ind:.0%} of 300 paths")

    plt.show()




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

 .. code-block:: none

    constant VaR: Kupiec rejects 50%, independence rejects 69% of 300 paths
    GARCH VaR   : Kupiec rejects 4%, independence rejects 1% of 300 paths





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

   **Total running time of the script:** (0 minutes 0.868 seconds)


.. _sphx_glr_download_api_gallery_risk_backtesting_plot_03_christoffersen.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_03_christoffersen.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_christoffersen.ipynb <plot_03_christoffersen.ipynb>`

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

      :download:`Download Python source code: plot_03_christoffersen.py <plot_03_christoffersen.py>`

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

      :download:`Download zipped: plot_03_christoffersen.zip <plot_03_christoffersen.zip>`


.. only:: html

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

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