.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/copulas/plot_01_sklar_gaussian_copula.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_copulas_plot_01_sklar_gaussian_copula.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_copulas_plot_01_sklar_gaussian_copula.py:


Sklar's theorem and the Gaussian copula
=======================================

Sklar (1959) showed that every joint distribution is its marginals
glued together by a copula :math:`C`:

.. math:: F(x_1, x_2) = C\big(F_1(x_1), F_2(x_2)\big).

Risk managers can therefore model each asset's return on its own and
choose the dependence separately. Here a Gaussian copula with
correlation 0.7 joins a fat-tailed Student-t equity return to a skewed
credit loss, and the copula is recovered from the joint sample by
ranks alone.

.. GENERATED FROM PYTHON SOURCE LINES 18-28

.. code-block:: Python

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

    from financialkit.risk import apply_marginals, copula_kendall_tau, gaussian_copula_sample, pseudo_observations
    from financialkit.risk.visualizers import plot_copula

    corr = np.array([[1.0, 0.7], [0.7, 1.0]])
    u = gaussian_copula_sample(corr, 3000, rng=0)








.. GENERATED FROM PYTHON SOURCE LINES 29-31

One copula, any marginals
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 31-45

.. code-block:: Python


    equity = stats.t(4, scale=0.01)
    credit = stats.lognorm(0.8, scale=0.005)
    x = apply_marginals(u, [equity.ppf, credit.ppf])

    fig, axes = plt.subplots(1, 3, figsize=(13, 4))
    plot_copula(u, ax=axes[0], title="Gaussian copula, rho = 0.7")
    axes[1].scatter(x[:, 0], x[:, 1], s=2, alpha=0.4)
    axes[1].set_xlabel("equity return (t, 4 df)")
    axes[1].set_ylabel("credit loss (lognormal)")
    axes[1].set_title("joint distribution")
    plot_copula(pseudo_observations(x), ax=axes[2], title="copula recovered from ranks")
    fig.tight_layout()




.. image-sg:: /api/gallery/risk/copulas/images/sphx_glr_plot_01_sklar_gaussian_copula_001.png
   :alt: Gaussian copula, rho = 0.7, joint distribution, copula recovered from ranks
   :srcset: /api/gallery/risk/copulas/images/sphx_glr_plot_01_sklar_gaussian_copula_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 46-51

Rank correlation is a property of the copula
--------------------------------------------

Pearson correlation changes with the marginals; Kendall's tau does
not, and for the Gaussian copula it equals :math:`\tfrac{2}{\pi}\arcsin\rho`.

.. GENERATED FROM PYTHON SOURCE LINES 51-57

.. code-block:: Python


    print(f"Pearson, uniforms {np.corrcoef(u.T)[0, 1]:.3f}  vs  Pearson, t/lognormal {np.corrcoef(x.T)[0, 1]:.3f}")
    print(f"Kendall, uniforms {stats.kendalltau(*u.T).statistic:.3f}  vs  Kendall, t/lognormal {stats.kendalltau(*x.T).statistic:.3f}")
    print(f"theory 2/pi arcsin(0.7) = {copula_kendall_tau(0.7):.3f}")

    plt.show()




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

 .. code-block:: none

    Pearson, uniforms 0.677  vs  Pearson, t/lognormal 0.570
    Kendall, uniforms 0.486  vs  Kendall, t/lognormal 0.486
    theory 2/pi arcsin(0.7) = 0.494





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

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_risk_copulas_plot_01_sklar_gaussian_copula.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/copulas/plot_01_sklar_gaussian_copula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_sklar_gaussian_copula.ipynb <plot_01_sklar_gaussian_copula.ipynb>`

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

      :download:`Download Python source code: plot_01_sklar_gaussian_copula.py <plot_01_sklar_gaussian_copula.py>`

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

      :download:`Download zipped: plot_01_sklar_gaussian_copula.zip <plot_01_sklar_gaussian_copula.zip>`


.. only:: html

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

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