.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/portfolio_loss/plot_03_li_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_credit_portfolio_loss_plot_03_li_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_credit_portfolio_loss_plot_03_li_gaussian_copula.py:


Li's Gaussian copula for default times
======================================

Li (2000) took each name's default-time distribution from its CDS curve
and joined them with a Gaussian copula: correlated normals become
uniforms, and each uniform is mapped through that name's survival
curve. The marginals stay exactly as the market implies; the copula
correlation controls how often names default together, which is what a
first-to-default basket or a CDO tranche is really priced on.

.. GENERATED FROM PYTHON SOURCE LINES 14-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import HazardCurve, copula_default_times, default_correlation, joint_default_probability, nth_to_default_probability

    curve = HazardCurve.flat(0.02)
    T = 5.0
    p = float(curve.default_probability(T))

    fig, ax = plt.subplots(1, 2, figsize=(11, 4.5))
    for a, rho in zip(ax, (0.0, 0.7), strict=True):
        tau = copula_default_times(curve, rho, 3000, n_names=2, rng=0)
        a.scatter(np.minimum(tau[:, 0], 60), np.minimum(tau[:, 1], 60), s=3, alpha=0.4)
        a.axvline(T, color="C3", lw=0.8)
        a.axhline(T, color="C3", lw=0.8)
        a.set_xlim(0, 60)
        a.set_ylim(0, 60)
        a.set_title(f"rho = {rho}: P(both by {T:g}y) = {joint_default_probability(p, p, rho):.4f}")
        a.set_xlabel("default time, name 1")
    ax[0].set_ylabel("default time, name 2")




.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_03_li_gaussian_copula_001.png
   :alt: rho = 0.0: P(both by 5y) = 0.0091, rho = 0.7: P(both by 5y) = 0.0439
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_03_li_gaussian_copula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(104.22222222222221, 0.5, 'default time, name 2')



.. GENERATED FROM PYTHON SOURCE LINES 36-38

Default correlation is much smaller than copula correlation
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python


    rhos = np.linspace(0, 0.95, 20)
    print("copula rho -> default-indicator correlation:", [f"{r:.2f}->{default_correlation(p, p, r):.3f}" for r in rhos[::5]])





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

 .. code-block:: none

    copula rho -> default-indicator correlation: ['0.00->-0.000', '0.25->0.102', '0.50->0.246', '0.75->0.455']




.. GENERATED FROM PYTHON SOURCE LINES 43-45

A ten-name basket
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 45-57

.. code-block:: Python


    horizons = np.linspace(0.5, 10, 20)
    fig, ax = plt.subplots(figsize=(7, 4))
    for rho in (0.0, 0.3, 0.7):
        tau = copula_default_times(curve, rho, 20000, n_names=10, rng=1)
        ax.plot(horizons, nth_to_default_probability(tau, 1, horizons), label=f"first to default, rho = {rho}")
        ax.plot(horizons, nth_to_default_probability(tau, 3, horizons), ls="--", color=ax.lines[-1].get_color(), label=f"third to default, rho = {rho}")
    ax.set_xlabel("horizon (years)")
    ax.set_ylabel("probability")
    ax.legend(fontsize=8)

    plt.show()



.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_03_li_gaussian_copula_002.png
   :alt: plot 03 li gaussian copula
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_03_li_gaussian_copula_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.104 seconds)


.. _sphx_glr_download_api_gallery_credit_portfolio_loss_plot_03_li_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/credit/portfolio_loss/plot_03_li_gaussian_copula.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_li_gaussian_copula.ipynb <plot_03_li_gaussian_copula.ipynb>`

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

      :download:`Download Python source code: plot_03_li_gaussian_copula.py <plot_03_li_gaussian_copula.py>`

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

      :download:`Download zipped: plot_03_li_gaussian_copula.zip <plot_03_li_gaussian_copula.zip>`


.. only:: html

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

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