.. 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_05_tranche_expected_loss.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_05_tranche_expected_loss.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_05_tranche_expected_loss.py:


Hull and White: pricing CDO tranches without Monte Carlo
========================================================

A tranche of a synthetic CDO absorbs portfolio losses between its
attachment and detachment points. Hull and White (2004) priced tranches
in Li's one-factor Gaussian copula by conditioning on the common factor:
given it, defaults are independent and the pool loss is binomial, so the
expected tranche loss at each date is one integral over the factor.
Raising the correlation moves risk from the equity tranche, which loses
less, to the senior tranches, which lose more.

.. GENERATED FROM PYTHON SOURCE LINES 15-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import HazardCurve, expected_tranche_loss, tranche_spread
    from financialkit.credit.visualizers import plot_tranche_losses

    curve = HazardCurve.flat(0.012)
    tranches = [(0.0, 0.03), (0.03, 0.07), (0.07, 0.10), (0.10, 0.15), (0.15, 0.30)]
    res = [tranche_spread(a, d, curve, 0.25, 5.0, 0.03, n_names=125) for a, d in tranches]
    ax = plot_tranche_losses(res)
    ax.set_title("125 names, 5 years, copula correlation 0.25")




.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_05_tranche_expected_loss_001.png
   :alt: 125 names, 5 years, copula correlation 0.25
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_05_tranche_expected_loss_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, '125 names, 5 years, copula correlation 0.25')



.. GENERATED FROM PYTHON SOURCE LINES 28-30

Spreads against correlation
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-40

.. code-block:: Python


    rhos = np.linspace(0.02, 0.9, 15)
    fig, ax = plt.subplots(figsize=(7, 4))
    for a, d in tranches:
        s = [tranche_spread(a, d, curve, rho, 5.0, 0.03).par_spread for rho in rhos]
        ax.semilogy(rhos, 1e4 * np.array(s), label=f"{100 * a:g}-{100 * d:g}%")
    ax.set_xlabel("copula correlation")
    ax.set_ylabel("par spread (bp, large pool)")
    ax.legend()




.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_05_tranche_expected_loss_002.png
   :alt: plot 05 tranche expected loss
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_05_tranche_expected_loss_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1163b38c0>



.. GENERATED FROM PYTHON SOURCE LINES 41-43

Finite pool versus the large-pool limit
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-50

.. code-block:: Python


    p = float(curve.default_probability(5.0))
    for n in (25, 125, 1000, None):
        el = expected_tranche_loss(0.03, 0.07, p, 0.25, n_names=n)
        print(f"{'LHP' if n is None else n:>5}: expected 3-7% tranche loss at 5y = {el:.4f}")

    plt.show()




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

 .. code-block:: none

       25: expected 3-7% tranche loss at 5y = 0.2634
      125: expected 3-7% tranche loss at 5y = 0.2454
     1000: expected 3-7% tranche loss at 5y = 0.2409
      LHP: expected 3-7% tranche loss at 5y = 0.2398





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

   **Total running time of the script:** (0 minutes 0.528 seconds)


.. _sphx_glr_download_api_gallery_credit_portfolio_loss_plot_05_tranche_expected_loss.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_05_tranche_expected_loss.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_tranche_expected_loss.ipynb <plot_05_tranche_expected_loss.ipynb>`

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

      :download:`Download Python source code: plot_05_tranche_expected_loss.py <plot_05_tranche_expected_loss.py>`

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

      :download:`Download zipped: plot_05_tranche_expected_loss.zip <plot_05_tranche_expected_loss.zip>`


.. only:: html

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

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