.. 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_01_vasicek_lhp.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_01_vasicek_lhp.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_01_vasicek_lhp.py:


Vasicek's large homogeneous portfolio
=====================================

Vasicek (1987) let every loan default when its asset return
:math:`\sqrt\rho M + \sqrt{1-\rho}\,\varepsilon_i` falls below
:math:`N^{-1}(p)`. Given the common factor :math:`M`, defaults are
independent, so in a large pool the default rate is the deterministic
:math:`p(M)`, and its distribution is known in closed form. Correlation
does not change the expected loss, only how it is spread out: at high
correlation most years are quiet and a few are disasters.

.. GENERATED FROM PYTHON SOURCE LINES 15-30

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import simulate_one_factor_defaults, vasicek_cdf, vasicek_pdf, vasicek_quantile

    p = 0.02
    x = np.linspace(1e-4, 0.12, 400)
    fig, ax = plt.subplots(figsize=(7, 4))
    for rho in (0.05, 0.12, 0.24):
        ax.plot(x, vasicek_pdf(x, p, rho), label=f"rho = {rho}, 99.9% quantile {vasicek_quantile(0.999, p, rho):.1%}")
    ax.axvline(p, color="k", ls=":", label="mean default rate p")
    ax.set_xlabel("portfolio default rate")
    ax.set_ylabel("density")
    ax.legend()




.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_01_vasicek_lhp_001.png
   :alt: plot 01 vasicek lhp
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_01_vasicek_lhp_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x11659a660>



.. GENERATED FROM PYTHON SOURCE LINES 31-33

Finite pools converge to the limit
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 33-46

.. code-block:: Python


    rho = 0.12
    fig, ax = plt.subplots(figsize=(7, 4))
    for n in (50, 500, 5000):
        frac = np.sort(simulate_one_factor_defaults(p, rho, n, 20000, rng=0) / n)
        ax.step(frac, np.arange(1, frac.size + 1) / frac.size, where="post", label=f"{n} loans (simulated)")
    ax.plot(x, vasicek_cdf(x, p, rho), "k", lw=2, label="Vasicek limit")
    ax.set_xlim(0, 0.08)
    ax.set_xlabel("portfolio default rate")
    ax.set_ylabel("cumulative probability")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_01_vasicek_lhp_002.png
   :alt: plot 01 vasicek lhp
   :srcset: /api/gallery/credit/portfolio_loss/images/sphx_glr_plot_01_vasicek_lhp_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.089 seconds)


.. _sphx_glr_download_api_gallery_credit_portfolio_loss_plot_01_vasicek_lhp.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_01_vasicek_lhp.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_vasicek_lhp.ipynb <plot_01_vasicek_lhp.ipynb>`

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

      :download:`Download Python source code: plot_01_vasicek_lhp.py <plot_01_vasicek_lhp.py>`

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

      :download:`Download zipped: plot_01_vasicek_lhp.zip <plot_01_vasicek_lhp.zip>`


.. only:: html

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

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