.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/migration/plot_02_generator_matrix.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_migration_plot_02_generator_matrix.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_migration_plot_02_generator_matrix.py:


Jarrow, Lando and Turnbull: generator matrices
==============================================

Jarrow, Lando and Turnbull (1997) put rating migration in continuous
time. A generator :math:`\Lambda`, with non-negative off-diagonal
intensities and rows summing to zero, gives the transition matrix for
every horizon, :math:`P(t) = e^{\Lambda t}`. Default probabilities by
rating, and risky bond prices with recovery of treasury, follow for
any maturity, including fractions of a year that an annual matrix
cannot provide.

.. GENERATED FROM PYTHON SOURCE LINES 15-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import RATINGS, creditmetrics_matrix, generator_from_transition, jlt_risky_zero, regularize_generator, transition_from_generator
    from financialkit.credit.visualizers import plot_transition_matrix

    G = regularize_generator(generator_from_transition(creditmetrics_matrix()))
    plot_transition_matrix(G, labels=RATINGS, fmt="{:.1f}")
    plt.title("generator (% per year)")




.. image-sg:: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_001.png
   :alt: generator (% per year)
   :srcset: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'generator (% per year)')



.. GENERATED FROM PYTHON SOURCE LINES 26-28

The term structure of default probabilities
-------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-38

.. code-block:: Python


    t = np.linspace(0.0, 10.0, 81)
    Pt = transition_from_generator(G, t)
    fig, ax = plt.subplots(figsize=(7, 4))
    for i in range(7):
        ax.semilogy(t[1:], Pt[1:, i, -1], label=RATINGS[i])
    ax.set_xlabel("years")
    ax.set_ylabel("cumulative default probability")
    ax.legend(ncol=2)




.. image-sg:: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_002.png
   :alt: plot 02 generator matrix
   :srcset: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1165667b0>



.. GENERATED FROM PYTHON SOURCE LINES 39-41

exp(G t) interpolates the annual matrix powers
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-48

.. code-block:: Python


    P1 = transition_from_generator(G, 1.0)
    years = np.arange(1, 11)
    powers = [np.linalg.matrix_power(P1, n)[3, -1] for n in years]
    ax.plot(years, powers, "kx", label="P(1)^n, BBB")
    ax.legend(ncol=2)





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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116565940>



.. GENERATED FROM PYTHON SOURCE LINES 49-51

Risky zero-coupon spreads by rating
-----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 51-63

.. code-block:: Python


    T = np.linspace(0.25, 10, 40)
    fig, ax = plt.subplots(figsize=(7, 4))
    for i in range(1, 7):
        P = jlt_risky_zero(T, 0.03, G, i, recovery=0.4)
        ax.plot(T, 1e4 * (-np.log(P) / T - 0.03), label=RATINGS[i])
    ax.set_xlabel("maturity")
    ax.set_ylabel("spread (bp)")
    ax.set_yscale("log")
    ax.legend()

    plt.show()



.. image-sg:: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_003.png
   :alt: plot 02 generator matrix
   :srcset: /api/gallery/credit/migration/images/sphx_glr_plot_02_generator_matrix_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.138 seconds)


.. _sphx_glr_download_api_gallery_credit_migration_plot_02_generator_matrix.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/migration/plot_02_generator_matrix.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_generator_matrix.ipynb <plot_02_generator_matrix.ipynb>`

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

      :download:`Download Python source code: plot_02_generator_matrix.py <plot_02_generator_matrix.py>`

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

      :download:`Download zipped: plot_02_generator_matrix.zip <plot_02_generator_matrix.zip>`


.. only:: html

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

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