.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/structural/plot_04_distance_to_default.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_structural_plot_04_distance_to_default.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_structural_plot_04_distance_to_default.py:


Crosbie and Bohn: the distance to default
=========================================

Moody's KMV (Crosbie and Bohn 2003) turned Merton's model into a
default-prediction tool. Asset values are backed out from a daily equity
series by iterating: guess the asset volatility, invert the equity call
day by day, re-estimate the volatility from the implied asset returns,
and repeat. The default point is short-term debt plus half the
long-term debt, and the number of standard deviations between the
assets and that point,

.. math:: DD = \frac{\ln(V/D) + (\mu - \sigma_V^2/2)T}{\sigma_V\sqrt{T}},

ranks firms by default risk.

.. GENERATED FROM PYTHON SOURCE LINES 19-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import kmv_default_point, kmv_iterate
    from financialkit.integrators import gbm_paths
    from financialkit.options import black_scholes_price

    D = kmv_default_point(short_term_debt=40.0, long_term_debt=60.0)
    _, V = gbm_paths(100.0, -0.10, 0.25, T=2.0, n_steps=504, n_paths=1, rng=3)
    V = V[0]
    E = np.asarray(black_scholes_price(V, D, 1.0, 0.03, 0.25))

    res = kmv_iterate(E, D, r=0.03)
    print(f"default point {D:.0f}; converged in {res.n_iter} iterations to sigma_V = {res.asset_volatility:.3f}")





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

 .. code-block:: none

    default point 70; converged in 11 iterations to sigma_V = 0.250




.. GENERATED FROM PYTHON SOURCE LINES 35-37

The iteration
-------------

.. GENERATED FROM PYTHON SOURCE LINES 37-50

.. code-block:: Python


    fig, ax = plt.subplots(1, 2, figsize=(11, 4))
    ax[0].plot(res.volatility_history, "o-")
    ax[0].set_xlabel("iteration")
    ax[0].set_ylabel("asset volatility")
    ax[0].set_title("from the levered guess to the fixed point")
    days = np.arange(E.size)
    ax[1].plot(days, E, label="equity (observed)")
    ax[1].plot(days, res.asset_values, label="assets (implied)")
    ax[1].axhline(D, color="C3", ls="--", label="default point")
    ax[1].set_xlabel("trading day")
    ax[1].legend()




.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_04_distance_to_default_001.png
   :alt: from the levered guess to the fixed point
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_04_distance_to_default_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1163b2cf0>



.. GENERATED FROM PYTHON SOURCE LINES 51-56

A rolling distance to default
-----------------------------

Re-running the estimate on a one-year window each month tracks how the
firm drifts toward its default point.

.. GENERATED FROM PYTHON SOURCE LINES 56-65

.. code-block:: Python


    ends = np.arange(252, E.size + 1, 21)
    dd = [kmv_iterate(E[e - 252 : e], D, r=0.03).distance_to_default for e in ends]
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(ends, dd, "o-")
    ax.set_xlabel("trading day")
    ax.set_ylabel("distance to default")

    plt.show()



.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_04_distance_to_default_002.png
   :alt: plot 04 distance to default
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_04_distance_to_default_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.102 seconds)


.. _sphx_glr_download_api_gallery_credit_structural_plot_04_distance_to_default.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/structural/plot_04_distance_to_default.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_distance_to_default.ipynb <plot_04_distance_to_default.ipynb>`

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

      :download:`Download Python source code: plot_04_distance_to_default.py <plot_04_distance_to_default.py>`

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

      :download:`Download zipped: plot_04_distance_to_default.zip <plot_04_distance_to_default.zip>`


.. only:: html

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

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