.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/credit/structural/plot_03_leland.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_03_leland.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_03_leland.py:


Leland's endogenous default and optimal leverage
================================================

Leland (1994) priced perpetual debt on a firm whose coupons are tax
deductible and whose bankruptcy destroys half its assets. Shareholders
keep paying coupons as long as equity is worth something and default at
the asset level :math:`V_B` where equity hits zero with zero slope.
Firm value, unlevered assets plus tax shields minus expected bankruptcy
costs, is maximised at a coupon that sets the optimal leverage; in
Leland's base case this is about 75% with a 75 bp spread.

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.credit import leland_model, leland_optimal_coupon

    r, sigma, tau, alpha = 0.06, 0.20, 0.35, 0.50
    C_star, best = leland_optimal_coupon(100.0, r, sigma, tau, alpha)
    print(f"optimal coupon {C_star:.2f}: leverage {best.leverage:.1%}, spread {1e4 * best.credit_spread:.0f} bp, V_B = {best.default_barrier:.1f}")





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

 .. code-block:: none

    optimal coupon 6.50: leverage 75.0%, spread 75 bp, V_B = 52.8




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Equity is an option that shareholders exercise by defaulting
------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-40

.. code-block:: Python


    V = np.linspace(30, 160, 300)
    fig, ax = plt.subplots(figsize=(7, 4))
    for VB, ls in ((best.default_barrier, "-"), (0.8 * best.default_barrier, "--"), (1.2 * best.default_barrier, ":")):
        E = [leland_model(v, C_star, r, sigma, tau, alpha, VB).equity if v > VB else 0.0 for v in V]
        ax.plot(V, E, ls=ls, label=f"default at V_B = {VB:.1f}")
    ax.axhline(0, color="0.5", lw=0.8)
    ax.set_xlabel("asset value V")
    ax.set_ylabel("equity")
    ax.set_ylim(-10, 80)
    ax.set_title("only the endogenous barrier meets zero smoothly")
    ax.legend()




.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_03_leland_001.png
   :alt: only the endogenous barrier meets zero smoothly
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_03_leland_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1165641a0>



.. GENERATED FROM PYTHON SOURCE LINES 41-43

The trade-off of capital structure
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 43-59

.. code-block:: Python


    coupons = np.linspace(0.2, 14, 120)
    res = [leland_model(100.0, c, r, sigma, tau, alpha) for c in coupons]
    fig, ax = plt.subplots(1, 2, figsize=(11, 4))
    ax[0].plot(coupons, [x.firm_value for x in res], label="firm value")
    ax[0].plot(coupons, [x.debt for x in res], label="debt")
    ax[0].plot(coupons, [x.equity for x in res], label="equity")
    ax[0].axvline(C_star, color="k", ls=":")
    ax[0].set_xlabel("coupon")
    ax[0].legend()
    ax[1].plot([x.leverage for x in res], [1e4 * x.credit_spread for x in res])
    ax[1].axvline(best.leverage, color="k", ls=":")
    ax[1].set_xlabel("leverage")
    ax[1].set_ylabel("credit spread (bp)")

    plt.show()



.. image-sg:: /api/gallery/credit/structural/images/sphx_glr_plot_03_leland_002.png
   :alt: plot 03 leland
   :srcset: /api/gallery/credit/structural/images/sphx_glr_plot_03_leland_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.085 seconds)


.. _sphx_glr_download_api_gallery_credit_structural_plot_03_leland.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_03_leland.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_leland.ipynb <plot_03_leland.ipynb>`

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

      :download:`Download Python source code: plot_03_leland.py <plot_03_leland.py>`

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

      :download:`Download zipped: plot_03_leland.zip <plot_03_leland.zip>`


.. only:: html

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

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