.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/sdf/plot_02_hansen_jagannathan.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_asset_pricing_sdf_plot_02_hansen_jagannathan.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_asset_pricing_sdf_plot_02_hansen_jagannathan.py:


The Hansen-Jagannathan bound
============================

Lars Peter Hansen and Ravi Jagannathan (1991) asked what asset returns
alone reveal about the stochastic discount factor. Any :math:`m` that
prices the returns, :math:`E[m R] = 1`, must satisfy

.. math::

   \frac{\sigma(m)}{E(m)} \ge \frac{|E[R^e]|}{\sigma(R^e)}

for every excess return :math:`R^e`. With US stocks' Sharpe ratio of
about 0.4 a year, the discount factor must vary by about 40% a year.
The power-utility SDF :math:`\beta (C_{t+1}/C_t)^{-\gamma}` varies by
about :math:`\gamma \sigma_c`, roughly :math:`0.036\gamma`, so it enters
the admissible region only for :math:`\gamma` around 25, and on the way
it passes through riskless rates :math:`1/E(m) - 1` above 10%.

.. GENERATED FROM PYTHON SOURCE LINES 22-47



.. image-sg:: /api/gallery/asset_pricing/sdf/images/sphx_glr_plot_02_hansen_jagannathan_001.png
   :alt: plot 02 hansen jagannathan
   :srcset: /api/gallery/asset_pricing/sdf/images/sphx_glr_plot_02_hansen_jagannathan_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    stock Sharpe ratio over bills: 0.36
     CRRA, gamma = 2: E[m] = 0.957, sigma(m) = 0.069, admissible: False
     CRRA, gamma = 5: E[m] = 0.920, sigma(m) = 0.167, admissible: False
    CRRA, gamma = 10: E[m] = 0.882, sigma(m) = 0.328, admissible: False
    CRRA, gamma = 15: E[m] = 0.874, sigma(m) = 0.509, admissible: False
    CRRA, gamma = 20: E[m] = 0.895, sigma(m) = 0.738, admissible: False
    CRRA, gamma = 25: E[m] = 0.946, sigma(m) = 1.057, admissible: True






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import LucasTree, hansen_jagannathan_bound, max_sharpe_ratio
    from financialkit.asset_pricing.visualizers import plot_hj_bound

    # Annual gross returns of stocks and bills, Mehra-Prescott sample moments.
    mu = np.array([1.0698, 1.0080])
    vol = np.array([0.1654, 0.0567])
    cov = np.array([[1.0, 0.1], [0.1, 1.0]]) * np.outer(vol, vol)
    bound = hansen_jagannathan_bound(mu, cov, np.linspace(0.85, 1.05, 200))
    print(f"stock Sharpe ratio over bills: {max_sharpe_ratio([mu[0] - mu[1]], [[cov[0, 0] - 2 * cov[0, 1] + cov[1, 1]]]):.2f}")

    models = {}
    for g in (2, 5, 10, 15, 20, 25):
        models[f"CRRA, gamma = {g}"] = LucasTree(0.99, g, 0.018, 0.036).moments()

    fig, ax = plt.subplots(figsize=(7, 4.5))
    plot_hj_bound(bound, models, ax=ax)
    for name, m in models.items():
        print(f"{name:>16}: E[m] = {m.sdf_mean:.3f}, sigma(m) = {m.sdf_vol:.3f}, admissible: {bound.admits(m.sdf_mean, m.sdf_vol)}")
    ax.set_ylim(0, 1.2)
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.048 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_sdf_plot_02_hansen_jagannathan.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/asset_pricing/sdf/plot_02_hansen_jagannathan.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_hansen_jagannathan.ipynb <plot_02_hansen_jagannathan.ipynb>`

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

      :download:`Download Python source code: plot_02_hansen_jagannathan.py <plot_02_hansen_jagannathan.py>`

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

      :download:`Download zipped: plot_02_hansen_jagannathan.zip <plot_02_hansen_jagannathan.zip>`


.. only:: html

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

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