.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/var/plot_02_roy_safety_first.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_risk_var_plot_02_roy_safety_first.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_risk_var_plot_02_roy_safety_first.py:


Roy's safety-first criterion
============================

The same year as Markowitz, A. D. Roy (1952) asked which portfolio
minimises the chance of disaster, :math:`P(R < d)`. With only means and
variances known, Chebyshev's inequality bounds it by :math:`1 / s^2`,
where :math:`s = (\mu - d)/\sigma`, so the safest portfolio maximises
:math:`s`: the point where a line from :math:`(0, d)` touches the
efficient frontier. It is the first downside risk measure and an early
form of the Sharpe ratio.

.. GENERATED FROM PYTHON SOURCE LINES 15-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import safety_first_portfolio, safety_first_ratio, shortfall_probability

    mu = np.array([0.04, 0.07, 0.10, 0.14])
    vol = np.array([0.06, 0.12, 0.18, 0.28])
    corr = np.array([[1.0, 0.2, 0.1, 0.0], [0.2, 1.0, 0.4, 0.3], [0.1, 0.4, 1.0, 0.5], [0.0, 0.3, 0.5, 1.0]])
    cov = corr * np.outer(vol, vol)








.. GENERATED FROM PYTHON SOURCE LINES 26-28

The frontier and Roy's tangent
------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-47

.. code-block:: Python


    rng = np.random.default_rng(0)
    W = rng.dirichlet(np.ones(4), 4000)
    port_mu, port_sd = W @ mu, np.sqrt(np.einsum("ij,jk,ik->i", W, cov, W))
    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.scatter(port_sd, port_mu, s=2, c="0.75", label="random long-only portfolios")
    for d, color in [(-0.05, "C0"), (0.0, "C1"), (0.02, "C3")]:
        w = safety_first_portfolio(mu, cov, d)
        m, s = w @ mu, np.sqrt(w @ cov @ w)
        sr = safety_first_ratio(m, s, d)
        xs = np.linspace(0, 0.3, 2)
        ax.plot(xs, d + sr * xs, color=color, lw=1)
        ax.plot(s, m, "o", color=color, label=f"d = {d:+.0%}: s = {sr:.2f}, P(R<d) <= {shortfall_probability(m, s, d):.2f}")
    ax.set_xlabel("volatility")
    ax.set_ylabel("expected return")
    ax.set_xlim(0, 0.3)
    ax.legend(fontsize=8, loc="lower right")
    fig.tight_layout()




.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_02_roy_safety_first_001.png
   :alt: plot 02 roy safety first
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_02_roy_safety_first_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-53

Chebyshev's bound against the normal probability
------------------------------------------------

Roy's bound holds for any distribution, which is why it is loose: for
normal returns the true shortfall probability is far smaller.

.. GENERATED FROM PYTHON SOURCE LINES 53-64

.. code-block:: Python


    s = np.linspace(0.5, 4, 100)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.semilogy(s, shortfall_probability(s, 1.0, 0.0), label="Chebyshev bound 1/s^2")
    ax.semilogy(s, shortfall_probability(s, 1.0, 0.0, "normal"), label="normal: Phi(-s)")
    ax.set_xlabel("safety-first ratio s")
    ax.set_ylabel("P(R < d)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_02_roy_safety_first_002.png
   :alt: plot 02 roy safety first
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_02_roy_safety_first_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.096 seconds)


.. _sphx_glr_download_api_gallery_risk_var_plot_02_roy_safety_first.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/risk/var/plot_02_roy_safety_first.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_roy_safety_first.ipynb <plot_02_roy_safety_first.ipynb>`

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

      :download:`Download Python source code: plot_02_roy_safety_first.py <plot_02_roy_safety_first.py>`

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      :download:`Download zipped: plot_02_roy_safety_first.zip <plot_02_roy_safety_first.zip>`


.. only:: html

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

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