.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/stylized_facts/plot_01_mandelbrot_fat_tails.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_time_series_stylized_facts_plot_01_mandelbrot_fat_tails.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_time_series_stylized_facts_plot_01_mandelbrot_fat_tails.py:


Mandelbrot: fat tails
=====================

Benoit Mandelbrot (1963) plotted the frequency of large changes in cotton
prices on doubly logarithmic paper and found straight lines: the tail
probability :math:`P(|X| > u)` decays like a power :math:`u^{-\alpha}`,
not like the Gaussian :math:`e^{-u^2/2}`. Five-sigma days, which a normal
distribution makes once-in-several-millennia events, happen every few
years. He proposed the stable (Lévy) laws, with :math:`\alpha \approx 1.7`
for cotton; later work settled on tail indices between 3 and 5 for most
returns, but the log-log plot remains the standard diagnostic.

.. GENERATED FROM PYTHON SOURCE LINES 16-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import stats

    from financialkit.time_series import return_statistics
    from financialkit.time_series.visualizers import plot_tail








.. GENERATED FROM PYTHON SOURCE LINES 24-28

Three samples with the same scale
---------------------------------
Daily "returns" from a normal, a Student t with 3 degrees of freedom, and
a symmetric stable law with Mandelbrot's :math:`\alpha = 1.7`.

.. GENERATED FROM PYTHON SOURCE LINES 28-47

.. code-block:: Python


    rng = np.random.default_rng(1963)
    n = 20_000
    samples = {
        "normal": rng.standard_normal(n),
        "Student t(3)": stats.t.rvs(3, size=n, random_state=rng) / np.sqrt(3),
        "stable (alpha 1.7)": stats.levy_stable.rvs(1.7, 0.0, size=n, random_state=rng) / np.sqrt(2),
    }

    fig, ax = plt.subplots()
    for name, x in samples.items():
        plot_tail(x, ax=ax, label=name)
    u = np.logspace(0.5, 2, 20)
    ax.loglog(u, 0.6 * u**-1.7, "k--", lw=1, label=r"slope $-1.7$")
    ax.set_ylim(1 / n, 1)
    ax.set_xlim(0.1, 200)
    ax.legend()
    ax.set_title("Log-log tails: straight lines are power laws")




.. image-sg:: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_01_mandelbrot_fat_tails_001.png
   :alt: Log-log tails: straight lines are power laws
   :srcset: /api/gallery/time_series/stylized_facts/images/sphx_glr_plot_01_mandelbrot_fat_tails_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Log-log tails: straight lines are power laws')



.. GENERATED FROM PYTHON SOURCE LINES 48-53

Moments
-------
The normal sample has zero excess kurtosis; the fat-tailed ones have
huge, unstable sample kurtosis (the stable law's variance is infinite),
and Jarque-Bera rejects normality at any level.

.. GENERATED FROM PYTHON SOURCE LINES 53-58

.. code-block:: Python


    for name, x in samples.items():
        s = return_statistics(x)
        print(f"{name:22s} excess kurtosis {s.excess_kurtosis:9.1f}  JB p-value {s.jarque_bera_pvalue:.2g}")





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

 .. code-block:: none

    normal                 excess kurtosis       0.0  JB p-value 0.23
    Student t(3)           excess kurtosis      86.8  JB p-value 0
    stable (alpha 1.7)     excess kurtosis    1268.7  JB p-value 0




.. GENERATED FROM PYTHON SOURCE LINES 59-61

How often are there 5-sigma days?
---------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 61-68

.. code-block:: Python


    for name, x in samples.items():
        freq = np.mean(np.abs(x) > 5 * x.std())
        years = np.inf if freq == 0 else 1 / (freq * 252)
        print(f"{name:22s} one |move| > 5 sd every {years:,.1f} years")
    print(f"{'normal theory':22s} one every {1 / (2 * stats.norm.sf(5) * 252):,.0f} years")
    plt.show()




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

 .. code-block:: none

    normal                 one |move| > 5 sd every inf years
    Student t(3)           one |move| > 5 sd every 1.3 years
    stable (alpha 1.7)     one |move| > 5 sd every 1.6 years
    normal theory          one every 6,922 years





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

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


.. _sphx_glr_download_api_gallery_time_series_stylized_facts_plot_01_mandelbrot_fat_tails.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/time_series/stylized_facts/plot_01_mandelbrot_fat_tails.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_mandelbrot_fat_tails.ipynb <plot_01_mandelbrot_fat_tails.ipynb>`

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

      :download:`Download Python source code: plot_01_mandelbrot_fat_tails.py <plot_01_mandelbrot_fat_tails.py>`

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

      :download:`Download zipped: plot_01_mandelbrot_fat_tails.zip <plot_01_mandelbrot_fat_tails.zip>`


.. only:: html

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

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