.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/growth/plot_01_kelly_criterion.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_portfolio_growth_plot_01_kelly_criterion.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_portfolio_growth_plot_01_kelly_criterion.py:


The Kelly criterion
===================

John Kelly (1956), at Bell Labs, asked how a gambler with a private wire
should bet. Staking the fraction :math:`f` of wealth on a bet won with
probability :math:`p` at odds :math:`b`, wealth grows at the rate
:math:`g(f) = p\log(1 + bf) + q\log(1 - f)` per bet, maximized by
:math:`f^* = p - q/b`. The Kelly bettor eventually outgrows any other
fixed-fraction strategy; betting more than twice Kelly makes growth
negative even though every bet is favourable.

.. GENERATED FROM PYTHON SOURCE LINES 15-24

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import kelly_fraction, kelly_growth_rate, kelly_wealth_paths

    p = 0.55
    f_star = kelly_fraction(p)
    print(f"Kelly fraction for an even-money bet with p = {p}: {f_star:.2f}")





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

 .. code-block:: none

    Kelly fraction for an even-money bet with p = 0.55: 0.10




.. GENERATED FROM PYTHON SOURCE LINES 25-27

Growth rate against the fraction staked
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 27-39

.. code-block:: Python


    f = np.linspace(0, 0.25, 300)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(f, kelly_growth_rate(f, p))
    ax.axvline(f_star, color="C2", ls="--", label="Kelly")
    ax.axvline(2 * f_star, color="C3", ls=":", label="twice Kelly")
    ax.axhline(0, color="0.3", lw=0.8)
    ax.set_xlabel("fraction of wealth staked")
    ax.set_ylabel("expected log growth per bet")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/portfolio/growth/images/sphx_glr_plot_01_kelly_criterion_001.png
   :alt: plot 01 kelly criterion
   :srcset: /api/gallery/portfolio/growth/images/sphx_glr_plot_01_kelly_criterion_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 40-45

A thousand bets, three bettors
------------------------------

The same coin flips for each bettor. Half Kelly gives up a quarter of
the growth for much smaller swings; double Kelly goes nowhere.

.. GENERATED FROM PYTHON SOURCE LINES 45-58

.. code-block:: Python


    n_bets, n_paths = 1000, 200
    fig, ax = plt.subplots(figsize=(7, 4))
    for mult, color in [(0.5, "C0"), (1.0, "C2"), (2.0, "C3")]:
        W = kelly_wealth_paths(mult * f_star, p, n_bets, n_paths, rng=0)
        ax.semilogy(np.median(W, axis=0), color=color, lw=2, label=f"{mult:g} x Kelly (median)")
        ax.fill_between(np.arange(n_bets + 1), *np.percentile(W, [10, 90], axis=0), color=color, alpha=0.15)
        print(f"{mult:g} x Kelly: median final wealth {np.median(W[:, -1]):.3g}, P(lose money) {np.mean(W[:, -1] < 1):.2f}")
    ax.set_xlabel("bet")
    ax.set_ylabel("wealth")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/growth/images/sphx_glr_plot_01_kelly_criterion_002.png
   :alt: plot 01 kelly criterion
   :srcset: /api/gallery/portfolio/growth/images/sphx_glr_plot_01_kelly_criterion_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    0.5 x Kelly: median final wealth 47.1, P(lose money) 0.01
    1 x Kelly: median final wealth 183, P(lose money) 0.04
    2 x Kelly: median final wealth 1.31, P(lose money) 0.47





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

   **Total running time of the script:** (0 minutes 0.131 seconds)


.. _sphx_glr_download_api_gallery_portfolio_growth_plot_01_kelly_criterion.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/portfolio/growth/plot_01_kelly_criterion.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_kelly_criterion.ipynb <plot_01_kelly_criterion.ipynb>`

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      :download:`Download Python source code: plot_01_kelly_criterion.py <plot_01_kelly_criterion.py>`

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      :download:`Download zipped: plot_01_kelly_criterion.zip <plot_01_kelly_criterion.zip>`


.. only:: html

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

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