.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/risk_based/plot_02_most_diversified.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_risk_based_plot_02_most_diversified.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_risk_based_plot_02_most_diversified.py:


Choueifaty's most-diversified portfolio
=======================================

Yves Choueifaty and Yves Coignard (2008) defined the diversification
ratio of a long-only portfolio as its weighted average volatility over
its actual volatility,

.. math::

   D(w) = \frac{w'\sigma}{\sqrt{w'\Sigma w}},

which equals one for a single asset and :math:`\sqrt{d}` for :math:`d`
uncorrelated, equally risky assets held equally. The portfolio that
maximizes it has a neat property: every asset it holds has the same
correlation with it, and every asset it leaves out has a higher one.

.. GENERATED FROM PYTHON SOURCE LINES 20-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import diversification_ratio, inverse_volatility_portfolio, min_variance_portfolio, most_diversified_portfolio
    from financialkit.portfolio.visualizers import plot_weights

    names = ["govt", "IG credit", "HY credit", "US eq", "EU eq", "EM eq", "gold", "REITs"]
    vol = np.array([0.05, 0.06, 0.10, 0.16, 0.18, 0.22, 0.16, 0.20])
    corr = np.array(
        [
            [1.0, 0.7, 0.1, -0.2, -0.2, -0.1, 0.2, 0.1],
            [0.7, 1.0, 0.5, 0.2, 0.2, 0.3, 0.2, 0.3],
            [0.1, 0.5, 1.0, 0.6, 0.6, 0.6, 0.1, 0.6],
            [-0.2, 0.2, 0.6, 1.0, 0.85, 0.75, 0.0, 0.7],
            [-0.2, 0.2, 0.6, 0.85, 1.0, 0.8, 0.05, 0.65],
            [-0.1, 0.3, 0.6, 0.75, 0.8, 1.0, 0.15, 0.6],
            [0.2, 0.2, 0.1, 0.0, 0.05, 0.15, 1.0, 0.1],
            [0.1, 0.3, 0.6, 0.7, 0.65, 0.6, 0.1, 1.0],
        ]
    )
    cov = corr * np.outer(vol, vol)








.. GENERATED FROM PYTHON SOURCE LINES 43-45

Three risk-based portfolios
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-56

.. code-block:: Python


    ports = {
        "minimum variance": min_variance_portfolio(cov, long_only=True),
        "inverse volatility": inverse_volatility_portfolio(cov),
        "most diversified": most_diversified_portfolio(cov),
    }
    fig, axes = plt.subplots(3, 1, figsize=(7, 7), sharex=True)
    for ax, (label, p) in zip(axes, ports.items(), strict=True):
        plot_weights(p.weights, names, ax=ax).set_title(f"{label}: D = {diversification_ratio(p.weights, cov):.2f}, vol {p.volatility:.2%}")
    fig.tight_layout()




.. image-sg:: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_02_most_diversified_001.png
   :alt: minimum variance: D = 1.53, vol 4.40%, inverse volatility: D = 1.54, vol 6.98%, most diversified: D = 1.78, vol 5.12%
   :srcset: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_02_most_diversified_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 57-59

The core property
-----------------

.. GENERATED FROM PYTHON SOURCE LINES 59-68

.. code-block:: Python


    mdp = ports["most diversified"]
    rho = (cov @ mdp.weights) / (vol * mdp.volatility)
    fig, ax = plt.subplots(figsize=(7, 3))
    ax.bar(names, rho, color=["C0" if w > 1e-6 else "0.7" for w in mdp.weights])
    ax.set_ylabel("correlation with the MDP")
    ax.set_title("held assets (blue) share one correlation; excluded ones (grey) exceed it")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_02_most_diversified_002.png
   :alt: held assets (blue) share one correlation; excluded ones (grey) exceed it
   :srcset: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_02_most_diversified_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.123 seconds)


.. _sphx_glr_download_api_gallery_portfolio_risk_based_plot_02_most_diversified.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
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        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_most_diversified.ipynb <plot_02_most_diversified.ipynb>`

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      :download:`Download Python source code: plot_02_most_diversified.py <plot_02_most_diversified.py>`

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

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