.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/mean_variance/plot_01_markowitz_frontier.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_mean_variance_plot_01_markowitz_frontier.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_mean_variance_plot_01_markowitz_frontier.py:


Markowitz's mean-variance frontier
==================================

Harry Markowitz (1952) argued that an investor should look at a
portfolio's expected return and its variance together, and hold only
portfolios with the least variance for their mean. Diversification then
has a precise meaning: combining imperfectly correlated assets moves the
frontier to the left of every asset on its own. With short sales the
frontier is a hyperbola; without them, every point is a quadratic
program, solved here with :func:`scipy.optimize.minimize`.

.. GENERATED FROM PYTHON SOURCE LINES 15-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import efficient_frontier, min_variance_portfolio
    from financialkit.portfolio.visualizers import plot_frontier, plot_weights

    names = ["bonds", "credit", "US equity", "intl equity", "emerging"]
    mu = np.array([0.03, 0.045, 0.07, 0.075, 0.09])
    vol = np.array([0.05, 0.08, 0.16, 0.18, 0.25])
    corr = np.array(
        [
            [1.0, 0.6, -0.1, -0.1, 0.0],
            [0.6, 1.0, 0.3, 0.3, 0.4],
            [-0.1, 0.3, 1.0, 0.8, 0.7],
            [-0.1, 0.3, 0.8, 1.0, 0.75],
            [0.0, 0.4, 0.7, 0.75, 1.0],
        ]
    )
    cov = corr * np.outer(vol, vol)








.. GENERATED FROM PYTHON SOURCE LINES 36-38

Random portfolios fill the region the frontier bounds
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-49

.. code-block:: Python


    rng = np.random.default_rng(0)
    W = rng.dirichlet(np.ones(5) * 0.5, 3000)
    ax = plot_frontier(efficient_frontier(mu, cov, 40, long_only=True), labels=names)
    plot_frontier(efficient_frontier(mu, cov, 40, max_return=0.1), ax=ax, ls="--", label="with short sales")
    ax.scatter(np.sqrt(np.einsum("ij,jk,ik->i", W, cov, W)), W @ mu, s=2, color="0.8", zorder=0)
    gmv = min_variance_portfolio(cov, mu, long_only=True)
    ax.plot(gmv.volatility, gmv.expected_return, "D", color="C2", label="minimum variance")
    ax.legend(fontsize=8)
    print(f"minimum-variance portfolio: {gmv.volatility:.2%} volatility, below the safest asset's {vol.min():.0%}")




.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_01_markowitz_frontier_001.png
   :alt: plot 01 markowitz frontier
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_01_markowitz_frontier_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    minimum-variance portfolio: 4.61% volatility, below the safest asset's 5%




.. GENERATED FROM PYTHON SOURCE LINES 50-56

The composition along the long-only frontier
--------------------------------------------

The weights move smoothly from mostly bonds to all emerging markets.
Note that the minimum-variance portfolio holds some equity: its
negative correlation with bonds lowers risk.

.. GENERATED FROM PYTHON SOURCE LINES 56-62

.. code-block:: Python


    fr = efficient_frontier(mu, cov, 40, long_only=True)
    ax = plot_weights(fr.weights, labels=names, x=fr.volatilities)
    ax.set_xlabel("volatility")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_01_markowitz_frontier_002.png
   :alt: plot 01 markowitz frontier
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_01_markowitz_frontier_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_portfolio_mean_variance_plot_01_markowitz_frontier.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/mean_variance/plot_01_markowitz_frontier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_markowitz_frontier.ipynb <plot_01_markowitz_frontier.ipynb>`

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

      :download:`Download Python source code: plot_01_markowitz_frontier.py <plot_01_markowitz_frontier.py>`

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

      :download:`Download zipped: plot_01_markowitz_frontier.zip <plot_01_markowitz_frontier.zip>`


.. only:: html

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

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