.. 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_03_merton_analytic_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_03_merton_analytic_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_03_merton_analytic_frontier.py:


Merton's analytic frontier
==========================

Robert Merton (1972) derived the unconstrained frontier in closed form.
Four scalars,

.. math::

   A = \mathbf 1'\Sigma^{-1}\mu,\quad B = \mu'\Sigma^{-1}\mu,\quad
   C = \mathbf 1'\Sigma^{-1}\mathbf 1,\quad D = BC - A^2,

give the least variance at every mean, :math:`\sigma^2(m) = (Cm^2 - 2Am
+ B)/D`: a hyperbola with vertex at the minimum-variance portfolio
:math:`(1/\sqrt C, A/C)` and asymptotes of slope :math:`\pm\sqrt{D/C}`.
Every frontier portfolio is a mix of two of them (two-fund separation).

.. GENERATED FROM PYTHON SOURCE LINES 20-33

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import efficient_portfolio, frontier_constants
    from financialkit.portfolio.utils import solve_qp

    mu = np.array([0.04, 0.06, 0.08, 0.11])
    vol = np.array([0.08, 0.12, 0.18, 0.26])
    corr = np.array([[1.0, 0.3, 0.2, 0.1], [0.3, 1.0, 0.5, 0.4], [0.2, 0.5, 1.0, 0.6], [0.1, 0.4, 0.6, 1.0]])
    cov = corr * np.outer(vol, vol)
    fc = frontier_constants(mu, cov)
    print(f"A = {fc.A:.3f}, B = {fc.B:.4f}, C = {fc.C:.2f}, D = {fc.D:.4f}")





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

 .. code-block:: none

    A = 8.271, B = 0.4556, C = 180.40, D = 13.7838




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Closed form against a numerical optimizer
-----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-53

.. code-block:: Python


    m = np.linspace(-0.05, 0.2, 200)
    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.plot(fc.volatility(m), m, lw=2, label="Merton's hyperbola")
    targets = np.linspace(-0.02, 0.18, 9)
    numeric = [np.sqrt(w @ cov @ w) for w in (solve_qp(cov, np.vstack([np.ones(4), mu]), [1.0, t], bounds=(-10.0, 10.0)) for t in targets)]
    ax.plot(numeric, targets, "o", color="C3", label="SLSQP")
    s = np.linspace(0, fc.volatility(0.2), 2)
    for sign in (1, -1):
        ax.plot(s, fc.gmv_return + sign * fc.asymptote_slope() * s, ":", color="0.4")
    ax.plot(fc.gmv_volatility, fc.gmv_return, "D", color="C2", label="vertex: minimum variance")
    ax.plot(vol, mu, "o", color="0.3")
    ax.set_xlabel("volatility")
    ax.set_ylabel("expected return")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_03_merton_analytic_frontier_001.png
   :alt: plot 03 merton analytic frontier
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_03_merton_analytic_frontier_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 54-59

Two-fund separation
-------------------

The weights are linear in the target mean, so any two frontier
portfolios span the whole frontier.

.. GENERATED FROM PYTHON SOURCE LINES 59-64

.. code-block:: Python


    w1, w2 = efficient_portfolio(mu, cov, 0.05).weights, efficient_portfolio(mu, cov, 0.10).weights
    w3 = efficient_portfolio(mu, cov, 0.15).weights
    print("w(15%) = 2 w(10%) - w(5%):", np.allclose(w3, 2 * w2 - w1))
    plt.show()




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

 .. code-block:: none

    w(15%) = 2 w(10%) - w(5%): True





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

   **Total running time of the script:** (0 minutes 0.035 seconds)


.. _sphx_glr_download_api_gallery_portfolio_mean_variance_plot_03_merton_analytic_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_03_merton_analytic_frontier.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_merton_analytic_frontier.ipynb <plot_03_merton_analytic_frontier.ipynb>`

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

      :download:`Download Python source code: plot_03_merton_analytic_frontier.py <plot_03_merton_analytic_frontier.py>`

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

      :download:`Download zipped: plot_03_merton_analytic_frontier.zip <plot_03_merton_analytic_frontier.zip>`


.. only:: html

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

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