.. 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_02_tobin_tangency.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_02_tobin_tangency.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_02_tobin_tangency.py:


Tobin separation and the tangency portfolio
===========================================

James Tobin (1958) added a riskless asset to Markowitz's problem.
Combinations of cash and any risky portfolio lie on a straight line, and
the steepest such line touches the frontier at the *tangency portfolio*.
Every mean-variance investor holds that same mix of risky assets and
differs only in how much cash goes with it: the risky choice separates
from the attitude to risk. The line is the capital market line, and its
slope is the highest attainable Sharpe ratio.

.. GENERATED FROM PYTHON SOURCE LINES 15-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import efficient_frontier, tangency_portfolio, two_fund_allocation
    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)
    rf = 0.02








.. GENERATED FROM PYTHON SOURCE LINES 37-39

The capital market line
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 39-44

.. code-block:: Python


    t = tangency_portfolio(mu, cov, rf)
    ax = plot_frontier(efficient_frontier(mu, cov, 60, max_return=0.12), labels=names, rf=rf)
    print(f"tangency portfolio: mean {t.expected_return:.2%}, volatility {t.volatility:.2%}, Sharpe {t.sharpe_ratio(rf):.3f}")




.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_001.png
   :alt: plot 02 tobin tangency
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    tangency portfolio: mean 4.56%, volatility 6.22%, Sharpe 0.411




.. GENERATED FROM PYTHON SOURCE LINES 45-50

Three investors, one risky portfolio
------------------------------------

The optimal allocation :math:`\Sigma^{-1}(\mu - r_f)/\gamma` scales with
risk tolerance; normalized, it is the tangency portfolio every time.

.. GENERATED FROM PYTHON SOURCE LINES 50-62

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 3.5))
    for i, gamma in enumerate([4.0, 8.0, 16.0]):
        res = two_fund_allocation(mu, cov, rf, gamma)
        ax.bar(np.arange(6) + 0.27 * (i - 1), np.append(res.weights, 1 - res.weights.sum()), width=0.27, label=f"gamma = {gamma:g}")
        print(f"gamma = {gamma:g}: risky mix {np.round(res.weights / res.weights.sum(), 3)}, cash {1 - res.weights.sum():+.2f}")
    ax.set_xticks(range(6), names + ["cash"])
    ax.axhline(0, color="0.3", lw=0.8)
    ax.set_ylabel("weight")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_002.png
   :alt: plot 02 tobin tangency
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    gamma = 4: risky mix [0.484 0.227 0.168 0.12  0.   ], cash -0.65
    gamma = 8: risky mix [0.484 0.227 0.168 0.12  0.   ], cash +0.17
    gamma = 16: risky mix [0.484 0.227 0.168 0.12  0.   ], cash +0.59




.. GENERATED FROM PYTHON SOURCE LINES 63-67

.. code-block:: Python


    plot_weights(t.weights, labels=names).set_title("tangency portfolio")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_003.png
   :alt: tangency portfolio
   :srcset: /api/gallery/portfolio/mean_variance/images/sphx_glr_plot_02_tobin_tangency_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.089 seconds)


.. _sphx_glr_download_api_gallery_portfolio_mean_variance_plot_02_tobin_tangency.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_02_tobin_tangency.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_tobin_tangency.ipynb <plot_02_tobin_tangency.ipynb>`

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

      :download:`Download Python source code: plot_02_tobin_tangency.py <plot_02_tobin_tangency.py>`

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

      :download:`Download zipped: plot_02_tobin_tangency.zip <plot_02_tobin_tangency.zip>`


.. only:: html

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

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