.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/capm/plot_03_zero_beta.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_asset_pricing_capm_plot_03_zero_beta.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_asset_pricing_capm_plot_03_zero_beta.py:


Black's zero-beta CAPM
======================

Sharpe's CAPM assumes unlimited riskless borrowing. Fischer Black
(1972) dropped it: every frontier portfolio :math:`m` still prices all
assets linearly, with the expected return of :math:`z`, the
minimum-variance portfolio uncorrelated with :math:`m`, in place of the
riskless rate,

.. math::

   E[r_i] = E[r_z] + \beta_{im} (E[r_m] - E[r_z]).

Since :math:`E[r_z]` lies above the lending rate, the security market
line is flatter than Sharpe's, which matched Black, Jensen and Scholes's
(1972) finding that low-beta stocks earn more, and high-beta ones less,
than the CAPM predicts.

.. GENERATED FROM PYTHON SOURCE LINES 22-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import zero_beta_capm

    mu = np.array([0.05, 0.07, 0.09, 0.11, 0.13])
    vol = np.array([0.12, 0.16, 0.20, 0.25, 0.32])
    cov = (0.3 + 0.7 * np.eye(5)) * np.outer(vol, vol)
    res = zero_beta_capm(mu, cov, zero_beta_rate=0.02)
    wz, wm = res.zero_beta_weights, res.market_weights
    print(f"E[r_z] = {res.zero_beta_rate:.4f},  cov(z, m) = {wz @ cov @ wm:.2e}")





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

 .. code-block:: none

    E[r_z] = 0.0200,  cov(z, m) = 1.52e-18




.. GENERATED FROM PYTHON SOURCE LINES 35-37

The frontier, the market and its zero-beta portfolio
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 37-53

.. code-block:: Python


    ones = np.ones(5)
    Ci = np.linalg.inv(cov)
    A, B, C = ones @ Ci @ mu, mu @ Ci @ mu, ones @ Ci @ ones
    targets = np.linspace(0.02, 0.14, 200)
    sd = np.sqrt((C * targets**2 - 2 * A * targets + B) / (B * C - A**2))
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(sd, targets, "k", lw=1, label="minimum-variance frontier")
    axes[0].plot(np.sqrt(wm @ cov @ wm), res.market_return, "s", color="C3", label="market m")
    axes[0].plot(np.sqrt(wz @ cov @ wz), res.zero_beta_rate, "o", color="C0", label="zero-beta z")
    x = np.linspace(0, 0.3, 2)
    axes[0].plot(x, res.zero_beta_rate + (res.market_return - res.zero_beta_rate) / np.sqrt(wm @ cov @ wm) * x, "C0", ls=":", lw=1)
    axes[0].set_xlabel("volatility")
    axes[0].set_ylabel("expected return")
    axes[0].legend(fontsize=8)




.. image-sg:: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_03_zero_beta_001.png
   :alt: plot 03 zero beta
   :srcset: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_03_zero_beta_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116ab46e0>



.. GENERATED FROM PYTHON SOURCE LINES 54-56

The security market line starts at E[r_z]
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 56-67

.. code-block:: Python


    b = np.linspace(0, 1.6, 2)
    axes[1].plot(res.betas, mu, "o", color="C1", label="assets")
    axes[1].plot(b, res.zero_beta_rate + b * (res.market_return - res.zero_beta_rate), "C0", label="zero-beta SML")
    rf = 0.005
    axes[1].plot(b, rf + b * (res.market_return - rf), "C3", ls="--", label=f"Sharpe SML with r_f = {rf:.1%}")
    axes[1].set_xlabel("beta on m")
    axes[1].legend(fontsize=8)
    fig.tight_layout()

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.055 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_capm_plot_03_zero_beta.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/asset_pricing/capm/plot_03_zero_beta.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_zero_beta.ipynb <plot_03_zero_beta.ipynb>`

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

      :download:`Download Python source code: plot_03_zero_beta.py <plot_03_zero_beta.py>`

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

      :download:`Download zipped: plot_03_zero_beta.zip <plot_03_zero_beta.zip>`


.. only:: html

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

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