.. 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_02_jensen_alpha.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_02_jensen_alpha.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_02_jensen_alpha.py:


Jensen's alpha
==============

Michael Jensen (1968) measured a mutual fund's skill as the intercept
of its excess return regressed on the market's,

.. math::

   r_{pt} - r_{ft} = \alpha_p + \beta_p (r_{mt} - r_{ft}) + \varepsilon_{pt}.

A manager who takes more market risk earns more on average but no
alpha; only returns beyond the security market line count. Over 1945-1964
Jensen found the average alpha of 115 funds to be negative after fees,
and no more funds with significant positive alpha than chance allows.

.. GENERATED FROM PYTHON SOURCE LINES 19-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import jensen_alpha, simulate_factor_model

    rng = np.random.default_rng(1968)
    n_funds, T = 115, 240
    betas = rng.uniform(0.7, 1.3, n_funds)
    # Fees of 1% a year and no skill, except five genuinely skilled funds.
    true_alpha = np.full(n_funds, -0.01 / 12)
    true_alpha[:5] = 0.04 / 12
    panel = simulate_factor_model(T, betas, [0.006], [[0.0020]], idio_vol=0.02, alphas=true_alpha, rng=rng)
    res = jensen_alpha(panel.excess_returns, panel.factors[:, 0])
    print(f"mean annual alpha: {12 * res.alphas.mean():+.2%}")
    print(f"funds with alpha t > 2: {(res.alpha_t > 2).sum()} (true skilled: 5)")





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

 .. code-block:: none

    mean annual alpha: -0.70%
    funds with alpha t > 2: 4 (true skilled: 5)




.. GENERATED FROM PYTHON SOURCE LINES 36-51

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].hist(res.alpha_t, bins=25, color="0.7")
    axes[0].axvline(2, color="C3", ls="--")
    axes[0].set_xlabel("t-statistic of alpha")
    axes[0].set_title("most funds: no skill after fees")
    axes[1].scatter(res.betas[:, 0], 12 * res.alphas, s=10)
    axes[1].scatter(res.betas[:5, 0], 12 * res.alphas[:5], s=25, color="C3", label="skilled funds")
    axes[1].axhline(0, color="k", lw=0.8)
    axes[1].set_xlabel("estimated beta")
    axes[1].set_ylabel("annual alpha")
    axes[1].set_title("alpha is unrelated to beta")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_02_jensen_alpha_001.png
   :alt: most funds: no skill after fees, alpha is unrelated to beta
   :srcset: /api/gallery/asset_pricing/capm/images/sphx_glr_plot_02_jensen_alpha_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.090 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_capm_plot_02_jensen_alpha.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_02_jensen_alpha.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_jensen_alpha.ipynb <plot_02_jensen_alpha.ipynb>`

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

      :download:`Download Python source code: plot_02_jensen_alpha.py <plot_02_jensen_alpha.py>`

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

      :download:`Download zipped: plot_02_jensen_alpha.zip <plot_02_jensen_alpha.zip>`


.. only:: html

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

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