.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/performance/plot_02_fundamental_law.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_performance_plot_02_fundamental_law.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_performance_plot_02_fundamental_law.py:


Grinold's fundamental law of active management
==============================================

Richard Grinold (1989) decomposed an active manager's information ratio
into skill and breadth:

.. math::

   IR \approx IC \sqrt{BR},

where the information coefficient :math:`IC` is the correlation between
forecasts and outcomes and the breadth :math:`BR` is the number of
independent forecasts a year. A tiny edge applied to many independent
bets beats a large edge applied to a few: an IC of 0.05 on 400 stocks
gives the same IR as an IC of 0.5 on four markets.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import fundamental_law, simulate_active_management








.. GENERATED FROM PYTHON SOURCE LINES 26-32

Simulated managers against the law
----------------------------------

Each manager makes ``breadth`` independent bets a year for 400 years.
The exact IR of this model, :math:`IC\sqrt{BR}/\sqrt{1 + IC^2}`, is
barely below Grinold's first-order law.

.. GENERATED FROM PYTHON SOURCE LINES 32-46

.. code-block:: Python


    breadths = np.array([4, 16, 64, 256, 1024])
    fig, ax = plt.subplots(figsize=(7, 4))
    for k, ic in enumerate([0.02, 0.05, 0.1]):
        sims = [simulate_active_management(ic, int(b), 400, rng=k * 10 + i) for i, b in enumerate(breadths)]
        ax.errorbar(breadths, [s.ir for s in sims], yerr=[2 * s.ir_se for s in sims], fmt="o", color=f"C{k}", capsize=3)
        bb = np.logspace(0.5, 3.1, 100)
        ax.plot(bb, fundamental_law(ic, bb), color=f"C{k}", label=f"IC = {ic}")
    ax.set_xscale("log")
    ax.set_xlabel("breadth (independent bets per year)")
    ax.set_ylabel("information ratio")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/portfolio/performance/images/sphx_glr_plot_02_fundamental_law_001.png
   :alt: plot 02 fundamental law
   :srcset: /api/gallery/portfolio/performance/images/sphx_glr_plot_02_fundamental_law_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 47-49

Same IR, different businesses
-----------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-54

.. code-block:: Python


    for ic, br in [(0.5, 4), (0.1, 100), (0.05, 400), (0.02, 2500)]:
        res = simulate_active_management(ic, br, 2000, rng=1)
        print(f"IC {ic:<4} x sqrt({br:>4}) = {fundamental_law(ic, br):.2f}; simulated IR {res.ir:.2f} +/- {res.ir_se:.2f}")
    plt.show()




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

 .. code-block:: none

    IC 0.5  x sqrt(   4) = 1.00; simulated IR 0.90 +/- 0.03
    IC 0.1  x sqrt( 100) = 1.00; simulated IR 0.98 +/- 0.03
    IC 0.05 x sqrt( 400) = 1.00; simulated IR 0.96 +/- 0.03
    IC 0.02 x sqrt(2500) = 1.00; simulated IR 1.00 +/- 0.03





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

   **Total running time of the script:** (0 minutes 0.143 seconds)


.. _sphx_glr_download_api_gallery_portfolio_performance_plot_02_fundamental_law.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/performance/plot_02_fundamental_law.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_fundamental_law.ipynb <plot_02_fundamental_law.ipynb>`

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

      :download:`Download Python source code: plot_02_fundamental_law.py <plot_02_fundamental_law.py>`

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

      :download:`Download zipped: plot_02_fundamental_law.zip <plot_02_fundamental_law.zip>`


.. only:: html

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

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