.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/performance/plot_01_sharpe_ratio.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_01_sharpe_ratio.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_01_sharpe_ratio.py:


The Sharpe ratio
================

William Sharpe (1966) ranked 34 mutual funds by their
*reward-to-variability ratio*, the mean excess return divided by its
standard deviation. By Tobin's separation it is the right yardstick for
an investor's whole portfolio: leverage changes the return but not the
ratio. It is also an estimate, and Andrew Lo (2002) showed how noisy:
with independent returns its standard error is
:math:`\sqrt{(1 + SR^2/2)/n}` per period, so years of data separate good
managers from lucky ones only slowly.

.. GENERATED FROM PYTHON SOURCE LINES 16-23

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import sharpe_ratio, sharpe_ratio_se

    rng = np.random.default_rng(7)








.. GENERATED FROM PYTHON SOURCE LINES 24-26

Leverage does not change the ratio
----------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 26-33

.. code-block:: Python


    r = rng.normal(0.01, 0.04, 120)
    rf = 0.002
    for lev in [0.5, 1.0, 2.0]:
        levered = rf + lev * (r - rf)
        print(f"leverage {lev:g}: mean {12 * levered.mean():.1%}, annual Sharpe {sharpe_ratio(levered, rf, 12):.3f}")





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

 .. code-block:: none

    leverage 0.5: mean 3.7%, annual Sharpe 0.214
    leverage 1: mean 5.0%, annual Sharpe 0.214
    leverage 2: mean 7.5%, annual Sharpe 0.214




.. GENERATED FROM PYTHON SOURCE LINES 34-39

Sampling distribution of the estimate
-------------------------------------

Ten years of monthly returns from a fund whose true annual Sharpe
ratio is 0.5, repeated 4000 times.

.. GENERATED FROM PYTHON SOURCE LINES 39-54

.. code-block:: Python


    true_sr = 0.5
    R = rng.normal(true_sr / np.sqrt(12) * 0.04, 0.04, size=(120, 4000))
    est = sharpe_ratio(R, 0.0, 12)
    se = sharpe_ratio_se(R[:, 0], 0.0, 12)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.hist(est, bins=60, density=True, color="0.75", label="estimates")
    x = np.linspace(est.min(), est.max(), 200)
    ax.plot(x, np.exp(-0.5 * ((x - true_sr) / se) ** 2) / (se * np.sqrt(2 * np.pi)), "C3", label=f"Lo's normal approximation, se = {se:.2f}")
    ax.axvline(true_sr, color="k", ls="--")
    ax.set_xlabel("estimated annual Sharpe ratio")
    ax.legend()
    fig.tight_layout()
    print(f"P(estimate < 0) = {np.mean(est < 0):.1%}")




.. image-sg:: /api/gallery/portfolio/performance/images/sphx_glr_plot_01_sharpe_ratio_001.png
   :alt: plot 01 sharpe ratio
   :srcset: /api/gallery/portfolio/performance/images/sphx_glr_plot_01_sharpe_ratio_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    P(estimate < 0) = 5.6%




.. GENERATED FROM PYTHON SOURCE LINES 55-57

Years needed to tell a Sharpe ratio from zero
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 57-68

.. code-block:: Python


    years = np.arange(1, 41)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    for s in [0.25, 0.5, 1.0]:
        ax.plot(years, s / np.sqrt((1 + 0.5 * (s / np.sqrt(12)) ** 2) / (12 * years)) / np.sqrt(12), label=f"SR = {s:g}")
    ax.axhline(1.96, color="k", ls=":")
    ax.set_xlabel("years of monthly data")
    ax.set_ylabel("t-statistic")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/performance/images/sphx_glr_plot_01_sharpe_ratio_002.png
   :alt: plot 01 sharpe ratio
   :srcset: /api/gallery/portfolio/performance/images/sphx_glr_plot_01_sharpe_ratio_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_portfolio_performance_plot_01_sharpe_ratio.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_01_sharpe_ratio.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_sharpe_ratio.ipynb <plot_01_sharpe_ratio.ipynb>`

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

      :download:`Download Python source code: plot_01_sharpe_ratio.py <plot_01_sharpe_ratio.py>`

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

      :download:`Download zipped: plot_01_sharpe_ratio.zip <plot_01_sharpe_ratio.zip>`


.. only:: html

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

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