.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/covariance/plot_01_single_index.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_covariance_plot_01_single_index.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_covariance_plot_01_single_index.py:


Sharpe's single-index model
===========================

Markowitz's method needed every pairwise covariance, :math:`d(d+1)/2`
numbers, which no 1960s computer (or analyst) could supply for hundreds
of stocks. William Sharpe (1963) assumed that stocks move together only
through a common market index,
:math:`R_i = \alpha_i + \beta_i R_m + \varepsilon_i`, so the covariance
is :math:`\sigma_m^2\beta\beta' + \mathrm{diag}(\sigma_{\varepsilon}^2)`,
with :math:`3d + 1` parameters. The structure also filters noise: with
short histories, the single-index estimate is closer to the truth than
the sample covariance.

.. GENERATED FROM PYTHON SOURCE LINES 17-35

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import sample_covariance, single_index_covariance

    rng = np.random.default_rng(5)
    d, n = 30, 36
    beta = rng.uniform(0.5, 1.5, d)
    resid = rng.uniform(0.04, 0.10, d)
    sig_m = 0.045
    true = sig_m**2 * np.outer(beta, beta) + np.diag(resid**2)


    def simulate(n_months):
        m = sig_m * rng.standard_normal(n_months)
        return np.outer(m, beta) + resid * rng.standard_normal((n_months, d)), m









.. GENERATED FROM PYTHON SOURCE LINES 36-38

Fitted betas from three years of monthly data
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-49

.. code-block:: Python


    X, m = simulate(n)
    si = single_index_covariance(X, m)
    fig, ax = plt.subplots(figsize=(5, 4))
    ax.plot(beta, si.betas, "o")
    ax.plot([0.4, 1.6], [0.4, 1.6], "k:")
    ax.set_xlabel("true beta")
    ax.set_ylabel("estimated beta")
    fig.tight_layout()
    print(f"parameters: sample {sample_covariance(X).n_params}, single index {si.n_params}")




.. image-sg:: /api/gallery/portfolio/covariance/images/sphx_glr_plot_01_single_index_001.png
   :alt: plot 01 single index
   :srcset: /api/gallery/portfolio/covariance/images/sphx_glr_plot_01_single_index_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    parameters: sample 465, single index 91




.. GENERATED FROM PYTHON SOURCE LINES 50-52

Estimation error against the length of history
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-71

.. code-block:: Python


    months = [24, 36, 60, 120, 240]
    err_s, err_si = [], []
    for T in months:
        es, esi = [], []
        for _ in range(30):
            X, m = simulate(T)
            es.append(np.linalg.norm(sample_covariance(X).cov - true))
            esi.append(np.linalg.norm(single_index_covariance(X, m).cov - true))
        err_s.append(np.mean(es))
        err_si.append(np.mean(esi))
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.loglog(months, err_s, "o-", label="sample covariance")
    ax.loglog(months, err_si, "s-", label="single-index model")
    ax.set_xlabel("months of data")
    ax.set_ylabel("Frobenius error")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/covariance/images/sphx_glr_plot_01_single_index_002.png
   :alt: plot 01 single index
   :srcset: /api/gallery/portfolio/covariance/images/sphx_glr_plot_01_single_index_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.108 seconds)


.. _sphx_glr_download_api_gallery_portfolio_covariance_plot_01_single_index.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/covariance/plot_01_single_index.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_single_index.ipynb <plot_01_single_index.ipynb>`

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

      :download:`Download Python source code: plot_01_single_index.py <plot_01_single_index.py>`

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

      :download:`Download zipped: plot_01_single_index.zip <plot_01_single_index.zip>`


.. only:: html

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

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