.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/covariance/plot_02_ledoit_wolf.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_02_ledoit_wolf.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_02_ledoit_wolf.py:


Ledoit-Wolf shrinkage
=====================

When the number of assets is comparable to the number of observations,
the sample covariance's eigenvalues spread out far more than the true
ones: the smallest are too small, so the minimum-variance optimizer
loads up on spurious "riskless" combinations. Olivier Ledoit and
Michael Wolf (2004) shrank the sample matrix towards a structured target,
:math:`\delta F + (1 - \delta)S`, with the intensity :math:`\delta`
estimated from the data to minimize the expected squared error. Their
two targets are a scaled identity and the constant-correlation matrix.

.. GENERATED FROM PYTHON SOURCE LINES 16-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import ledoit_wolf, min_variance_portfolio, sample_covariance

    rng = np.random.default_rng(6)
    d = 40
    vol = rng.uniform(0.04, 0.1, d)
    true = (0.3 + 0.7 * np.eye(d)) * np.outer(vol, vol)
    L = np.linalg.cholesky(true)








.. GENERATED FROM PYTHON SOURCE LINES 28-30

Eigenvalue dispersion with 60 observations
------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-42

.. code-block:: Python


    X = rng.standard_normal((60, d)) @ L.T
    S, lw = sample_covariance(X), ledoit_wolf(X, "constant_correlation")
    fig, ax = plt.subplots(figsize=(7, 3.5))
    for C, label in [(true, "true"), (S.cov, "sample"), (lw.cov, f"Ledoit-Wolf, delta = {lw.shrinkage:.2f}")]:
        ax.semilogy(np.sort(np.linalg.eigvalsh(C))[::-1], "o-", ms=3, label=label)
    ax.set_xlabel("eigenvalue rank")
    ax.set_ylabel("eigenvalue")
    ax.legend()
    fig.tight_layout()
    print(f"condition numbers: sample {S.condition_number:.0f}, shrunk {lw.condition_number:.0f}")




.. image-sg:: /api/gallery/portfolio/covariance/images/sphx_glr_plot_02_ledoit_wolf_001.png
   :alt: plot 02 ledoit wolf
   :srcset: /api/gallery/portfolio/covariance/images/sphx_glr_plot_02_ledoit_wolf_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    condition numbers: sample 761, shrunk 65




.. GENERATED FROM PYTHON SOURCE LINES 43-49

Shrinkage intensity and out-of-sample risk
------------------------------------------

The true volatility of the minimum-variance portfolio built from each
estimate. Shrinkage helps most when data are scarce and fades as they
accumulate.

.. GENERATED FROM PYTHON SOURCE LINES 49-74

.. code-block:: Python


    ns = [45, 60, 90, 150, 300, 600]
    rows = []
    for n in ns:
        r = []
        for _ in range(20):
            X = rng.standard_normal((n, d)) @ L.T
            ests = [sample_covariance(X), ledoit_wolf(X), ledoit_wolf(X, "constant_correlation")]
            r.append([np.sqrt(w @ true @ w) for w in (min_variance_portfolio(e.cov).weights for e in ests)] + [ests[2].shrinkage])
        rows.append(np.mean(r, axis=0))
    rows = np.array(rows)
    best = min_variance_portfolio(true).volatility
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.5))
    for k, label in enumerate(["sample", "LW identity", "LW constant correlation"]):
        ax1.plot(ns, rows[:, k] / best, "o-", label=label)
    ax1.set_xscale("log")
    ax1.set_xlabel("observations")
    ax1.set_ylabel("true vol / optimal vol")
    ax1.legend(fontsize=8)
    ax2.plot(ns, rows[:, 3], "o-")
    ax2.set_xscale("log")
    ax2.set_xlabel("observations")
    ax2.set_ylabel("shrinkage intensity")
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/covariance/images/sphx_glr_plot_02_ledoit_wolf_002.png
   :alt: plot 02 ledoit wolf
   :srcset: /api/gallery/portfolio/covariance/images/sphx_glr_plot_02_ledoit_wolf_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.162 seconds)


.. _sphx_glr_download_api_gallery_portfolio_covariance_plot_02_ledoit_wolf.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_02_ledoit_wolf.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_ledoit_wolf.ipynb <plot_02_ledoit_wolf.ipynb>`

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

      :download:`Download Python source code: plot_02_ledoit_wolf.py <plot_02_ledoit_wolf.py>`

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

      :download:`Download zipped: plot_02_ledoit_wolf.zip <plot_02_ledoit_wolf.zip>`


.. only:: html

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

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