.. _sphx_glr_api_gallery_portfolio_covariance:

Covariance estimation
---------------------

Structured and shrunk estimates of the covariance matrix, the input that
mean-variance optimizers are most sensitive to after the means.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Markowitz&#x27;s method needed every pairwise covariance, 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, R_i = \alpha_i + \beta_i R_m + \varepsilon_i, so the covariance is \sigma_m^2\beta\beta&#x27; + \mathrm{diag}(\sigma_{\varepsilon}^2), with 3d + 1 parameters. The structure also filters noise: with short histories, the single-index estimate is closer to the truth than the sample covariance.">

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  .. image:: /api/gallery/portfolio/covariance/images/thumb/sphx_glr_plot_01_single_index_thumb.png
    :alt:

  :doc:`/api/gallery/portfolio/covariance/plot_01_single_index`

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      <div class="sphx-glr-thumbnail-title">Sharpe's single-index model</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="When the number of assets is comparable to the number of observations, the sample covariance&#x27;s eigenvalues spread out far more than the true ones: the smallest are too small, so the minimum-variance optimizer loads up on spurious &quot;riskless&quot; combinations. Olivier Ledoit and Michael Wolf (2004) shrank the sample matrix towards a structured target, \delta F + (1 - \delta)S, with the intensity \delta estimated from the data to minimize the expected squared error. Their two targets are a scaled identity and the constant-correlation matrix.">

.. only:: html

  .. image:: /api/gallery/portfolio/covariance/images/thumb/sphx_glr_plot_02_ledoit_wolf_thumb.png
    :alt:

  :doc:`/api/gallery/portfolio/covariance/plot_02_ledoit_wolf`

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      <div class="sphx-glr-thumbnail-title">Ledoit-Wolf shrinkage</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/portfolio/covariance/plot_01_single_index
   /api/gallery/portfolio/covariance/plot_02_ledoit_wolf