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.. "examples/Builtin/portfolio.py"
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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_examples_Builtin_portfolio.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_Builtin_portfolio.py:


Distributionally Robust Portfolio optimization
==============================================

The objective and instances are based on [Bertsimas et al. (2018), Sec. 7.5] (for the original robust portfolio problem) and [Esfahani et al. (2018), Sec. 7] for the WDRO counterpart.

We consider a market of m = 10 assets whose yearly returns are modeled as the sum of i) a common risk factor following a normal distribution of mean 0 and variance 0.02; and ii) a specific factor following a normal distribution of mean 0.03*i and variance 0.025*i for asset i (i=1,..,m).

We solve the WDRO version of the portfolio optimization problem

.. math::

    \mathbb{E}[ - \langle x ; \xi \rangle ] + \eta \mathrm{CVar}_\alpha[- \langle x ; \xi \rangle]

which amounts to using the following loss function

.. math::

    \ell(x,\tau;\xi) =  - \langle x ; \xi \rangle + \eta \tau + \frac{1}{\alpha} \max( - \langle x ; \xi \rangle - \tau ; 0)

where :math:`\tau` is an extra real parameter accounting for the threshold of the CVaR (see [Rockafellar and Uryasev (2000)]). The parameter :math:`x` is constrained to live in the simplex (This is encoded in the constraints of the problem in [Esfahani et al. (2018)] and by an exponential reparametrization for the entropy-regularized version). 

We set the confidence level :math:`\alpha` of the CVaR to 0.2 and the robustness coefficient :math:`\eta` to 10.

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.. code-block:: Python

    import numpy as np
    import matplotlib.pyplot as plt
    from skwdro.operations_research._portfolio import Portfolio


    ## PROBLEM SETUP
    N = 30 # Number of observations
    m = 10  # Number of assets

    common_var = 0.02               # Variance of the common part
    specific_mean_factor = 0.03     # Mean factor of the common part
    specific_var_factor  = 0.025    # Variance factor of the specific part

    psi = np.random.randn(N, 1) * np.sqrt(common_var)
    zeta = np.random.randn(N, m) * np.sqrt(specific_var_factor) + specific_mean_factor * np.arange(1,m+1)[None, :]
    returns_by_asset = psi + zeta

    ## RESOLUTION FOR DIFFERENT RADII
    tested_radii = np.logspace(-3, 0, 10)

    pbrs = [Portfolio(rho = i, alpha=.2, eta=10.).fit(returns_by_asset) for i in tested_radii]


    data = np.vstack([m.coef_ for m in pbrs]).T
    data = data[np.argsort(data[:, -1]), :]








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Robustness illustration
~~~~~~~~~~~~~~~~~~~~~~~

We plot the portfolio repartition for different values of the robustness radius. For small values of :math:`\rho`, the portfolio is concentrated on the most rewarding assets. Over a critical radius, the WDRO portfolio choice is equally divided among all assets which is the most robust strategy (regardless of the assets payoffs).

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.. code-block:: Python



    ## Plotting
    fig, ax = plt.subplots()
    _ = plt.stackplot([m.rho for m in pbrs], data)
    plt.xlabel("Robustness radius")
    plt.xlim([np.min(tested_radii),np.max(tested_radii)])
    ax.set_xscale('log')
    plt.ylabel("Portfolio repartition")
    plt.ylim([0,1])

    plt.show()





.. image-sg:: /examples/Builtin/images/sphx_glr_portfolio_001.png
   :alt: portfolio
   :srcset: /examples/Builtin/images/sphx_glr_portfolio_001.png
   :class: sphx-glr-single-img





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References
~~~~~~~~~~

[Bertsimas et al. (2018)] Bertsimas, Dimitris, Vishal Gupta, and Nathan Kallus. "Robust sample average approximation." Mathematical Programming 171 (2018): 217-282.

[Esfahani et al. (2018)] Mohajerin Esfahani, Peyman, and Daniel Kuhn. "Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations." Mathematical Programming 171.1 (2018): 115-166.

[Rockafellar and Uryasev (2000)] Rockafellar, R. Tyrrell, and Stanislav Uryasev. "Optimization of conditional value-at-risk." Journal of risk 2 (2000): 21-42.


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

   **Total running time of the script:** (0 minutes 2.561 seconds)


.. _sphx_glr_download_examples_Builtin_portfolio.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

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

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

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

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

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


.. only:: html

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

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