.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/risk_based/plot_01_cvar_optimization.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_risk_based_plot_01_cvar_optimization.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_risk_based_plot_01_cvar_optimization.py:


Rockafellar-Uryasev CVaR optimization
=====================================

Value at risk is not convex in the portfolio weights, so minimizing it
is hard. R. Tyrrell Rockafellar and Stanislav Uryasev (2000) showed that
conditional value at risk (expected shortfall) is, and better: over a
set of scenarios, minimizing it is a linear program,

.. math::

   \min_{w,\zeta,u}\; \zeta + \frac{1}{(1-\alpha)n}\sum_k u_k
   \quad\text{s.t.}\quad u_k \ge -r_k'w - \zeta,\; u_k \ge 0,

and the optimal :math:`\zeta` is the VaR as a by-product. With fat-tailed,
skewed scenarios, the CVaR-optimal portfolio avoids the asset whose
variance looks modest but whose crashes are large.

.. GENERATED FROM PYTHON SOURCE LINES 21-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import cvar_optimization, efficient_portfolio
    from financialkit.portfolio.visualizers import plot_weights
    from financialkit.risk import expected_shortfall

    rng = np.random.default_rng(9)
    names = ["bonds", "equity", "credit carry", "commodities"]
    n = 2000
    R = rng.multivariate_normal([0.003, 0.006, 0.0, 0.005], np.diag([0.01, 0.045, 0.008, 0.05]) ** 2, size=n)
    # The carry trade earns a steady premium but loses heavily in crash months, alongside equities.
    crash = rng.random(n) < 0.04
    R[:, 2] += np.where(crash, -0.10, 0.009)
    R[:, 1] += np.where(crash, -0.05, 0.0)








.. GENERATED FROM PYTHON SOURCE LINES 38-40

Minimum-CVaR versus minimum-variance at the same mean
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 40-53

.. code-block:: Python


    target = 0.004
    cv = cvar_optimization(R, 0.95, target_return=target)
    mv = efficient_portfolio(R.mean(axis=0), np.cov(R, rowvar=False), target, long_only=True)
    fig, axes = plt.subplots(1, 2, figsize=(10, 3.5), sharey=True)
    plot_weights(mv.weights, names, ax=axes[0]).set_title("minimum variance")
    plot_weights(cv.weights, names, ax=axes[1]).set_title("minimum 95% CVaR")
    fig.tight_layout()
    for label, w in [("min variance", mv.weights), ("min CVaR", cv.weights)]:
        pnl = R @ w
        print(f"{label:>12}: vol {pnl.std():.4f}, 95% ES {expected_shortfall(pnl, 0.95):.4f}")
    print(f"LP: CVaR {cv.cvar:.4f}, VaR (zeta) {cv.var:.4f}")




.. image-sg:: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_01_cvar_optimization_001.png
   :alt: minimum variance, minimum 95% CVaR
   :srcset: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_01_cvar_optimization_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    min variance: vol 0.0111, 95% ES 0.0291
        min CVaR: vol 0.0124, 95% ES 0.0226
    LP: CVaR 0.0226, VaR (zeta) 0.0170




.. GENERATED FROM PYTHON SOURCE LINES 54-56

The mean-CVaR frontier
----------------------

.. GENERATED FROM PYTHON SOURCE LINES 56-66

.. code-block:: Python


    targets = np.linspace(R.mean(axis=0).min() + 1e-4, R.mean(axis=0).max() - 1e-4, 15)
    fig, ax = plt.subplots(figsize=(7, 3.5))
    for a in [0.9, 0.95, 0.99]:
        ax.plot([cvar_optimization(R, a, t).cvar for t in targets], targets, "o-", ms=3, label=f"alpha = {a}")
    ax.set_xlabel("CVaR (monthly loss)")
    ax.set_ylabel("expected return")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_01_cvar_optimization_002.png
   :alt: plot 01 cvar optimization
   :srcset: /api/gallery/portfolio/risk_based/images/sphx_glr_plot_01_cvar_optimization_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 1.732 seconds)


.. _sphx_glr_download_api_gallery_portfolio_risk_based_plot_01_cvar_optimization.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/risk_based/plot_01_cvar_optimization.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_cvar_optimization.ipynb <plot_01_cvar_optimization.ipynb>`

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      :download:`Download Python source code: plot_01_cvar_optimization.py <plot_01_cvar_optimization.py>`

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