.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/risk/var/plot_03_parametric_var.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_risk_var_plot_03_parametric_var.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_risk_var_plot_03_parametric_var.py:


RiskMetrics: parametric (delta-normal) VaR
==========================================

In October 1994 J.P. Morgan published *RiskMetrics*, its methodology and
daily covariance data, free. A portfolio with money exposures :math:`w`
to factors with covariance :math:`\Sigma` has, under joint normality,

.. math:: \mathrm{VaR}_\alpha = z_\alpha \sqrt{h\, w^\top \Sigma w},

computable in a spreadsheet for a whole bank. Diversification is
explicit: the portfolio VaR is less than the sum of the stand-alone
VaRs unless every correlation is one.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.risk import delta_normal_var, parametric_var








.. GENERATED FROM PYTHON SOURCE LINES 23-29

Hull's two-asset example
------------------------

Hull (*Options, Futures, and Other Derivatives*, 10th ed., Ch. 22):
$10 million of Microsoft (2% daily volatility) and $5 million of AT&T
(1%), correlation 0.3.

.. GENERATED FROM PYTHON SOURCE LINES 29-38

.. code-block:: Python


    w = np.array([10e6, 5e6])
    vols = np.array([0.02, 0.01])
    cov = np.array([[1.0, 0.3], [0.3, 1.0]]) * np.outer(vols, vols)
    stand_alone = [parametric_var(0.0, wi * si, 0.99, horizon=10) for wi, si in zip(w, vols, strict=True)]
    total = delta_normal_var(w, cov, 0.99, horizon=10)
    print(f"ten-day 99% VaR: Microsoft {stand_alone[0]:,.0f}, AT&T {stand_alone[1]:,.0f}")
    print(f"portfolio {total:,.0f}; diversification benefit {sum(stand_alone) - total:,.0f}")





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

 .. code-block:: none

    ten-day 99% VaR: Microsoft 1,471,312, AT&T 367,828
    portfolio 1,620,114; diversification benefit 219,026




.. GENERATED FROM PYTHON SOURCE LINES 39-41

Diversification and the square-root-of-time rule
------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-59

.. code-block:: Python


    rho = np.linspace(-1, 1, 81)
    var_rho = [delta_normal_var(w, np.array([[1.0, c], [c, 1.0]]) * np.outer(vols, vols), 0.99, 10) for c in rho]
    h = np.arange(1, 31)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4))
    axes[0].plot(rho, np.array(var_rho) / 1e6)
    axes[0].axhline(sum(stand_alone) / 1e6, color="k", ls="--", lw=1, label="sum of stand-alone VaRs")
    axes[0].set_xlabel("correlation")
    axes[0].set_ylabel("ten-day 99% VaR ($m)")
    axes[0].legend()
    axes[1].plot(h, [delta_normal_var(w, cov, 0.99, k) / 1e6 for k in h], label="99%")
    axes[1].plot(h, [delta_normal_var(w, cov, 0.95, k) / 1e6 for k in h], label="95%")
    axes[1].set_xlabel("horizon (days)")
    axes[1].set_ylabel("VaR ($m)")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/risk/var/images/sphx_glr_plot_03_parametric_var_001.png
   :alt: plot 03 parametric var
   :srcset: /api/gallery/risk/var/images/sphx_glr_plot_03_parametric_var_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_risk_var_plot_03_parametric_var.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/risk/var/plot_03_parametric_var.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_parametric_var.ipynb <plot_03_parametric_var.ipynb>`

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

      :download:`Download Python source code: plot_03_parametric_var.py <plot_03_parametric_var.py>`

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

      :download:`Download zipped: plot_03_parametric_var.zip <plot_03_parametric_var.zip>`


.. only:: html

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

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