.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/time_series/volatility/plot_06_riskmetrics_ewma.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_time_series_volatility_plot_06_riskmetrics_ewma.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_time_series_volatility_plot_06_riskmetrics_ewma.py:


RiskMetrics EWMA
================

In 1994 J.P. Morgan published RiskMetrics, a free methodology and data set
for Value-at-Risk. Its volatility model was deliberately simple: an
exponentially weighted moving average

.. math:: \sigma_t^2 = \lambda\,\sigma_{t-1}^2 + (1 - \lambda)\,r_{t-1}^2,

with one decay factor for every asset, :math:`\lambda = 0.94` for daily
and :math:`0.97` for monthly data, chosen to minimize forecast errors
across hundreds of series. It is GARCH(1,1) with :math:`\omega = 0` and
:math:`\alpha + \beta = 1`: no estimation, no long-run level, and a flat
term structure of forecasts.

.. GENERATED FROM PYTHON SOURCE LINES 19-27

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.constants import TRADING_DAYS_PER_YEAR
    from financialkit.time_series import EWMA, GARCH

    ann = TRADING_DAYS_PER_YEAR








.. GENERATED FROM PYTHON SOURCE LINES 28-33

Weights on past squared returns
-------------------------------
A 0.94 decay puts 6% weight on yesterday and has an effective memory of
about 1/(1 - 0.94), roughly 17 days, against an equally weighted window's
hard cut-off.

.. GENERATED FROM PYTHON SOURCE LINES 33-43

.. code-block:: Python


    k = np.arange(1, 101)
    fig, ax = plt.subplots()
    for lam in (0.94, 0.97):
        ax.plot(k, (1 - lam) * lam ** (k - 1), label=rf"EWMA $\lambda$ = {lam}")
    ax.step(k, np.where(k <= 30, 1 / 30, 0.0), where="post", color="k", lw=1, label="30-day window")
    ax.set_xlabel("days ago")
    ax.set_ylabel("weight")
    ax.legend()




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_06_riskmetrics_ewma_001.png
   :alt: plot 06 riskmetrics ewma
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_06_riskmetrics_ewma_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116d7dbe0>



.. GENERATED FROM PYTHON SOURCE LINES 44-49

Tracking a volatility shock
---------------------------
Returns from a GARCH process with a crisis in the middle. EWMA follows the
true conditional volatility closely without any fitting; the 30-day
window lags it and drops abruptly 30 days after the shock.

.. GENERATED FROM PYTHON SOURCE LINES 49-68

.. code-block:: Python


    g = GARCH(omega=0.15**2 / ann * 0.03, alpha=0.09, beta=0.88)
    path = g.simulate(1000, rng=1994)
    r = path.returns.copy()
    r[500:505] *= 6  # five days of crisis
    true_var = g.variance(r, v0=path.variance[0])
    ewma = EWMA(0.94).variance(r, v0=path.variance[0])
    window = np.convolve(r**2, np.ones(30) / 30)[: r.size]

    fig, ax = plt.subplots(figsize=(8, 3.5))
    sl = slice(430, 650)
    t = np.arange(r.size)[sl]
    ax.plot(t, 100 * np.sqrt(ann * true_var[sl]), "k", lw=1.5, label="true (GARCH)")
    ax.plot(t, 100 * np.sqrt(ann * ewma[sl]), label="RiskMetrics EWMA")
    ax.plot(t, 100 * np.sqrt(ann * window[sl]), label="30-day window")
    ax.set_xlabel("day")
    ax.set_ylabel("volatility (% a year)")
    ax.legend()




.. image-sg:: /api/gallery/time_series/volatility/images/sphx_glr_plot_06_riskmetrics_ewma_002.png
   :alt: plot 06 riskmetrics ewma
   :srcset: /api/gallery/time_series/volatility/images/sphx_glr_plot_06_riskmetrics_ewma_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116d874d0>



.. GENERATED FROM PYTHON SOURCE LINES 69-75

Estimating lambda
-----------------
Maximum likelihood picks the decay that best fits this one series.
RiskMetrics instead
fixed 0.94 for every series, trading a little fit for simplicity and
comparability across a whole portfolio.

.. GENERATED FROM PYTHON SOURCE LINES 75-78

.. code-block:: Python


    print(f"maximum-likelihood lambda: {EWMA.fit(r).params['lam']:.3f}")
    plt.show()




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

 .. code-block:: none

    maximum-likelihood lambda: 0.896





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

   **Total running time of the script:** (0 minutes 0.054 seconds)


.. _sphx_glr_download_api_gallery_time_series_volatility_plot_06_riskmetrics_ewma.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/time_series/volatility/plot_06_riskmetrics_ewma.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_06_riskmetrics_ewma.ipynb <plot_06_riskmetrics_ewma.ipynb>`

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

      :download:`Download Python source code: plot_06_riskmetrics_ewma.py <plot_06_riskmetrics_ewma.py>`

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

      :download:`Download zipped: plot_06_riskmetrics_ewma.zip <plot_06_riskmetrics_ewma.zip>`


.. only:: html

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

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