.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/valuation/plot_04_shiller_excess_volatility.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_asset_pricing_valuation_plot_04_shiller_excess_volatility.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_asset_pricing_valuation_plot_04_shiller_excess_volatility.py:


Shiller's excess volatility
===========================

If a price is the rational forecast of discounted future dividends,
:math:`p_t = E_t[p^*_t]`, then the ex-post rational price :math:`p^*_t`
computed from the dividends actually paid equals the price plus an
unforecastable error. An optimal forecast varies less than the thing it
forecasts, so :math:`\sigma(p) \le \sigma(p^*)`. Robert Shiller (1981)
found US stock prices five to thirteen times too volatile for this
bound.

Here dividends follow an AR(1) around a constant. The rational price
respects the bound; a market that treats every dividend shock as
permanent violates it.

.. GENERATED FROM PYTHON SOURCE LINES 19-37

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import ar1_dividend_price, variance_bound
    from financialkit.asset_pricing.visualizers import plot_excess_volatility

    rng = np.random.default_rng(1981)
    r, rho, Dbar, n = 0.05, 0.9, 1.0, 400
    D = np.empty(n)
    D[0] = Dbar
    for t in range(1, n):
        D[t] = Dbar + rho * (D[t - 1] - Dbar) + 0.05 * rng.standard_normal()

    rational = variance_bound(ar1_dividend_price(D, r, rho, Dbar), D, r, terminal_price=Dbar / r)
    overreacting = variance_bound(D / r, D, r, terminal_price=Dbar / r)
    print(f"rational price: sd ratio {rational.ratio:.2f}  (bound violated: {rational.violated})")
    print(f"overreacting:   sd ratio {overreacting.ratio:.2f}  (bound violated: {overreacting.violated})")





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

 .. code-block:: none

    rational price: sd ratio 0.70  (bound violated: False)
    overreacting:   sd ratio 2.33  (bound violated: True)




.. GENERATED FROM PYTHON SOURCE LINES 38-46

.. code-block:: Python

    fig, axes = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
    plot_excess_volatility(rational, ax=axes[0], label="rational price")
    plot_excess_volatility(overreacting, ax=axes[1], label="price = D / r")
    axes[0].set_title("the rational forecast is smoother than p*")
    axes[1].set_title("treating shocks as permanent violates the bound")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/valuation/images/sphx_glr_plot_04_shiller_excess_volatility_001.png
   :alt: the rational forecast is smoother than p*, treating shocks as permanent violates the bound
   :srcset: /api/gallery/asset_pricing/valuation/images/sphx_glr_plot_04_shiller_excess_volatility_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.065 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_valuation_plot_04_shiller_excess_volatility.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/asset_pricing/valuation/plot_04_shiller_excess_volatility.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_shiller_excess_volatility.ipynb <plot_04_shiller_excess_volatility.ipynb>`

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

      :download:`Download Python source code: plot_04_shiller_excess_volatility.py <plot_04_shiller_excess_volatility.py>`

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

      :download:`Download zipped: plot_04_shiller_excess_volatility.zip <plot_04_shiller_excess_volatility.zip>`


.. only:: html

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

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