.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/consumption/plot_03_equity_premium_puzzle.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_consumption_plot_03_equity_premium_puzzle.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_consumption_plot_03_equity_premium_puzzle.py:


The equity premium puzzle
=========================

Rajnish Mehra and Edward Prescott (1985) calibrated a two-state version
of Lucas's economy to US consumption growth from 1889 to 1978 (mean
1.8%, standard deviation 3.6%, autocorrelation -0.14). Over the same
period stocks earned 6.98% a year and short-term bills 0.80%: a premium
of 6.18%. Searching over every time preference :math:`\beta < 1` and
risk aversion :math:`\gamma \le 10`, they could produce a premium of at
most 0.35% while keeping the riskless rate below 4%. Consumption is
simply too smooth for stocks to be that risky to a power-utility agent.

.. GENERATED FROM PYTHON SOURCE LINES 16-43

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import MehraPrescott, implied_risk_aversion

    points = []
    for g in np.linspace(0.1, 10, 40):
        for b in np.linspace(0.85, 0.9999, 40):
            try:
                m = MehraPrescott(b, g).moments()
            except ValueError:
                continue
            points.append((m.risk_free - 1, m.equity_premium))
    pts = np.array(points)
    ok = (pts[:, 0] >= 0) & (pts[:, 0] <= 0.04)
    print(f"largest premium with 0 <= r_f <= 4%: {pts[ok, 1].max():.2%}")

    fig, ax = plt.subplots(figsize=(7, 4.5))
    ax.scatter(100 * pts[:, 0], 100 * pts[:, 1], s=4, c="0.6", label="model, gamma <= 10, beta < 1")
    ax.scatter(100 * pts[ok, 0], 100 * pts[ok, 1], s=4, c="C0", label="... with 0 <= r_f <= 4%")
    ax.plot(0.80, 6.18, "*", color="C3", ms=14, label="US data 1889-1978")
    ax.set_xlim(-2, 20)
    ax.set_xlabel("average riskless rate (%)")
    ax.set_ylabel("equity premium (%)")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_03_equity_premium_puzzle_001.png
   :alt: plot 03 equity premium puzzle
   :srcset: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_03_equity_premium_puzzle_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    largest premium with 0 <= r_f <= 4%: 0.35%




.. GENERATED FROM PYTHON SOURCE LINES 44-51

How much risk aversion would it take?
-------------------------------------

With the observed 16.5% equity volatility and 3.6% consumption
volatility, the consumption CAPM needs :math:`\gamma` above 10 even
if stocks and consumption were perfectly correlated; with a realistic
correlation of about 0.4 it needs about 26.

.. GENERATED FROM PYTHON SOURCE LINES 51-66

.. code-block:: Python


    for rho in (1.0, 0.4):
        print(f"correlation {rho}: gamma = {implied_risk_aversion(0.0618, 0.1654, 0.036, rho):.0f}")

    gammas = np.linspace(1, 40, 40)
    prem = [MehraPrescott(0.99, g).moments().equity_premium for g in gammas]
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(gammas, 100 * np.array(prem))
    ax.axhline(6.18, color="C3", ls="--", label="observed premium")
    ax.set_xlabel("risk aversion gamma (beta = 0.99)")
    ax.set_ylabel("model premium (%)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_03_equity_premium_puzzle_002.png
   :alt: plot 03 equity premium puzzle
   :srcset: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_03_equity_premium_puzzle_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    correlation 1.0: gamma = 10
    correlation 0.4: gamma = 26





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

   **Total running time of the script:** (0 minutes 0.105 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_consumption_plot_03_equity_premium_puzzle.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/consumption/plot_03_equity_premium_puzzle.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_equity_premium_puzzle.ipynb <plot_03_equity_premium_puzzle.ipynb>`

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

      :download:`Download Python source code: plot_03_equity_premium_puzzle.py <plot_03_equity_premium_puzzle.py>`

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

      :download:`Download zipped: plot_03_equity_premium_puzzle.zip <plot_03_equity_premium_puzzle.zip>`


.. only:: html

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

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