.. 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_02_breeden_ccapm.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_02_breeden_ccapm.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_02_breeden_ccapm.py:


Breeden's consumption CAPM
==========================

Merton's intertemporal CAPM needs a beta for every state variable that
shifts investment opportunities. Douglas Breeden (1979) showed that
they all collapse into one: since optimal consumption already responds
to every state variable, an asset's risk premium is proportional to its
covariance with aggregate consumption growth,

.. math::

   E[R_i] - R_f \approx \gamma\,\mathrm{Cov}(R_i, \Delta c) = \gamma\,\beta_{i,c}\,\sigma_c^2.

Assets that pay off when consumption is low are insurance and earn
less.

.. GENERATED FROM PYTHON SOURCE LINES 20-49



.. image-sg:: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_02_breeden_ccapm_001.png
   :alt: plot 02 breeden ccapm
   :srcset: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_02_breeden_ccapm_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    price of consumption risk: 0.00242 (true gamma sigma_c^2 = 0.00300)
    implied gamma: 24.2






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import ccapm_expected_excess_return, estimate_beta

    rng = np.random.default_rng(1979)
    T, N, gamma, sigma_c = 400, 25, 30.0, 0.01  # quarterly data
    dc = 0.005 + sigma_c * rng.standard_normal(T)
    beta_c = np.linspace(-1.0, 5.0, N)
    premia = ccapm_expected_excess_return(beta_c, sigma_c, gamma)
    R = premia + np.outer(dc - dc.mean(), beta_c) + 0.06 * rng.standard_normal((T, N))

    b_hat = estimate_beta(R, dc)
    slope, intercept = np.polyfit(b_hat, R.mean(axis=0), 1)
    print(f"price of consumption risk: {slope:.5f} (true gamma sigma_c^2 = {gamma * sigma_c**2:.5f})")
    print(f"implied gamma: {slope / sigma_c**2:.1f}")

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(b_hat, R.mean(axis=0), "o", label="assets")
    bb = np.linspace(b_hat.min(), b_hat.max(), 2)
    ax.plot(bb, intercept + slope * bb, "C3", label="cross-sectional fit")
    ax.plot(bb, ccapm_expected_excess_return(bb, sigma_c, gamma), "k:", label="CCAPM")
    ax.axvline(0, color="0.6", lw=0.8)
    ax.set_xlabel("consumption beta")
    ax.set_ylabel("average excess return per quarter")
    ax.legend()
    fig.tight_layout()

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.040 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_consumption_plot_02_breeden_ccapm.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_02_breeden_ccapm.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_breeden_ccapm.ipynb <plot_02_breeden_ccapm.ipynb>`

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

      :download:`Download Python source code: plot_02_breeden_ccapm.py <plot_02_breeden_ccapm.py>`

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

      :download:`Download zipped: plot_02_breeden_ccapm.zip <plot_02_breeden_ccapm.zip>`


.. only:: html

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

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