.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/asset_pricing/factors/plot_04_carhart_momentum.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_factors_plot_04_carhart_momentum.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_factors_plot_04_carhart_momentum.py:


Carhart's momentum factor
=========================

Jegadeesh and Titman (1993) found that stocks with the highest returns
over the past year keep outperforming for months, which the
Fama-French factors cannot explain. Mark Carhart (1997) added a fourth
factor, UMD (up minus down), long last year's winners and short its
losers, skipping the most recent month. With it, most of the apparent
persistence in mutual fund performance disappeared: winning funds had
simply held winning stocks.

In this synthetic market each stock's expected return drifts slowly,
so past winners tend to keep winning.

.. GENERATED FROM PYTHON SOURCE LINES 18-42

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import long_short_factor, momentum_signal, sort_portfolios, time_series_regression
    from financialkit.asset_pricing.visualizers import plot_alphas

    rng = np.random.default_rng(1997)
    T, N, phi = 372, 500, 0.95
    size, value = rng.standard_normal(N), rng.standard_normal(N)
    loadings = np.column_stack([rng.uniform(0.8, 1.2, N), -0.6 * size, 0.6 * value])
    f_true = rng.multivariate_normal([0.006, 0.0025, 0.003], np.diag([0.045, 0.03, 0.03]) ** 2, T)
    drift = np.zeros((T, N))
    for t in range(1, T):
        drift[t] = phi * drift[t - 1] + 0.012 * np.sqrt(1 - phi**2) * rng.standard_normal(N)
    R = f_true @ loadings.T + drift + 0.06 * rng.standard_normal((T, N))

    mom = momentum_signal(R, lookback=12, skip=1)
    keep = slice(12, None)
    mkt = R.mean(axis=1)[keep]
    smb = long_short_factor(-size, R)[keep]
    hml = long_short_factor(value, R)[keep]
    umd = long_short_factor(mom, R)[keep]
    print(f"mean monthly UMD: {umd.mean():.4f} (t = {umd.mean() / umd.std() * np.sqrt(umd.size):.1f})")





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

 .. code-block:: none

    mean monthly UMD: 0.0102 (t = 11.9)




.. GENERATED FROM PYTHON SOURCE LINES 43-45

Momentum deciles under three and four factors
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 45-56

.. code-block:: Python


    deciles = sort_portfolios(mom, R, 10)[keep]
    ff3 = time_series_regression(deciles, np.column_stack([mkt, smb, hml]))
    c4 = time_series_regression(deciles, np.column_stack([mkt, smb, hml, umd]))
    fig, ax = plt.subplots(figsize=(9, 4))
    plot_alphas(ff3, ax=ax, label="Fama-French 3", offset=-0.2, width=0.4)
    plot_alphas(c4, ax=ax, label="Carhart 4", offset=0.2, width=0.4)
    ax.set_xlabel("past-return decile (losers to winners)")
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_04_carhart_momentum_001.png
   :alt: plot 04 carhart momentum
   :srcset: /api/gallery/asset_pricing/factors/images/sphx_glr_plot_04_carhart_momentum_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_factors_plot_04_carhart_momentum.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/factors/plot_04_carhart_momentum.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_carhart_momentum.ipynb <plot_04_carhart_momentum.ipynb>`

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

      :download:`Download Python source code: plot_04_carhart_momentum.py <plot_04_carhart_momentum.py>`

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

      :download:`Download zipped: plot_04_carhart_momentum.zip <plot_04_carhart_momentum.zip>`


.. only:: html

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

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