.. 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_01_lucas_tree.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_01_lucas_tree.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_01_lucas_tree.py:


Lucas's tree
============

Robert Lucas (1978) priced assets in the simplest economy that has
them: a representative agent with power utility owns a tree whose fruit
is the only consumption good. Prices must make the agent content to
hold the tree and eat its dividend, which gives the Euler equation

.. math::

   P_t = E_t\left[\beta \left(\frac{C_{t+1}}{C_t}\right)^{-\gamma} (P_{t+1} + D_{t+1})\right], \qquad C = D.

With i.i.d. lognormal growth the price-dividend ratio is constant, so
the price inherits the dividend's random walk. The riskless rate and
the equity premium follow from just :math:`\beta`, :math:`\gamma` and the
mean and volatility of consumption growth.

.. GENERATED FROM PYTHON SOURCE LINES 21-41

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.asset_pricing import LucasTree

    tree = LucasTree(beta=0.97, gamma=2.0, mu=0.02, sigma=0.036)
    m = tree.moments()
    print(f"P/D = {m.price_dividend:.1f},  R_f = {m.risk_free - 1:.2%},  E[R_e] = {m.equity_return - 1:.2%},  premium = {m.equity_premium:.2%}")

    path = tree.simulate(100, rng=1978)
    D = np.cumprod(np.concatenate([[1.0], path.growth]))
    fig, ax = plt.subplots(figsize=(8, 3.5))
    ax.semilogy(D * m.price_dividend, label="price P_t")
    ax.semilogy(D, "C3", lw=1, label="dividend D_t")
    ax.set_xlabel("year")
    ax.set_ylabel("price")
    ax.set_title("the price is a constant multiple of the dividend")
    ax.legend()
    fig.tight_layout()




.. image-sg:: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_01_lucas_tree_001.png
   :alt: the price is a constant multiple of the dividend
   :srcset: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_01_lucas_tree_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    P/D = 19.6,  R_f = 7.02%,  E[R_e] = 7.30%,  premium = 0.28%




.. GENERATED FROM PYTHON SOURCE LINES 42-48

Prices and returns across risk aversion
---------------------------------------

Higher :math:`\gamma` raises the riskless rate (the agent wants to
borrow against future growth) but adds only :math:`\gamma\sigma^2` to
the log premium.

.. GENERATED FROM PYTHON SOURCE LINES 48-62

.. code-block:: Python


    gammas = np.linspace(0.5, 10, 60)
    ms = [LucasTree(0.97, g, 0.02, 0.036).moments() for g in gammas]
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(gammas, [x.price_dividend for x in ms])
    axes[0].set_xlabel("risk aversion gamma")
    axes[0].set_ylabel("price-dividend ratio")
    axes[1].plot(gammas, [x.risk_free - 1 for x in ms], label="riskless rate")
    axes[1].plot(gammas, [x.equity_premium for x in ms], label="equity premium")
    axes[1].set_xlabel("risk aversion gamma")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_01_lucas_tree_002.png
   :alt: plot 01 lucas tree
   :srcset: /api/gallery/asset_pricing/consumption/images/sphx_glr_plot_01_lucas_tree_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.181 seconds)


.. _sphx_glr_download_api_gallery_asset_pricing_consumption_plot_01_lucas_tree.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_01_lucas_tree.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_lucas_tree.ipynb <plot_01_lucas_tree.ipynb>`

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

      :download:`Download Python source code: plot_01_lucas_tree.py <plot_01_lucas_tree.py>`

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

      :download:`Download zipped: plot_01_lucas_tree.zip <plot_01_lucas_tree.zip>`


.. only:: html

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

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