.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/portfolio/growth/plot_02_merton_portfolio.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_portfolio_growth_plot_02_merton_portfolio.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_portfolio_growth_plot_02_merton_portfolio.py:


Merton's portfolio problem
==========================

Robert Merton (1969) solved the lifetime consumption and investment
problem in continuous time. With a stock following geometric Brownian
motion, a riskless rate :math:`r` and constant relative risk aversion
:math:`\gamma`, the optimal policy is to keep the constant fraction

.. math::

   \pi^* = \frac{\mu - r}{\gamma\sigma^2}

of wealth in the stock, rebalancing continuously, and to consume a
constant fraction of wealth. The Kelly rule is the case :math:`\gamma = 1`.

.. GENERATED FROM PYTHON SOURCE LINES 19-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.portfolio import merton_certainty_equivalent, merton_consumption_rate, merton_fraction, merton_wealth_paths

    mu, r, sigma = 0.07, 0.02, 0.18








.. GENERATED FROM PYTHON SOURCE LINES 27-29

The certainty-equivalent return of each constant mix
----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-44

.. code-block:: Python


    pi = np.linspace(-0.5, 2.5, 300)
    fig, ax = plt.subplots(figsize=(7, 4))
    for gamma in [1.0, 2.0, 4.0, 8.0]:
        star = merton_fraction(mu, r, sigma, gamma)
        ax.plot(pi, merton_certainty_equivalent(pi, mu, r, sigma, gamma), label=f"gamma = {gamma:g}: pi* = {star:.2f}")
        ax.plot(star, merton_certainty_equivalent(star, mu, r, sigma, gamma), "ko", ms=4)
    ax.set_ylim(0.0, 0.06)
    ax.set_xlabel("fraction of wealth in the stock")
    ax.set_ylabel("certainty-equivalent return")
    ax.legend(fontsize=8)
    fig.tight_layout()
    for gamma in [2.0, 4.0]:
        print(f"gamma = {gamma:g}: consume {merton_consumption_rate(mu, r, sigma, gamma, 0.03):.2%} of wealth a year")




.. image-sg:: /api/gallery/portfolio/growth/images/sphx_glr_plot_02_merton_portfolio_001.png
   :alt: plot 02 merton portfolio
   :srcset: /api/gallery/portfolio/growth/images/sphx_glr_plot_02_merton_portfolio_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    gamma = 2: consume 3.46% of wealth a year
    gamma = 4: consume 2.97% of wealth a year




.. GENERATED FROM PYTHON SOURCE LINES 45-51

Monte Carlo check: no other mix gives a higher expected utility
---------------------------------------------------------------

Terminal wealth after 10 years for :math:`\gamma = 4`, simulated
exactly through :func:`financialkit.integrators.gbm_paths` on common
random numbers.

.. GENERATED FROM PYTHON SOURCE LINES 51-67

.. code-block:: Python


    gamma, T = 4.0, 10.0
    mixes = np.linspace(0.0, 1.0, 11)
    eu = []
    for m in mixes:
        # The same seed for every mix: common random numbers.
        _, W = merton_wealth_paths(m, mu, r, sigma, T, 1, 100_000, rng=0)
        eu.append(np.mean(W[:, -1] ** (1 - gamma)) / (1 - gamma))
    fig, ax = plt.subplots(figsize=(7, 3.5))
    ax.plot(mixes, eu, "o-", label="simulated expected CRRA utility")
    ax.axvline(merton_fraction(mu, r, sigma, gamma), color="C2", ls="--", label="Merton's fraction")
    ax.set_xlabel("fraction in the stock")
    ax.set_ylabel("E[U(W_T)]")
    ax.legend()
    fig.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/portfolio/growth/images/sphx_glr_plot_02_merton_portfolio_002.png
   :alt: plot 02 merton portfolio
   :srcset: /api/gallery/portfolio/growth/images/sphx_glr_plot_02_merton_portfolio_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.098 seconds)


.. _sphx_glr_download_api_gallery_portfolio_growth_plot_02_merton_portfolio.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/portfolio/growth/plot_02_merton_portfolio.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_merton_portfolio.ipynb <plot_02_merton_portfolio.ipynb>`

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

      :download:`Download Python source code: plot_02_merton_portfolio.py <plot_02_merton_portfolio.py>`

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      :download:`Download zipped: plot_02_merton_portfolio.zip <plot_02_merton_portfolio.zip>`


.. only:: html

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

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