.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "generated_examples\savings\plot_ex1_rand.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_generated_examples_savings_plot_ex1_rand.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_generated_examples_savings_plot_ex1_rand.py:


Account value distribution
===========================================

10,000 random numbers drawn from the standard normal distribution
are generated for each time step.
The graph shows how well the 10,000 random numbers for ``t=0``
fit the PDF of the standard normal distribution.

.. GENERATED FROM PYTHON SOURCE LINES 11-30

.. code-block:: default

    import modelx as mx
    import pandas as pd
    import matplotlib.pyplot as plt
    from scipy.stats import norm, lognorm
    import numpy as np

    model = mx.read_model("CashValue_ME_EX1")
    rand_nums = model.Projection.std_norm_rand()
    pv_avs = model.Projection.pv_claims_from_av('MATURITY')
    num_bins = 100
    S0 = 45000000
    sigma = 0.03
    T = 10

    fig, ax = plt.subplots()
    n, bins, patches = ax.hist(rand_nums[:, 0], bins=num_bins, density=True)
    ax.plot(bins, norm.pdf(bins), '-')
    fig.suptitle('Standard normal distribution for t=0')




.. image-sg:: /generated_examples/savings/images/sphx_glr_plot_ex1_rand_001.png
   :alt: Standard normal distribution for t=0
   :srcset: /generated_examples/savings/images/sphx_glr_plot_ex1_rand_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 31-47

The distibution of the account value at ``t=120`` follows a log normal distribution.
In the expression below, :math:`S_{T}` and :math:`S_{0}` denote the account value
at ``t=T=120`` and ``t=0`` respectively.

.. math::

  \ln\frac{S_{T}}{S_{0}}\sim\phi\left[\left(r-\frac{\sigma^{2}}{2}\right)T, \sigma\sqrt{T}\right]

The graph shows how well the distribution of :math:`e^{-rT}S_{T}`, the present
values of the account value at `t=0`, fits the PDF of a log normal ditribution.

Reference: *Options, Futures, and Other Derivatives* by John C.Hull

.. seealso::

    * :doc:`/libraries/notebooks/savings/savings_example1` notebook in the :mod:`~savings` library

.. GENERATED FROM PYTHON SOURCE LINES 47-56

.. code-block:: default



    fig, ax = plt.subplots()
    n, bins, patches = ax.hist(pv_avs, bins=num_bins, density=True)
    ax.plot(bins, lognorm.pdf(bins, sigma * T**0.5, scale=S0), '-')
    fig.suptitle('PV of account value at t=120')






.. image-sg:: /generated_examples/savings/images/sphx_glr_plot_ex1_rand_002.png
   :alt: PV of account value at t=120
   :srcset: /generated_examples/savings/images/sphx_glr_plot_ex1_rand_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  2.264 seconds)


.. _sphx_glr_download_generated_examples_savings_plot_ex1_rand.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


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

      :download:`Download Python source code: plot_ex1_rand.py <plot_ex1_rand.py>`

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

      :download:`Download Jupyter notebook: plot_ex1_rand.ipynb <plot_ex1_rand.ipynb>`


.. only:: html

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

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