.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fraud/schemes/plot_01_ponzi_scheme.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_fraud_schemes_plot_01_ponzi_scheme.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_fraud_schemes_plot_01_ponzi_scheme.py:


Ponzi's scheme: paying old investors with new money (1920)
==========================================================

In 1920 Charles Ponzi promised Boston investors 50% in 45 days, supposedly
from arbitrage on international reply coupons. There was no arbitrage:
each payout came from later deposits. Such a scheme pays its promises only
while new money grows fast enough. A deposit :math:`D` made :math:`\tau`
periods ago is owed :math:`(1 + r) D` today; if deposits grow by
:math:`g` per period and the operator keeps a fraction :math:`s` of each,
the scheme keeps up exactly when

.. math::

   (1 - s)(1 + g)^\tau \ge 1 + r .

For Ponzi's terms, deposits must grow by half every 45 days. No population
of investors grows like that for long: once growth slows, the payout
outruns the cash, and the latest investors lose everything they paid in.

.. GENERATED FROM PYTHON SOURCE LINES 23-28

.. code-block:: Python

    import matplotlib.pyplot as plt

    import blockchainkit as bk
    from blockchainkit.fraud.visualizers import plot_ponzi








.. GENERATED FROM PYTHON SOURCE LINES 29-33

Growth that cannot last
-----------------------
Deposits grow by 60% a period, faster than the breakeven 50%, until the
pool of new investors saturates at 20 times the first period's money.

.. GENERATED FROM PYTHON SOURCE LINES 33-46

.. code-block:: Python


    needed = bk.fraud.breakeven_growth(0.5, term=1)
    print(f"breakeven growth: {needed:.0%} per 45 days")
    deposits = [min(100 * 1.6**t, 2_000) for t in range(15)]
    run = bk.fraud.simulate_ponzi(deposits, promised_return=0.5, term=1)
    print("collapsed in period", run.collapse, f"after raising {run.raised:,.0f}")
    print(f"still owed to investors: {run.owed[-1]:,.0f}")
    assert run.collapse is not None and run.owed[-1] > 2_000

    fig, ax = plt.subplots(figsize=(8, 4))
    plot_ponzi(run, ax=ax)
    fig.tight_layout()




.. image-sg:: /api/gallery/fraud/schemes/images/sphx_glr_plot_01_ponzi_scheme_001.png
   :alt: Ponzi scheme: collapsed in period 7
   :srcset: /api/gallery/fraud/schemes/images/sphx_glr_plot_01_ponzi_scheme_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    breakeven growth: 50% per 45 days
    collapsed in period 7 after raising 6,307
    still owed to investors: 3,154




.. GENERATED FROM PYTHON SOURCE LINES 47-49

When it collapses, against how fast deposits grow
-------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 49-68

.. code-block:: Python


    growths = [0.3 + 0.05 * k for k in range(9)]
    lifetimes = []
    for g in growths:
        flows = [min(100 * (1 + g) ** t, 2_000) for t in range(40)]
        result = bk.fraud.simulate_ponzi(flows, promised_return=0.5)
        lifetimes.append(len(result.deposits))
    assert lifetimes[0] < lifetimes[-1] < 40  # Faster growth only postpones the end.

    fig, ax = plt.subplots(figsize=(7, 4))
    ax.plot(growths, lifetimes, "o-", color="#dc2626")
    ax.axvline(needed, color="#64748b", linestyle="--", label="breakeven growth")
    ax.set(xlabel="growth of deposits per period", ylabel="periods until collapse")
    ax.set_title("A saturating market ends every Ponzi scheme")
    ax.legend()
    fig.tight_layout()

    plt.show()




.. image-sg:: /api/gallery/fraud/schemes/images/sphx_glr_plot_01_ponzi_scheme_002.png
   :alt: A saturating market ends every Ponzi scheme
   :srcset: /api/gallery/fraud/schemes/images/sphx_glr_plot_01_ponzi_scheme_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 69-75

Exercise
--------
Ponzi kept part of every deposit for himself. How fast must deposits grow
for a scheme promising 50% in 45 days if the operator skims 20%? Check
your answer with :func:`~blockchainkit.fraud.systems.ponzi.breakeven_growth`.
A worked solution is in :doc:`/exercises/fraud`.


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

   **Total running time of the script:** (0 minutes 0.075 seconds)


.. _sphx_glr_download_api_gallery_fraud_schemes_plot_01_ponzi_scheme.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/fraud/schemes/plot_01_ponzi_scheme.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_ponzi_scheme.ipynb <plot_01_ponzi_scheme.ipynb>`

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

      :download:`Download Python source code: plot_01_ponzi_scheme.py <plot_01_ponzi_scheme.py>`

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

      :download:`Download zipped: plot_01_ponzi_scheme.zip <plot_01_ponzi_scheme.zip>`


.. only:: html

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

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