.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fixed_income/curves/plot_01_lutz_forward_rates.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_fixed_income_curves_plot_01_lutz_forward_rates.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_fixed_income_curves_plot_01_lutz_forward_rates.py:


Lutz: forward rates
===================

Friedrich Lutz (1940) read the term structure as a sequence of expected
future short rates. The forward rate for the period from :math:`t_1` to
:math:`t_2` is what the zero curve locks in today,

.. math:: f(t_1, t_2) = \frac{r_2 t_2 - r_1 t_1}{t_2 - t_1},

and each zero rate is the time average of the forwards before it. A rising
zero curve therefore has forwards above it, and a falling one has forwards
below it.

.. GENERATED FROM PYTHON SOURCE LINES 17-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import ZeroCurve, forward_rates








.. GENERATED FROM PYTHON SOURCE LINES 23-25

Hull's Table 4.5
----------------

.. GENERATED FROM PYTHON SOURCE LINES 25-32

.. code-block:: Python


    tenors = np.array([1, 2, 3, 4, 5.0])
    zeros = np.array([0.03, 0.04, 0.046, 0.05, 0.053])
    fwd = forward_rates(tenors, zeros)
    for T, r, f in zip(tenors, zeros, fwd, strict=True):
        print(f"year {T:.0f}: zero {100 * r:.2f}%  forward for year {T:.0f} {100 * f:.2f}%")





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

 .. code-block:: none

    year 1: zero 3.00%  forward for year 1 3.00%
    year 2: zero 4.00%  forward for year 2 5.00%
    year 3: zero 4.60%  forward for year 3 5.80%
    year 4: zero 5.00%  forward for year 4 6.20%
    year 5: zero 5.30%  forward for year 5 6.50%




.. GENERATED FROM PYTHON SOURCE LINES 33-35

Forwards above a rising curve, below a falling one
--------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-48

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 4), sharey=True)
    t = np.arange(1, 11, dtype=float)
    shapes = {"rising": 0.02 + 0.03 * (1 - np.exp(-t / 3)), "inverted": 0.05 - 0.02 * (1 - np.exp(-t / 2))}
    for ax, (name, r) in zip(axes, shapes.items(), strict=True):
        f = forward_rates(t, r)
        ax.step(np.r_[0, t], 100 * np.r_[f, f[-1]], where="post", label="one-year forwards")
        ax.plot(t, 100 * r, "o-", label="zero rates")
        ax.set_title(name)
        ax.set_xlabel("maturity (years)")
    axes[0].set_ylabel("rate (%)")
    axes[0].legend()




.. image-sg:: /api/gallery/fixed_income/curves/images/sphx_glr_plot_01_lutz_forward_rates_001.png
   :alt: rising, inverted
   :srcset: /api/gallery/fixed_income/curves/images/sphx_glr_plot_01_lutz_forward_rates_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116d9cc20>



.. GENERATED FROM PYTHON SOURCE LINES 49-51

Zero rates are time averages of forwards
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 51-56

.. code-block:: Python


    curve = ZeroCurve(t, shapes["rising"])
    f = forward_rates(t, shapes["rising"])
    print("max |average of forwards - zero rate| =", np.abs(np.cumsum(f) / t - curve.zero_rate(t)).max())
    plt.show()




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

 .. code-block:: none

    max |average of forwards - zero rate| = 0.0





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

   **Total running time of the script:** (0 minutes 0.034 seconds)


.. _sphx_glr_download_api_gallery_fixed_income_curves_plot_01_lutz_forward_rates.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/fixed_income/curves/plot_01_lutz_forward_rates.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_lutz_forward_rates.ipynb <plot_01_lutz_forward_rates.ipynb>`

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

      :download:`Download Python source code: plot_01_lutz_forward_rates.py <plot_01_lutz_forward_rates.py>`

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

      :download:`Download zipped: plot_01_lutz_forward_rates.zip <plot_01_lutz_forward_rates.zip>`


.. only:: html

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

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