.. 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_04_nelson_siegel.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_04_nelson_siegel.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_04_nelson_siegel.py:


Nelson-Siegel
=============

Nelson and Siegel (1987) described the whole yield curve with four numbers:

.. math:: r(t) = \beta_0 + \beta_1 \frac{1-e^{-t/\tau}}{t/\tau}
          + \beta_2\left(\frac{1-e^{-t/\tau}}{t/\tau} - e^{-t/\tau}\right).

The three loadings are a level, a slope that decays from 1 to 0, and a
hump. The script plots them and fits the model to noisy yields.

.. GENERATED FROM PYTHON SOURCE LINES 15-21

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import NelsonSiegelCurve, fit_nelson_siegel
    from financialkit.fixed_income.visualizers import plot_yield_curve








.. GENERATED FROM PYTHON SOURCE LINES 22-24

The three loadings
------------------

.. GENERATED FROM PYTHON SOURCE LINES 24-33

.. code-block:: Python


    t = np.linspace(0.01, 30, 300)
    fig, ax = plt.subplots()
    for name, curve in (("level", NelsonSiegelCurve(1, 0, 0, 2.0)), ("slope", NelsonSiegelCurve(0, 1, 0, 2.0)), ("curvature", NelsonSiegelCurve(0, 0, 1, 2.0))):
        ax.plot(t, curve.zero_rate(t), label=name)
    ax.set_xlabel("maturity (years)")
    ax.set_ylabel(r"loading ($\tau$ = 2)")
    ax.legend()




.. image-sg:: /api/gallery/fixed_income/curves/images/sphx_glr_plot_04_nelson_siegel_001.png
   :alt: plot 04 nelson siegel
   :srcset: /api/gallery/fixed_income/curves/images/sphx_glr_plot_04_nelson_siegel_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x116f0d6a0>



.. GENERATED FROM PYTHON SOURCE LINES 34-36

Fit to noisy yields
-------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-47

.. code-block:: Python


    rng = np.random.default_rng(1987)
    tenors = np.array([0.25, 0.5, 1, 2, 3, 5, 7, 10, 15, 20, 30])
    true = NelsonSiegelCurve(0.048, -0.022, 0.025, 1.6)
    observed = true.zero_rate(tenors) + rng.normal(0, 0.0005, tenors.size)
    fit = fit_nelson_siegel(tenors, observed)
    print("fitted [b0, b1, b2, tau]:", fit.params.round(4), f"RMSE {1e4 * fit.rmse:.1f} bp")
    ax = plot_yield_curve(fit.curve, label="fitted")
    ax.plot(tenors, 100 * observed, "ko", label="observed")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/fixed_income/curves/images/sphx_glr_plot_04_nelson_siegel_002.png
   :alt: plot 04 nelson siegel
   :srcset: /api/gallery/fixed_income/curves/images/sphx_glr_plot_04_nelson_siegel_002.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    fitted [b0, b1, b2, tau]: [ 0.0485 -0.0225  0.0233  1.5472] RMSE 2.4 bp





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

   **Total running time of the script:** (0 minutes 0.132 seconds)


.. _sphx_glr_download_api_gallery_fixed_income_curves_plot_04_nelson_siegel.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_04_nelson_siegel.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_nelson_siegel.ipynb <plot_04_nelson_siegel.ipynb>`

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

      :download:`Download Python source code: plot_04_nelson_siegel.py <plot_04_nelson_siegel.py>`

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

      :download:`Download zipped: plot_04_nelson_siegel.zip <plot_04_nelson_siegel.zip>`


.. only:: html

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

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