.. 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_02_mcculloch_spline.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_02_mcculloch_spline.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_02_mcculloch_spline.py:


McCulloch's spline discount function
====================================

Huston McCulloch (1971) fitted the discount function itself with a
spline, :math:`d(t) = 1 + \sum_j a_j g_j(t)`. Because every bond price is
linear in :math:`d`, it is also linear in the coefficients, so the fit is
ordinary least squares on a cross-section of bond prices. His 1971 paper
used quadratic splines; the cubic version (1975) gives smooth forward
rates.

.. GENERATED FROM PYTHON SOURCE LINES 14-22

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from financialkit.fixed_income import NelsonSiegelCurve, bond_cash_flows, fit_mcculloch_spline

    rng = np.random.default_rng(1971)
    true = NelsonSiegelCurve(0.05, -0.03, 0.02, 1.5)








.. GENERATED FROM PYTHON SOURCE LINES 23-25

A noisy cross-section of 40 coupon bonds
----------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 25-31

.. code-block:: Python


    maturities = np.sort(rng.uniform(0.3, 25, 40))
    coupons = np.round(rng.uniform(0.01, 0.07, 40), 3)
    bonds = [bond_cash_flows(100.0, c, T) for c, T in zip(coupons, maturities, strict=True)]
    prices = np.array([b.present_value(true) for b in bonds]) + rng.normal(0, 0.05, 40)








.. GENERATED FROM PYTHON SOURCE LINES 32-34

Quadratic and cubic fits
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-53

.. code-block:: Python


    knots = [1.0, 3.0, 7.0, 15.0]
    t = np.linspace(0.1, 25, 300)
    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].plot(t, true.discount(t), "k", lw=3, alpha=0.3, label="true")
    axes[1].plot(t, 100 * true.zero_rate(t), "k", lw=3, alpha=0.3, label="true")
    for degree in (2, 3):
        res = fit_mcculloch_spline(bonds, prices, knots=knots, degree=degree)
        print(f"degree {degree}: price RMSE {res.rmse:.4f}")
        axes[0].plot(t, res.curve.discount(t), label=f"degree {degree}")
        axes[1].plot(t, 100 * res.curve.zero_rate(t), label=f"degree {degree}")
    for k in knots:
        axes[0].axvline(k, color="0.85", lw=0.8)
    axes[0].set_ylabel("discount factor d(t)")
    axes[1].set_ylabel("zero rate (%)")
    for ax in axes:
        ax.set_xlabel("maturity (years)")
        ax.legend()
    plt.show()



.. image-sg:: /api/gallery/fixed_income/curves/images/sphx_glr_plot_02_mcculloch_spline_001.png
   :alt: plot 02 mcculloch spline
   :srcset: /api/gallery/fixed_income/curves/images/sphx_glr_plot_02_mcculloch_spline_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    degree 2: price RMSE 0.0513
    degree 3: price RMSE 0.0434





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

   **Total running time of the script:** (0 minutes 0.043 seconds)


.. _sphx_glr_download_api_gallery_fixed_income_curves_plot_02_mcculloch_spline.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_02_mcculloch_spline.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_mcculloch_spline.ipynb <plot_02_mcculloch_spline.ipynb>`

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

      :download:`Download Python source code: plot_02_mcculloch_spline.py <plot_02_mcculloch_spline.py>`

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

      :download:`Download zipped: plot_02_mcculloch_spline.zip <plot_02_mcculloch_spline.zip>`


.. only:: html

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

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