.. _sphx_glr_api_gallery_fixed_income_duration:

Duration and convexity
----------------------

From Macaulay's average time to payment to Ho's key rate durations.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Frederick Macaulay (1938) measured a bond&#x27;s &quot;length&quot; not by its maturity but by the present-value-weighted average time to its payments,">

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  .. image:: /api/gallery/fixed_income/duration/images/thumb/sphx_glr_plot_01_macaulay_duration_thumb.png
    :alt:

  :doc:`/api/gallery/fixed_income/duration/plot_01_macaulay_duration`

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      <div class="sphx-glr-thumbnail-title">Macaulay duration</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="John Hicks (1939) found Macaulay&#x27;s average period again as the elasticity of a capital value with respect to the discount factor. Measured per unit of yield, that elasticity is modified duration,">

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  .. image:: /api/gallery/fixed_income/duration/images/thumb/sphx_glr_plot_02_hicks_modified_duration_thumb.png
    :alt:

  :doc:`/api/gallery/fixed_income/duration/plot_02_hicks_modified_duration`

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      <div class="sphx-glr-thumbnail-title">Hicks: duration as an elasticity (modified duration)</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fisher and Weil (1971) computed Macaulay&#x27;s weights with each cash flow discounted on its own zero rate instead of the bond&#x27;s single yield,">

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  .. image:: /api/gallery/fixed_income/duration/images/thumb/sphx_glr_plot_03_fisher_weil_duration_thumb.png
    :alt:

  :doc:`/api/gallery/fixed_income/duration/plot_03_fisher_weil_duration`

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      <div class="sphx-glr-thumbnail-title">Fisher-Weil duration</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Duration is a tangent, but the price-yield curve bends. Bierwag, Kaufman and Toevs (1983) brought the second-order term into portfolio practice:">

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  .. image:: /api/gallery/fixed_income/duration/images/thumb/sphx_glr_plot_04_convexity_thumb.png
    :alt:

  :doc:`/api/gallery/fixed_income/duration/plot_04_convexity`

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      <div class="sphx-glr-thumbnail-title">Bierwag-Kaufman-Toevs: convexity</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="A single duration assumes the whole curve moves in parallel. Thomas Ho (1992) shocked the curve one key rate at a time, with triangular shapes that sum to a parallel shift, so a bond&#x27;s risk becomes a vector whose entries add up to its effective duration. Two portfolios with the same duration can have very different key rate profiles.">

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  .. image:: /api/gallery/fixed_income/duration/images/thumb/sphx_glr_plot_05_ho_key_rate_durations_thumb.png
    :alt:

  :doc:`/api/gallery/fixed_income/duration/plot_05_ho_key_rate_durations`

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      <div class="sphx-glr-thumbnail-title">Ho's key rate durations</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/fixed_income/duration/plot_01_macaulay_duration
   /api/gallery/fixed_income/duration/plot_02_hicks_modified_duration
   /api/gallery/fixed_income/duration/plot_03_fisher_weil_duration
   /api/gallery/fixed_income/duration/plot_04_convexity
   /api/gallery/fixed_income/duration/plot_05_ho_key_rate_durations