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Transforming Zero-Rate Sensitivities into Par-Rate Sensitivities

Article Quant Q&A · Author: Gourav Poddar
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Summary

The document describes how to convert a bond’s pillar-wise sensitivity to zero rates into sensitivity to observable par or coupon rates. It frames the conversion as a Jacobian transformation: multiply the vector of sensitivities with respect to zero rates by the rate of change of zero rates with respect to par rates. Since zero rates are typically derived rather than directly observed, par-rate sensitivities can be more practical for reporting or hedging.

To obtain the transformation matrix, the answer suggests a finite-difference approach: perturb a par rate, rebuild the zero-rate curve, and measure how the zero rates change. The document offers no Excel-specific instructions, numerical example, or guidance on perturbation size and numerical stability. The method therefore outlines the sensitivity conversion concept but leaves implementation details to the practitioner.

Key ideas

  • Zero-rate sensitivities can be converted to par-rate sensitivities using a Jacobian.
  • The zero-rate sensitivity vector can be obtained from the bond pricing formula.
  • The Jacobian describes how zero rates respond to changes in par rates.
  • Finite differences can estimate the Jacobian by perturbing par rates and rebuilding the zero curve.
  • The document does not specify numerical settings or spreadsheet implementation details.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Jacobian transformation


# Jacobian transformation












I am trying to calculate pillar-wise sensitivity of a fixed coupon bond using par rates (given pillar-wise zero coupon sensitivities). I came across the formula pv01(par) = pv01(zero) * dz/dr, where dz/dr is the rate of change of the zero rates w.r.t the par rates. I am however unable to understand how to use this formula in excel. Any help is appreciated. Thanks.

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/18481

As zero rates are usually not observable, people tend to use the sensitivity with respect to par, or coupon, rates. Here, pv01(zero) is a vector, which cen be computed using the pricing formula that is usually expressed in terms of the zero rates. To compute $dz/dr$, you may need to use a finite difference scheme, for example, to shift the par rate $r$, and then re-compute the zero rates.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.