# 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.

## 1 Answer

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.

• The finite difference scheme is very low on precision and I am looking for more than three digit precision. Any idea on how to achieve the same through Jacobian matrix computation in MS Excel? Thanks. – Gourav Poddar Jun 22 '15 at 20:26
• @GouravPoddar, To have the Jacobian, you need the analytical expressions between $z$ and $r$. Once you have those formulas, then you can do it in Excel. – Gordon Jun 22 '15 at 21:14