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I have come across a Q&A about the calculation of the duration of an interest rate swap on this site.

In the Q&A, the derivative is calculated as:

$\frac{\partial PV}{\partial r}=t_nD(t_n)+q\sum_j\Delta_j^{fix} t_jD(t_j)$.

I don't understand what $r$ is in the equation above? Shouldn't the derivative be calculated as $\frac{\partial PV}{\partial q}$, assuming $q$ is the swap rate?

I have also read somewhere that sensitivity of an interest rate swap w.r.t. its swap rate is approximately life of the swap.

So, I should expect that $\frac{\partial PV}{\partial q}\approx t_n$, but it is stated in that Q&A that $\frac{\partial PV}{\partial r}\approx t_n$. Are they both equal? If they are indeed equal, why so?

Alper
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augustine
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    https://quant.stackexchange.com/questions/49582/interest-rate-swap-pv01-vs-dv01/49585#49585 does this help? – Attack68 Aug 06 '23 at 06:24
  • @Attack68 Thanks for this reference. However still I failed to understand why only $t_n$ term stays in above expression. Am I missing something very trivial? – augustine Aug 06 '23 at 08:05
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    The duration of any stream of cashflows is the derivative w.r.t. a flat interest rate that you use for discounting. The most prominent example is the duration of a bond which is the derivative of the price w.r.t. the yield. In your swap we take the derivatve w.r.t. $r$ and not w.r.t. the fixed swap rate $q,.$ On top of that we divide that derivative by the price: duration$=\partial_r PV/PV,.$ Exercise: what is the duration of a single fixed cashflow when you discount with a continously compounded rate $r,?$ – Kurt G. Aug 07 '23 at 07:09
  • @kurt to answer your question, I think that duration of such cash flow should be the time of that cashflow (e.g. ZCB) Hope this answer is correct – augustine Aug 08 '23 at 18:51
  • It is. And it explains why one calls it duration. – Kurt G. Aug 08 '23 at 18:57
  • But how to fit that answer to my original question? In Swap, I have Fixed and Floating cashflows. For floating cashflow, I have duration as next reset date (from Floating rate bond). But for fixed cashflow, I have some other duration. Why then the net duration for Swap will be the same as life of the swap? – augustine Aug 08 '23 at 19:14
  • nobody claimed that duration is always the life of the trade. this is only the case for a single fixed cashflow. – Kurt G. Aug 09 '23 at 05:31

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