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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?

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    $\begingroup$ quant.stackexchange.com/questions/49582/… does this help? $\endgroup$
    – Attack68
    Aug 6, 2023 at 6:24
  • $\begingroup$ @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? $\endgroup$
    – augustine
    Aug 6, 2023 at 8:05
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    $\begingroup$ 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 $r\,?$ $\endgroup$
    – Kurt G.
    Aug 7, 2023 at 7:09
  • $\begingroup$ @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 $\endgroup$
    – augustine
    Aug 8, 2023 at 18:51
  • $\begingroup$ It is. And it explains why one calls it duration. $\endgroup$
    – Kurt G.
    Aug 8, 2023 at 18:57

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