1
$\begingroup$

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?

Improve this question
$\endgroup$
7
  • 1
    $\begingroup$ quant.stackexchange.com/questions/49582/… does this help? $\endgroup$
    – Attack68 ♦
    Commented 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
    Commented Aug 6, 2023 at 8:05
  • 1
    $\begingroup$ 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.
    Commented 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
    Commented Aug 8, 2023 at 18:51
  • $\begingroup$ It is. And it explains why one calls it duration. $\endgroup$
    – Kurt G.
    Commented Aug 8, 2023 at 18:57

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.