# Does a 100mio 10y swap have the same dv01 when rates are at 1% and 10%?

If not how come, whats the right way to look at it and have a quick rule of thumb to work out what dv01 is 100mio 10yr?

Thanks!

Swaps are more sensitive to interest rate movements when rates are low.

An intuitive way to see this is to realise that the change in present value of the swap comes, mainly, from the change in expected value of the floating payments that are yet to be received. These are discounted using the appropriate discount rate, so a \$1 change in the future expected value of a floating rate payment is worth less than \$1 now, and is worth even less the more it is discounted.

A rule of thumb for the DV01 of a swap with $$n$$ years to maturity and a \$1,000,000 face value, when the swap rate is $$r$$, is $${\rm DV01} \approx 100\sum_{i=1}^n e^{-ri} = 100\frac{e^{-r}(1- e^{-rn})}{1-e^{-r}}$$ That is, when rates are 10%, the DV01 of a 10 year swap is about \$600, whereas when rates are 1% it is about \$945. The approximation doesn't work when rates are exactly zero, but in that case the DV01 for a \$1m notional $$n$$-year swap is $$100\times n$$

Note that the major simplification here is the use of a single variable $$r$$ (the current swap rate) to discount the future payments, rather than discounting using a term structure of discount rates. The approximation works well for reasonably flat curves, but will be worse the steeper the curve gets.

• Why do you treat the swap rate as continuous compounded? I am not sure of the point of that. Also you might add that.your approx breaks at exact zero interest.rates. Mar 26 '20 at 18:11
• The compounding convention of the swap rate is a long way from being the biggest source of error, and it's easy to switch to semi-annual or annual compounding by replacing $e^{-r}$ with $1/(1+r)$ Apr 1 '20 at 12:06

No, there is a material convexity (interest rate gamma).

An intuitive way to see this: if the rates are 100 bps, then a 1 bp change is a much bigger deal than the same change when the rates are 1,000 bps.

A quick way to estimate what the new IR delta would be if the rates move a lot is to start with the IR delta (dv01) now and adjust it by the convexity. But this might not be accurate enough if the rates move so much. I'd reprice the swap under various scenarios and not use shortcuts.

There is zero libor gamma (for a usual CSA, ie ois discounting) Ie if you are trading a libor swap, then you have gamma only in ois

And for a rule of thumb, approx dvo1 is years to maturity X notional / 10,000 , this approx worsens as maturity recedes and as rates go away from zero

why do i differentiate between libor and ois? well, look at the market now , we see that they do not necessarily move in parallel at all! so when one talks about dv01, one should distinguish between the 2.

PS, if the person that downgraded my answer could give a comment why, that would be nice!

• I didn't downvote, but I guess whoever did thought that the comments about LIBOR vs OIS gamma, and indeed the distinction between LIBOR and OIS in general, were not really helpful for answering the question. It is a pretty basic question, it needs a basic (and jargon-free) answer. Apr 1 '20 at 12:13
• to me, it seemed the person asking wanted a quick rule of thumb, which is what i gave - its a calc a person can do in their head. these other formulas are much more serious approximations, i wouldnt call them a 'quick rule of thumb' . i described libor (projection curve), and ois (discount curve) in order to show the gamma comes just from the discount curve, which i think is relevent ? anyways thanks for your comment! Apr 2 '20 at 20:38