How Swap DV01 Changes with Rates and Discounting
Summary
The document explains why the DV01 of a fixed income swap is not constant across interest rate levels. A simplified rule of thumb discounts each future annual payment using a single rate, then sums the discounted sensitivities. Under that approximation, a swap is more sensitive when rates are low because distant cash flows retain more present value; the examples compare a ten-year swap at low and higher rates. At zero rates, the rule reduces to notional-scaled maturity sensitivity.
The answer stresses that this shortcut assumes a flat curve and should be less reliable when the curve is steep. Another response points to convexity as a reason that sensitivity changes with rate moves and recommends repricing under scenarios when rates shift substantially. A further answer distinguishes sensitivity to different rate curves, such as LIBOR and OIS, and cautions that they may not move in parallel. The discussion offers intuition and approximations, not a full swap valuation method; actual DV01 depends on cash flows, curve construction, discounting, and conventions.
Key ideas
- Swap DV01 generally increases as rates fall because future cash flows are discounted less heavily.
- A discounted sum of future sensitivities provides a rough rule of thumb under a flat curve assumption.
- The simple approximation becomes less reliable when the yield curve is steep.
- Convexity affects how DV01 changes after substantial interest rate movements.
- Specify which rate curve defines the sensitivity because different curves may move independently.
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Full text
# Does a 100mio 10y swap have the same dv01 when rates are at 1% and 10%?
# 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!
## Answer by Chris Taylor (score 4)
https://quant.stackexchange.com/a/51756
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.
## Answer by Dimitri Vulis (score 2)
https://quant.stackexchange.com/a/51753
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.
## Answer by Randor (score -1)
https://quant.stackexchange.com/a/51801
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!Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.