# duration of a cms swap

it says the following: A Swedish company has recently embraced the concept of duration and is keen to manage the duration of its debt portfolio. In the past, the company has used the Interest Rate Swap market to convert LIBOR based funding into fixed rate and as swap transactions mature has sought to replace them with new 3, 5 and 7yr swaps. The debt duration of the company is therefore quite volatile as it continues to shorten until new transactions are booked when it jumps higher. The Constant Maturity Swap can be used to alleviate this problem. If the company is seeking to maintain duration at the same level as say a 5 year swap, instead of entering into a 5 yr swap, they can enter the following Constant Maturity swap:   The tenor of the swap is not as relevant, and in this case could be for say 5 years. The "duration" of the transaction is almost always at the same level as a 5yr swap and as time goes by, the duration remains the same unlike the traditional swap. So here, the duration will remain around 5yrs for the life of the Constant Maturity Swap, regardless of the tenor of the transaction.

it is not clear to me how this can be true. i think that both the net (outright) duration, and the key rate duration on the 5y rate will still be proportionate to how much life is left in the swap, and so on a swap with term of 10y, it would be 10 times higher than on a swap with term 1y.

Any explanation much appreciated!

• i am thinking maybe that since a 3m cms swaplet on a 10y cms index has similar P&L, ie similar risk (ignoring the different convexities) to a 40x deleveraged 10y vanilla swap , then they must have same risk , ie same duration. therefore i suppose here that the usual duration formula is modified by a factor of 40x (= ratio of swap annuity vs cms swaplet annuity). But when i try to properly prove this mathematically by taking dP/dy/P for a floating rate note whose coupon is the cms , i do not get that factor. please help with that! Mar 8 '17 at 19:52

Your intuition is correct and the paper seems to misunderstand the exposure of a swap based on CMS. The term "Constant Maturity Swap" or CMS, refers to the name of an index (the prevailing swap rate at the time of observation). A swap based on the CMS can be versus either a fixed rate or Libor. In the context of this question, consider a USD100mm 5yr swap consisting of a fixed rate versus 5 year CMS. Each quarter, the company will pay fixed and receive 5 yr CMS. For a given swaplet (say the one with the next upcoming reset date), indeed the exposure is approximately the same as a 5 year swap with a notional equal to 0.25/5 *USD100mm. However the swap also has future reset dates. Looking at the reset date in 5 years' time, this is economically equivalent to a forward swap, in 5 years for 5 years, on the same adjusted notional. But this forward swap is equivalent to being short 10 year duration versus long 5 year duration, which does not necessarily behave like a simple 5 year liability at all. Representing the exposure of a swap on 5yr CMS as wholly in the 5 year bucket is wrong.

• i have my doubts that the paper is completely wrong... Mar 27 '17 at 8:52

The cash flows on CMS leg reset based on some index, let's say the 5y swap rate. Duration measures sensitivity to different parts of the yield curve. The amount paid is based on that rate as of the last reset date, so essentially your duration should be close to 5 as your exposure is concentrated at the 5yr point on the yield curve (could be a bit higher/lower as the swap gains/loses premium).

• aah ok so the cms leg is just like floating rate note - there is risk just on the current coupon... but i am still not seeing how technically this duration would be calculated!!!! Mar 28 '17 at 15:49