Constant Maturity Swap Rates and Their Derivative Exposures

# What is a Constant Maturity Swap (CMS) rate?


# What is a Constant Maturity Swap (CMS) rate?












I have been searching in books and on the internet for a basic definition and explanation of CMS rates, but I cannot find anything clear and simple. Can you explain (maybe with an example) what a CMS rate is?

And how is it used in derivatives, such as CMS swaps, CMS caps/floors, CMS spread options?

## Answer by compilation-error (score 8, accepted)

https://quant.stackexchange.com/a/24782

A constant maturity swap (CMS) rate for a given tenor is referenced as a point on the Swap curve. A swap curve itself is a term structure wherein every point on the curve is the effective par swap rate for that tenor. This is analogous to a 3m LIBOR curve represents 3m forward rates for a given tenor.

A swap rate can be considered as a weighted-average of forward rates. e.g. a two year par swap rate would be the fixed rate that makes a swap on (assume) LIBOR have NPV zero at inception. Usually, a LIBOR curve (or more generically a forward curve) would be bootstrapped using swap rates in the market (usually from 2y on-wards).

For almost all derivatives you mentioned (best of my knowledge) you can liken them to their LIBOR counterpart where the reference curve is the par swap curve (effective swap rates per tenor) in lieu of the 3m LIBOR curve. e.g. a CMS swap's floating leg will (on fixing day) not refer the 3m LIBOR but the swap rate for the tenor instead.

Moreover, one could also have the other leg floating and refer to LIBOR underlying curve. E.g. a 6m LIBOR v/s 2Y CMS swap will have one leg will pay 6m LIBOR for any fixing date v/s the other leg which will pay par 2Y swap rate for the fixing date.

Wikipedia has an example mentioned as well.

## Answer by Alex C (score 7)

https://quant.stackexchange.com/a/24784

In simple terms: An ordinary swap might be a 10 year swap of Libor vs a fixed rate; this fixed rate is determined in the marketplace every day and is published by Reuters, Bloomberg etc. as the '10 year swap rate'. Once you enter into the swap this rate remains fixed for you, of course, that is why it is called a fixed rate. But every day Reuters publishes a new number for 5 year swaps, 10 year swaps, etc.

A CMS swap is a kind of second order swap where you swap a rate of your choice against the above mentioned '10 year swap rate'. Every once in a while the rate is changed by referencing whatever Reuters says on that date the '10 year swap rate' is. Because it is always the 10 year rate that is referenced, it is called a constant maturity (in this case 10 year maturity) swap. Your payments however vary depending on developments in the market for ordinary swaps.

## Answer by quant360 (score 6)

https://quant.stackexchange.com/a/28072

In a vanilla swap, the IR on the floating leg usually depends on the reset period/swap frequency. If frequency is 6m, 6m LIBOR is used for reset, 3m LIBOR for quarterly resets etc. In a floating CMS leg, the rate used is the CMS rate, regardless of the reset frequency e.g: 10yr CMS leg will use the 10 yr CMS rate, regardless of whether the reset happens semi-annually or quarterly (of course, the rate will be multiplied by the accruing factor to make the dollar interest proportional to the length of the accruing period)

## Answer by RiskNeutral (score 2)

https://quant.stackexchange.com/a/54651

The other answers explain the structure but they do not appear to address your follow up questions in the comments regarding the naming of a constant maturity swap (CMS). While it is true that the tenor of the floating leg of both a plain vanilla swap and a CMS are constant in length, only the floating leg of the CMS has a a tenor (and therefore an exposure to risk) which extends beyond the maturity date of the structure.

To illustrate, let's compare a 2 year (maturity) plain vanilla fixed rate versus 6 month Libor swap with a 2 year (maturity) fixed rate versus 10 year swap rate CMS (both with semi-annual resets). Now let's assume that the structures were entered into just under 1.5 years ago so that there is just over 0.5 years left to maturity of both. The risk exposure of the plain vanilla structure extends out to about 6 months b/c the next and last reset will be to 6 month Libor. However, the risk of the CMS swap extends out to about 10 years because the next and last reset will be to the 10 year swap rate. So even though both swaps mature in about 6 months, only the CMS still has an exposure to the 10 year swap rate.

In other words, your exposure to the swap curve at each reset date is somewhat constant for the CMS whereas for the plain vanilla swap your risk exposure is reduced with each reset and never extends beyond the maturity date.