Constant Maturity Swap
Also known as: CMS, CMS swap
Pays a long-term rate every quarter — which sounds simple and is where the convexity adjustment was invented.
1 · SnapshotThe one idea to remember
2 · BeginnerWhat is it, really?
An ordinary interest rate swap exchanges a fixed rate for a short-term floating rate. The floating side resets every three months to a three-month rate.
A constant maturity swap changes one thing. The floating side still resets every three months, but it resets to a long rate — the ten-year swap rate, say. Every quarter you are paid whatever ten-year money costs on that day.
That is useful if you have a long-dated liability and want your income to follow long rates rather than short ones. Insurers and pension funds have exactly that problem.
It is also the reason this instrument is not simple. You are being paid a ten-year rate for three months, and those two horizons do not agree with each other.
- Asset class
- Rates derivatives
- Instrument type
- Swap referencing a swap rate
- Traded
- OTC, cleared where mandated
- Typical users
- Insurers, structured note issuers, rates desks
Which risks decide the outcome
Not how risky this is, and not a rating — there is deliberately no total. It says which of five failure modes drives what happens here, in the same order on all 129 products so they can be compared. This publication's own reading; see the notice below.
- Marketdecides it
- Creditbarely applies
- Liquiditymatters
- Fundingdecides it
- Operationalbarely applies
What decides it here. The level and shape of the curve decide it, and volatility enters through the convexity adjustment. Cleared positions demand cash daily long before the view is right.
3 · IntermediateHow it works in practice
What a CMS is really a bet on
- The level of long rates, reset frequently — not the level of short rates.
- The shape of the curve. A CMS against a short floating rate is a steepener: it pays when the curve is steep and costs when it is flat or inverted.
- Volatility. This is the surprising one, and it is the whole subject of the next level.
Where it turns up
Rarely on its own. It is the engine inside retail structured notes — CMS steepeners, CMS-linked coupons, range accruals on the curve — because "pays the ten-year rate" is a sentence a distributor can say. The convexity adjustment underneath it is not.
4 · AdvancedPricing & valuation
The convexity adjustment
The forward swap rate is a martingale under the annuity measure, not under the forward measure the payment is settled in. Paying a swap rate at a single date therefore is not worth its forward:
What the symbols mean
- Ean expected value
- Tmaturity, in years
- Sthe price of the underlying today
- Gammahow fast Delta itself changes
- sigmavolatility, the standard deviation of returns
The adjustment is positive, grows with the square of volatility and with maturity, and vanishes only if the payment is made on the swap's own annuity schedule. Ignoring it prices a CMS as a plain forward, which is wrong by an amount that is small in calm markets and material in volatile ones.
Replication
The clean way to value it is not to model the adjustment but to replicate the payoff with a strip of swaptions across strikes, which prices the whole smile rather than a single volatility:
What the symbols mean
- wa weight in a portfolio
- Kthe strike: the price written into the contract
This makes a CMS book a position in the swaption surface, which is why CMS and swaption desks are the same desk.
Why it matters beyond the instrument
The convexity adjustment here is the same effect that makes a futures rate differ from a forward rate, and that makes a quanto payoff differ from its unadjusted twin. Learning it once on a CMS is the cheapest way to see it, because the sign and the driver are both unambiguous.
The formulas above are standard textbook formulations, simplified for teaching. They explain the mechanism — they are not a valuation tool, and they will not reproduce a dealer’s price.
5 · Desk notesHow practitioners think about it
Now say it back
Close the page and give Constant Maturity Swap in four sentences. It takes a minute and it is the only way to find out whether reading it was enough.
- Who wants what — two parties wanted opposite things badly enough to write it down.
- What the contract obliges, and when — not the payoff; the obligation.
- Where the money comes from — name the source, or you have described a hope.
- What makes it lose — the ordinary way, not the dramatic one.