Constant Maturity Swaps And CMS Convexity

An ordinary floating-rate swap leg resets to a short rate — three-month SOFR, say — which is a small, easily hedged sliver of the curve. A constant maturity swap leg instead resets, every period, to the current 10-year swap rate itself, no matter how many periods have already passed. It's the difference between a mortgage that resets to the overnight rate every month versus one that resets every month to whatever the going 10-year rate happens to be that month — the second is a much bigger, curve-shape-dependent bet, and it turns out you can't price it by naively taking the forward of that long rate; you have to correct for a subtle bias called CMS convexity.

Why a correction is needed at all

The forward swap rate you'd read off today's curve is not, on average, what the future swap rate turns out to be, once you account for the fact that a swap rate and its own discounting are correlated. The correction is

$$\text{CMS rate} = \text{Forward swap rate} + \text{Convexity adjustment},$$

where the adjustment is positive and grows with the swap rate's volatility, the payment's time horizon, and — critically — the tenor of the underlying swap, because a longer swap has more convexity in how its price responds to rate moves (like a bond's convexity). In plain English: naively paying the forward 10-year rate understates what a CMS payer should actually receive on average, because the payoff is a convex function of the rate, and convex payoffs are worth more than their value at the average outcome (Jensen's inequality, applied to a swap rate instead of a stock).

Worked example 1 — small correction, short leg

A CMS leg pays the 2-year swap rate in 1 year. The 1-year-forward 2-year swap rate today is 4.20%. Given typical volatility and the short tenor, the convexity adjustment might be roughly 3 basis points. Fair CMS rate: $4.20\% + 0.03\% = 4.23\%$. On a $50,000,000 notional annual payment, that 3bp difference is worth 50{,}000{,}000 \times 0.0003 = \15{,}000$ — small in relative terms, but it is real money that a naive "just use the forward" pricer would leave on the table for the counterparty receiving it.

Worked example 2 — larger correction, longer leg and longer horizon

A CMS leg instead pays the 30-year swap rate in 10 years — a much longer underlying tenor and a much longer horizon, both of which increase the adjustment. Suppose the forward 30-year swap rate 10 years out is 5.00%, and the convexity adjustment for this combination of long tenor and long horizon comes out to 45 basis points. Fair CMS rate: $5.00\% + 0.45\% = 5.45\%$. On the same $50,000,000 notional, that's 50{,}000{,}000 \times 0.0045 = \225{,}000$ of value — fifteen times larger than the first example purely because both the tenor and horizon grew, illustrating how quickly convexity effects compound for long-dated CMS products.

The convexity adjustment grows faster than linearly in both horizon and tenor — much like the quadratic shape above — which is why short-dated CMS legs are near-trivial to price and 30-year CMS products demand a full term-structure model.

What this means in practice

CMS products are popular in structured notes and pension-fund hedges because they let an investor take a direct view on the long end of the curve without duration risk from holding an actual long bond. Pricing them correctly requires the same swaption vol cube used elsewhere on the desk, since the convexity adjustment is itself derived by replicating the CMS payoff with a strip of swaptions across all strikes.