When Floating-Rate and CMS Cash Flows Need Convexity Adjustments
Summary
The discussion explains why convexity adjustments depend on the structure and timing of a rate derivative’s cash flows, rather than simply applying to every underlying rate in the same way. It contrasts a spread option with a linear derivative and notes that adjustments can arise when the rate’s fixing tenor differs from the payment period, or when the fixing and payment timing are shifted from the standard arrangement.
Examples distinguish a compounded SOFR rate fixed in advance and paid quarterly from a payment at a different frequency, and compare a vanilla swap with a security whose payments do not follow the swap’s usual cash-flow schedule. Another answer emphasizes that CMS-linked payoffs may require convexity treatment even when the payoff is linear, especially when CMS tenor and payment frequency materially differ. The replies offer qualitative guidance rather than a formula or worked valuation. The precise adjustment depends on the instrument’s conventions and cash-flow design, so the examples should not be taken as a universal rule for every rate product.
Key ideas
- Convexity adjustments are tied to derivative payoff structure and cash-flow timing.
- A mismatch between the rate fixing tenor and payment period can require an adjustment.
- Standard timing for a rate and its payment may avoid the adjustment in some cases.
- CMS-linked payoffs may need convexity treatment when payment frequency differs materially from the CMS tenor.
- The discussion gives qualitative examples but no adjustment formula or valuation calculation.
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Full text
# Convexity adjustment doubt # Convexity adjustment doubt So this the question and the answer to the first one states that only the 5 year swap rate will be adjusted for convexity and the answer to the second one states that neither of the rates will be adjusted for convexity. My question is why? Why won't both the rates (ie the the sofr and treasury rate) go under the adjustment as well? I've cracked my head on this question and I still can't figure it out so any help would be highly appreciated! Source : Options, Futures, and other Derivatives : Hull, John C ## Answer by Bob Jansen (score 1) https://quant.stackexchange.com/a/71818 I think you've missed the fact that the first question deals with a spread option and the second one with a linear derivative. ## Answer by user35980 (score 1) https://quant.stackexchange.com/a/75404 The first question relates a 5y tenor security paid quarterly while the second is a (spread of) quarterly securities paid quarterly - fixed in advance, paid in arrears. Timing/convexity adjustments are necessary when there is a differential between the fixing/payment period tenors or if the fixing is at any time other than advanced during the period. E.g. 3m compounded SOFR rate (fixed in adv) paid in arrears quarterly wouldn't require an adj, while paid semi-annually it would. Similarly if the 5y swap rate is paid in any way other than as a 5y set of fixed/float cashflows (i.e. a vanilla swap), it would have convexity. All of this has to do with messing around with the discount factors used to value your security free of arbitrage, and these guys don't react well to moving fixing or payment times around because they're a non-linear function of rates. ## Answer by Nobodys Fool (score 0) https://quant.stackexchange.com/a/80800 Convexity adjustments are required on any CMS payoff, even if it’s linear. So even a structure that swaps CMS for SOFR (no floor/cap), this requires a convexity adjustment. Whenever there is a material difference between payment frequency and the tenor of the CMS rate, there’s a need for a convexity adjustment.
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