“Bond futures convexity refers to the non-linear curvature of the relationship between bond prices and interest rates, which causes a future contract’s price to rise faster when yields drop than it falls when yields increase. Because bond futures are tied to an underlying basket of physical bonds with varying maturities and coupon rates, this curvature determines which specific bond becomes the ‘cheapest-to-deliver’ as market rates fluctuate.” – Bond futures convexity – Fixed income

Pricing risk in a bond futures contract cannot be reduced to a single duration figure. The contract gives the short position a choice among eligible securities, while the long position receives whichever security is tendered. That delivery flexibility means the futures price is linked not to one permanent bond, but to the deliverable issue that offers the most economical settlement at a particular point in the rate cycle. Convexity makes this choice dynamic because each bond’s price responds differently as yields move, and those differences can alter both the hedge ratio and the identity of the cheapest-to-deliver security.

For an option-free fixed-rate bond, price and yield have a positively curved relationship. Duration supplies the first-order approximation: a small rise in yield normally produces a fall in price, while a small decline produces a rise. Convexity supplies the second-order correction, so the gain from a given fall in yield is larger than the loss from an equally sized rise in yield. The standard approximation is \frac{\Delta P}{P}\approx -D_{mod}\Delta y+\frac{1}{2}C(\Delta y)^2, where D_{mod} is modified duration, C is convexity and \Delta y is the change in yield. The effect becomes more material for longer maturities, lower coupons and larger yield movements.⁵

A futures contract adds another layer because its deliverable basket contains bonds with different coupons and remaining maturities. Exchanges use conversion factors to make those securities broadly comparable. For each eligible issue, the factor approximates the price at which the security would trade at a 6 percent yield to maturity during the delivery month. The delivery invoice is then calculated from the futures settlement price, the relevant conversion factor, accrued interest and the contract multiplier.^2,3 In simplified form, the invoice amount is I_i=\left(F\times CF_i+AI_i\right)\times M, where F is the futures price, CF_i is the conversion factor for bond i, AI_i is accrued interest and M is the contract multiplier.

The short chooses the security that minimises the economic cost of delivery, not necessarily the bond with the lowest market price. A useful comparison is the net basis, commonly expressed as the cash price less the conversion-factor-adjusted futures price. The issue with the lowest relevant net basis is generally the cheapest to deliver, although financing costs, accrued interest, delivery timing and embedded delivery options must also be considered. Another equivalent perspective is the implied repo rate: the CTD usually offers the highest return from buying the cash bond, carrying it and delivering it against the futures contract.^6,10 This is why the futures price tends to track the CTD more closely than the average characteristics of the entire basket.

Why convexity changes the CTD

Two eligible bonds can have similar duration at the current yield but different convexity. A longer-maturity, lower-coupon bond generally has greater convexity, while a higher-coupon or shorter-maturity bond generally has less. If yields fall, the more convex issue may appreciate faster than its competitor. Its cash price can therefore move relative to the futures invoice value, changing the net basis. If yields rise, the relative movement can reverse. The result is a switching boundary: one bond is cheapest in one rate environment, while another becomes cheapest after a sufficiently large movement in yields.

Conversion factors do not eliminate this effect. They standardise deliverable securities against a notional 6 percent coupon, but they are calculated using simplified assumptions and discrete maturity conventions. They do not perfectly equalise each bond’s forward value across all yield levels. A bond with a coupon far below the standard may be advantaged or disadvantaged as market yields move away from the conversion-factor assumption, depending on its duration, convexity and financing economics. The exchange mechanism creates comparability for invoicing, not identical exposure to every possible curve movement.^2,3

Practical implications for hedging

A portfolio manager hedging a cash bond with futures must therefore distinguish between price sensitivity and delivery sensitivity. The usual hedge begins with a duration or DV01 comparison, such as h=\frac{DV01_{cash}}{DV01_{futures}}, where h is the approximate number of futures contracts required. That estimate is local: it describes risk around the current yield and current CTD. If the CTD changes, the futures DV01 changes as well, so a hedge that was initially close to neutral can acquire a material residual exposure.

Convexity also matters when the hedge is intended to perform across a range of outcomes rather than at one small rate move. A cash portfolio and a futures position can have matching duration but different second-order behaviour. After a sizeable rally, the cash portfolio may gain more than the hedge loses, or the reverse may occur after a sell-off, depending on the relative convexity of the portfolio and the active CTD. Curve shape adds another complication: a futures contract may respond primarily to the yield of its CTD, whereas the hedged portfolio may be exposed to several maturities and to changes in slope or curvature.

Delivery options and competing interpretations

The short’s delivery choice is economically valuable. The short can select among eligible securities, choose a permitted delivery date and, in some contracts, benefit from timing and financing flexibility. These features mean that the futures contract contains an embedded option even though the traded instrument is not usually described as an option. The value of that option grows when the deliverable basket is diverse, volatility is high or the relative pricing of eligible bonds is unstable. A static CTD assumption can then understate basis risk.

Market practitioners sometimes describe the CTD as if it were a permanent property of the contract. That is convenient for daily risk management, but it is not a complete description. The CTD is conditional on futures price, cash prices, repo rates, accrued interest, delivery timing and expected rate paths. The bond with the lowest current basis may not remain cheapest after a rate shock. Models that project CTD switches therefore compare forward invoice economics across the whole basket rather than extrapolating from a single observed bond.

Why the concept still matters

Bond futures convexity matters because it connects a textbook curvature measure with an operational trading decision. It explains why equal-sized yield moves do not produce equal price moves, why duration-only hedges can drift, and why the short’s delivery choice can alter the contract’s effective exposure. For investors, the practical discipline is to monitor the CTD, conversion factor, basis, implied repo rate, delivery calendar and the relative convexity of the eligible issues. In calm markets, the distinction may appear small; during sharp rate moves, it can determine whether a hedge protects the intended risk or merely transfers it into basis and delivery uncertainty.

 

References

1. Mortgage risk and the yield curve – https://www.bis.org/publications/working-paper-532-mortgage-risk-and-yield-curve.pdf

2. Futures & Options Trading for Risk Management – CME Group – https://www.cmegroup.com/trading/interest-rates/files/treasury-futures-conversion-factor-look-up-tables.xls

3. Treasury Futures Delivery Options, Basis Spreads, and … – https://www.cmegroup.com/content/dam/cmegroup/education/files/treasury-futures-basis-spreads.pdf

4. Interest Rate Futures – https://fatih.ai/sjtu/derivatives/files/slides/lecture6.pdf

5. Yield-Based Bond Convexity and Portfolio Properties | CFA … – https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/yield-based-bond-convexity-and-portfolio-properties

6. [PDF] A Fat Cat in a Small Door – The Convexity Maven – https://www.convexitymaven.com/wp-content/uploads/2026/10/Convexity-Maven-Fat-Cat.pdf

7. Managing Bond Futures – 2006-05-09 – https://www.investopedia.com/terms/b/bondfutures.asp

8. Key Financial Market Concepts, 2nd Edition – 2012-09-07 – https://www.oreilly.com/library/view/key-financial-market/9780273750123/html/chapter-024.html

9. Convexity and prepayment risk – https://download.dws.com/download/asset/54bad0e6-e644-4e8e-830d-5047d0e67393?tenant

10. [PDF] CBOT® US Treasury Futures and Options Reference Guide – https://insigniafutures.com/Docs/CBOT_Treasuries.pdf