Reader question: A Treasury futures contract can often be settled by delivering any one of several eligible Treasury securities. If the bonds have different coupons and maturities, how can one futures price represent the whole basket — and how does the short decide which security is economically cheapest to deliver?
The answer is a delivery algorithm built around conversion factors. Each eligible Treasury security receives a factor that approximately standardises its price to a common 6% yield convention. At delivery, the futures settlement price is multiplied by the chosen security’s conversion factor and accrued interest is added to determine the invoice amount. The short then compares eligible securities after accounting for cash price, conversion factor, coupon carry, financing and delivery timing. The security with the most favourable delivery economics is the cheapest-to-deliver (CTD).
The mathematics is a useful lesson in optimisation under optionality: the contract does not identify one underlying bond in advance. It defines a basket and gives the short a choice.
What this page owns — and what it does not
This page owns the delivery-basket problem:
eligible Treasuries + conversion factors + futures price + financing/carry → invoice economics → CTD selection.
It does not replace bond yield inversion, duration and convexity, repo-pricing algorithms, or exchange matching. Those are adjacent owners.
This is fixed-income mechanism education, not an execution or trading recommendation.
Step 1: construct the deliverable basket
Each CME Treasury futures contract defines which U.S. Treasury notes or bonds are eligible for physical delivery. The maturity window depends on the contract. For example, CME’s current educational material lists different remaining-maturity bands for 2-year, 5-year, 10-year, bond and ultra-bond futures.
The delivery algorithm begins by filtering the Treasury security master:
DeliverableSet = {security satisfying contract grade and maturity rules}.
A security outside the contract’s grade specifications should never enter CTD optimisation, no matter how attractive its market price looks.
Step 2: apply the conversion factor
CME defines the conversion factor as the approximate decimal price at which one dollar of par value of an eligible Treasury security would trade if it yielded 6% to maturity under the contract’s convention.
For eligible security i:
CFi = conversion factor assigned by the exchange.
Higher-coupon bonds tend to have conversion factors above 1 when the coupon exceeds the 6% benchmark, while lower-coupon securities generally have factors below 1.
The factor is not a market price and does not change every second. It is a contract conversion parameter used to standardise deliverable securities.
Why a conversion factor is needed
Without conversion factors, a futures contract deliverable into both a low-coupon and high-coupon Treasury would systematically favour one simply because their clean prices differ for mechanical coupon reasons.
The 6% standard attempts to put securities on a common basis. It is only an approximation, which is why one bond can still be cheaper to deliver than another.
If the conversion-factor system perfectly equalised every bond at every yield-curve shape and financing condition, CTD choice would be economically irrelevant. The fact that CTD matters is evidence that the standardisation is intentionally practical rather than exact.
Step 3: calculate the delivery invoice
CME states the core invoice relationship as:
Invoice price per unit par = Futures settlement price × Conversion Factor + Accrued Interest.
For a contract face amount, the dollar invoice scales accordingly.
If the futures price is 112.00, conversion factor 0.9000 and accrued interest 0.70 per 100 par, then the stylised invoice is:
112.00 × 0.9000 + 0.70 = 101.50 per 100 par.
The short delivers the Treasury and receives the contractually determined invoice amount.
The gross basis
A first comparison between cash and futures is the gross basis:
Gross Basis = Cash clean price − Futures price × Conversion Factor.
CME’s Treasury basis material uses this form.
If two eligible bonds have different conversion factors, the raw cash prices are not directly comparable. The converted futures price places them into the futures-delivery framework.
Gross basis is only a starting point
Suppose one candidate has a lower gross basis but pays a coupon before delivery, while another has more favourable repo financing. The true carry economics can reverse the ranking.
A fuller comparison considers:
- cash purchase price;
- financing cost until delivery;
- coupon payments received before delivery;
- reinvestment or coupon carry;
- accrued interest paid on purchase and received on delivery;
- conversion-factor-adjusted futures proceeds;
- delivery date.
This leads to net basis or implied-repo calculations.
Implied repo: turn delivery into a financing return
CME describes implied repo as the rate of return obtained by borrowing to buy the appropriate cash Treasury while simultaneously selling the related Treasury futures and delivering the cash security into the futures contract.
Conceptually:
cash outflow today → financed Treasury purchase → coupons/carry → futures delivery invoice → financing return.
For each eligible security, the algorithm solves for the annualised financing rate that makes those dated cash flows reconcile.
The candidate with the highest implied repo rate is often the cheapest-to-deliver under the specified financing and delivery assumptions.
Net basis and implied repo tell the same economics from different directions
Net basis subtracts carry from the gross cash-futures price difference. Implied repo converts the same cash-and-carry economics into an annualised financing return.
Under consistent assumptions:
- lower net basis tends to indicate a more attractive delivery candidate;
- higher implied repo tends to indicate the same candidate.
If the two methods identify different CTDs, the implementation likely uses inconsistent coupon, settlement, financing or accrued-interest assumptions.
A simple candidate comparison
Suppose two securities A and B are deliverable.
| A | B | |
|---|---|---|
| Cash clean price | 98.20 | 104.10 |
| Conversion factor | 0.8700 | 0.9200 |
| Futures price | 112.00 | 112.00 |
Converted futures values are:
A: 112 × 0.8700 = 97.44.
B: 112 × 0.9200 = 103.04.
Gross bases:
A = 98.20 − 97.44 = 0.76.
B = 104.10 − 103.04 = 1.06.
On gross basis alone, A looks cheaper. But coupon income and repo financing can still alter the final CTD choice.
The short owns the delivery choice
CME’s public delivery material states that the short position has the delivery optionality: it chooses which eligible security to deliver and, subject to contract rules, when during the delivery period to deliver.
The long is assigned delivery rather than choosing the bond.
This creates embedded quality and timing options for the short. CTD analysis is therefore not merely a static price comparison; it is part of an option-bearing delivery mechanism.
Why CTD can switch
Conversion factors are anchored to a 6% yield convention, while actual Treasury yields can be far above or below 6% and the yield curve is not flat.
As yields change:
- low- and high-coupon bond prices respond differently;
- durations differ;
- curve shape changes relative valuations;
- repo specialness changes financing economics.
A bond that is CTD today can therefore lose that status tomorrow.
This is CTD switch risk.
Why the futures contract often behaves like the CTD
CME notes that Treasury futures tend to trade economically like their CTD because the delivery option ties futures value most tightly to the cheapest candidate.
That relationship is not permanent. If two securities are near the CTD boundary, futures risk can reflect the possibility of a switch.
A risk engine that hard-codes one CTD for the life of a contract can therefore misstate futures duration and basis sensitivity.
DV01 and the conversion factor
DV01 measures the dollar price change for a one-basis-point yield movement.
Because the futures contract maps to the CTD through the conversion factor, a common approximate relationship is:
Futures DV01 ≈ CTD cash DV01 / Conversion Factor,
scaled to contract size and under the current CTD assumption.
The result is an approximation because CTD can switch and the yield curve can move non-parallel.
This connects to duration, convexity and scenario shocks.
Repo specialness can change CTD economics
Some Treasury securities trade “special” in repo, meaning they can be financed at unusually low repo rates because borrowers particularly want that collateral.
A security that is expensive in cash-price terms can still have attractive futures-delivery economics if its financing benefit is sufficiently valuable.
This links CTD selection directly to the repo-pricing algorithm.
Inputs and outputs
A CTD engine can require:
- futures contract and delivery month;
- exchange deliverable-grade rules;
- eligible Treasury identifiers;
- published conversion factors;
- cash clean prices;
- coupon rates and coupon dates;
- accrued interest;
- futures settlement price;
- repo/financing rates by security;
- candidate delivery dates;
- settlement calendars and contract face amount.
Outputs can include:
- invoice price for each candidate;
- gross basis;
- net basis;
- implied repo rate;
- ranked delivery candidates;
- current CTD;
- CTD switch thresholds or scenario sensitivities;
- CTD-adjusted futures DV01.
Evidence polarity: what supports confidence?
Evidence for a CTD result includes a deliverable basket matching CME contract specifications, conversion factors matching CME’s published tables, invoice amounts reproducing exchange examples, gross-basis calculations reconciling independently, consistent implied-repo/net-basis rankings and realistic financing inputs.
Evidence against confidence includes an ineligible bond appearing as CTD, a conversion factor from the wrong delivery month, clean/dirty-price confusion, missing coupon cash flows before delivery, use of one generic repo rate when a candidate is special, or a CTD that never changes under extreme yield scenarios despite near-tied alternatives.
Counterexample: lowest cash price is not necessarily CTD
Bond A can have a lower clean price than Bond B but also a much lower conversion factor. The relevant comparison is not the raw cash price. It is delivery economics after conversion factor, carry and financing.
Counterexample: lowest gross basis is not always final CTD
If two bonds have similar basis but one earns a coupon before delivery or finances significantly cheaper in repo, the implied-repo ranking can differ from gross basis.
Therefore gross basis is a useful screen, not a complete delivery optimiser.
Counterexample: a 6% conversion factor does not mean the bond yields 6% today
The conversion factor is calculated under a standardised 6% yield convention. It is not a market-yield observation.
A bond can have a conversion factor based on 6% while its current yield is 4%, 5% or 7%.
Counterexample: CTD is conditional on delivery date
Coupon accrual, coupon receipts and financing days change across the delivery month. A candidate that is CTD on one assumed delivery date may not remain CTD on another.
The short’s timing option is therefore economically meaningful.
Weak links in implementation
Wrong delivery basket. Maturity eligibility is calculated from today rather than the specified delivery-month reference date.
Stale conversion factors. Factors from another contract month are reused.
Price-unit error. Treasury 32nds are parsed incorrectly before decimal calculations.
Accrued-interest omission. Invoice and cash purchase are compared on inconsistent clean/dirty bases.
Coupon-timing omission. A coupon received before delivery is ignored.
Flat financing assumption. Repo specialness is missed.
Fixed CTD assumption. Futures risk is mapped to one bond even after the CTD changes.
calendar error. Delivery or coupon dates fall on the wrong business day.
Diagnostics: how to test the engine
- CME factor replay: compare every candidate conversion factor with the exchange’s official lookup table.
- invoice replay: reproduce CME’s futures-price × factor + accrued-interest examples.
- basket test: construct securities just inside and just outside the maturity window.
- price-format test: convert Treasury 32nds to decimals and back exactly.
- gross-basis test: independently calculate cash price minus converted futures price.
- implied-repo test: solve the dated cash flows independently and compare annualised returns.
- coupon-boundary test: move delivery across a coupon payment date.
- repo-specialness test: change one candidate’s financing rate and verify ranking response.
- yield-shock test: apply parallel and curve-shape shocks to observe possible CTD switches.
- delivery-date test: evaluate every permitted delivery date rather than one arbitrary date when timing optionality matters.
What would falsify confidence?
Confidence should be withdrawn if the CTD candidate is not actually deliverable; if invoice prices disagree with exchange methodology; if net basis and implied repo imply contradictory rankings under the same assumptions; if coupon or repo cash flows are missing; or if an independent revaluation under the same data produces a different CTD.
Alternatives and limits
For basic education, gross basis may be enough to show why conversion factors are necessary. For risk management, implied repo and CTD-switch scenarios are stronger. More complete futures valuation can model the full embedded delivery option and future repo/yield-curve states rather than treating today’s CTD as permanent.
No CTD algorithm can turn uncertain future financing rates and yield curves into certainty. The result is conditional on the market data and delivery assumptions supplied.
How this connects to the surrounding knowledge estate
Bond-yield algorithms provide candidate yield analytics. Accrued-interest algorithms supply clean/dirty conversion at purchase and delivery. Repo algorithms determine financing carry. Duration and DV01 connect CTD to futures risk. The CTD layer combines those owners into a delivery optimisation problem.
Verification and update triggers
Preserve contract specifications, delivery-month basket, exchange conversion-factor file, price timestamps, coupon schedules, repo inputs, accrued-interest convention and delivery-date assumptions. Revalidate after CME contract-specification changes, new Treasury issuance alters the deliverable basket, conversion-factor updates, major repo-specialness shifts, or any CTD change that materially alters futures DV01.
Primary and high-quality references
- CME Group, U.S. Treasury Futures Conversion Factor Look-Up Tables.
- CME Group, Calculating U.S. Treasury Futures Conversion Factors.
- CME Group, Learn about the Treasuries Delivery Process.
- CME Group, Treasury Analytics User Guide, including CTD, conversion-factor, DV01 and implied-repo definitions.
- CME Group, The Basics of Treasuries Basis.
Educational boundary: This article explains Treasury-futures delivery mathematics. It does not recommend a futures, cash-Treasury, repo or basis position and does not provide personalized financial advice.