Treasury Futures and the Cheapest-to-Deliver: Conversion Factors, Basis, and Implied Repo

A Treasury futures contract is a promise to deliver $100,000 face value of any one of a basket of eligible Treasury notes or bonds, chosen by the seller, at an invoice price equal to the futures price times that security’s conversion factor, plus accrued interest. Because the conversion factors cannot make every eligible security equally attractive, one of them is cheapest for the seller to buy and deliver: the cheapest-to-deliver, or CTD, which the futures price tracks.

In short: the conversion factor is the price of $1 of a security at a 6% yield, set by a fixed CME formula. The gross basis is cash price minus futures price times conversion factor. Subtract the carry (coupon income minus repo financing) and you get the net basis, which measures the value of the seller’s delivery options. Turn the same trade into an annual rate and you get the implied repo rate. The CTD is the deliverable with the highest implied repo rate, which is usually also the one with the lowest net basis. When yields move, the CTD can switch, which is why Treasury futures behave like a bond with an option sold against it.

This article assumes you know the contract basics in interest rate futures explained and the general difference between physical and cash settlement. It works from the CBOT rulebook, CME’s conversion factor documentation, and CME’s September 2016 paper on delivery options and basis spreads.

The misconception: “10-year futures track the 10-year note”

The CBOT contract everyone calls “the 10-year” (ticker ZN) is governed by CBOT Rulebook Chapter 19, whose title is “U.S. Treasury Note Futures (6½ to 8-Year).” Its deliverable grade is notes with an original term of not more than 10 years and a remaining term “of not less than 6 years 6 months and less than 8 years” from the first day of the delivery month. The on-the-run 10-year note is not deliverable. The futures price follows whichever 6.5-to-8-year note is cheapest to deliver, and its interest rate sensitivity is that of a roughly seven-year note, adjusted by a conversion factor.

The contract terms that matter

From CBOT Chapter 18 (U.S. Treasury Bond futures, ZB) and Chapter 19 (ZN), and the deliverable-grade table in CME’s 2024 conversion factor guide:

Contract	Deliverable grade (remaining term from first day of delivery month)	Rounding for eligibility and CF	Rulebook chapter
Ultra Bond (UB)	at least 25 years	three-month	40
Treasury Bond (ZB)	at least 15 years, less than 25 years	three-month	18
Ultra 10-Year (TN)	original 10-year notes, 9 years 5 months to 10 years	three-month	26
10-Year Note (ZN)	6 years 6 months to less than 8 years (original term not over 10 years)	three-month	19
5-Year Note (ZF)	original term not over 5 years 3 months, remaining at least 4 years 2 months	one-month	20

Other terms, from the rulebook text:

Size and price. $100,000 face value; par is 100 points, each point $1,000. ZB’s minimum tick is 1/32 of a point ($31.25); ZN’s is one-half of 1/32 ($15.625).
Invoice. Invoice Amount = ($1,000 x P x c) + accrued interest, where P is the daily settlement price on the day the seller gives notice of intention to deliver, and c is the conversion factor. Rule 19101.B. gives an example: P of 100 and 25.5/32nds and c of 0.9633 give a product of $97,097.6296875, rounded to $97,097.63.
Delivery window. Delivery may be made on any business day of the delivery month. Trading in the expiring contract stops for the last 7 business days of the month, but delivery continues to the last business day.
Notice. The short must tender a notice of intention “no later than 6:00 p.m. on the second business day preceding the intended day of delivery” (Rule 18104.A., and the same in 19104.A.).

Those last two details create the delivery options discussed below.

Conversion factors

What they are

CME’s “Calculating U.S. Treasury Futures Conversion Factors” describes the factor as “the estimated decimal price at which $1 par value of the security would trade if it had a yield to maturity of 6%.” A note with a coupon above 6% has a factor above 1; below 6%, below 1. Conversion factors for every deliverable are fixed before a contract month lists and do not change during its life.

The formula

From the CME document, with coupon rounded to the nearest 1/8 of a percent, n the whole years from the first day of the delivery month to maturity, and z the extra months (rounded down to the quarter for ZN and ZB, to the month for the shorter contracts):

v = z if z < 7; otherwise v = 3 (for ZN and ZB) or z - 6 (for the 2-, 3- and 5-year)
a = 1 / 1.03^(v/6)
b = (coupon / 2) x (6 - v) / 6
c = 1 / 1.03^(2n) if z < 7; otherwise 1 / 1.03^(2n + 1)
d = (coupon / 0.06) x (1 - c)
factor = a x [(coupon / 2) + c + d] - b, rounded to four decimals

We implemented the formula in Python and checked it against the worked examples in the CME document; it reproduces them to four decimals (for instance, 0.9003 for the 4-1/8s of August 31, 2030 against December 2023 10-year futures, and 0.9135 for the 3-5/8s of May 31, 2028 against March 2024 5-year futures).

Why they are imperfect

The factor system implicitly assumes that every deliverable yields exactly 6%. CME’s “Understanding Treasury Futures” (Labuszewski, 2013) spells out the consequence: when yields are well below 6%, the system favours delivery of short-duration (high-coupon, shorter-maturity) securities; when yields are well above 6%, it favours long-duration (low-coupon, longer-maturity) ones. The intuition: at yields below 6%, all deliverables trade above their factor-implied value, and the longest-duration ones are inflated the most, so they are the most expensive to deliver. Curve shape adds a second bias: in an upward-sloping curve, longer deliverables yield more, which makes them cheaper to deliver.

Gross basis, carry, net basis and implied repo

CME’s September 2016 paper “Treasury Futures Delivery Options, Basis Spreads, and Delivery Tails” gives the standard definitions:

Gross basis = P - (F x cf), where P is the cash clean price and F the futures price. Usually quoted in 32nds.
Carry = coupon income minus repo interest on the cash bond from settlement to the assumed delivery date.
Net basis = gross basis - carry. CME describes the net basis as “the imputed value of the futures contract’s embedded delivery options.”
Implied repo rate. CME writes the ratio of invoice amount to purchase cost as Ratio = [($1,000 x F x cf) + AI(d)] / [($1,000 x P) + AI(t+1)], and Implied Repo Rate = (Ratio - 1) x 360 / (d - (t+1)). That form assumes no coupon between purchase and delivery; when one falls in the window, the coupon is added to the proceeds and the repo arithmetic adjusted, which is what we do below.

Buying the basis means buying the cash security and selling futures; selling the basis is the reverse. The convention for the delivery date is set by carry: if carry is positive, a basis buyer plans to deliver on the last delivery day to collect as much of it as possible; if negative, on the first.

CME’s own example in the same paper shows how small the net basis of a CTD can be: for the Ultra 10-year September 2016 contract, it calculates a net basis of 0.19/32nds, “slightly less than one fifth of one tick.”

Worked example: finding the CTD

This is an illustrative basket, not a live market. Five hypothetical notes are eligible for a March 2027 ZN contract. The trade settles on Tuesday, November 10, 2026, and the assumed delivery is the last delivery day, Wednesday, March 31, 2027: 141 days. Term repo is assumed at 3.90% (actual/360). Futures trade at 111-26 (111.8125). Each note has one coupon falling inside the window, which we include in carry and the implied repo rate.

Step 1: eligibility and conversion factors

Remaining term is measured from March 1, 2027, and rounded down to the quarter. For Note A (4.50% of November 15, 2033), the actual remaining term is 6 years 8 months, rounded to 6 years 6 months: n = 6, z = 6, v = 6. Then a = 1/1.03 = 0.970874, b = 0, c = 1/1.03^12 = 0.701380, d = 0.75 x (1 - 0.701380) = 0.223965, and the factor is 0.970874 x (0.0225 + 0.701380 + 0.223965) = 0.9202.

Note	Coupon	Maturity	Remaining (rounded)	Conversion factor
A	4.500%	Nov 15, 2033	6y 6m	0.9202
B	4.000%	Feb 15, 2034	6y 9m	0.8902
C	4.375%	May 15, 2034	7y 0m	0.9082
D	3.875%	Aug 15, 2034	7y 3m	0.8765
E	4.250%	Nov 15, 2034	7y 6m	0.8955

Note B shows the z of 7 or more branch: n = 6, z = 9, so v = 3 and c uses 2n + 1 periods, giving 0.8902.

Step 2: basis and implied repo

Cash prices come from assumed yields of 3.98% to 4.11%, rising with maturity. Carry runs to March 31, 2027.

Note	Yield	Clean price	Gross basis (32nds)	Carry (32nds)	Net basis (32nds)	Implied repo
A	3.98%	103-04.95	8.47	5.62	2.85	3.680%
B	4.02%	99-27.84	10.71	0.32	10.39	3.073%
C	4.05%	102-02.78	17.24	4.59	12.65	2.912%
D	4.08%	98-20.57	20.45	-0.61	21.06	2.202%
E	4.11%	100-30.31	26.21	3.59	22.62	2.113%

Note A is the CTD. It has both the lowest net basis and the highest implied repo rate, as CME’s documentation says is normally the case. It is the shortest-duration note in the basket, which is what the 6% rule predicts with yields near 4%.

In dollars, for $10 million face of Note A:

Gross basis: 103.1546 - (111.8125 x 0.9202) = 103.1546 - 102.8899 = 0.2647 points, or $26,470.
Carry: the November 15 coupon ($225,000) plus accrued interest at delivery ($169,061), minus accrued paid at purchase ($218,886), minus 141 days’ repo on the $10,534,342 purchase price, net of reinvesting the coupon: about $17,578.
Net basis: $26,470 - $17,578 = $8,892, or 2.85/32nds.

The implied repo rate of 3.68% sits 22 basis points below the 3.90% repo rate. A basis buyer who held Note A to delivery and delivered it would earn 3.68% on the money: less than financing it costs. That gap is the price of the delivery options the basis buyer gets for being short the future. Labuszewski’s 2013 CME paper states the general rule, that the CTD’s implied repo rate “tends to run at a level that is a bit inferior to the returns associated with comparable short-term investment alternatives,” and gives a real data point: on January 10, 2013, the CTD for March 2013 ZN had an implied repo rate of 0.121%, against an effective fed funds rate of 0.160% and a 13-week bill yield of 0.051%. At those near-zero rates the paper judged the CTD’s return “essentially equivalent” to the alternatives, a reminder that the gap is small when the delivery options are worth little.

Step 3: the invoice and the hedge ratio

If the short delivers Note A on March 31 at a settlement price of 111-26, the invoice per contract is $1,000 x 111.8125 x 0.9202 = $102,889.86, plus accrued interest of $1,690.61, for $104,580.47.

The futures behave like the CTD divided by its conversion factor. Note A’s DV01 is about $61.79 per $100,000 face, so the contract’s DV01 is about $61.79 / 0.9202 = $67.15. A conversion-factor hedge of $10 million of Note A is 100 x 0.9202 = 92 contracts, which is also the DV01-neutral ratio for the CTD, a point CME’s 2016 paper makes explicitly.

The delivery options

CME’s 2016 paper describes the short’s position as holding “an embedded American-style rainbow option” with two sets of privileges: timing and quality.

The carry (timing) option. The short picks the delivery day. With positive carry, waiting pays; with sufficiently negative carry, delivering early pays.
The quality (CTD) option. The short picks which eligible security to deliver, and can switch if relative values change before delivery.
The end-of-month option. For UB, ZB, TN and ZN, the end-of-month interval “spans six business days, from the seventh-to-last through the second-to-last business days of the delivery month.” Invoice prices are frozen at the final settlement price while cash Treasuries keep trading, so the short can react to moves in relative value with a known invoice.
The wild card option. On business days before the last trading day, the futures settlement price is published “shortly after 2:00 p.m. CT,” but a short can declare intent to deliver until 6:00 p.m. CT. For nearly four hours, the invoice is known while cash bonds keep trading until about 4:30 p.m. CT. A large late-day move can make delivery at the 2:00 p.m. price valuable.

All four belong to the short and all four make the futures contract cheaper, which is why the CTD’s net basis is positive and its implied repo rate sits below market repo. The basis buyer has, in effect, bought these options; the basis seller has sold them.

When the CTD switches

The futures price is set by the deliverable that is cheapest at delivery, not today. Hold the basket fixed and shift every yield in parallel, repricing the futures each time so that the cheapest net basis stays at 3/32nds:

Yield shift	Futures price	Implied repo: A	C	E	CTD
-150 bp	122.59	3.688%	1.443%	-0.807%	A
0 bp	111.81	3.668%	2.900%	2.101%	A
+50 bp	108.46	3.661%	3.377%	3.055%	A
+80 bp	106.50	3.650%	3.653%	3.617%	C (barely)
+100 bp	105.08	3.299%	3.494%	3.646%	E
+200 bp	98.22	1.418%	2.547%	3.629%	E

As yields rise, the CTD rotates from the short-duration Note A to the longest Note E, passing through C. In this basket the crossover arrives near +80 basis points, with yields around 4.8%, well before 6%. That is the curve-slope bias at work: Note E yields 13 basis points more than Note A to begin with, which makes it cheaper to deliver as soon as the conversion factor bias weakens.

Two consequences follow:

Futures duration changes with the CTD. Below the crossover, futures track a 6.5-year note; above it, a 7.5-year note. A hedge sized to Note A’s DV01 will be wrong after a switch.
Negative convexity. Because the futures track whichever bond is cheapest, they lag the CTD when yields rise toward the crossover and do not capture the full gain of the longer bonds when yields fall. The short has the option; the long has sold it. The table assumes the option is worth a flat 3/32nds. In practice its value peaks near a crossover, when switching is most likely, so the net basis widens there.

The delivery tail

A basis position built with conversion-factor weights does not match the delivery obligation one for one. CME’s 2016 paper works through the example: $10 million of a CTD with a factor of 0.6928 is hedged with 69 contracts, but delivery is 69 x $100,000 = $6.9 million, leaving an unhedged $3.1 million “delivery tail.” In our example, 92 contracts against $10 million of Note A leaves an $800,000 tail. Only a security with a factor of exactly 1.0000 (a 6% coupon) has no tail.

What can go wrong

CTD switches. A basis buyer long the old CTD can see the basis widen sharply if another issue becomes cheaper. That is the option paying off for the basis buyer. For a basis seller, it is a loss with no fixed limit.
Repo specials. The calculation assumes general collateral repo. If the CTD becomes scarce and trades “special” in repo (lent cash at a lower rate), its carry rises and the basis moves, sometimes violently near delivery.
The tail and delivery logistics. Book-entry delivery must be completed by 1:00 p.m. on delivery day under Rule 18104.B., through a qualified bank. Failures create real costs, and the tail must be hedged or sold.
Leverage. The basis is small, so funds that run it at scale finance the cash leg in repo and use leverage, which links futures to funding markets. That is the subject of the Treasury basis trade.
Margin. Futures losses are paid in cash daily while the cash bond’s offsetting gain is collateral elsewhere, the same liquidity mismatch described in how futures margin works.

The short version

Treasury futures are physically delivered contracts on a basket of eligible notes or bonds, invoiced at the futures price times a conversion factor fixed at a 6% yield. Because the factors are imperfect, one issue is cheapest to deliver; the futures price tracks it, and its DV01 divided by its conversion factor is the contract’s DV01. Gross basis minus carry leaves the net basis, which is the market value of the short’s timing, quality, end-of-month and wild card options. Expressed as a rate, the CTD’s implied repo sits a little below market repo for the same reason. When yields move far enough, the CTD switches, futures duration changes, and anyone holding the basis finds out what the options were worth. Professionals run these numbers for every deliverable, every day; the arithmetic above is the core of it.