---
title: "Bond Futures"
book: "Markets II: Rates, FX and Credit"
subject: quant
language: en
chapter: 6
exercises: 8
source: https://one-course.com/books/quant/2/en/chapter/6-bond-futures
---

# Chapter 6 — Bond Futures

The Chicago Board of Trade’s ten-year note future promises the delivery of a note that does not exist. Its terms describe a basket: any Treasury note with a remaining life between six and a half and eight years at the start of the delivery month may be delivered, and the seller, the short, chooses which one. To make notes with different coupons and maturities interchangeable, each is scaled by a factor that the exchange publishes; the scaling is exact only when yields are 6%, which they rarely are, so one note in the basket is always cheaper to deliver than the others, and the future trades as a forward on that note plus the value of the short’s right to change its mind. Hedge funds hold hundreds of billions of dollars of notes against short futures to earn the small difference between the two. This chapter prices the contract, identifies the note it is really written on, and values the choice.

## 6.1 The contract and its basket

**Definition 6.1 (Deliverable basket).**

The *deliverable basket* of a bond future is the set of securities the short may deliver, defined by the exchange’s rules by remaining (and sometimes original) maturity at the start of the delivery month.

**As of September 2026 — Two ten-year contracts.**

**CBOT ten-year note future** (the “6.5 to 8-year” contract, code `ZN`): USD 100 000 face; tick half of a 32nd, USD 15.625; deliverable notes with an original term of at most ten years and a remaining term of at least six years six months and less than eight years; from the March 2026 contract month, reopened notes are deliverable too. Last trading day: the seventh business day before the last business day of the delivery month; delivery until the last business day. **Eurex Euro-Bund future** (`FGBL`): EUR 100 000 of a notional 6% bond; deliverable German federal bonds with 8.5 to 10.5 years remaining.

## 6.2 Conversion factors and invoice price

**Definition 6.2 (Conversion factor and invoice price).**

The *conversion factor* of a deliverable note is, approximately, the price of one unit of par at a yield of 6%, computed by the exchange’s formula with the note’s coupon rounded to the nearest eighth and its remaining life from the first day of the delivery month rounded down, to a whole quarter for the ten-year and bond contracts. A short that delivers the note receives the *invoice price*

$$
\text{invoice} \;=\; F \times \mathrm{CF} \;+\; \text{accrued interest at delivery},
$$

per 100 of face, where $F$ is the futures settlement price.

**Method 6.3 (The exchange’s conversion factor).**

With the coupon $c$ in decimals, $n$ whole years and $z$ whole months from the first day of the delivery month to maturity ($z$ rounded down to a multiple of 3 for the ten-year and bond contracts), $v = z$ if $z < 7$ and $v = 3$ otherwise (for those contracts), set $a = 1.03^{-v/6}$, $b = \frac c2
\frac{6 - v}{6}$, $c' = 1.03^{-2n}$ if $z < 7$ and $1.03^{-(2n+1)}$ otherwise, $d = \frac{c}{0.06}(1 - c')$, and

$$
\mathrm{CF} \;=\; a\,\Bigl[\frac c2 + c' + d\Bigr] - b,
$$

rounded to four decimals. It is the price at 6% of a note whose next coupon is $v$ months away, less the part of that coupon already accrued.

**Example 6.4 (The exchange’s own check).**

For the 3.75% note of 15 November 2018 and the December 2008 contract, $n = 9$, the remaining 11 months are rounded down to $z = 9$, so $v =
3$: $a = 0.985329$, $b = 0.009375$, $c' = 0.570286$, $d = 0.268571$ and $\mathrm{CF} = 0.8357$, the factor the exchange published. The chapter’s build reproduces the exchange’s five worked examples to the fourth decimal.

The factor makes a note with a high coupon worth proportionally more in delivery, so that every note in the basket would be equally attractive to deliver if all yielded 6%. At any other yield they are not. When yields are below 6% a long note gains more than the factor allows, so the short prefers to deliver the note with the *least* duration; when yields are above 6% it prefers the most.

## 6.3 Cheapest-to-deliver

**Definition 6.5 (Cheapest-to-deliver).**

The *cheapest-to-deliver* (CTD) is the note in the basket that costs the short least to buy and deliver: the one with the lowest price relative to what it will be invoiced at, equivalently, the highest [implied repo rate](#def-m2-bond-futures-irr) ([Definition 6.7](#def-m2-bond-futures-irr)).

**Proposition 6.6 (At delivery).**

On the last delivery day, if no arbitrage is possible, $F = \min_i
P_i/\mathrm{CF}_i$, where $P_i$ is the clean price of note $i$: the futures price converges to the price of the cheapest note divided by its factor.

**Proof.** If $F > P_i/\mathrm{CF}_i$ for some $i$, sell the future, buy note $i$ and deliver it: the invoice $F\,\mathrm{CF}_i + A_i$ exceeds the cost $P_i + A_i$. If $F < \min_i P_i/\mathrm{CF}_i$, buy the future: the short must deliver some note $j$ worth $P_j + A_j > F\,\mathrm{CF}_j + A_j$, which the long can sell for more than it paid. ∎

Before delivery the future is priced off the CTD’s forward, the price at which buying the note today and financing it in repo breaks even at delivery, and it trades a little below that, because the short owns the options of [Section 6.5](#sec-6-5).

## 6.4 Gross basis, net basis and implied repo

**Definition 6.7 (Gross basis, net basis, implied repo, basis trade).**

For a deliverable note with clean price $P$, factor $\mathrm{CF}$ and future price $F$: the *gross basis* is $P - F\,\mathrm{CF}$; the *net basis* is the gross basis less the note’s carry to delivery ([Proposition 5.3](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#prop-m2-repo-and-specials-carry)); the *implied repo rate* is the financing rate at which buying the note, financing it and delivering it into the future exactly breaks even,

$$
\text{IRR} \;=\; \frac{F\,\mathrm{CF} + A_{\text{del}} + C - (P + A_0)}{P + A_0}\,\frac{360}{d},
$$

with accrued interest $A_0$ today and $A_{\text{del}}$ at delivery, coupons $C$ received in between and $d$ days. A *basis trade* buys the note and sells the future in the ratio of the factor (long the basis), or the reverse (short the basis).

The [implied repo rate](#def-m2-bond-futures-irr) is the futures market’s quote for financing: a trader who is long the basis earns it on the note’s price and pays the actual [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate), so the difference is the trade’s profit if the note is delivered. For the CTD it is usually a little below general collateral, and the gap, like the [net basis](#def-m2-bond-futures-irr), prices the short’s options.

**Example 6.8 (A December basket).**

On 25 September 2026 a December ten-year future trades at 111-24+ (111.765625); delivery is assumed on 31 December, 97 days away, and repo is 3.90%. The table below shows five deliverable notes. The 3.875% of August 2033, the shortest, has the smallest [net basis](#def-m2-bond-futures-irr), two thirds of a 32nd, and the highest implied repo, 3.82%, eight basis points below repo: it is the CTD. The longest note would deliver at an implied repo of 0.05%. These notes and prices are illustrative.

| Note | Clean price | Factor | [Gross basis](#def-m2-bond-futures-irr) | [Net basis](#def-m2-bond-futures-irr) | Implied repo |
| --- | --- | --- | --- | --- | --- |
| 3.875% Aug-33 | 99.132 | 0.8870 | $-0.004$ | 0.020 | 3.82% |
| 4.125% Nov-33 | 100.549 | 0.8971 | 0.284 | 0.261 | 2.95% |
| 4.25% Feb-34 | 101.262 | 0.9012 | 0.539 | 0.488 | 2.12% |
| 4.625% May-34 | 103.641 | 0.9200 | 0.817 | 0.695 | 1.45% |
| 4% Aug-34 | 99.462 | 0.8806 | 1.041 | 1.036 | 0.05% |

![The cash-and-carry basis trade. The trader buys the cheapest note, finances it in repo, sells futures in the ratio of the factor, and delivers the note at expiry against the invoice price. It earns the implied repo rate and pays the repo rate; with leverage from repo, a few basis points on a large position is the business.](https://one-course.com/images/onecourse/chapters/quant-2/m2-bond-futures/fig-92dcbd5a5c80.svg)

***Figure 6.1.** The cash-and-carry [basis trade](#def-m2-bond-futures-irr). The trader buys the cheapest note, finances it in repo, sells futures in the ratio of the factor, and delivers the note at expiry against the [invoice price](#def-m2-bond-futures-cf). It earns the [implied repo rate](#def-m2-bond-futures-irr) and pays the [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate); with leverage from repo, a few basis points on a large position is the business.*

**As of September 2026 — The size of the basis trade.**

A Federal Reserve note of June 2026 estimated aggregate Treasury cash-futures basis positions at about USD 830 billion in September 2025, roughly twice the early-2020 peak and 3.5% of privately held Treasuries by market value. Published estimates range from USD 350 billion to 1.5 trillion; hedge funds’ cash Treasury holdings reached USD 2 trillion at the end of 2025.

## 6.5 The delivery options

**Definition 6.9 (Quality option and wildcard option).**

The short’s right to choose which note to deliver is the *quality option*. The *wildcard option* is its right, during the delivery period, to decide whether to deliver after the futures settlement price has been fixed for the day, while the notes still trade; after the last trading day the [invoice price](#def-m2-bond-futures-cf) is fixed and the short may deliver on any of the remaining days. The right to choose the day within the delivery month is the *timing* option.

![The delivery month of the ten-year note future. The short chooses the note (quality option), the day (timing option), and, every day until trading stops, whether to deliver after the day’s settlement price is known (wildcard option). Trading stops on the seventh business day before the last business day of the month; the invoice price is then fixed, and delivery may take place on any remaining business day up to the last.](https://one-course.com/images/onecourse/chapters/quant-2/m2-bond-futures/fig-5e4d1ea8724c.svg)

***Figure 6.2.** The delivery month of the ten-year note future. The short chooses the note ([quality option](#def-m2-bond-futures-options)), the day (timing option), and, every day until trading stops, whether to deliver after the day’s settlement price is known ([wildcard option](#def-m2-bond-futures-options)). Trading stops on the seventh business day before the last business day of the month; the [invoice price](#def-m2-bond-futures-cf) is then fixed, and delivery may take place on any remaining business day up to the last.*

All three belong to the short, so the future is cheaper than the CTD’s forward by their value, and a trader long the basis, long the note and short the future, owns them. The [quality option](#def-m2-bond-futures-options) is worth most when the CTD could switch: when two notes’ prices per factor are close, or when rates are volatile enough to bring the switch within reach ([Figure 6.3](#fig-m2-bond-futures-switch)).

![At delivery, each note’s clean price divided by its factor, minus the lowest, as all yields move together. The note on zero is the CTD. From today’s yields the shortest note stays cheapest until yields have risen 147 basis points, to about 5.5%; beyond that the longest note takes over. Notes and yields are illustrative. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-bond-futures/fig-1ad4246863cc.svg)

***Figure 6.3.** At delivery, each note’s clean price divided by its factor, minus the lowest, as all yields move together. The note on zero is the CTD. From today’s yields the shortest note stays cheapest until yields have risen 147 basis points, to about 5.5%; beyond that the longest note takes over. Notes and yields are illustrative. Data: the chapter’s tutorial.*

![Implied repo rates of the five notes of . The CTD’s is highest and just below the repo rate; the others would lose money if bought and delivered. Data: the chapter’s tutorial.](https://one-course.com/images/onecourse/chapters/quant-2/m2-bond-futures/fig-267d0fcba347.svg)

***Figure 6.4.** [Implied repo rates](#def-m2-bond-futures-irr) of the five notes of [Example 6.8](#ex-m2-bond-futures-basket). The CTD’s is highest and just below the [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate); the others would lose money if bought and delivered. Data: the chapter’s tutorial.*

## 6.6 Tutorial: finding the cheapest-to-deliver

**Goal.** Compute the factors of a basket, its gross and net bases and [implied repo rates](#def-m2-bond-futures-irr), find the CTD, and locate the yield at which it switches. **End state:** the table of [Example 6.8](#ex-m2-bond-futures-basket) and Figures [6.3](#fig-m2-bond-futures-switch) and [6.4](#fig-m2-bond-futures-irr).

1. **The factor**, from the exchange’s formula. `def conversion_factor (coupon: float , first_day_of_delivery_month: dt.date, maturity: dt.date, quarter_rounding: bool = True ) -> float : """CBOT conversion factor, rounded to four decimals. coupon in decimals (0.0375).""" cpn = math.floor(coupon * 800 + 0.5 ) / 800 # nearest 1/8 of a percent, ties up months = whole_months(first_day_of_delivery_month, maturity) n, z = divmod (months, 12 ) if quarter_rounding: z -= z % 3 v = z if z < 7 else (3 if quarter_rounding else z - 6 ) a = 1.0 / 1.03 ** (v / 6 ) b = cpn / 2 * (6 - v) / 6 c = 1.0 / 1.03 ** (2 * n) if z < 7 else 1.0 / 1.03 ** (2 * n + 1 ) d = cpn / 0.06 * (1 - c) return round (a * (cpn / 2 + c + d) - b, 4 )` **Listing 6.1.** The conversion factor. code/firm/ctd/firm_ctd.py Check the exchange’s examples: 0.9229, 0.8747, 0.8653, 0.8357, 0.7943.
2. **Basis and carry.** `def gross_basis (d: Deliverable, futures: float ) -> float : return d.clean - futures * d.cf def carry_to_delivery (d: Deliverable, repo: float , days: int ) -> float : """Coupon income over the period less financing of the dirty price, per 100.""" accrual = d.accrued_delivery - d.accrued_now + d.coupon_income return accrual - (d.clean + d.accrued_now) * repo * days / 360.0 def net_basis (d: Deliverable, futures: float , repo: float , days: int ) -> float : return gross_basis(d, futures) - carry_to_delivery(d, repo, days)` **Listing 6.2.** Gross basis, carry to delivery and net basis. code/firm/ctd/firm_ctd.py
3. **Implied repo and the CTD.** `def implied_repo (d: Deliverable, futures: float , days: int ) -> float : """The financing rate that makes buying the bond and delivering it into the future break even.""" cost = d.clean + d.accrued_now proceeds = invoice_price(futures, d.cf, d.accrued_delivery) + d.coupon_income return (proceeds - cost) / cost * 360.0 / days def cheapest_to_deliver (basket: list [Deliverable], futures: float , days: int ) -> Deliverable: """The deliverable with the highest implied repo rate (equivalently, lowest net basis).""" return max (basket, key=lambda d: implied_repo(d, futures, days))` **Listing 6.3.** Implied repo rate, and the note that maximises it. code/firm/ctd/firm_ctd.py
4. **The switch.** `futures_demo.switch_shift()` scans parallel shifts and reports the CTD changing at $+147$ basis points; `fig_futures.py` writes the two charts.

**What to change next.** Steepen the curve by adding a basis point per year of maturity to each note’s yield and find the new switch point; then compute the value of the switch option for several volatilities ([Exercise 6.7](#exo-m2-bond-futures-7)).

## 6.7 Build: the basis analyser

**Purpose.** The miniature firm hedges its bond positions with futures and runs [basis trades](#def-m2-bond-futures-irr); both need the factor, the [invoice price](#def-m2-bond-futures-cf), the basis and the implied repo of every deliverable, every day.

**Interface.** `conversion_factor(coupon, first_day, maturity, quarter_rounding)`; `invoice_price(F, cf, accrued)`; `Deliverable(name, clean, accrued_now, accrued_delivery, cf, coupon_income)`; `gross_basis`; `carry_to_delivery`; `net_basis`; `implied_repo`; `cheapest_to_deliver`.

**Rules.** Factors by the exchange’s formula; prices per 100; actual/360 financing; accrued interest from the bond library ([Section 3.6](https://one-course.com/books/quant/2/en/chapter/3-government-bonds#bld-m2-government-bonds-bond)).

**Acceptance tests.** `code/firm/ctd/tests/`: the five published factors; whole-month counting; a 6% note maturing on the first day of the delivery month has a factor of 1; the CTD’s [net basis](#def-m2-bond-futures-irr) is zero at its own implied repo and every other note’s is positive.

**Stretch.** The Eurex factor formula (annual coupons); delivery-date choice from the timing option; a quality-option value from a curve model rather than parallel shifts.

Sources and further reading

- CBOT, submission 25-099 to the CFTC (contract grade of the ten-year note future), February 2025; CME Group, *Calculating U.S. Treasury Futures Conversion Factors* .
- M. Henrard, *Bond Futures: Description and Pricing* , OpenGamma Quantitative Research, 2011.
- Federal Reserve, FEDS Notes, “Decomposing Hedge Funds’ U.S. Treasury Exposures”, June 2026; Office of Financial Research, “Hedge Funds’ Cash Treasury Holdings Reach $2 Trillion”, August 2026.
- G. Burghardt and T. Belton, *The Treasury Bond Basis* , 3rd ed., McGraw-Hill, 2005.

## 6.8 Exercises

**Exercise 6.1 ★.**

The future settles at 111-24+. The CTD’s factor is 0.8870 and its accrued interest at delivery 1.453125. Give the [invoice price](#def-m2-bond-futures-cf) per 100 and per contract.

**Solution of Exercise 6.1.**

$111.765625 \times 0.8870 + 1.453125 = 100.589234$ per 100; USD 100 589.23 per contract of USD 100 000 face.

**Exercise 6.2 ★.**

The CTD’s clean price is 99.132. Give its [gross basis](#def-m2-bond-futures-irr) in price and in 32nds. Why can a [gross basis](#def-m2-bond-futures-irr) be negative?

**Solution of Exercise 6.2.**

$99.132 - 111.765625 \times 0.8870 = -0.004$, about $-0.14$ of a 32nd. The [gross basis](#def-m2-bond-futures-irr) compares today’s spot price with the futures price scaled by the factor; the future is a forward, so the gap also contains the carry to delivery. Here carry is negative (the note’s coupon, 3.875%, earns less than 3.90% repo on its price), so the forward is above spot and the [gross basis](#def-m2-bond-futures-irr) can be negative while the [net basis](#def-m2-bond-futures-irr) is positive.

**Exercise 6.3 ★.**

Compute the December 2026 factor of a 4% note maturing 15 November 2033, and of a 6% note with the same maturity. Why is the second not exactly 1?

**Solution of Exercise 6.3.**

From 1 December 2026 to 15 November 2033: 6 years 11 months, rounded down to 6 years 9 months, so $n = 6$, $z = 9$, $v = 3$. The 4% note: 0.8902. The 6% note: 0.9999, not 1, because the formula prices it as if its next coupon were three months away and its life rounded down; only a 6% note whose life is a whole number of quarters from the first day of the month has a factor of exactly 1.

**Exercise 6.4 ★★.**

The CTD’s implied repo is 3.82% and repo is 3.90%. You buy the basis in USD 100 million face and hold it to delivery with no switch. What do you lose? Why might you still do it?

**Solution of Exercise 6.4.**

Earning 3.8238% and paying 3.90% on the dirty price, USD 99 563 440, for 97 days loses $99\,563\,440 \times 0.000762 \times 97/360 = \text{USD}~20\,451$, if the note is delivered as the CTD. One still buys the basis to own the short’s options: if yields move far enough for the CTD to switch, or if the timing and [wildcard options](#def-m2-bond-futures-options) pay, the basis widens and the long profits; and if the note becomes special in repo, its financing costs less than 3.90%.

**Exercise 6.5 ★★.**

A fund holds USD 100 million face of the CTD and wants to hedge it with the future. How many contracts should it sell? Why does the answer involve the factor?

**Solution of Exercise 6.5.**

The future moves like the CTD divided by its factor, so its DV01 per contract is the CTD’s DV01 per USD 100 000 divided by the factor. Contracts $= \frac{100\,000\,000/100\,000 \times \mathrm{DV01}_{\text{CTD}}}{\mathrm{DV01}_{\text{CTD}}/\mathrm{CF}} = 1\,000 \times 0.8870 = 887$ contracts. The factor is the number of futures “per bond”.

**Exercise 6.6 ★★.**

Explain from duration why the shortest note is cheapest to deliver when yields are below 6%, and why the switch goes to the longest note when yields rise.

**Solution of Exercise 6.6.**

The factor prices every note at 6%. Below 6% all notes are worth more than their factor-scaled value, and the longer the duration, the more they have gained as yields fell from 6%: the long notes are the dearest to deliver, the shortest the cheapest. As yields rise towards 6% the long notes lose more, the gap shrinks, and at some yield (below 6% here, because the curve and the coupons differ) the longest note becomes the cheaper one.

**Exercise 6.7 ★★★.**

*Coding.* With `futures_demo.switch_option_value`, value the switch option for a normal volatility of 80 basis points a year over the 97 days, then three times that. Report both in 64ths and comment.

**Solution of Exercise 6.7.**

The standard deviation over 97 days is $80 \times \sqrt{97/365} = 41$ basis points. The switch option is worth 0.001 of a 64th: the switch is 3.6 standard deviations away. At three times the volatility, 124 basis points, it is worth 3.07 64ths. The [quality option](#def-m2-bond-futures-options) is worth nothing until the switch comes within reach, then grows fast: it is an out-of-the-money option on the level of yields.

**Exercise 6.8 ★★★.**

*Find the flaw.* “The ten-year future trades at 111 while the notes it delivers trade near 100: the future is 11% expensive, sell it and buy the notes.” Correct the reasoning.

**Solution of Exercise 6.8.**

The future is priced per 100 of a notional 6% note; the deliverable notes pay about 4% and are worth about 100 at 4% yields. A 6% coupon note of the same maturity would be worth about 112 at those yields. The factors (0.88 to 0.92) convert: $111.77 \times 0.887 = 99.14$, within a hundredth of the CTD’s 99.13. The comparison must be made after multiplying by the factor, where the gap is a few hundredths of a point, not 11%.

## 6.9 Problem: The Switch

**Problem 6.1.**

Weekend problem — when the cheapest-to-deliver changes

Take the basket and prices of [Example 6.8](#ex-m2-bond-futures-basket): the future at 111-24+, repo 3.90%, 97 days to delivery.

**Part I — The basket.**

1. Which notes are deliverable into the December contract, by the rule of [Box 6.1](#dat-m2-bond-futures-contracts) ? Check each of the five.
2. Give the factor of the 4% note of August 2034 and of the CTD.
3. Give the CTD’s [gross basis](#def-m2-bond-futures-irr) , carry and [net basis](#def-m2-bond-futures-irr) .
4. Give its [implied repo rate](#def-m2-bond-futures-irr) . How far below repo is it?
5. Why is its carry negative?

**Part II — The switch.**

6. At delivery, with yields unchanged, what is each note’s price per factor above the CTD’s?
7. By how much must all yields rise for the CTD to change? Which note takes over?
8. What yield does the CTD have at that point?
9. Why is the switch below 6%?
10. After the switch, how does the future’s duration change?

**Part III — The option.**

11. With a normal volatility of 80 basis points a year, what is the standard deviation of yields over the 97 days?
12. How many standard deviations away is the switch?
13. Give the value of the switch option at that volatility, in 64ths.
14. And at three times the volatility?
15. The future trades one 64th below the CTD’s forward in the example. What else could that 64th be paying for?

**Part IV — Judgement.**

16. A trader is long the basis of the CTD. Why is she long volatility?
17. Why does the switch matter to a hedger who uses futures, not only to a basis trader?
18. What would a reopening of the Aug-33 note, adding to its supply, do to the basis?
19. State the *named result* : the shift at which the CTD switches, and the value of the switch option at both volatilities.
20. In one sentence: what does a bond future really deliver?

**Solution of Problem 6.1.**

**1.** From 1 December 2026: Aug-33 6 years 8 months, Nov-33 6 years 11 months, Feb-34 7 years 2 months, May-34 7 years 5 months, Aug-34 7 years 8 months: all at least 6 years 6 months and less than 8 years, and all issued as notes of at most ten years: all deliverable. **2.** Aug-34: 0.8806. CTD (Aug-33): 0.8870. **3.** [Gross basis](#def-m2-bond-futures-irr) $-0.004$; carry $-0.025$; [net basis](#def-m2-bond-futures-irr) $-0.004 + 0.025 = 0.020$. **4.** 3.82%, eight basis points below repo. **5.** Its coupon income over the 97 days, 1.021 per 100, is less than the repo interest on its dirty price, 1.046: a coupon of 3.875% a year on face accrues more slowly than 3.90% charged on a 360-day year. **6.** Nov-33 0.270, Feb-34 0.527, May-34 0.742, Aug-34 1.171 points above the CTD’s 111.793. **7.** $+147$ basis points; the 4% note of August 2034, the longest, takes over. **8.** $4.02\% + 1.47\% = 5.49\%$ (the Aug-34 then yields 5.55%). **9.** The notes do not all yield the same: the longer ones yield more, as the curve slopes up, and they differ in coupon. The switch happens where the longer note’s extra yield compensates the factor’s bias, below the 6% where a flat curve would put it. **10.** The future’s DV01 follows the new CTD’s DV01 per factor: from about 586 to 658 per 100 of futures notional (in units of $10^{-4}$), 12% more. A hedger who holds a fixed number of contracts is suddenly over-hedged. **11.** $80\sqrt{97/365} = 41$ basis points. **12.** $147/41 = 3.6$ standard deviations. **13.** 0.001 of a 64th: worthless. **14.** 3.07 64ths. **15.** The timing and [wildcard options](#def-m2-bond-futures-options), the end-of-month option, and the market’s uncertainty about the [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate) the CTD will finance at: the CTD may turn special and its forward be lower than the general-collateral forward. **16.** Long the basis she is long the note and short the future, which is short the options the futures short owns: she owns them. Options gain from volatility: a large move either way can make another note cheaper, and her long note’s price then rises relative to the future. **17.** Because the future’s duration jumps when the CTD changes: a hedge ratio computed from the old CTD is wrong after the switch, just when yields have moved a lot. **18.** More supply of the CTD: it cheapens in the cash market and in repo (less special), its [net basis](#def-m2-bond-futures-irr) falls, and it becomes even more firmly the cheapest to deliver. **19.** Named result: *the switch point* is a parallel rise of 147 basis points (CTD yield 5.49%), when the 4% of August 2034 replaces the 3.875% of August 2033; the switch option is worth 0.001 of a 64th at 80 basis points of annual volatility and 3.07 64ths at 240. **20.** The cheapest note in its basket, divided by its factor, less the value of the short’s choices.

## 6.10 Interview questions

**Interview question 6.1 ★ trader, bank.**

What is a [conversion factor](#def-m2-bond-futures-cf), and why does a Treasury future need one?

**Solution of Interview question 6.1.**

The contract allows several notes with different coupons and maturities to be delivered. The factor, the note’s price at a 6% yield, scales the invoice so that each note is delivered at a price proportional to its value if yields were 6%. Without it the short would always deliver the lowest-priced note and the contract would be written on one issue, open to squeezes.

*What the interviewer is looking for: what the factor is (a 6% price) and why a basket needs it.*

**Interview question 6.2 ★ trader, researcher.**

How do you identify the [cheapest-to-deliver](#def-m2-bond-futures-ctd)?

**Solution of Interview question 6.2.**

For each deliverable, compute the [implied repo rate](#def-m2-bond-futures-irr) from its price, the futures price, the factor, accrued interest and coupons to delivery; the highest is the CTD. Equivalently, the lowest [net basis](#def-m2-bond-futures-irr). At delivery, the lowest price divided by factor.

*What the interviewer is looking for: implied repo, not the lowest price or lowest [gross basis](#def-m2-bond-futures-irr).*

**Interview question 6.3 ★★ trader, researcher.**

What is the [implied repo rate](#def-m2-bond-futures-irr), and what does it mean if the CTD’s implied repo is above the actual [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate)?

**Solution of Interview question 6.3.**

The financing rate at which buying the note, financing it and delivering it into the future breaks even. If the CTD’s implied repo exceeds actual repo, buying the basis and delivering locks in a profit: the future is rich. In practice the CTD’s implied repo sits a little below repo, the difference paying for the short’s delivery options.

*What the interviewer is looking for: the arbitrage interpretation and why the normal state is below repo.*

**Interview question 6.4 ★★ trader, bank.**

Explain the Treasury [basis trade](#def-m2-bond-futures-irr) and what can go wrong with it.

**Solution of Interview question 6.4.**

Buy the CTD, finance it in repo, sell futures in the ratio of the factor; earn the [implied repo rate](#def-m2-bond-futures-irr) against the [repo rate](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate) paid, a few basis points, levered many times through repo. What can go wrong: [repo rates](https://one-course.com/books/quant/2/en/chapter/5-repo-and-specials#def-m2-repo-and-specials-rate) spike or lenders raise haircuts, so financing costs more or must be reduced; futures margin calls on the short side during a rally must be met in cash while the gain on the notes is unrealised; the basis widens in a [dash for cash](https://one-course.com/books/quant/2/en/chapter/4-the-treasury-market#def-m2-the-treasury-market-dash); and many leveraged funds unwinding at once can move the market they unwind in.

*What the interviewer is looking for: the financing and margin mechanics, and the funding liquidity risk, not only the arbitrage.*

**Interview question 6.5 ★★ researcher, developer.**

How would you compute the DV01 of a Treasury future?

**Solution of Interview question 6.5.**

To first order, the CTD’s DV01 per 100 of face divided by its [conversion factor](#def-m2-bond-futures-cf), times 1 000 for a USD 100 000 contract. Better: reprice the future from the whole basket under bumped yields (the minimum of forward prices per factor, plus an option model), which captures the change in CTD and the optionality; the two agree only far from a switch.

*What the interviewer is looking for: DV01 over factor, and why the switch makes it state-dependent.*

**Interview question 6.6 ★★★ researcher, trader.**

Which options does the short in a Treasury future hold, and how would you value the most important one?

**Solution of Interview question 6.6.**

The [quality option](#def-m2-bond-futures-options) (which note), the timing option (which day of the delivery month), the [wildcard option](#def-m2-bond-futures-options) (deliver after the futures price is fixed for the day, or after trading ends at a fixed [invoice price](#def-m2-bond-futures-cf)), and the end-of-month option. The [quality option](#def-m2-bond-futures-options) is usually the largest: value it by simulating the yield curve to delivery (a factor model of parallel, slope and curvature moves), computing each note’s price per factor, and taking the expected minimum against the current CTD’s; volatility and the distance to the switch drive it.

*What the interviewer is looking for: naming the options and a concrete method for the [quality option](#def-m2-bond-futures-options).*