FRM Exam Part I · Futures Markets

Interest Rate Futures and Treasury Futures Explained

Q: How are Treasury futures quoted?

They are quoted per $100 of face value in points and 32nds. A quote of 118-16 means 118 and 16/32, or 118.5. One contract covers $100,000 face value, so this quote implies $118,500.

Q: How do I find the cheapest-to-deliver bond?

For each eligible bond, compute the quoted bond price minus the futures price times its conversion factor. The bond with the smallest result is the cheapest to deliver. The short expects to deliver that bond.

Q: What is a conversion factor?

It is the approximate price of the bond per $1 of face value if it yielded 6% with semiannual compounding. It uses the bond's maturity rounded down to the nearest three months for bond futures, or rounded down to the nearest one month for note futures. It makes bonds with different coupons and maturities comparable for delivery.

Q: Why is there a convexity adjustment for Eurodollar futures?

Futures are settled daily, and futures prices move against interest rates. A short futures position gains margin when rates rise and can invest it at those high rates, while its losses come when rates are low. The short's position is favourable, so the futures price is lower and the futures rate higher than the forward rate. You correct it with forward = futures − ½σ²t₁t₂.

Q: Do I short or buy futures to hedge a bond portfolio?

If you hold bonds and fear rising yields, you short futures. The gain on the futures offsets the loss on the bonds. You go long futures only to hedge a planned future purchase of bonds.

Updated 11 October 2026

Interest rate futures let you lock in or hedge rates. Treasury futures are quoted in 32nds, and the short delivers a bond and receives quoted futures price × conversion factor plus accrued interest. The cheapest-to-deliver bond minimises quoted price − futures price × CF. Hedge with N = (P × D_P) ÷ (F × D_F).

Understand Interest Rate Futures and Treasury Futures

An interest rate future is a standardised contract whose value moves with interest rates. Two families matter for the exam: Treasury bond and note futures, and short-term rate futures such as Eurodollar futures. Eurodollar futures were tied to LIBOR. Newer equivalents such as SOFR futures work the same way.

Treasury futures are quoted in dollars per $100 of face value, in 32nds. A quote of 118-16 means 118 and 16/32, which is 118.5. One contract covers $100,000 face value, so 118-16 means a contract value of $118,500. Prices fall when yields rise, so a short position gains when rates rise.

The short can deliver any of several eligible bonds. To make them comparable, the exchange uses a conversion factor (CF). The CF is roughly the price of the bond per $1 of face value if it yielded 6% with semiannual compounding. The bond's maturity is rounded down to the nearest three months for bond futures, and rounded down to the nearest one month for Treasury note futures, before the calculation. The short receives: quoted futures price × CF + accrued interest. The short chooses the bond that is cheapest to hand over. That is the cheapest-to-deliver (CTD) bond. It minimises quoted bond price − (quoted futures price × CF). When yields are above 6%, low-duration bonds (high-coupon, short-maturity) tend to be CTD. When yields are below 6%, high-duration bonds (low-coupon, long-maturity) tend to be CTD. Treat this as a tendency, not a rule.

Eurodollar futures are quoted as 100 − the annualised 3-month rate. A quote of 96.00 means 4%. The contract is on a $1 million deposit for three months, so one basis point is worth $1,000,000 × 0.0001 × 0.25 = $25. Futures are settled daily, while forward rates are settled at the end of the period. Futures prices also move against interest rates. A trader who is short the futures gains margin when rates rise and can invest it at those high rates. Losses arrive when rates are low and are cheap to finance. The long is the opposite: it is hurt by higher rates. The short's position is therefore favourable, so the futures price is lower and, under the standard model, the futures rate is higher than the equivalent forward rate. The convexity adjustment corrects for it: forward rate = futures rate − ½σ²t₁t₂. It grows with volatility and with maturity.

Duration-based hedging sizes a futures position so that its price change offsets the change in the portfolio. You match the dollar sensitivity of the portfolio to the dollar sensitivity of one contract. It assumes small, parallel shifts in yield. The CTD bond drives the futures' duration.

Key formulas to remember

Treasury futures quote: Price = whole points + 32nds ÷ 32. Contract value = price ÷ 100 × $100,000; 118-16 = 118.5, so one contract is worth $118,500.
Cash received by the short: Cash = (Quoted futures price × CF) + accrued interest; Per $100 face value of the delivered bond.
Cheapest-to-deliver: Delivery cost = Quoted bond price − (Quoted futures price × CF); Pick the bond with the lowest cost. Both prices are per $100 face value.
Eurodollar quote and tick value: Rate = 100 − quote. 1 bp = $1,000,000 × 0.0001 × 0.25 = $25; A higher quote means a lower rate. The contract is for a 3-month rate.
Convexity adjustment: Forward rate = Futures rate − ½ × σ² × t₁ × t₂; σ is the annual volatility of the short rate. t₁ is the futures maturity and t₂ is the maturity of the underlying rate, both in years.
Duration-based hedge ratio: N = (P × D_P) ÷ (F × D_F); P is portfolio value, F is the value of one contract, D_P and D_F are durations. Short if you hold bonds and fear rising rates.
DV01-based hedge ratio: N = DV01 of portfolio ÷ DV01 of one futures contract; Equivalent in spirit to the duration formula. Use it when DV01s are given.

How to solve Interest Rate Futures and Treasury Futures questions

Identify what the question asks: a quote, a delivery choice or a hedge. Then follow these steps.

1Read the quote. Convert 32nds to decimals (divide the fraction by 32) and scale to the $100,000 contract.
2For delivery questions, list each bond's quoted price and conversion factor. Check that all prices use the same basis (per $100).
3Compute quoted futures price × CF for each bond. Subtract it from the bond's quoted price. The lowest result is the CTD.
4For the cash received, use futures price × CF for the CTD, then add accrued interest. Multiply by face value ÷ 100 if needed.
5For Eurodollar contracts, convert the quote to a rate with 100 − quote. Use $25 per basis point for P&L.
6For convexity questions, apply forward = futures − ½σ²t₁t₂. Keep σ and times in years.
7For hedges, find the dollar duration of the portfolio (P × D_P) and of one contract (F × D_F). Divide, round to a whole number of contracts, and decide the direction.
8Check direction and size. Short futures hedge a long bond portfolio. A negative or absurdly large N signals an error.

Quickest way: CTD and hedge in under a minute

When to use it: Use it when the question gives bond prices, conversion factors and a futures price, or a portfolio value and two durations.

For CTD, compute futures price × CF for each bond in your head or on the calculator. Subtract from the quoted price.
The smallest number wins. Ignore accrued interest, because it does not decide the CTD in the standard setup.
For hedges, write N = (P × D_P) ÷ (F × D_F) once. Cancel zeros before dividing.
Convert the futures quote to dollars first: 95-16 is 95.5, so F is $95,500.
Sanity check: if D_F > D_P, then N < P ÷ F.

Common mistakes in Interest Rate Futures and Treasury Futures

Reading 118-16 as 118.16.
The dash looks like a decimal point.
Fix: The number after the dash is in 32nds. Divide it by 32: 16 ÷ 32 = 0.5, so 118.5.
Multiplying the bond price by the conversion factor instead of the futures price.
The CF belongs to the bond, so it feels natural to apply it to the bond.
Fix: The CF scales the futures price. The CTD cost is bond price − (futures price × CF).
Picking the CTD as the bond with the lowest price or highest coupon.
Students look for one simple feature of the bond.
Fix: Always compute the delivery cost for each bond. The lowest cost wins, whatever the price or coupon.
Using $25 per basis point for the wrong contract or forgetting the 100 − quote rule.
Eurodollar quoting differs from Treasury quoting.
Fix: Rate = 100 − quote. One bp is $25 on a 3-month, $1 million contract. A quote rising by 0.01 means a rate falling by 1 bp.
Applying the convexity adjustment with the wrong sign.
Students forget that the short gains margin when rates rise and can invest it at high rates, so the short's position is favourable. That makes the futures price lower and the futures rate higher than the forward rate.
Fix: Forward = futures − ½σ²t₁t₂. Under the standard model the forward rate is lower than the futures rate.
Going long futures to hedge a long bond portfolio.
Students think 'hedge' means 'hold the same position'.
Fix: A long bond portfolio loses when yields rise, so short futures. Go long only if you need to hedge a future purchase of bonds.

Worked examples

Example 1

Treasury bond futures are quoted at 120-00. Bond A has a quoted price of 125.00 and CF 1.0300. Bond B has a quoted price of 138.50 and CF 1.1500. Which bond is cheapest to deliver, and what amount per $100 face value (before accrued interest) does the short receive?

Show the solution

Futures price = 120-00 = 120.00.
Bond A: 120 × 1.0300 = 123.60. Cost = 125.00 − 123.60 = 1.40.
Bond B: 120 × 1.1500 = 138.00. Cost = 138.50 − 138.00 = 0.50.
The lower cost is Bond B (0.50 < 1.40), so B is CTD.
Cash received per $100 face = 120 × 1.1500 = 138.00, plus accrued interest.

Answer: Bond B is cheapest to deliver. The short receives 138.00 per $100 face value, i.e. $138,000 per $100,000 face, plus accrued interest.

Example 2

A fund holds a $10 million Treasury portfolio with modified duration 6.8. The Treasury futures price is 95-16 and one contract has face value $100,000 and duration 8.0. How many contracts should the fund trade to fully hedge against small parallel yield changes?

Show the solution

Convert the quote: 95-16 = 95 + 16/32 = 95.5, so F = 95.5% × $100,000 = $95,500.
Dollar duration of the portfolio: P × D_P = 10,000,000 × 6.8 = 68,000,000.
Dollar duration of one contract: F × D_F = 95,500 × 8.0 = 764,000.
N = 68,000,000 ÷ 764,000 = 89.01.
The fund holds bonds and is hurt by rising yields, so it should short futures.

Answer: Short about 89 contracts.

Exam tips

Convert the quote to a decimal and then to dollars before any other step. Many wrong options are built from a misread 32nd.
For CTD questions, show the delivery cost for every bond. The answer options often differ by a small amount, so careful arithmetic matters.
Know the direction of each trade: short futures to hedge long bonds, and the forward rate is below the futures rate after the convexity adjustment.
If a question gives DV01s, use the DV01 ratio. If it gives durations, use P × D_P ÷ (F × D_F). Do not mix the two.
Use the calculator memory for the hedge ratio so you do not retype large numbers.

Practice questions from Futures Markets

Interest Rate Futures and Treasury Futures: frequently asked questions

How are Treasury futures quoted?

They are quoted per $100 of face value in points and 32nds. A quote of 118-16 means 118 and 16/32, or 118.5. One contract covers $100,000 face value, so this quote implies $118,500.

How do I find the cheapest-to-deliver bond?

For each eligible bond, compute the quoted bond price minus the futures price times its conversion factor. The bond with the smallest result is the cheapest to deliver. The short expects to deliver that bond.

What is a conversion factor?

It is the approximate price of the bond per $1 of face value if it yielded 6% with semiannual compounding. It uses the bond's maturity rounded down to the nearest three months for bond futures, or rounded down to the nearest one month for note futures. It makes bonds with different coupons and maturities comparable for delivery.

Why is there a convexity adjustment for Eurodollar futures?

Futures are settled daily, and futures prices move against interest rates. A short futures position gains margin when rates rise and can invest it at those high rates, while its losses come when rates are low. The short's position is favourable, so the futures price is lower and the futures rate higher than the forward rate. You correct it with forward = futures − ½σ²t₁t₂.

Do I short or buy futures to hedge a bond portfolio?

If you hold bonds and fear rising yields, you short futures. The gain on the futures offsets the loss on the bonds. You go long futures only to hedge a planned future purchase of bonds.