FRM Exam Part I · Pricing Financial Forwards and Futures

Treasury Bond Futures and Eurodollar Futures Explained

Updated 11 October 2026 · Fact-checked

With Treasury bond futures, the short delivers a bond from an eligible basket and receives the futures quote times the conversion factor, plus accrued interest. The short picks the cheapest-to-deliver bond. Eurodollar futures quote 100 minus a three-month rate, and 1 bp is worth $25. A convexity adjustment converts futures rates into forward rates.

Understand Treasury Bond Futures and Eurodollar Futures

A Treasury bond futures contract covers $100,000 face value of long-term US Treasury bonds. The price is quoted in dollars and 32nds per $100 of face value. The short position can choose which bond to deliver from a basket of eligible bonds. These bonds have different coupons and maturities, so the contract needs a way to make them comparable.

That tool is the conversion factor (CF). It is the price of the delivered bond per $1 of par if it yielded 6% a year with semiannual compounding. The bond's maturity is rounded down to the nearest three months for this calculation. A bond with a coupon above 6% has a CF above 1. A bond with a coupon below 6% has a CF below 1. The short receives: (latest settlement quote × CF) + accrued interest.

The short wants to give up the bond that costs least relative to what it receives. That is the cheapest-to-deliver (CTD) bond. It minimises: quoted bond price − (futures quote × CF). As a rule of thumb, when yields are above 6% the CTD tends to be a low-coupon, long-maturity bond. When yields are below 6% it tends to be a high-coupon, short-maturity bond. The short also holds timing and quality options. These options have value, so they tend to push the futures price down.

Eurodollar futures were based on the 3-month USD LIBOR deposit rate. The quote is 100 minus the rate in percent, so a quote of 96.00 means 4.00%. The contract is for $1 million of notional and is cash settled. A 1 bp move in the rate changes the contract value by $25. Because LIBOR has ceased, 3-month SOFR futures replace them. They use the same $25 per bp convention, but they settle on the compounded overnight SOFR rate over the reference quarter. SOFR is a secured, nearly risk-free rate. LIBOR included bank credit risk.

Futures are marked to market daily, but forwards settle once at the end. A long futures position gains when rates fall, which is when the gains can be reinvested at lower rates. It loses when rates rise, which is when the losses must be funded at higher rates. This makes the futures rate higher than the equivalent forward rate. The convexity adjustment corrects for the difference. It grows with volatility and with the square of maturity, roughly.

Key formulas to remember

Cash received by the short (Treasury futures): Cash = (Futures quote × CF) + Accrued interest; Per $100 face value. Multiply by 1,000 for one $100,000 contract.
Conversion factor: CF = Price per $1 par of the bond at a 6% yield, semiannual compounding; Maturity rounded down to the nearest 3 months. Coupon above 6% gives CF above 1.
Cheapest-to-deliver: Delivery cost = Quoted bond price − (Futures quote × CF); The bond with the lowest delivery cost is the CTD. Use the quoted price, not the cash price.
Theoretical Treasury futures quote: F quote = [(S − I) × e^(rT) − Accrued at delivery] ÷ CF; S is the CTD cash price. I is the present value of coupons paid before delivery. r is the risk-free rate and T is the time to delivery.
Eurodollar (and 3-month SOFR) futures quote: Quote = 100 − Rate (in %); Rate is the 3-month annualised rate. A quote of 96.00 means 4.00%.
Eurodollar contract value: Value = $10,000 × [100 − 0.25 × (100 − Quote)]; 1 bp = $25 on $1 million notional. A 0.01 change in the quote is worth $25.
Convexity adjustment: Forward rate = Futures rate − ½ × σ² × T1 × T2; σ is the annual standard deviation of the short rate change. T1 is the futures maturity and T2 is the maturity of the underlying rate, both in years. Use decimals, e.g. 0.012 for 1.2%.

How to solve Treasury Bond Futures and Eurodollar Futures questions

Use this sequence for any question on bond futures or short-term rate futures.

1Identify the contract: Treasury bond futures (delivery, CF, CTD) or Eurodollar/SOFR futures (quote = 100 − rate).
2For Treasury futures, list each deliverable bond with its quoted price, CF and accrued interest if given.
3Compute quoted price − (futures quote × CF) for each bond. The lowest result is the CTD.
4To find the cash the short receives, compute futures quote × CF + accrued interest for the chosen bond. Scale to the $100,000 face value if asked.
5For Eurodollar or SOFR futures, convert the quote to a rate: Rate = 100 − Quote. Use $25 per bp for P&L.
6If the question asks for a forward rate, subtract ½ × σ² × T1 × T2 from the futures rate. Keep units consistent and in years and decimals.
7Check the sign and size of the result. The forward rate must be below the futures rate. The CF must be above 1 for coupons above 6%.

Quickest way: Fast CTD and tick-value shortcuts

When to use it: Use when the exam gives a small table of bonds or asks for a P&L on a rate futures position.

CTD: for each bond, multiply the futures quote by the CF and subtract that from the bond price. Pick the smallest value. Do not compute anything else.
Compare bonds by the gap alone. Ignore accrued interest, because the short receives it in either case.
Rate futures P&L: change in quote × 100 × $25 × number of contracts. A 0.01 quote change is 1 bp. A long position gains when the quote rises.
Convexity: compute 0.5 × σ² × T1 × T2 and convert to bp by multiplying by 10,000. Subtract from the futures rate.
Sanity check: if the adjustment is more than a few bp for ordinary inputs, recheck that σ was converted to a decimal.

Common mistakes in Treasury Bond Futures and Eurodollar Futures

Choosing the CTD by the lowest bond price.
It feels natural to deliver the cheapest-looking bond, but the short is paid on the CF-adjusted futures price.
Fix: Always compute quoted price − (futures quote × CF) and pick the smallest result.
Using the cash price instead of the quoted price in the CTD test.
Accrued interest is easy to include by habit.
Fix: Accrued interest is paid to the short on delivery whichever bond is chosen. Use quoted prices in the CTD comparison.
Forgetting to multiply the futures quote by the CF when computing cash received.
Students treat the futures quote as the price of the delivered bond.
Fix: Write Cash = quote × CF + accrued interest every time, and scale to $100,000 face value.
Adding the convexity adjustment to the futures rate instead of subtracting it.
The adjustment feels like a correction that should add to the number.
Fix: Remember that the futures rate is higher than the forward rate. Forward = futures − ½σ²T1T2.
Using σ as a percentage, such as 1.2, in the convexity formula.
The inputs are given as percentages and the formula is applied directly.
Fix: Convert σ to a decimal (0.012) before squaring. The result is also a decimal, so multiply by 10,000 to get bp.
Treating SOFR futures the same as Eurodollar futures in every respect.
Both quote 100 minus a rate and have $25 per bp.
Fix: Eurodollar futures referenced LIBOR, which carried bank credit risk. 3-month SOFR futures settle on compounded overnight SOFR, a secured, nearly risk-free rate.

Worked examples

Example 1

The June Treasury bond futures quote is 103.50. Two bonds are deliverable. Bond A has a quoted price of 125.00 and a CF of 1.2000. Bond B has a quoted price of 140.00 and a CF of 1.3500. Which bond is the CTD, and what cash (excluding accrued interest) does the short receive per $100,000 face value if Bond B is delivered?

Show the solution

Delivery cost for A = 125.00 − (103.50 × 1.2000) = 125.00 − 124.20 = 0.80.
Delivery cost for B = 140.00 − (103.50 × 1.3500) = 140.00 − 139.725 = 0.275.
B has the lower delivery cost, so B is the CTD.
Cash per $100 face for B = 103.50 × 1.3500 = 139.725, excluding accrued interest.
Scale to $100,000 face value: 139.725 × 1,000 = $139,725.

Answer: Bond B is the CTD. The short receives $139,725 per contract, plus accrued interest.

Example 2

A Eurodollar futures contract maturing in 2 years is quoted at 96.00. The underlying 3-month rate matures at 2.25 years. The annual standard deviation of the short rate change is 1.2%. Estimate the forward rate using the convexity adjustment. Also find the dollar loss to a long position if the quote falls to 95.90.

Show the solution

Futures rate = 100 − 96.00 = 4.00%.
Convert σ to a decimal: σ = 0.012, so σ² = 0.000144.
Adjustment = 0.5 × 0.000144 × 2 × 2.25 = 0.5 × 0.000144 × 4.5 = 0.000324.
In percentage terms the adjustment is 0.0324%, or 3.24 bp.
Forward rate = 4.00% − 0.0324% = 3.9676%.
Quote change = 96.00 − 95.90 = 0.10, which is 10 bp. Loss for a long position = 10 × $25 = $250 per contract.

Answer: The forward rate is about 3.9676%. The long position loses $250 per contract.

Exam tips

Questions on CTD usually give a small table of bonds. Practise computing quoted price − (quote × CF) quickly and in the same order each time.
Know the direction of each effect: convexity adjustment lowers the forward rate relative to the futures rate, and delivery options lower the futures price.
Memorise the rate-futures conventions: quote = 100 − rate, and 1 bp = $25 on a $1 million 3-month contract. These appear in P&L and hedging questions.
Expect a conceptual question on why LIBOR-based Eurodollar futures were replaced by SOFR futures. The key points are credit risk in LIBOR and the secured nature of SOFR.
Read the units in convexity questions. σ and times must be in decimals and years, and the answer is often wanted in basis points.

Practice questions from Pricing Financial Forwards and Futures

Treasury Bond Futures and Eurodollar Futures: frequently asked questions

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

For each deliverable bond, compute the quoted bond price minus the futures quote times its conversion factor. The bond with the smallest result is the cheapest to deliver. It costs the short the least relative to the cash received.

What is a conversion factor in Treasury futures?

It is the price per $1 of par of the deliverable bond if it yielded 6% with semiannual compounding, with maturity rounded down to the nearest three months. It adjusts the futures price so that bonds with different coupons and maturities can be delivered fairly. Coupons above 6% give a CF above 1.

Why is there a convexity adjustment for Eurodollar futures?

Futures are settled daily, while forward rate agreements settle once at the end. A long futures position gains when rates fall and loses when they rise, which makes the futures rate higher than the forward rate. You subtract ½σ²T1T2 from the futures rate to get the forward rate.

How are SOFR futures different from Eurodollar futures?

Both use a quote of 100 minus the rate and have $25 per bp for 3-month contracts. Eurodollar futures were based on LIBOR, which included bank credit risk. 3-month SOFR futures settle on the compounded overnight SOFR rate, which is a secured, nearly risk-free rate.