Interest Rate Futures: formula sheet

Key formulas

Accrued interest (actual/actual)

Accrued = (Coupon per period) × (Days since last coupon ÷ Days in coupon period)

Both day counts are actual calendar days. Coupon per period is the annual coupon ÷ 2 for semiannual bonds.

30/360 day count

Days = 360 × (Y2 − Y1) + 30 × (M2 − M1) + (D2 − D1)

Adjust first: if D1 = 31, set D1 = 30. If D2 = 31 and D1 is 30 or 31, set D2 = 30. Accrual fraction = days ÷ 360 for a year, or days ÷ 180 for a semiannual period.

Actual/360 accrual

Interest = Rate × (Actual days ÷ 360) × Principal

Gives slightly more interest than actual/365 for the same stated rate.

Cash (dirty) price

Cash price = Quoted (clean) price + Accrued interest

Quoted prices exclude accrued interest.

32nds quote

Price = whole dollars + (32nds ÷ 32)

Example: 102-08 = 102 + 8/32 = 102.25. One 32nd = 0.03125 per $100.

T-bill price from discount rate

P = 100 − Y × (n ÷ 360)

P per $100 face, Y is the quoted discount rate in percent, n is days to maturity.

Bond equivalent yield of a T-bill

BEY = [(100 − P) ÷ P] × (365 ÷ n)

Uses the actual price paid as the base and a 365-day year, so it exceeds the discount rate.

Cash received by short

Cash = (Quoted futures price × Conversion factor) + Accrued interest

Per $100 face value. The long pays this invoice amount.

Cost of delivery

Cost = Quoted bond price − (Quoted futures price × Conversion factor)

Choose the bond with the lowest cost as CTD. Ignore accrued interest, which cancels.

Conversion factor

CF = price of the bond at a 6% yield (semiannual), per $1 face, using maturity rounded down to the nearest 3 months

Coupon above 6% gives CF above 1. Coupon below 6% gives CF below 1.

Quoted futures price from CTD

Quoted futures price = (Forward cash price of CTD − Accrued interest at delivery) ÷ CF

Forward cash price = (Spot cash price − PV of coupons) × e^(rT), under continuous compounding.

Accrued interest

Accrued = Coupon ÷ 2 × (days since last coupon ÷ days in coupon period)

Treasury bonds use actual/actual day count.

Futures quote

Quote = 100 − annualized rate (in %)

Rate = 100 − Quote. A higher quote means a lower rate.

Eurodollar contract value

Price = 10,000 × [100 − 0.25 × (100 − Quote)]

For USD 1 million, 3-month contract. The 0.25 is the accrual fraction.

Value of one basis point

USD 1,000,000 × 0.0001 × 0.25 = USD 25

Daily gain or loss for a long = 25 × change in quote in basis points.

Convexity adjustment

Forward rate = Futures rate − ½ × σ² × T₁ × T₂

Use decimals for rates, σ and years. Futures rate is the higher of the two.

Implied forward rate

Forward rate = (1/τ) × (Z(T₁)/Z(T₂) − 1)

Simple rate over period τ from T₁ to T₂, using discount factors Z. Compare with the adjusted futures rate.

Number of futures contracts to hedge

N* = −(P × D_P) ÷ (F × D_F)

P = portfolio value, D_P = portfolio duration, F = contract price (value of one contract), D_F = duration of the futures underlying (cheapest-to-deliver). Negative means short.

DV01 form

N* = −DV01_P ÷ DV01_F

DV01 is the value change for a one basis point yield change. Use the same sign convention and units on both sides.

Change duration to a target

N = (D_T − D_P) × P ÷ (F × D_F)

D_T is the target duration. Positive N means buy futures. Negative N means sell.

Duration price approximation

ΔP ≈ −D × P × Δy

D is modified duration when Δy is the change in the periodic yield. Valid for small changes.

Futures contract value (U.S. Treasury bond)

F = quoted price ÷ 100 × ₹ or $ contract size

A US T-bond contract has a face value of $100,000, so a quote of 120 means F = $120,000. Convert 32nds carefully.

Quick revision

• Treasury notes and bonds use actual/actual; corporate and municipal bonds use 30/360 (180 days in a half-year coupon period); money market instruments use actual/360.
• Accrued interest = coupon per period × days since last coupon ÷ days in the coupon period, with days counted under the convention: actual days for Treasuries, 30/360 days (180 in a half-year period) for corporates and municipals.
• Treasury futures are quoted in points and 32nds of a point, so 110-16 means 110 and 16/32.
• Cash received by the short = (quoted futures price × conversion factor) + accrued interest.
• The short chooses the cheapest-to-deliver bond: compute quoted bond price − (futures quote × conversion factor) for each deliverable bond and pick the one with the smallest value.
• Short-term rate futures such as SOFR futures (formerly Eurodollar) are quoted as 100 − rate, so a higher price means a lower rate.
• When rates rise, a long position in rate futures loses and a short gains.
• The full-hedge ratio is N = −(D_P × V_P) ÷ (D_F × F), where the sign is set by the direction of risk and a negative N means short. To move to a target duration, use (D_target − D_P) in place of D_P, with the sign the same way.
• To hedge a bond portfolio against rising rates, you short futures; a party who will invest later and fears falling rates goes long.
• Duration hedging protects only against small parallel shifts in the yield curve, and basis risk remains.
• DV01 matching gives N = DV01 of portfolio ÷ DV01 of one futures contract.
• Sense check every answer: direction of the trade, sign of the P&L and size of the number.

Common mistakes

• Using 365 days in the denominator for Treasury accrued interest. Fix: Divide by the actual number of days in the coupon period (about 181 to 184 for a semiannual bond), not the year.
• Reading 95-16 as 95.16. Fix: The digits after the hyphen are 32nds. 95-16 = 95 + 16/32 = 95.50.
• Adding accrued interest when comparing bonds for the CTD. Fix: Accrued interest is paid by the long on whichever bond is delivered, so it cancels. Compare quoted price − futures price × CF.
• Choosing the CTD as the bond with the lowest quoted price. Fix: Cheapest means lowest net cost after the CF-adjusted futures receipt. Always do the subtraction.
• Using the quote as the interest rate. Fix: Always subtract from 100 before doing any rate work.
• Forgetting the 0.25 accrual factor in the contract value. Fix: The rate applies for three months, so multiply by 0.25. That gives USD 25 per basis point.
• Using the wrong sign and buying futures to hedge a long bond portfolio Fix: Ask who loses when rates rise. A long bond holder loses, so you need a position that gains: short futures.
• Using the quoted futures price instead of the contract value Fix: Multiply the quote ÷ 100 by the contract face value. A quote of 120 on a $100,000 contract is $120,000.

Exam tips

• Match the convention to the instrument first. Many questions are solved by this alone.
• Read whether the question asks for the clean (quoted) or cash (dirty) price. Distractor options often include the wrong one.
• Do the 32nds conversion before anything else. Mistakes here carry through the whole answer.
• A financial calculator handles the arithmetic, but it will not count days for you. Count the days by hand and check both day totals.
• Expect the T-bill discount rate versus yield contrast. The bond equivalent yield is always above the discount rate.
• Questions often hand you a table of bonds. Do the subtraction for each and pick the minimum.
• Know which direction the delivery options move the futures price: down, because they favour the short.
• Remember that the wild card option is about timing after the market close, and the quality option is about the choice of bond.