Derive the theoretical price of a futures contract on a Treasury bond.
The idea
The theoretical price of a Treasury bond futures contract treats the contract as a forward on the cheapest-to-deliver (CTD) bond, which pays coupons. First, take the CTD's cash price today, S, which is its quoted price plus accrued interest. Second, subtract the present value of the coupons paid before delivery, each discounted at the risk-free rate r over its time t: I = Σ coupon × e^(−rt). Third, compound what is left forward to the delivery date, at time T: Cash futures price = (S − I) × e^(rT). Fourth, subtract the accrued interest at delivery to get a quoted bond price, then divide by the CTD's conversion factor: Quoted futures price = (cash futures price − accrued at delivery) ÷ CF. Typical errors are starting from the quoted rather than the cash price, adding the coupon PV instead of subtracting it, forgetting the accrued interest at delivery and multiplying by CF instead of dividing. If the market futures price is above the theoretical value, sell futures and buy the CTD bond to deliver.
Formulas
Cash futures price of a Treasury bond futures contract
cash futures price = (S − I) × e^(r × T)
- S
- the cheapest-to-deliver bond's cash price, quoted plus accrued
- I
- the present value of coupons paid before delivery
- r
- the continuously compounded risk-free rate
- T
- the time to delivery, in years
Start from the cash price, not the quote, and subtract the coupon value. Then subtract accrued interest at delivery and divide by CF to get the quoted futures price.
Quoted Treasury futures price
quoted futures price = (cash futures price − accrued at delivery) ÷ CF
- cash futures price
- (S − I) × e^(r × T) for the cheapest-to-deliver bond
- accrued at delivery
- accrued interest on that bond at the delivery date
- CF
- its conversion factor
Divide by the conversion factor, never multiply. If the market price is above this, sell futures and buy the bond to deliver.
A practice question
Which expression gives the cash futures price implied by the cheapest-to-deliver bond, where S is its cash price, I the present value of coupons paid during the life of the contract, r the risk-free rate (continuous) and T the time to delivery?
- (S + I) x e^(rT)Why not: Adds the coupon PV; the holder of the futures does not receive the coupons, so they are subtracted.
- (S - I) x e^(-rT)Why not: Discounts instead of compounding; the futures price is a forward value at delivery.
- (S - I) x e^(rT) - correct
- S x e^(rT) - IWhy not: Subtracts the coupon PV without compounding it; the coupon income must be removed from the spot price before compounding.
Buying the bond and holding to delivery earns coupons whose PV is I. The futures price is the cost of carrying the bond net of those coupons: (S - I) e^(rT).
A harder one
The cheapest-to-deliver bond has a cash price of 123.00 and pays a coupon of 5.00 in four months. Delivery is in nine months and the risk-free rate is 6% (continuous, flat). The cash futures price is closest to:
- 123.53 - correct
- 128.66Why not: Ignores the coupon paid during the life of the contract.
- 133.79Why not: Adds the PV of the coupon instead of subtracting it.
- 123.76Why not: Subtracts the PV of the coupon after compounding the full price; the coupon PV must be removed before compounding.
I = 5 x e^(-0.06 x 4/12) = 4.9010. F = (123 - 4.9010) x e^(0.06 x 0.75) = 123.53.
The hardest tier
The theoretical quoted price of a Treasury bond futures contract is 98.60 and the market price is 99.45. The cheapest-to-deliver bond has a conversion factor of 1.1500 and the contract covers USD 100,000 face value. Which arbitrage is appropriate and what is the approximate profit per contract at delivery, ignoring financing mismatch?
- Buy futures and sell the bond: about USD 978Why not: Wrong direction: the futures are overpriced relative to the bond, so the arbitrageur sells futures.
- Sell futures and buy the bond: about USD 850Why not: Ignores the conversion factor when converting the contract price difference into the profit on the delivered bond.
- Sell futures and buy the bond: about USD 978 - correct
- Sell futures and buy the bond: about USD 739Why not: Divides by the conversion factor instead of multiplying: the contract price difference is on the standardised scale.
The market price exceeds the theoretical price by 0.85, so sell futures, buy and deliver the bond. Delivery of 100,000 face receives futures price x CF, so profit = 0.85 x 1.15 x 1,000 = 978.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Interest Rate Futures
- Recognise the most widely used day count conventions, the markets in which each normally applies, and how to use each in an interest calculation.
- Work out the price of a US Treasury bill given its discount rate.
- Distinguish a US Treasury bond's clean price from its dirty price, and compute its accrued interest and dirty price.
- Compute the conversion factor of a US Treasury bond futures contract and explain what it represents.
- Determine the expense of handing over a given bond to settle a Treasury bond futures contract.
- Describe how the yield curve's level and shape affect which Treasury bond is cheapest to deliver.
- Create a duration-based hedging strategy in interest rate futures, starting with working out the hedge ratio.
- Explain the limitations of a hedging strategy based on duration.