Bond futures: the basis, the implied repo and the cheapest to deliver
A bond future is a promise to deliver a government bond later at a price fixed now. Owning the bond against the future locks in a return, the implied repo rate; the bond with the highest one is the cheapest to deliver, and the future trades off it.
- Compute the invoice amount from the futures price and the conversion factor
- Split the gross basis into carry and net basis
- Compute the implied repo rate of a cash-and-carry, and compare it with actual repo
- Pick the cheapest-to-deliver bond and size a futures hedge off it
The intuition
The short in a bond future can deliver any of several bonds, so the exchange scales each one by a conversion factor to make them roughly comparable: the invoice the short receives is the futures price times that factor, per 100 of face. Roughly comparable is not exactly comparable, so one bond is always a little cheaper to deliver than the rest, and the future behaves like a forward on that bond.
Buy the bond today, sell the future, deliver at expiry. You pay the cash price, collect coupon while you wait, and receive the invoice price. That locks in a return: the implied repo rate. If it beats the rate you can borrow at, the future is rich and the cash-and-carry makes money; if not, the reverse. The gap between the cash price and the invoice price is the gross basis; take out the carry you earn while waiting and the net basis is what is left, which is the price of the short's delivery options. And the bond offering the highest implied repo is the cheapest to deliver.
Invoice = futures × CF. Gross basis = cash − futures × CF. Carry = coupon × days ÷ 365 − cash × repo × days ÷ 365. Net basis = gross − carry. Implied repo = (futures × CF + coupon accrual − cash) ÷ cash × 365 ÷ days. Fair futures = (cash − carry) ÷ CF. Futures DV01 = CTD DV01 ÷ CF, so contracts = portfolio DV01 ÷ that.
Why it works
- The conventions here: a contract on $100,000 face. Invoice price = futures price × conversion factor, accrued interest ignored throughout. Carry to delivery = coupon × days ÷ 365 − cash price × repo × days ÷ 365, simple, 365-day year for both. The cheapest to deliver is the bond with the higher implied repo. Fair futures price = (cash − carry) ÷ CF. Futures DV01 = CTD DV01 per $100,000 ÷ CF. Delivery options are ignored. Cash prices are printed to three decimals and every answer comes from the printed price.
- What the conversion factor is. Roughly the price per 1 of face at which the bond would yield the contract's notional coupon (6% for US Treasury futures). High-coupon bonds get a factor above 1, low-coupon bonds below. It is approximate, which is why a cheapest to deliver exists at all.
- Gross basis is not enough to pick the CTD. Bonds carry differently: a high-coupon bond earns more while you hold it, so it can have a larger gross basis and still be cheaper once carry is counted. Implied repo, or net basis at a common repo rate, puts carry in.
- Why higher repo raises the futures price. The future is a forward. Buying the future instead of the bond saves the financing and gives up the coupon; when financing costs more, that saving is worth more, so the future must cost more to keep the two routes equal.
- Hedging with futures. The future moves like the CTD divided by its conversion factor. Contracts = portfolio DV01 ÷ (CTD DV01 per contract ÷ CF), rounded to a whole number in practice. The hedge breaks if a large yield move changes which bond is cheapest.
- Run it backwards. The implied repo is the inverse of the fair-futures formula: given the cash price, the future and the coupon, what rate did the cash-and-carry lock in? Compare it with actual repo to decide which way to trade.
| Invoice price: 115.00 × 0.8792 | 101.108 per 100 |
| Cash price: 101.108 + 0.50 | 101.608 |
| Coupon income: 4.50 × 60 ÷ 365 | 0.740 |
| Repo cost: 101.608 × 3% × 60 ÷ 365 | 0.501 |
| Carry: 0.740 − 0.501 | 0.239; net basis 0.50 − 0.239 = 0.261 |
| Implied repo: (101.108 + 0.740 − 101.608) ÷ 101.608 × 365 ÷ 60 | 1.44%, below the 3% repo: the future is cheap |
| Fair futures: (101.608 − 0.239) ÷ 0.8792 | 115.297 |
| Hedge: CTD DV01 $80 per $100,000 ÷ 0.8792 = $91.0 a contract; $50,000 ÷ $91.0 | 549.5 contracts |
The formulas
What the short receives for delivering, accrued interest aside.
The bond's richness to the future, before and after the carry earned while waiting.
Coupon earned less financing paid, to delivery.
The annualized return a cash-and-carry locks in.
The forward price of the CTD scaled by its factor, and the hedge built on it.
Worked example
Futures times the conversion factor, the gross basis, the coupon earned and the repo paid to delivery, then carry and net basis. The follow-up asks what a positive net basis means.
See it move
Same contract and deliverable bond. Change the gross basis, the coupon, repo, the days to delivery, and the portfolio being hedged.
- Raise the gross basis. The cash price rises with it, the net basis rises, the implied repo falls and the fair futures price rises.
- Raise the coupon. Carry rises, so net basis falls; the implied repo rises and the fair futures price falls, because the bond pays you more to hold it.
- Raise repo. Carry falls, net basis rises and the fair futures price rises; the implied repo does not move, because it is built from the cash price, the future and the coupon alone.
- Change the days to delivery and watch: carry and the implied repo depend on how long the accrual runs, and the sign of the carry decides which way the net basis moves.
- Raise the portfolio DV01 and the contracts rise in proportion; raise the CTD's DV01 and the contracts fall.
Run it backwards
You buy the bond at its cash price, sell the future, and deliver in a stated number of days. What annualized return does that lock in?
Money in at delivery is the invoice price plus the coupon earned; money out today is the cash price. The return over the period, annualized on a 365-day year, is the implied repo rate.
If it is above the rate you can borrow at, buy the bond in repo and sell the future: the cash-and-carry. If below, the reverse, if you can borrow the bond. Either way the bond with the highest implied repo is the cheapest to deliver.
Traps
Say it in the interview
“What is the cheapest to deliver, and how do you find it?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Invoice = futures × CF. Gross basis = cash − invoice. Net basis = gross basis − carry.
- Implied repo is the return a cash-and-carry locks in; the highest one is the cheapest to deliver.
- Fair futures = (cash − carry) ÷ CF, so higher repo raises the futures price.
- Futures DV01 = CTD DV01 ÷ CF; contracts = portfolio DV01 ÷ futures DV01.