Hedging the market with index futures
A long book carries market beta. Short index futures to remove it, or to keep just the amount you want.
- Calculate a book's beta-dollars and what a market move does to it
- Work out how many index futures to short for a target beta
- Split a market move into the book, the futures and the hedged result
- Back out the beta a hedge assumed
The intuition
A farmer grows wheat and worries that the price will fall before harvest. Agreeing today to sell the crop at a fixed price removes that worry: whatever wheat prices do, the farm is paid for growing wheat, not for guessing prices.
A stock picker's long book has the same problem: its beta means it rises and falls with the market. Shorting index futures removes that. Each contract carries a fixed amount of market exposure, the index level × the contract's multiplier, so the number of contracts is the beta-dollars to remove divided by the exposure per contract.
Beta-dollars = beta × book value. Contract notional = index level × multiplier. Contracts to short = (beta − target beta) × book value ÷ contract notional. After a market move m, the hedged book gains or loses about target beta × m × book value.
Why it works
- The conventions here: one index future with a $50 multiplier. Contracts are rounded to whole contracts only when traded, which leaves a small leftover beta.
- Beta-dollars say how much money is on the line. A beta of 1.2 on a $100M book and on a $2B book is the same number and a very different risk.
- Hedge to zero and the P&L is stock selection alone; hedge to a target and you keep a chosen amount of market.
- Futures are the cheap way to hedge: they need only margin, not the full notional, and there is no borrow fee. They have to be rolled every quarter.
- The hedge is only as good as the beta estimate. Betas drift and positions change, so a hedge is re-struck, not set and forgotten.
- An index hedge removes the market, not everything. Sector, size and style exposures stay: a book of small-cap value names hedged with a large-cap future is still a factor bet.
| Contract notional: 5,000 × $50 | $250,000 |
| Beta-dollars to shed: (1.2 − 0.2) × 500 | $500M |
| Contracts to short: 500 ÷ 0.25 | 2,000 |
| Unhedged P&L: 1.2 × −5% × 500 | −$30M |
| Futures P&L: −2,000 × 0.25 × −5% | +$25M |
| Hedged P&L: −30 + 25 | −$5M, the 0.2 beta kept |
If a risk report says 2,000 contracts leave this book at a beta of 0.2, the beta it assumed is 0.2 + 2,000 × 0.25 ÷ 500 = 1.2.
The formulas
Market exposure in dollars.
Market exposure per contract.
The beta-dollars to shed, in contracts.
Only the beta you kept moves.
Run it backwards.
Worked example
One contract's notional first, then the beta-dollars to shed, then divide.
See it move
Same book and the same index level. Change the book's size and beta, the beta you want to keep, and the market move.
- Raise the book's beta. More contracts are needed.
- Raise the beta to keep. Fewer contracts are needed.
- Raise the book value. More contracts are needed.
- Move the market up. The book gains, the futures lose, and the hedged book moves only by the beta kept, plus a small rounding residual.
Run it backwards
Same book, reversed: a hedge of a known size is said to leave a target beta. What beta was the risk system assuming for the book?
Each contract removes notional ÷ book value of beta. Add the beta the futures removed back on to the target.
If that assumed beta is out of date, so is the hedge: positions change, prices move the weights and betas drift, which is why large books re-strike their hedges every day.
Traps
Say it in the interview
“You run a long book of stock picks and want to be paid only for the picks. What do you do?”
Check yourself
4 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.
- Beta-dollars = beta × book value.
- Contracts = (beta − target) × book ÷ (index × multiplier).
- Hedged P&L ≈ target beta × move × book.
- A beta hedge leaves sector and style bets in place.