Cutting Beta with Index Futures - and What the Hedge Does Not Do

Dollar exposure to remove = (1.25 − 0.80) × $60,000,000 = 0.45 × $60M = $27,000,000

Contract notional = 4,200 × $250 = $1,050,000

Contracts = 27,000,000 / 1,050,000 = 25.71 → short 26 contracts (25 if you prefer to round toward the target rather than past it).

Realised beta:

• 26 contracts: 1.25 − (26 × 1,050,000 / 60,000,000) = 1.25 − 0.455 = 0.795
• 25 contracts: 1.25 − 0.4375 = 0.8125

The rounding error is small (0.005 to 0.0125 of beta, i.e. roughly $0.3–0.75 million of market exposure), and with a 12-month horizon the hedge must be rolled at least once, because the contract expires in six months.

Imperfections that matter:

• Beta drift. The portfolio's realised beta is an estimate and moves with the holdings; a static contract count degrades over a year.
• Dividend and rate assumptions. The hedge assumes the index total return is what the futures track; a dividend cut or a rate move changes the correct ratio.
• Basis risk. The portfolio is not the index. Idiosyncratic and style moves are unhedged, and the futures basis (spot vs futures) can move against a roll.
• Margin and liquidity. Short futures require variation margin; in a sharp rally the fund posts cash precisely when it may not want to.
• Mandate floor. Cutting to 0.80 leaves ~80% net long, so the 60% floor is not breached - but a large further hedge would be.

Theoretical futures price (six months, continuous compounding):

F = S × e^((r − q)T) = 4,100 × e^((0.05 − 0.018) × 0.5) = 4,100 × e^(0.016) = 4,100 × 1.016129 = 4,166.1

The market quotes 4,200, i.e. the futures are rich by 33.9 index points (about 0.81%).

Arbitrage (cash-and-carry), per index unit:

1. Sell the futures at 4,200.
2. Borrow 4,100 at 5.0% for six months and buy the index basket (or a cheap index ETF).
3. Hold for six months, collecting dividends at 1.8%.
4. Deliver into the short futures at 4,200.

Terminal accounting per index unit:

• Basket value at delivery = 4,200 (sold forward)
• Dividends accumulated = 4,100 × (e^(0.018×0.5) − 1) ≈ 4,100 × 0.009041 = 37.07
• Loan repayment = 4,100 × e^(0.05×0.5) = 4,100 × 1.025315 = 4,203.79
• Net = 4,200 + 37.07 − 4,203.79 = +33.28 per index unit

Per contract: 33.28 × $250 ≈ $8,320 (≈ 0.81% of the $1,050,000 notional) before transaction costs, financing spread, and the tracking error of the substitute basket.

Caveat for the hedger. A rich futures price is good for the manager who is short futures: the hedge is being placed at a favourable level versus fair value, which adds a small positive carry. It is not free money unless the arbitrage is executable - for a large fund, index substitution (ETF tracking error, borrow costs, dividend timing) can consume most of an 81 bp edge.

The three tools solve different problems:

Tool	Removes downside?	Keeps upside?	Cost	Beta outcome
Short futures	Yes, linearly	No - gives up upside 1:1	Margin, roll, basis	Deterministic
Protective put	Yes, below strike	Yes, above strike	Premium (a real cash outflow)	Non-linear
Collar (buy put, sell call)	Yes, below put strike	Only up to the call strike	Near zero, but capped upside	Non-linear

Fit to the stated objectives:

• The committee wants protection against a 15% drawdown and participation in further upside. Short futures fail the second condition: they convert the position into a near-synthetic bond. If the index rises 10%, the hedge loses roughly $27M × 10% ≈ $2.7M of the gain.
• A put gives exactly the described payoff shape. The manager must then decide whether the premium is worth it - with implied volatility at the time of purchase the dominant input, and a 12-month horizon that requires a longer-dated option or a rolling programme.
• A collar finances the put but caps upside, which could conflict with a mandate that emphasises participation; it also introduces the risk that the written call is exercised in a sharp rally.

Mandate constraint. Every structure must keep the fund at least 60% net long. Short futures at 26 contracts leave ~80% net long - fine. A collar that sells calls is still long equities and is fine. A large outright put plus futures could breach the floor and must be sized carefully.

Recommendation. If the committee genuinely wants convexity, use a put (or a collar sized so the cap is well above the fund's target return); accept that it costs money and that the cost is not a hedging 'loss'. If the objective is to engineer a beta of 0.80, futures are the correct and cheapest instrument, and the committee should be told plainly that the upside they are giving up is the price of that certainty.