10.4 Using Futures for Hedging
Key Takeaways
- Short hedges protect long underlying exposures; long hedges protect prospective purchases or short underlying positions
- Arguments for hedging include risk reduction and focus on core business; arguments against include basis risk, cost, and shareholders’ ability to diversify
- Basis risk arises when the futures hedging instrument imperfectly matches the hedged spot exposure in grade, location, or timing
- Cross hedges use related underlyings; optimal hedge ratios use spot-futures volatility and correlation; effectiveness is the variance reduction achieved
- Tailing the hedge adjusts contract count for daily settlement present-value effects; stock-index futures can change portfolio beta; stack-and-roll hedges face calendar and liquidity risks
Using Futures for Hedging
FMP–8 is where futures become a risk tool. The exam tests whether you can choose long vs short, quantify basis risk, compute a hedge ratio and number of contracts, explain tailing, and adjust equity beta with index futures—without confusing hedging with speculation.
Long and Short Hedges
| Hedge type | Futures position | Typical exposure being hedged |
|---|---|---|
| Short hedge | Short futures | Long the asset (inventory, receivables in asset terms, anticipated sale) |
| Long hedge | Long futures | Short the asset or planning to buy (refiner needing crude; portfolio manager expecting inflows) |
Short hedge example: A copper producer will sell copper in three months. Short copper futures locks in a selling price path (subject to basis). If spot copper falls, futures gains offset; if spot rises, futures losses offset higher cash-market sale proceeds.
Long hedge example: An airline will buy jet fuel in six months. Long heating-oil or jet-fuel-linked futures (a cross hedge if the contract is imperfect) offsets price increases.
Arguments For and Against Hedging
For hedging
- Reduces earnings and cash-flow volatility
- Lets management focus on operational comparative advantage
- May lower financial distress costs and improve debt capacity
- Protects solvency through price shocks
Against / limits
- Shareholders can diversify commodity or market risk themselves
- Hedging has costs (bid–ask, margin funding, basis risk)
- Competitors who do not hedge may gain when prices move favorably
- Agency issues: managers may hedge to protect jobs/bonuses rather than shareholder value
- Accounting and disclosure complexity
FRM answers should match the vignette’s corporate-finance framing: distress costs and comparative advantage support hedging; pure Modigliani–Miller frictionless arguments weaken the case.
Basis and Basis Risk
Define basis = Spot price of asset to be hedged − Futures price used (confirm sign convention in the stem).
Strengthening basis: spot rises relative to futures (basis increases under this definition). Weakening basis: spot falls relative to futures.
For a short hedger, a strengthening basis helps; a weakening basis hurts (classic textbook result under Spot − Futures). Always re-derive with numbers if the definition is flipped.
Basis risk is uncertainty in the basis at hedge lift. Sources:
- Asset mismatch (hedge asset ≠ futures underlying)—cross hedge
- Location mismatch
- Timing mismatch (hedge horizon ≠ futures expiry)—may require rolling
- Grade / quality mismatch
Worked basis P&L (short hedge)
Spot today S0 = $4.00; futures F0 = $4.20; basis = −0.20. Short futures. At hedge end, S1 = $3.70; F1 = $3.75; basis = −0.05.
Cash-market loss on inventory = 3.70 − 4.00 = −$0.30. Futures gain (short) = 4.20 − 3.75 = +$0.45. Net = +$0.15 per unit.
Equivalently, net selling price ≈ F0 + final basis = 4.20 + (−0.05) = $4.15, versus initial spot $4.00—the change in basis drove the residual.
Cross Hedge, Hedge Ratio, and Effectiveness
When the available futures underlying differs from the spot exposure, use a cross hedge and estimate the optimal hedge ratio.
Minimum-variance hedge ratio:
h* = ρ × (σ_S / σ_F)
where σ_S and σ_F are standard deviations of spot and futures price changes (or returns—be consistent with the regression used), and ρ is their correlation.
Number of futures contracts:
N* = h* × (Value of position / Value of one futures contract)
Sometimes written with β from regressing spot changes on futures changes: h* = β.
Hedge effectiveness ≈ R² of that regression, or equivalently the fraction of spot variance eliminated: effectiveness = ρ² in the simple one-factor case when using the optimal h*.
Worked hedge-ratio example
Airline exposure: 2,000,000 gallons of jet fuel. Heating oil futures: 42,000 gallons per contract. σ_jet = 0.028, σ_HO = 0.025, ρ = 0.80 (monthly changes).
h* = 0.80 × (0.028 / 0.025) = 0.80 × 1.12 = 0.896.
N* = 0.896 × (2,000,000 / 42,000) ≈ 0.896 × 47.62 ≈ 42.7 → 43 contracts long (fuel buyer).
If ρ were only 0.40, h* halves in the correlation term and effectiveness ρ² falls from 0.64 to 0.16—cross hedging becomes much noisier.
| Metric | Formula / meaning |
|---|---|
| h* | ρ σ_S / σ_F |
| N* | h* × (position size / contract size) |
| Effectiveness | ≈ ρ² (optimal hedge, simple setting) |
P&L of the Hedged Position
Net P&L ≈ change in spot position + futures P&L (sign by long/short) − costs.
Futures P&L per contract ≈ multiplier × (exit futures − entry futures) for longs; opposite for shorts. Stack this with inventory or anticipated purchase P&L.
Combined numeric
Long inventory of 100 oz gold bought at $2,000. Short 1 gold futures (100 oz) at $2,020. Later spot $1,950; futures $1,960.
Inventory: −$50 × 100 = −$5,000. Short futures: (2,020 − 1,960) × 100 = +$6,000. Net ≈ +$1,000 (basis moved favorably for the short hedger).
Optimal Contracts and Tailing the Hedge
Daily settlement means futures gains/losses arrive early relative to a forward’s terminal settlement. Tailing the hedge reduces the futures position roughly by a present-value factor so that the future value of VM matches the exposure at the hedge horizon.
Stylized tail:
N_tailed ≈ N* × e^(−rT)
(or divide by (1+r)^T in discrete time). For short horizons and low rates, tailing is a small adjustment; for long-dated hedges with material rates, exam questions may expect the e^(−rT) haircut.
Worked tail
Optimal untailed N* = 100 contracts; r = 4% continuous; T = 0.75 years.
N_tailed = 100 × e^(−0.04 × 0.75) = 100 × e^(−0.03) ≈ 100 × 0.9704 = 97 contracts.
Stock-Index Futures to Change Beta
To adjust a portfolio’s beta from β_current to β_target:
N = (β_target − β_current) × (V_portfolio / V_futures)
Sign: if target < current, N is negative → short index futures to reduce beta. If target > current, long futures to increase beta.
Worked beta overlay
Portfolio value $50 million; β = 1.20. Futures on index at 4,000 with multiplier $50 → one contract notional = 4,000 × 50 = $200,000. Target β = 0.80.
N = (0.80 − 1.20) × (50,000,000 / 200,000) = (−0.40) × 250 = −100 → short 100 contracts.
To create a temporary synthetic cash position (β ≈ 0), short β × (V_p / V_f) contracts.
Stack-and-Roll Risks
When hedge horizon exceeds liquid nearby contract maturity, hedgers stack positions in the nearby (or a strip) and roll into later contracts as expiry approaches.
Risks:
- Calendar basis / roll risk: spreads between expiry months move against the hedger (contango roll-down for longs; reverse for shorts—context matters)
- Liquidity risk in deferred months
- Gap risk around roll dates
- Margin liquidity across repeated rolls
- Volume/open interest cliffs in far contracts
Metallgesellschaft-style disasters are the teaching story: stacked short-dated energy hedges against long-dated customer exposures suffered when nearby prices and strip relationships moved and margin calls drained liquidity—even if “economic” long-run hedge logic seemed sound.
| Strategy piece | Risk to watch |
|---|---|
| Stack in nearby | Calendar basis vs long-haul exposure |
| Roll forward | Slippage; spread jumps |
| Cross-commodity stack | Correlation breakdown |
| Levered margin | Funding squeeze in stress |
Synthesis
Pick the hedge direction from the cash exposure. Measure residual basis risk. Size with h* and N*. Tail when daily settlement PV matters. Use index futures as a beta dial. Treat stack-and-roll as a liquidity and calendar-basis strategy, not a perfect substitute for a matching-maturity forward. That is the FMP–8 toolkit.
A portfolio manager who will receive a large cash inflow in two months and fears equities will rise before the money can be invested should most likely:
σ_S = 0.20, σ_F = 0.25, ρ = 0.60. The minimum-variance hedge ratio h* is closest to:
A $40m equity portfolio with β = 1.1 should be hedged to β = 0 using index futures with $200,000 notional per contract. The appropriate trade is closest to:
Stack-and-roll hedging primarily introduces which incremental risk relative to a single matching-maturity futures hedge?