Hedge Ratio

The hedge ratio measures how much of an investment is protected by a hedge relative to the total position. It is used to quantify exposure management for assets, currencies, commodities, or other positions where an offsetting instrument — such as futures or options — is used to reduce risk.

What the hedge ratio means

• Hedge ratio = hedged position / total position.
• A hedge ratio of 0.5 means half the position is hedged; 1.0 means fully hedged.
• In cross-hedging or futures hedging, the hedge ratio compares the size or value of futures contracts to the size or value of the cash position.

How to calculate a straightforward hedge ratio

Divide the value (or notional) covered by the hedge by the total value of the exposure.

Example:
– $5,000 hedged on a $10,000 position → hedge ratio = 5,000 / 10,000 = 0.5.

Minimum variance (optimal) hedge ratio

The minimum variance hedge ratio (also called the optimal hedge ratio) identifies the number of futures contracts that minimizes the variance of the hedged position. It is especially useful in cross-hedging when the hedge instrument and the underlying are not identical.

Formula:
h* = ρ × (σ_spot / σ_futures)

Where:
– h* = minimum variance hedge ratio
– ρ = correlation coefficient between spot and futures price changes
– σ_spot = standard deviation of spot price changes
– σ_futures = standard deviation of futures price changes

To convert the hedge ratio into a number of futures contracts:
N = (h* × Q_spot) / Q_contract

Where:
– N = number of futures contracts
– Q_spot = total quantity (or notional) of the spot exposure
– Q_contract = quantity represented by one futures contract

Note: h* can be between 0 and 1 for partial hedges, equal 1 for a one-for-one hedge, or exceed 1 when the optimal hedge implies using more futures than the nominal units of the spot exposure.

Example: airline hedging jet fuel

An airline expects to buy 15,000,000 gallons of jet fuel over the next year and wants to hedge fuel-price risk using a correlated crude oil futures contract.

Given:
– Correlation (ρ) = 0.95
– σ_spot (jet fuel) = 3%
– σ_futures (crude oil) = 6%
– One futures contract covers 42,000 gallons (e.g., 1,000 barrels = 42,000 gallons)

Calculate h:
h = 0.95 × (3% / 6%) = 0.95 × 0.5 = 0.475

Calculate number of contracts:
N = (0.475 × 15,000,000) / 42,000 ≈ 170 contracts

The airline would buy about 170 crude oil futures contracts to minimize variance in its fuel-cost exposure.

Practical considerations and limitations

• Basis risk: The hedge instrument may not move perfectly with the exposure (price relationship can change).
• Liquidity and contract size: Contract sizes may force rounding and partial hedges.
• Transaction and carrying costs: Margins, commissions, and financing affect net outcomes.
• Over- or under-hedging: Using a hedge ratio >1 or <1 can be optimal statistically but may not match business needs.
• Rebalancing: Hedge ratios may need updating as volatility, correlation, or exposure changes.

Quick FAQs

• How do you calculate a hedge ratio? Divide the hedged portion by the total position; for an optimal hedge, use h* = ρ × (σ_spot / σ_futures).
• Why is the minimum variance hedge ratio important? It identifies the number of contracts that minimizes the variance of the combined (spot + hedge) position.
• Is the minimum variance hedge ratio called anything else? Yes — it’s often called the optimal hedge ratio.

Key takeaways

• The hedge ratio quantifies the portion of an exposure protected by a hedge.
• The simple hedge ratio is a direct proportion; the minimum variance hedge ratio uses correlation and volatilities to find an optimal hedge.
• Consider basis risk, contract granularity, transaction costs, and changing market dynamics when implementing a hedge.