Calculate the minimum-variance hedge ratio, optimal number of futures contracts, hedge effectiveness, and variance reduction with correlation sensitivity analysis.
The optimal hedge ratio determines exactly how many futures contracts you need to minimize the variance of a hedged position. Rather than blindly selling one futures contract per unit of spot exposure, the minimum-variance approach adjusts for the imperfect correlation between spot and futures prices.
The formula h* = ρ × (σ_S / σ_F) considers three critical factors: the correlation between spot and futures price changes, the volatility of the spot asset, and the volatility of the futures contract. When correlation is perfect (ρ = 1) and volatilities are equal, the optimal ratio is 1.0 (a full hedge). In reality, cross-hedging with imperfect correlation requires ratios that differ significantly from 1.0.
This calculator computes the optimal hedge ratio, the number of contracts needed, hedge effectiveness (ρ²), and shows how portfolio volatility changes across different hedge ratios. The correlation sensitivity table reveals how hedge quality deteriorates as the relationship between spot and futures weakens — crucial for cross-hedge decisions.
Over-hedging or under-hedging both cost money. This calculator pinpoints the exact hedge ratio that minimizes portfolio risk, shows how many contracts to trade, and quantifies the basis risk you can't eliminate. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain. Use this clarification to avoid ambiguous interpretation.
h* = ρ × (σ_S / σ_F) Optimal Contracts = Round(h* × Position Qty / Contract Size) Hedge Effectiveness = ρ² Portfolio Variance = σ_S² − 2h·ρ·σ_S·σ_F + h²·σ_F² Basis = Futures Price − Spot Price
Result: Optimal ratio: 0.986, Contracts: ~659, Effectiveness: 84.6%
With 0.92 correlation, 30% spot vol and 28% futures vol, the optimal hedge ratio is 0.986 (near 1:1). Hedge effectiveness of 84.6% means the hedge eliminates 84.6% of variance, with the remaining 15.4% being unhedgeable basis risk.
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The proportion of your spot exposure to hedge with futures that minimizes overall portfolio variance. It equals ρ × (σ_spot / σ_futures).
A 1:1 hedge is only optimal when ρ=1 and σ_S=σ_F. Otherwise, over-hedging increases variance because the futures position adds more risk than it removes.
ρ² — the proportion of variance eliminated by the optimal hedge. A hedge effectiveness of 0.85 means 85% of risk is removed.
The risk that spot and futures prices don't move perfectly together. Basis risk = unhedgeable portion = (1 − ρ²) of total variance.
When no futures contract exists for your specific asset. Cross-hedging uses a correlated futures contract (e.g., heating oil futures for jet fuel).
Calculate the Pearson correlation of historical percentage price changes. Use at least 50 data points. Re-estimate periodically as correlations shift.