Questions: Optimal Hedging Ratios and Hedge Effectiveness

Question 1

An airline holds jet fuel exposure (spot volatility σ_S = 12%) and hedges using crude oil futures (σ_F = 20%). The correlation between jet fuel and crude oil price changes is ρ = 0.90. What is the optimal hedge ratio?

• A. 0.90 — the hedge ratio equals the correlation between the assets
• B. 1.00 — one futures contract per unit of spot exposure is always optimal
• C. 0.54 — calculated as ρ × (σ_S / σ_F) = 0.90 × (12/20)
• D. 1.67 — you need more futures contracts because crude is more volatile than jet fuel

Answer: (C)

The minimum-variance hedge ratio is h* = ρ × (σ_S / σ_F) = 0.90 × (0.12/0.20) = 0.54. Because crude oil futures are more volatile than jet fuel, each futures contract provides more price movement than one unit of spot exposure, so you need fewer contracts per unit hedged. The common mistake (option B) assumes 1:1 is always correct; that only holds when spot and futures have identical volatility AND perfect correlation. Option A confuses correlation with the hedge ratio.

Question 2

A risk manager uses a regression of daily spot price changes on futures price changes and finds an R² of 0.64. What does this tell her about the hedge?

• A. The hedge eliminates 64% of spot price variance — correlation between spot and futures is 0.80
• B. The hedge eliminates 80% of spot price variance — the hedge ratio itself is 0.80
• C. The hedge is unreliable — an R² below 0.9 indicates the futures contract is the wrong hedging instrument
• D. The remaining 36% of variance is due to systematic market risk that no hedge can reduce

Answer: (A)

Hedge effectiveness equals ρ², which is the R² from the regression of ΔS on ΔF. R² = 0.64 means the correlation is ρ = √0.64 = 0.80, and the hedge eliminates 64% of spot price variance. The remaining 36% is basis risk — the portion of spot price movement uncorrelated with futures — which cannot be eliminated regardless of the number of contracts held. R² values in the 0.6–0.8 range are common for cross-hedges and still represent meaningful risk reduction.

Question 3

A cross-hedge using crude oil futures to hedge jet fuel price exposure can eliminate most jet fuel price risk if enough futures contracts are held.

• T. True
• F. False

Answer: False

No number of futures contracts eliminates all basis risk when the correlation ρ < 1. The minimum-variance hedge ratio is optimal — it reduces variance as much as the available instrument allows — but residual variance equal to (1 − ρ²) × Var(ΔS) remains. For a cross-hedge where two commodities are related but distinct, ρ < 1 always, so basis risk is unavoidable. The only way to eliminate all price risk is a perfect hedge: ρ = 1, matched maturity, and identical underlying.

Question 4

If the futures contract used for hedging is more volatile than the spot asset being hedged, the optimal hedge ratio will be less than 1.

• T. True
• F. False

Answer: True

The formula h* = ρ × (σ_S / σ_F) shows this directly: when σ_F > σ_S, the ratio σ_S/σ_F < 1, and multiplying by ρ ≤ 1 only reduces it further. Intuitively, each futures contract moves more than one unit of spot exposure, so you need fewer contracts to offset a given spot position. Holding one-for-one would over-hedge, actually introducing more variance than the original unhedged position relative to the minimum-variance solution.

Question 5

Why is the optimal hedge ratio derived from a regression of spot price changes on futures price changes, and what does the R² of that regression tell you about the quality of the hedge?

Model answer: The OLS regression of ΔS on ΔF directly delivers the minimum-variance hedge ratio as the slope coefficient β = Cov(ΔS, ΔF) / Var(ΔF) = h*. This is not a coincidence — OLS minimizes the sum of squared residuals, which is equivalent to minimizing the variance of the hedged P&L (ΔS − h·ΔF). The R² measures what fraction of spot price variance is explained by the futures instrument and equals ρ², directly quantifying hedge effectiveness — the fraction of spot risk the hedge eliminates.

The regression framework is powerful because it simultaneously gives you the hedge ratio (slope), the hedge effectiveness (R²), and statistical confidence intervals — all from a single estimation. It also allows you to detect instability: if the hedge ratio has changed significantly between estimation periods, you need to rebalance your futures position.