What is hedging effectiveness (R-squared)?

Hedging effectiveness is measured by R-squared (rho-squared), which represents the proportion of the variance of the hedged position that is eliminated by the hedge under the minimum-variance model. An R-squared of 0.80 means 80% of the price variance is removed in-sample. R-squared ranges from 0 to 1, with values above 0.80 generally considered excellent for cross-hedging applications. Note that R-squared reflects historical explanatory power and is not a guarantee of future hedge performance.

What is the difference between a perfect hedge and an optimal hedge?

A perfect hedge eliminates all price risk, requiring correlation of exactly 1 and near-zero basis risk — this is rare in practice. An optimal hedge minimizes variance by accounting for imperfect correlation, resulting in h* = rho x (sigma_S / sigma_F). When correlation is less than 1, the optimal hedge ratio is less than the naive 1:1 ratio because hedging the full exposure would introduce unnecessary futures-side variance. The optimal hedge balances risk reduction against the cost of imperfect hedging.

Futures Hedge Ratio Calculator: Minimum Variance Optimal Hedge

Enter Values

Spot Price Volatility (σ_S)

e.g., 25 for 25% annualized

Futures Price Volatility (σ_F)

e.g., 20 for 20% annualized

Correlation (ρ)

Decimal from -1 to 1 (e.g., 0.85 = 85% correlated)

Portfolio/Exposure Value (Q_A)

Dollar value of position to hedge

Futures Contract Size (Q_F)

Dollar value of one futures contract

Hedge Ratio Formulas

h* = ρ × (σ_S / σ_F)

N* = h* × (Q_A / Q_F)

ρ = Correlation | σ_S = Spot volatility | σ_F = Futures volatility
Q_A = Exposure value | Q_F = Contract size

Calculator by Ryan O'Connell, CFA

Calculation Results

Optimal Hedge Ratio (h*) 1.0625 Over-Hedge

Futures Position Short 43 contracts

Exact N* 42.5000

Hedging Effectiveness (R²) 72.25%

Dollar Amount Hedged $10,750,000

Hedge Coverage 107.50%

Formula Breakdown

h* = ρ × (σ_S / σ_F) | N* = h* × (Q_A / Q_F)

Step-by-step calculation with your inputs

Hedge Ratio Interpretation

Hedge Ratio	Interpretation	Meaning
h* > 1	Over-Hedge	Futures position exceeds spot exposure
h* ≈ 1	Perfect Hedge	Futures exactly offset spot changes
0 < h* < 1	Under-Hedge	Partial variance reduction
h* ≈ 0	Hedge Likely Ineffective	Zero correlation — hedging adds no benefit
h* < 0	Negative Hedge	Long futures position required (negative correlation)

Model Assumptions

Constant volatilities and correlation over the hedge period
Linear relationship between spot and futures price changes
Residual basis risk not captured by correlation is assumed negligible for this model
Hedge ratio is static (not dynamically adjusted)
Hedge minimizes variance of the hedged position (minimum variance criterion)
R² = ρ² is in-sample explanatory power, not guaranteed future protection
Positive N* = short futures (hedging long exposure); Negative N* = long futures

For educational purposes. Not financial advice. Market conventions simplified.

Understanding the Minimum Variance Hedge Ratio

Video Explanation

Video: Futures Hedging Strategies Explained

What is the Optimal Hedge Ratio?

The optimal hedge ratio (h*) determines the proportion of a position that should be hedged with futures contracts to minimize the variance of the hedged position. It comes from regressing changes in spot prices against changes in futures prices (Hull Chapter 3, Equation 3.1).

When the asset being hedged differs from the futures contract’s underlying asset, a naive 1:1 hedge is not optimal. The minimum variance approach accounts for imperfect correlation and differing volatilities between the two instruments.

Minimum Variance Hedge Ratio

h* = ρ × (σ_S / σ_F)
where ρ = correlation, σ_S = spot volatility, σ_F = futures volatility

How Many Futures Contracts?

Once you know h*, the number of futures contracts is straightforward (Hull Equation 3.2):

Number of Contracts

N* = h* × (Q_A / Q_F)
Rounded to nearest integer. Positive = short futures, Negative = long futures.

Both Q_A (exposure) and Q_F (contract size) must be on the same basis — both in dollar notional, or both in physical units.

Hedging Effectiveness and R²

Hedging effectiveness is measured by R² = ρ², representing the proportion of variance eliminated by the hedge under the minimum-variance model. An R² of 0.80 means 80% of price variance is removed in-sample.

Note that R² reflects historical explanatory power. Actual future hedge performance depends on the stability of the correlation and volatility estimates over the hedge period.

Basis Risk: When the hedging instrument differs from the asset being hedged (cross-hedging), basis risk remains. Basis risk is time-varying and driven by multiple factors beyond a single correlation estimate. See also: Cross-Hedging and Equity Futures.

Key Assumptions

Constant volatilities and correlation over the hedge period
Linear relationship between spot and futures price changes
Static hedge ratio (not dynamically adjusted)
No transaction costs or margin requirements
Futures contracts are perfectly divisible (though we round N* in practice)

Frequently Asked Questions

The optimal hedge ratio (h*) is the proportion of a position's exposure that should be hedged using futures contracts to minimize the variance of the hedged position. It is calculated as h* = ρ × (σ_S / σ_F), where ρ is the correlation between spot and futures price changes, σ_S is the standard deviation of spot price changes, and σ_F is the standard deviation of futures price changes. This is also called the minimum variance hedge ratio (Hull Chapter 3, Equation 3.1).

The number of futures contracts needed is N* = h* × (Q_A / Q_F), where h* is the optimal hedge ratio, Q_A is the total exposure value, and Q_F is the size of one futures contract. The result is rounded to the nearest integer since you cannot trade fractional contracts. A positive N* means short (sell) futures; a negative N* means long (buy) futures.

Hedging effectiveness is measured by R² (ρ²), which represents the proportion of the variance of the hedged position that is eliminated by the hedge under the minimum-variance model. An R² of 0.80 means 80% of the price variance is removed in-sample. R² ranges from 0 to 1, with values above 0.80 generally considered excellent for cross-hedging applications. Note that R² reflects historical explanatory power and is not a guarantee of future hedge performance.

The hedge ratio exceeds 1 when the spot asset is more volatile than the futures contract relative to their correlation. Specifically, h* > 1 when ρ × (σ_S / σ_F) > 1. This means you need a futures position that is larger than your underlying exposure. This is called an over-hedge and can occur in cross-hedging situations where the hedging instrument has lower volatility than the asset being hedged.

Basis risk is the risk that the spot price and futures price do not move in perfect lockstep. It arises from cross-hedging (different assets), maturity mismatches, and changing market dynamics. Basis risk is reflected in a correlation less than 1, which reduces hedging effectiveness. The optimal hedge ratio accounts for basis risk through the correlation coefficient, but in practice basis risk is time-varying and driven by multiple factors beyond a single static correlation estimate.

A perfect hedge eliminates all price risk, requiring correlation of exactly 1 and near-zero basis risk — this is rare in practice. An optimal hedge minimizes variance by accounting for imperfect correlation, resulting in h* = ρ × (σ_S / σ_F). When correlation is less than 1, the optimal hedge ratio is less than the naive 1:1 ratio because hedging the full exposure would introduce unnecessary futures-side variance. The optimal hedge balances risk reduction against the cost of imperfect hedging.

Disclaimer

This calculator is for educational purposes only and uses the minimum variance hedge ratio framework from Hull's "Options, Futures, and Other Derivatives." Actual hedging decisions involve additional factors including transaction costs, margin requirements, liquidity, basis risk dynamics, and the specific futures contracts available. This tool should not be used for trading decisions without professional consultation.

Futures Hedge Ratio Calculator

Enter Values

Hedge Ratio Formulas

Calculation Results

Formula Breakdown

Hedge Ratio Interpretation

Model Assumptions

Understanding the Minimum Variance Hedge Ratio

Video Explanation

What is the Optimal Hedge Ratio?

How Many Futures Contracts?

Hedging Effectiveness and R²

Key Assumptions

Frequently Asked Questions

What is the optimal hedge ratio?

How do you calculate the number of futures contracts needed?

What is hedging effectiveness (R²)?

When is the hedge ratio greater than 1?

What is basis risk and how does it affect hedging?

What is the difference between a perfect hedge and an optimal hedge?

Disclaimer

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