Enter Values

%
e.g., 25 for 25% annualized
%
e.g., 20 for 20% annualized
Decimal from -1 to 1 (e.g., 0.85 = 85% correlated)
$
Dollar value of position to hedge
$
Dollar value of one futures contract
Hedge Ratio Formulas
h* = ρ × (σS / σF)
N* = h* × (QA / QF)
ρ = Correlation | σS = Spot volatility | σF = Futures volatility
QA = Exposure value | QF = Contract size
Ryan O'Connell, CFA
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 × (σS / σF)   |   N* = h* × (QA / QF)
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 × (σS / σF)
where ρ = correlation, σS = spot volatility, σF = 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* × (QA / QF)
Rounded to nearest integer. Positive = short futures, Negative = long futures.

Both QA (exposure) and QF (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 × (σS / σF), where ρ is the correlation between spot and futures price changes, σS is the standard deviation of spot price changes, and σF 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* × (QA / QF), where h* is the optimal hedge ratio, QA is the total exposure value, and QF 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 × (σS / σF) > 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 × (σS / σF). 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.

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