How to use the hedge ratio calculator
The page has three tools that build on each other.
Minimum variance finds the best hedge ratio. Choose how to give the inputs:
- Volatilities and correlation: the volatility of the exposure (spot), the volatility of the hedging instrument and their correlation, all measured on the same basis. The correlation calculator estimates the correlation from returns.
- Paste price changes: two columns of changes over the same periods, ideally periods as long as the hedge will last. The example is 24 fictional monthly changes in a spot fuel price and a related futures price, in cents per gallon. The calculator runs the regression for you.
Index futures (beta) sizes a stock index futures hedge. Enter the portfolio value, its beta, the target beta, the futures price and the contract. The presets are the CME E-mini S&P 500 ($50 per point), Micro E-mini S&P 500 ($5), E-mini Nasdaq-100 ($20) and Micro E-mini Nasdaq-100 ($2). The beta calculator measures the portfolio’s beta against the index.
Commodity hedge turns a hedge ratio into contracts for a physical exposure. Enter the quantity, the contract size, the hedge ratio and whether you will buy the commodity or already own it.
The formulas
σhedged = √(σS2 + h2σF2 − 2hρσSσF), at h*: σS × √(1 − ρ2)σS and σF are the standard deviations of spot and futures changes over the hedge horizon and ρ their correlation. From pasted data, h* is the least-squares slope of spot changes on futures changes.
A worked example
Minimum variance. An exposure has 20% volatility, the futures 24%, and they correlate at 0.9. The hedge ratio is 0.750 dollars of futures per dollar of exposure. It removes 81.0% of the variance and cuts volatility from 20% to 8.72%. A one-for-one hedge would leave 10.58%.
From the pasted fuel data, the regression gives a hedge ratio of 0.778 with a correlation of 0.900, and R² of 81.0%.
Index futures. A $5,000,000 portfolio has a beta of 1.2 and the manager wants 0.5. One E-mini contract at 6,500 points controls $325,000. The exact answer is -10.77 contracts (minus means sell), so the manager sells 11, which brings the beta to 0.485. In a 10% market fall, the expected market-driven P&L improves from -$600,000 to -$242,500.
Commodity hedge. A buyer needs 2,000,000 units of fuel and hedges with contracts of 42,000 units at a ratio of 0.75. That is 35.71 contracts, rounded to 36, covering 1,512,000 units.
Why the best hedge is often less than one to one
A hedge adds a second source of risk: the futures position. If the futures were a perfect copy of the exposure, the risk added would exactly cancel the risk removed, and a 1:1 hedge would leave nothing. In practice the hedging instrument differs in grade, location, maturity or composition, so part of its movement is unrelated to the exposure.
The minimum-variance ratio balances the two. It is the correlation times the ratio of the volatilities, so it falls below 1 whenever the futures are at least as volatile as the exposure and the correlation is imperfect. It can exceed 1 when the futures are much calmer than the exposure. The chart on the first tool shows the trade-off: hedged volatility falls as the ratio rises toward h*, then climbs again as the futures position starts to dominate.
The correlation matters most:
| Correlation | Hedge ratio | Variance removed | Volatility left | With a 1:1 hedge |
|---|---|---|---|---|
| 0.5 | 0.417 | 25.0% | 17.32% | 22.27% |
| 0.7 | 0.583 | 49.0% | 14.28% | 17.44% |
| 0.9 | 0.750 | 81.0% | 8.72% | 10.58% |
Effectiveness falls quickly as correlation drops: at 0.5 the best hedge removes only a quarter of the variance, and the 1:1 hedge leaves the position more volatile than the 20% it started with.
Hedging beta with index futures
For an equity portfolio, beta plays the role of the hedge ratio. A portfolio with a beta of 1.2 moves, on average, 1.2% for each 1% move in the index, so hedging it needs futures worth 1.2 times its value, not 1 times. Setting a target beta between zero and the current beta reduces market exposure without selling any stocks, which keeps the stock selection, avoids realizing gains, and is fast and cheap to reverse.
Two practical points change the count. Rounding: with E-minis, each contract moves the beta of the example portfolio by 0.065; the Micro contract cuts that tenfold, and 108 MES contracts land at a beta of 0.498. Matching the index: the beta must be measured against the index the futures track. A technology-heavy portfolio hedged with S&P 500 futures keeps the risk of technology moving differently from the broad market.
A beta hedge removes market risk, not stock-specific risk: the hedged portfolio still gains or loses on how its stocks do relative to the index.
What this calculator does not include
- Changing ratios. Volatilities, correlations and betas drift; hedge ratios need re-estimating and hedges re-sizing over time.
- Futures pricing details. The basis between futures and spot, dividends, financing and the roll from one contract month to the next are not modeled.
- Margin and cash flows. Futures are marked to market daily, so a hedge that works over its life can still need cash for variation margin along the way.
- Accounting and tax. Hedge accounting tests and the tax treatment of futures gains and losses are separate questions.
- Non-linear exposures. Options, or quantities that themselves depend on price, need more than a single ratio.