This hedge ratio calculator computes the minimum-variance hedge ratio — the correlation between spot and futures price changes multiplied by the ratio of their volatilities — and then converts that ratio into the number of futures contracts a stated exposure requires, given the exchange's contract size. Both of those numbers are routinely called "the hedge ratio", which is why this page names which one it is producing at every step. The first is a dimensionless slope; the second is a whole number of contracts.
Arb Digital builds free tools that make the assumption behind a number visible rather than burying it. Every volatility, correlation, price and contract size on this page is something you type in. The page publishes no market data of any kind, names no commodity, currency or contract as worth hedging, and recommends no position. It computes a published statistical relationship from your own estimates, and it is deliberately blunt about how fragile those estimates are.
What This Hedge Ratio Calculator Does
A hedge takes an existing exposure and offsets it with a position in a related instrument, usually a futures contract. The question the minimum-variance ratio answers is narrow and precise: given how the two instruments have moved together historically, what size of futures position makes the variance of the combined position as small as it can be? The answer is a slope, and it is the same slope you would get by regressing spot price changes on futures price changes.
The calculator produces that slope and then does the arithmetic that follows from it, reporting both the exact contract figure and the whole number you can actually trade. It also reports two diagnostics that matter more than the ratio itself: the proportion of variance the hedge removes, and the volatility that survives it.
Setting the mode to a naive one-for-one hedge fixes the ratio at 1.00 and leaves the contract arithmetic intact. The gap between the two counts is the entire practical value of the statistical work: where correlation is high and the volatilities are similar it is negligible, and where they diverge it is not.
How to Use It
- Estimate both volatilities on the same basis. If one is a monthly standard deviation of price changes, the other must be too. Our standard deviation calculator computes them from a series you already hold.
- Estimate the correlation over the same window. The correlation coefficient calculator takes paired observations, and the covariance calculator gives the unstandardised version of the same quantity.
- Enter the exposure in the contract's own unit. A mismatch between the exposure unit and the contract unit is the single most common error, and it produces a contract count that is wrong by a factor rather than a rounding.
- Choose the contract basis. The quantity basis divides units by units. The tailed value basis divides the exposure's value by the contract's value, which differs whenever the spot and futures prices are not equal.
- Read the residual volatility, not just the ratio. It tells you what proportion of the original risk the hedge leaves behind, and that is the number a hedge is actually judged on.
The Formula: How the Minimum-Variance Hedge Ratio Is Calculated
Write σS for the standard deviation of spot price changes, σF for the standard deviation of futures price changes, and ρ for the correlation between them. The minimum-variance hedge ratio is h* = ρ × (σS ÷ σF). Equivalently, it is the covariance of the two divided by the variance of the futures, which is why it is identical to the slope coefficient of a regression of spot changes on futures changes.
The contract count on a quantity basis is N* = h* × QA ÷ QF, where QA is the exposure in units and QF is the units per contract. On a tailed value basis it is N* = h* × VA ÷ VF, where VA is the exposure's value at the spot price and VF is one contract's value at the futures price.
The proportion of variance the hedge removes is ρ², the coefficient of determination of that same regression, and the volatility that remains is σS × √(1 − ρ²). Our R-squared calculator computes the first of those from a fitted relationship directly.
Work the defaults. Spot volatility 3.2 per cent, futures volatility 4.0 per cent, correlation 0.85. The ratio is 0.85 × (3.2 ÷ 4.0) = 0.85 × 0.8 = 0.6800. With two million units of exposure and a 42,000-unit contract, the quantity basis gives 0.68 × 2,000,000 ÷ 42,000 = 32.3810 contracts, rounding to 32. Variance removed is 0.85² = 72.25 per cent, and the residual volatility is 3.2 × √(1 − 0.7225) = 3.2 × 0.526783 = 1.6857 per cent.
On the tailed value basis with a spot price of 2.50 and a futures price of 2.55, the exposure is worth 5,000,000 and one contract is worth 107,100, so the count is 0.68 × 5,000,000 ÷ 107,100 = 31.7460 contracts, still rounding to 32 but for a different reason. Change the two prices and the two bases separate.
Two Different Numbers Both Called the Hedge Ratio
The ambiguity in the term causes real confusion. In a textbook treatment of futures hedging, "hedge ratio" means the dimensionless slope h*, a number typically somewhere near one. In a trading desk conversation it often means the contract count, an integer that depends on the exposure and the contract specification and has nothing to do with statistics. In options work it means something else again: the delta of the position, which changes continuously with the underlying price and is not what this page computes.
This page always shows both, with the ratio in the headline and the count in the grid, and it labels the count's basis explicitly. If you are sizing a directional position rather than offsetting an existing one, that is a different exercise entirely, and our position size calculator handles it. The contract specifications, notional value and margin arithmetic behind the futures leg sit in the futures contract calculator.
Basis Risk: What This Arithmetic Does Not Capture
The basis is the difference between the spot price of what you actually hold and the futures price of the contract you are hedging with. A hedge removes price risk and replaces it with basis risk. That is the trade, and it is not a footnote — it is the substance of what remains after the hedge is on.
Basis moves for reasons the historical correlation cannot anticipate. The grade, location or delivery month of the contract differs from the exposure, so a local supply disruption moves one price and not the other. The contract expires before the exposure does, so the hedge has to be rolled and the roll happens at whatever basis prevails on the day. Storage costs, transport constraints and convenience yields all move the relationship between a physical holding and a standardised contract. None of that appears in σS, σF or ρ.
What the residual volatility figure captures is only the part of the divergence that showed up in the estimation window. The MIT OpenCourseWare lecture on forward and futures contracts sets out the pricing relationship and the qualifications on it, and the qualifications are where hedging actually lives. A hedge that worked for three years can fail in a month when the basis behaves in a way it had not previously been observed to behave.
Hedge Effectiveness and the Volatility That Survives
Hedge effectiveness, reported here as the proportion of variance removed, equals ρ². That squaring is the reason correlations that sound high are less impressive than they read. A correlation of 0.9 removes 81 per cent of variance and leaves 19 per cent. A correlation of 0.8 removes 64 per cent and leaves 36 per cent — nearly double the residual. A correlation of 0.7 leaves just over half. In volatility terms rather than variance terms, a 0.85 correlation leaves 52.7 per cent of the original standard deviation standing.
Both statements describe the same hedge. The variance figure flatters it, because the square root pulls small numbers upward; the volatility figure is the one you actually experience.
The practical use of this is deciding whether a cross-hedge is worth putting on at all. If the only available contract has a correlation of 0.5 with your exposure, the hedge removes a quarter of the variance, leaves 86.6 per cent of the volatility, and introduces margin obligations, roll costs and operational risk in exchange. The MIT lectures on portfolio theory develop the underlying variance arithmetic, which is the same mathematics used to combine any two correlated positions.
Tailing, Rounding, and Why Contract Counts Are Whole Numbers
Futures trade in whole contracts. An exact count of 32.381 has to become 32 or 33, and neither is the minimum-variance position. Rounding down leaves the exposure slightly under-hedged; rounding up over-hedges it and leaves a small speculative position in the opposite direction. For a large exposure that gap is trivial. For a small one it can be a substantial fraction of the hedge, which is why the calculator reports the unrounded figure alongside the rounded one — the size of the discrepancy is information.
Tailing is the second adjustment. Because futures settle daily and the resulting cash flows are financed at prevailing rates, an untailed hedge is slightly oversized relative to what it protects. The value-basis option scales the count by the ratio of the exposure's value to a contract's value, which moves the default example from 32.38 to 31.75 — the same rounded answer, but a different distance from the boundary.
What the Estimate Actually Depends On
Every input is a historical measurement, and the ratio inherits every weakness of the window it came from. A short window is noisy; a long one averages across regimes that no longer apply; and a window containing one large shared shock shows a correlation far above the ordinary relationship, because a single common move dominates the covariance.
The periodicity matters as much as the length. Daily price changes, weekly changes and monthly changes give different correlations for the same pair, generally rising as the interval lengthens, because short-horizon noise averages out. A ratio estimated on daily data and applied to a hedge held for six months is answering a different question from the one being asked. Match the estimation interval to the horizon over which the hedge will actually be carried.
Correlations also move with market conditions, and they tend to move at the worst time. Relationships that are stable in calm periods can break in stressed ones, which is when a hedge is being relied on. Professional practice is to re-estimate periodically rather than to fix a ratio once, and to treat the number as a working estimate under review. The CFA Institute's professional learning resources cover risk-measurement practice for practitioners in more depth than a calculator page can.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Mixing units between the exposure and the contract — barrels against gallons or tonnes against bushels produces a contract count wrong by a factor, and the ratio itself gives no clue that it happened.
- Estimating the volatilities and the correlation over different windows — the three inputs are parts of one estimate, and mixing periods makes the resulting ratio meaningless.
- Reading variance removed as risk removed — 72 per cent of variance removed still leaves 53 per cent of the volatility, and it is the volatility you actually live with.
- Treating the hedge as fixed — correlations drift, contracts expire and have to be rolled, and the exposure itself changes size, so the ratio needs re-estimating rather than setting once.
- Forgetting that the hedge has its own cash flows — futures are marked to market daily and can require variation margin even while the hedged exposure is performing exactly as intended.
Related Free Tools From Arb Digital
The three statistical inputs each have their own page. The standard deviation calculator produces the two volatilities, the correlation coefficient calculator produces the correlation from paired observations, and the covariance calculator gives the unstandardised form used in the alternative statement of the formula. The R-squared calculator reports the effectiveness figure directly from a fitted relationship.
On the position side, the futures contract calculator covers notional value, tick value and margin for the contracts this hedge would use, and the margin call calculator shows how variation margin can bite while a hedge is running. For a sensitivity measure rather than a hedge ratio, the stock beta calculator computes the equity-market equivalent of the same slope. Everything else is on the free tools hub.
Frequently Asked Questions
It computes the minimum-variance hedge ratio, equal to the correlation between spot and futures price changes multiplied by the ratio of their standard deviations, and then converts that ratio into a futures contract count using the exposure and the contract size. Both figures are commonly called the hedge ratio, so the page labels each one.
A one-for-one hedge is only optimal when the spot and futures prices move identically. In practice their volatilities differ and their correlation is below one, so the variance-minimising position is scaled by the ratio of the volatilities and shrunk by the correlation. The naive setting on this page fixes the ratio at 1.00 for comparison.
Basis risk is the risk that the spot price of what you hold and the futures price you hedged with fail to move together as the estimate assumed. It arises from grade, location, delivery month, storage costs and roll timing. Those are not statistical quantities, so no correlation estimate captures them.
The proportion of variance removed is the correlation squared. Because volatility is the square root of variance, a hedge that removes 72 per cent of variance still leaves about 53 per cent of the original volatility standing. The calculator reports both so the difference is visible.
Tailing scales the contract count by the ratio of the exposure's value to a contract's value rather than by unit quantities. It accounts for the price difference between spot and futures and for the fact that futures settle daily, so an untailed hedge is slightly oversized relative to what it protects.
It is a measurement of one past window, not a property of the position. Short windows are noisy, long ones average across regimes, and correlations tend to shift in stressed conditions, which is exactly when a hedge is being relied upon. Practitioners re-estimate periodically rather than fixing a ratio once.
No. Every volatility, correlation, price, quantity and contract size is a figure you type in. The page publishes no market data, names no commodity, currency or contract, and takes no view on whether any exposure should be hedged at all.
This tool is provided for educational and estimating use only. It is not investment advice and does not recommend any contract, commodity, position or hedging strategy. Futures positions carry substantial risk, require margin, and can lose more than the amount deposited; anyone considering one should take advice from a qualified financial adviser.
