Futures Hedge Ratios: Sizing Risk When Production Is Uncertain

A futures hedge should be sized against the exposure it is meant to protect, not against a round percentage that looks reassuring. Quantity, price sensitivity, contract units and the relationship between the physical asset and the futures instrument all matter. A position described as “fully hedged” can still contain basis risk, uncertain production and a substantial unintended directional position.

Reference report · No directional market call

This report consolidates the foundation, applied and advanced futures material with the practical chart exercises. The worked figures are retained teaching scenarios. They are not current contract specifications, client positions or recommendations to take a commodity-market view.

Establish the exposure and the sign

Match hedge direction to the commercial exposure

For a simple long physical position hedged with short futures, write:

$$\Delta\Pi=Q\Delta S-Nq\Delta F-C.$$

Here $\Delta\Pi$ is change in combined value over the measurement period, $Q$ is physical quantity, $\Delta S$ the spot-price change per physical unit, $N$ the number of short futures contracts, $q$ units per contract, $\Delta F$ the futures-price change in matching units, and $C$ implementation costs. Currency conversion and grade adjustments must be included if the prices are not directly comparable.

If $h=Nq/Q$, the hedge ratio is futures-equivalent quantity divided by physical quantity. It is not automatically the percentage reduction in risk.

A numerical hedge can be correct without producing zero profit and loss

The foundation example holds 1,000 tonnes of copper and shorts futures equivalent to 950 tonnes. Spot rises from $8,000 to $8,500 per tonne while futures rise from $8,050 to $8,520.

Physical and futures changes in the copper illustration

The remaining change comes from incomplete quantity coverage and the difference between spot and futures movements. It is not proof that the hedge failed. A hedge is assessed against its objective and measurement boundary, including cash flows and costs. For a future purchase, a favourable futures profit is offset against a higher physical purchase bill; looking only at the derivative would misstate the economic result.

Figure 01 · Conceptual framework

Start with the commercial loss before choosing the hedge sign

How the parts connect

1. ACommercial exposurerevenue at riskFuture seller
2. BCommercial exposurecost at riskFuture buyer
3. CFuture sellershort directionContract sizing
4. DFuture buyerlong directionContract sizing
5. EContract sizingimplementFunding and monitoring

Derive the minimum-variance ratio

Suppose price changes are measured on a consistent basis. The variance of $\Delta S-h\Delta F$ is:

$$\operatorname{Var}(\Delta S)-2h\operatorname{Cov}(\Delta S,\Delta F)+h^2\operatorname{Var}(\Delta F).$$

In words: Combined variability reflects physical-price changes, their shared movement with futures, and the size of the futures hedge.

Minimising this expression gives:

$$h^*=\frac{\operatorname{Cov}(\Delta S,\Delta F)}{\operatorname{Var}(\Delta F)} =\rho\frac{\sigma_S}{\sigma_F}.$$

In words: The least-variable linear hedge scales the shared price movement by the variability of futures-price changes.

Here $\rho$ is the correlation of the two price changes, and $\sigma_S$ and $\sigma_F$ are their standard deviations. In a simple regression of spot-price changes on futures-price changes with an intercept, the estimated slope provides the corresponding sample hedge ratio.

The formula minimises variance within this linear model. It does not automatically minimise expected shortfall, premium cost, liquidity demand or accounting volatility. Nor does it preserve every source of expected upside the investor may wish to retain.

The copper calculation and its residual risk

Inputs to the applied copper hedge

The ratio is:

$$h^*=0.94\times\frac{420}{400}=0.987.$$

The implied futures-equivalent quantity is 9,870 tonnes. Divide that quantity by the actual contract unit to obtain contract count, then assess rounding and execution constraints. The teaching inputs do not establish a current exchange lot size.

Under the model, residual variance per tonne is $420^2(1-0.94^2)=20,532.96$ dollars squared. The source rounds this to about 20,534. Residual standard deviation is approximately $143.29 per tonne. The hedge removes about 88.36% of the original variance and leaves 11.64%.

Variance and standard deviation must not be interchanged. Retaining 11.64% of variance means retaining approximately 34.12% of standard deviation, because the square root connects them. It does not mean 11.64% of the dollar loss in every scenario remains.

Figure 02 · Qualitative comparison

Three sizing quantities that should not be confused

Three sizing quantities that should not be confused

Quantity	Inputs and units	Required conversion
Price-change hedge ratio	Inputs and unitsCovariance of comparable price changes divided by futures-price variance	Required conversionMultiply by exposed physical units, then apply contract unit size
Return sensitivity	Inputs and unitsCovariance of percentage returns divided by futures-return variance	Required conversionScale by portfolio value and contract notional
Contract count	Inputs and unitsSigned tradable number of futures contracts	Required conversionDocument rounding, expiry and remaining basis

Return-based and quantity-based ratios need reconciliation

A regression using percentage returns produces a return sensitivity. Translating it into a futures trade requires value scaling. If a portfolio of market value $V$ has beta $\beta$ to a futures return and each contract has notional value $Fq$, a first-order contract count is approximately $\beta V/(Fq)$, with the trade sign determined by whether risk is being reduced or added.

This differs from directly multiplying a price-change slope by physical units. The two approaches can be consistent if prices, returns, values and units are handled correctly, but combining pieces from each can create a large sizing error. Every hedge ticket should show the transformation from exposure units to contract units.

For processing margins, mapping must extend to multiple legs. A stylised three-unit crude input, two-unit gasoline output and one-unit distillate output structure represents a production recipe. A facility producing 42% gasoline and 28% diesel does not have the same sensitivities as that convenient ratio. Yield, energy use, quality and local prices affect the actual margin.

A hedge ratio is conditional on a regime

The sources compare a copper producer’s original hedge ratio of 0.97 with a later estimate of 0.84 on 50,000 tonnes. The original short position is 48,500 tonnes; the revised position is 42,000 tonnes. The difference is 6,500 tonnes, equivalent to $55.25 million at the example price of $8,500 per tonne.

A second calculation weights a high-volatility ratio of 0.79 by probability 72% and a low-volatility ratio of 0.97 by 28%. The exact weighted result is 0.8404, not exactly 0.84. It corresponds to 42,020 tonnes and a reduction of 6,480 tonnes. The close results illustrate agreement between two teaching estimates; they do not independently prove either model is accurate.

Dynamic conditional correlation, rolling regression, exponentially weighted estimates and regime-switching models can all update the inputs. More responsiveness can also create turnover and estimation noise. A sensible policy needs limits on how quickly positions change, stress estimates and a test of whether incremental risk reduction justifies trading costs.

Transaction cost cannot be compared directly with variance

The 6,500-tonne adjustment at $8,500 has notional value $55.25 million. Multiplying by a quoted 0.03% spread gives $16,575. Whether this represents a round-trip spread or the cost of a single trade depends on the quotation convention and execution price. A trade near mid-market may incur roughly half the full bid-ask spread before other costs, but actual execution can differ.

The source narrative compares dollar transaction cost with a variance quantity and declares the adjustment economically justified. Those units are incompatible. A defensible decision converts risk reduction into the objective being optimised: expected utility, expected shortfall limits, capital consumption, funding protection or another explicit criterion. It also includes commissions, market impact and model uncertainty.

Figure 03 · Conceptual framework

When lower production turns protection into an extra short

How the parts connect

1. AForecast outputinitial sizingFixed hedge quantity
2. BForecast outputoperating uncertaintyProduction shortfall
3. CFixed hedge quantityposition remainsExcess short exposure
4. DProduction shortfalloffset becomes smallerExcess short exposure
5. EExcess short exposurecontrol before stressQuantity policy

Production uncertainty turns nominal coverage into directional risk

The chart exercises use an expected 100,000 barrels and a hedge price of $80. Ignoring basis and costs, the combined result relative to that baseline is:

$$\Pi=(Q-Q_H)(S_T-80),$$

where $Q_H$ is hedged quantity and $S_T$ the terminal price.

Coverage changes both the size and sign of exposure

At a terminal price of $100, the 125% position loses $500,000 relative to the baseline; at $60 it gains $500,000. This is directional exposure created by excess hedging, not additional safety.

If actual production is only 80,000 barrels while the hedge remains 100,000, the $100 outcome is a $400,000 loss relative to the same baseline. The supposed full hedge is now 125% of actual production. Operational forecasts belong inside the risk process because equipment failure, weather and delivery changes can alter the exposure after the trade.

One source exercise states that the 125% hedge has greater scenario standard deviation than the 50% hedge. Under its stated fixed-quantity linear model, the sensitivities are 25% and 50% of the unhedged position, so the 125% hedge actually has half that standard deviation, with opposite sign. Its danger is the reversed exposure and mismatch with the producer’s objective, not that incorrect numerical comparison.

Choose a quantity policy before the delivery forecast changes

Forecast volume is often treated as a single number even when the commercial team already knows that production is uncertain. Replace that number with a documented distribution or at least several operational scenarios. Distinguish contracted deliveries, dependable base production and optimistic incremental production. The hedge programme can then assign different treatment to each component.

Price and quantity can also be related. A widespread crop failure may reduce output while increasing the selling price, creating a partial natural offset in revenue. A production outage specific to one facility can reduce volume without improving its selling price. The same expected tonnes and average price therefore imply different revenue risk depending on the joint distribution.

Options can be useful for uncertain increments because a purchased option need not create the same unconditional delivery-like exposure as a futures position. That flexibility has a price. Compare the option premium with the consequence of excess futures exposure under plausible shortfalls rather than assuming partial hedging or options are always preferable.

Assign responsibility for updating the physical forecast and transmitting changes to the hedge desk. A sophisticated ratio model cannot help if the quantity used in it is months out of date. Reconciliation between commercial forecasts, outstanding contracts and derivative positions is part of hedge sizing, not an administrative task separate from it.

A practical sizing discipline

Start with the cash flow or market value to protect. Map currency, quantity, maturity and payoff sensitivities. Select an instrument for economic fit and liquidity. Estimate a ratio using consistent units, then stress the estimate and the physical quantity. Translate the result into contracts, document rounding and maintain resources for margin.

Review the hedge when the exposure changes, the relationship weakens or liquidity costs exceed the budget. Partial hedging can be rational under uncertain volume or an explicit desire to retain upside. It should be a documented choice, not a substitute for understanding the uncovered risk.

The strongest hedge is one whose size can be traced from the underlying exposure through its assumptions to its expected behaviour in adverse scenarios. A round coverage percentage is only the beginning of that explanation. A reviewer should be able to reproduce the trade count, identify the physical forecast it protects and explain what would make that count change. That reproducibility is a practical safeguard against both accidental speculation and stale exposure estimates.