Futures curve: expiries, slope and movements

Let t be the observation time and Tᵢ the expiries. A snapshot can be written as:

Every point therefore uses the same observation time t, while expiry Tᵢ changes. The horizontal axis must state calendar expiries or time remaining; the vertical axis must state currency, quotation unit and contract specification.

What must remain comparable

A valid curve is not made by joining tickers that merely look similar. The contracts must refer to the same economic underlying and have comparable specifications: grade, quantity, currency, location and delivery or settlement method. Time matters too. Quotes taken at different moments can turn an intraday move into a false slope.

For every point, record at least:

1. snapshot date and timestamp;
2. contract code and month;
3. last price, bid-ask or settlement price used;
4. unit and multiplier;
5. time remaining to expiry;
6. liquidity, spread and any anomaly near delivery.

A continuous futures chart answers a different question: it links different contracts through time under a roll rule and may apply price adjustments. It does not replace the curve snapshot and cannot reconstruct it without the prices of the individual expiries.

Local slope and shape

Between two adjacent expiries Tᵢ and Tᵢ₊₁, the raw calendar spread is:

If prices are in euros per unit, ΔFᵢ is also in euros per unit. To compare different time intervals, a linearized slope can be stated:

Here τ is time remaining, expressed for example in years. If all prices are positive, an alternative is the logarithmic slope ln(Fᵢ₊₁/Fᵢ) / (τᵢ₊₁ − τᵢ). This measure is invalid for zero or negative prices and must not be presented as a realized return.

A curve can be upward sloping, downward sloping or approximately flat only on the stated segment. It can also be mixed: upward between the first two expiries, downward farther out, with local humps or dips. This is why contango and backwardation describe a relationship between specific expiries, not necessarily the whole curve.

Verifiable example

At 16:00, three comparable contracts are quoted at 80.00, 81.20 and 80.60 currency units, with expiries three months apart. The spreads are:

• second minus first: 81.20 − 80.00 = +1.20;
• third minus second: 80.60 − 81.20 = −0.60.

With 0.25-year intervals, the linearized slopes are respectively +4.80 and −2.40 currency units per unit of underlying per year. The curve is therefore mixed, with a local maximum at the second expiry. The calculations do not say where spot will trade in three or six months.

How a curve can move

Comparing two dates requires a consistent convention. A fixed-expiry comparison follows the same contracts as they move closer to expiry; a constant-maturity comparison reconstructs nodes with similar remaining terms, often through a selection or interpolation rule. Mixing the two views creates artificial signals.

• Shift: many expiries rise or fall together; “parallel” is an approximation to be checked against the data.
• Steepening or flattening: the difference in slope between selected nodes increases or decreases.
• Twist: one part of the curve rises while another falls.
• Curvature: the relationship between adjacent segments changes. With equally spaced nodes, F₃ − 2F₂ + F₁ is a simple second difference; with unequal spacing, normalized slopes are preferable.

These labels describe the observed movement. They do not explain its cause by themselves and do not justify extrapolating it.

Why shape differs across assets

Drivers depend on the contract. For storable commodities, financing, storage, insurance, inventories, local availability and the economic value of immediate possession can matter. Seasonality, grade and constraints at the delivery point can create irregularities between months.

For equity index futures, financing and expected cash flows from the basket enter the relationship; for currency futures, rates in both currencies and funding conditions matter; for interest-rate contracts, quotation convention, reference period and settlement rule matter. In every case, liquidity, hedging demand, limits to arbitrage and specifications can separate the observed price from a simplified theoretical formula.

The curve is therefore not a yield curve: a yield curve associates yields or rates with different maturities, whereas the vertical axis here contains futures prices. Nor is it a pure forecast. The price reflects contract terms, carry, hedging supply and demand, and risk premia that can change before expiry.

Curve, roll and result

A roll closes or reduces one expiry and opens the chosen replacement. The difference between the two prices is observable on the curve, but it is not automatically the total profit or loss on the operation. The result depends on actual entry and exit prices, subsequent contract evolution, quantity, multiplier, bid-ask spread and other costs.

Roll yield also requires an explicit methodology. Instantaneous slope, calendar spread, index return and account P&L are not synonyms. An adjusted continuous chart can also conceal level jumps between contracts while presenting a visually smooth series.

Reading checklist

1. Do all quotations belong to the same snapshot?
2. Are underlying, grade, location, unit and settlement comparable?
3. Does the axis use expiry dates or remaining term?
4. Is the price last, mid, bid-ask or official settlement?
5. Which segment do slope and contango or backwardation describe?
6. Are measures absolute, percentage or annualized, and under which formula?
7. Is the time comparison fixed-expiry or constant-maturity?
8. Are illiquid expiries or contracts near delivery distorting the view?
9. Have roll and its return been separated from curve shape alone?

Sources

• U.S. Commodity Futures Trading Commission, Futures Glossary — terminology for contracts, months, basis and the futures market.
• U.S. Commodity Futures Trading Commission, Customer Advisory: Learn About Risks Before Investing in Commodity ETPs or Funds — differences between spot exposure and products that renew futures contracts.
• Financial Industry Regulatory Authority, Regulatory Notice 10-51 — structure and risks of commodity-futures-linked products.
• CME Group, Understanding Futures Expiration & Contract Roll — expiry, contract replacement and the roll calendar.
• Campbell & Company, Deconstructing Futures Returns: The Role of Roll Yield, white paper hosted by CME Group — curve, convergence and decomposition of futures returns.
• U.S. Commodity Futures Trading Commission, Economic Requirements — Policy Statement on Price Differentials — differentials, grades, locations and the economic conditions of deliverable contracts.

Related pages

• Futures
• Futures basis
• Contango and backwardation
• Futures roll
• Roll yield
• Futures expiry and settlement