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Define the basis and compute it, then discuss where basis risk comes from and why it arises when futures are used as the hedge.

FRM® Part I · Financial Markets and Products · Using Futures for Hedging · FRM1.FM.8.c

The idea

The basis is the spot price of the asset being hedged minus the futures price. Basis = S − F. This note uses that convention, but some texts define it as F − S, so read the stem. For a hedge that ends before the delivery month the basis is not zero and is not known in advance. Effective price = S2 + F1 − F2 = F1 + b2. Here S2 and F2 are the spot and futures prices when the hedge ends, F1 is the futures price when it starts, and b2 is the final basis. Only b2 is uncertain. A rising basis (strengthening) helps a short hedger and hurts a long hedger. A falling basis (weakening) does the opposite. Basis risk comes from asset mismatch, a gap between the hedge end date and delivery month, and the closing date being uncertain. Choose the delivery month just after the hedge ends.

Formulas

Basis

basis = S − F

S
the spot price of the asset being hedged
F
the futures price of the contract used

The convention used here; some texts use F − S, so read the stem. It tends to zero at expiry but not before, and a rising basis helps a short hedger and hurts a long one.

Effective price locked in by a futures hedge

effective price = F₁ + b₂

F₁
the futures price when the hedge is set up
b₂
the basis, spot minus futures, when the hedge is lifted

The same for a short hedge's sale price and a long hedge's cost. Only the final basis is uncertain; held to expiry it is zero and the price is F₁.

A practice question

From the question bank, with the reason each wrong option is wrong.

A hedger observes a spot index of 2,310 and a futures price of 2,325. Using basis = spot minus futures, the basis in index points is:

  1. +15 index pointsWhy not: Sign reversed: this is futures minus spot, the opposite of the stated definition.
  2. -2,310 index pointsWhy not: Reports the spot level with a negative sign instead of the spot-minus-futures difference.
  3. -15 index points - correct
  4. Zero, because futures prices converge to spot at maturityWhy not: Convergence applies only at maturity; before then the basis is generally non-zero.

Basis = S - F = 2,310 - 2,325 = -15. Prior to delivery it is not zero.

A harder one

A producer opens a short hedge when spot is 60.50 and the futures price is 61.00. When the hedge is closed, spot is 58.40 and the futures price is 59.10. Using basis = spot minus futures, which statement is correct?

  1. The basis weakened by 0.20; the short hedger is worse off by 0.20 per unit - correct
  2. The basis strengthened by 0.20; the short hedger is better off by 0.20 per unitWhy not: Sign error on the basis change: it moved from -0.50 to -0.70, a decline.
  3. The basis weakened by 0.20; the short hedger is better off by 0.20 per unitWhy not: Right direction for the basis but wrong beneficiary: a weakening basis lowers the price a short hedger realises.
  4. The basis weakened by 0.70; the short hedger is worse off by 0.70 per unitWhy not: Treats the closing basis level as its change and ignores the opening basis of -0.50.

b1 = 60.50 - 61.00 = -0.50; b2 = 58.40 - 59.10 = -0.70. The change is -0.20. Effective price = S2 + F1 - F2 = 60.30, versus initial spot 60.50: -0.20.

The hardest tier

A jet-fuel producer sells heating oil futures to hedge a jet-fuel sale. The futures price is 2.40 USD per gallon at the start. When the hedge is closed, jet fuel sells for 2.15, heating oil spot is 2.20 and the futures price is 2.23 (all USD per gallon). The effective price per gallon realised is closest to:

  1. USD 2.37Why not: Counts only the basis of the futures' own underlying (heating oil spot minus futures) and ignores the jet-fuel versus heating-oil gap.
  2. USD 2.40Why not: Assumes a perfect hedge at the opening futures price, ignoring basis risk entirely.
  3. USD 2.48Why not: Subtracts the final basis instead of adding it: effective price is F1 plus b2, not F1 minus b2.
  4. USD 2.32 - correct

Effective price = S2 + F1 - F2 = 2.15 + 2.40 - 2.23 = 2.32. Equivalent to F1 + (S2 - S2*) + (S2* - F2) = 2.40 - 0.05 - 0.03. The mismatch between jet fuel and heating oil adds basis risk beyond the usual futures basis.

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