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Derive the number of futures contracts that best hedges an exposure, and clarify how tailing the hedge adjusts that number, including how to compute it.

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

The idea

The number of futures contracts that minimises the variance of a hedged position is N = h × QA ÷ QF. Here QA is the size of the position being hedged, QF the size of one contract and h the optimal hedge ratio, which is 1 if the hedge is exact. Otherwise, with ρ the correlation between changes in the spot and futures prices and σS and σF their standard deviations, h = ρ × σS ÷ σF. The result is rounded to a whole number. Tailing the hedge adjusts for daily settlement, which pays futures gains and losses immediately. With VA the spot value of the exposure and VF the value of one futures contract (futures price × contract size), the tailed number is N = h × VA ÷ VF. The tailed number is usually slightly smaller than the untailed one when the futures price exceeds spot. Check units: gallons against barrels is a common slip.

Formulas

Minimum-variance hedge ratio

h = ρ × σS ÷ σF

ρ
the correlation between changes in the spot and futures prices
σS
the standard deviation of changes in the spot price
σF
the standard deviation of changes in the futures price

Feeds the number of contracts; it is 1 when the futures are on the asset hedged and the hedge is exact. Inverting the volatility ratio is the usual slip.

Optimal number of futures contracts

N = h × QA ÷ QF

h
the optimal hedge ratio
QA
the size of the position being hedged, in units
QF
the size of one futures contract, in the same units

Round to a whole number. Check units before dividing: gallons against barrels is a common slip.

Tailed number of futures contracts

N = h × VA ÷ VF

h
the optimal hedge ratio
VA
the spot value of the exposure
VF
the value of one futures contract, futures price × contract size

Adjusts for daily settlement. When the futures price exceeds spot it gives slightly fewer contracts than the untailed number.

A practice question

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

A grain handler holds 750,000 bushels and hedges with futures on 5,000 bushels per contract using h* = 0.72. The number of contracts needed is closest to:

  1. 108 - correct
  2. 150Why not: Sets the hedge ratio to one and ignores h*.
  3. 208Why not: Divides by the hedge ratio instead of multiplying.
  4. 42Why not: Uses 1 - h* rather than h*.

N = h* x QA / QF = 0.72 x 750,000 / 5,000 = 108 contracts.

A harder one

A firm holds 20,000 tonnes of a commodity (spot USD 840 per tonne) and hedges with futures (USD 860 per tonne, 100 tonnes per contract) using h* = 0.9. The tailed number of contracts is closest to:

  1. 180Why not: Gives the untailed number and omits the adjustment for daily settlement.
  2. 176 - correct
  3. 184Why not: Inverts the tailing factor, using futures value over spot value instead of spot over futures.
  4. 195Why not: Tails correctly but omits the hedge ratio h*.

Untailed N = 0.9 x 20,000 / 100 = 180. Tailed N* = h* x VA / VF = 180 x 840/860 = 175.8, about 176 contracts.

The hardest tier

A refiner will sell 300,000 barrels of a product (spot USD 74.20) and hedges with crude futures (USD 75.60, 1,000 barrels per contract). Price-change volatilities are 2.6 percent for the product and 2.1 percent for the futures, with a correlation of 0.82. The tailed number of contracts, using the minimum-variance hedge ratio, is closest to:

  1. 305Why not: Computes the hedge ratio correctly but omits tailing for daily settlement.
  2. 299 - correct
  3. 310Why not: Inverts the tailing factor (futures over spot instead of spot over futures).
  4. 199Why not: Inverts the volatility ratio when computing h* (and ignores tailing).

h* = 0.82 x 0.026 / 0.021 = 1.0152. Untailed N = 1.0152 x 300 = 304.6. Tailed N* = 304.6 x 74.2/75.6 = 298.9, about 299.

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