Explain what a cross hedge is, compute its hedge ratio and interpret how effective the hedge is.

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

The idea

A cross hedge uses futures on a different asset from the one being hedged, because no futures exist on that asset or the available ones are more liquid. The minimum-variance hedge ratio depends on the correlation and the two volatilities. Hedge ratio = ρ × σS ÷ σF. Here σS and σF are the standard deviations of changes in the spot price and the futures price, and ρ is their correlation. The ratio is also the slope from regressing spot changes on futures changes. Effectiveness is the proportion of variance removed, which is ρ², the regression R². A correlation of 0.9 removes 81 percent of variance, not 90 percent. The residual standard deviation of the hedged position follows from it. Residual = σS × √(1 − ρ²). Do not invert the volatility ratio.

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

The slope from regressing spot changes on futures changes. Inverting the volatility ratio is the usual slip.

Hedge effectiveness

effectiveness = ρ²

ρ: the correlation between changes in the spot and futures prices

The share of variance removed by the minimum-variance hedge, the regression R². A correlation of 0.9 removes 81 percent, not 90 percent.

Standard deviation of a hedged position

σhedged = σS × √(1 − ρ²)

σS: the standard deviation of the unhedged exposure
ρ: the correlation between changes in the spot and futures prices

Holds for the minimum-variance hedge ratio only. A high correlation still leaves a sizeable residual: at 0.80 the standard deviation falls only to 0.6 of its unhedged level.

A practice question

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

The minimum-variance hedge ratio for a cross hedge is:

h* = ρ × σF / σSWhy not: Inverts the volatility ratio; a more volatile spot price requires more futures, not fewer.
h* = ρ × σS / σF, where σS and σF are the standard deviations of spot and futures price changes - correct
h* = σS / σFWhy not: Omits the correlation, which scales the hedge down when the two prices move imperfectly together.
h* = ρ² × σS / σFWhy not: Confuses the hedge ratio with the effectiveness measure; ρ² is the proportion of variance removed.

The hedge ratio is the regression slope of spot changes on futures changes, rho times sigma_S over sigma_F.

A harder one

Weekly price changes of an exposure have a standard deviation of 3.2 percent, those of the futures contract used have 4.0 percent, and their correlation is 0.78. The minimum-variance hedge ratio is closest to:

0.975Why not: Inverts the volatility ratio (sigma_F over sigma_S).
0.800Why not: Omits the correlation and uses only the volatility ratio.
0.487Why not: Uses rho² where rho is required.
0.624 - correct

h* = 0.78 x 0.032 / 0.04 = 0.624. Hedge 0.624 units of futures per unit of exposure.

The hardest tier

The weekly profit-and-loss of an unhedged exposure has a standard deviation of USD 400,000. A minimum-variance cross hedge is set up using futures whose price changes have a correlation of 0.80 with the exposure. The standard deviation of the hedged weekly P&L is closest to:

USD 144,000Why not: Applies the variance reduction factor (1 - rho²) directly to the standard deviation.
USD 80,000Why not: Scales the standard deviation by 1 - rho, mixing up correlation with the variance factor.
USD 240,000 - correct
USD 320,000Why not: Multiplies the standard deviation by rho, which is not the residual risk of the hedge.

Residual variance = sigma² (1 - rho²), so residual standard deviation = 400,000 x sqrt(1 - 0.64) = 400,000 x 0.6 = 240,000.

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