Charta is on Google Play: practise on the go. Get the app
Charta

Explain the reasons for, and the mechanics of, the difference between a regression hedge and one derived from regressing in the opposite direction.

FRM® Part II · Market Risk Measurement and Management · Regression Hedging and Principal Component Analysis · FRM2.MR.11.g

The idea

The regression hedge regresses the target yield change on the hedge yield change. Let σT and σH be the volatilities of the two yield changes, and ρ their correlation. The slope of that regression, which scales the DV01-neutral hedge, is as follows. Beta = ρ × (σT ÷ σH). Regressing the other way, hedge on target, gives a slope of ρ × σH ÷ σT, and the reciprocal of that slope is another candidate hedge ratio. The hedge ratio implied by that reverse regression is therefore the following one. Implied = σT ÷ (ρ × σH). This equals beta divided by ρ². Unless ρ is plus or minus one, the reverse regression therefore gives a larger hedge, an over-hedge. The reason is that ordinary regression minimizes squared errors in the dependent variable, so only the target-on-hedge regression minimizes the variance of the hedged P&L. The product of the two slopes equals R². Exam point: the position being hedged is the dependent variable.

Formulas

Hedge ratio implied by the reverse regression

hreverse = σT ÷ (ρ × σH)

σT
the volatility of the target's yield changes
σH
the volatility of the hedging instrument's yield changes
ρ
the correlation between the two yield changes

It equals the correct β ÷ ρ², so it over-hedges whenever the correlation is below one in absolute value. Use the target-on-hedge slope instead.

A practice question

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

To compute a regression hedge for a position, which regression should be run?

  1. Regress yield changes of the hedging instrument on yield changes of the position and use that slope directly.Why not: Reverses the direction and also uses the slope without inverting it; the slope is the wrong quantity.
  2. Regress yield changes of the hedging instrument on the position's changes and use the reciprocal of that slope.Why not: The reciprocal of the reverse slope over-hedges unless the correlation is one.
  3. Regress yield changes of the position being hedged on yield changes of the hedging instrument and use that slope. - correct
  4. Either direction, since the hedge ratio comes out the same from the same data.Why not: The two directions give the same ratio only if the correlation is plus or minus one.

The hedged position is the dependent variable. The regression finds the line that minimizes squared errors in the target's yield change.

A harder one

Yield changes of a target and a hedging instrument have standard deviations of 6 bp and 8 bp, with correlation 0.75. By what factor does the hedge ratio implied by the reverse regression (hedge on target, inverted) exceed the correct regression hedge ratio?

  1. 1.78. - correct
  2. 1.33.Why not: Uses 1/rho instead of 1/rho-squared.
  3. 0.56.Why not: Reports rho-squared, which is the product of the two slopes, not the ratio of the two hedge ratios.
  4. 1.00.Why not: Reports the reverse-implied hedge ratio, 6/(0.75 x 8) = 1.0, rather than its ratio to the correct beta of 0.5625.

Correct beta = 0.75 x 6/8 = 0.5625. Reverse-implied ratio = 6/(0.75 x 8) = 1.0. Factor = 1/0.75² = 1.78.

The hardest tier

Yield changes have standard deviations of 5 bp (position) and 6 bp (hedging instrument), with correlation 0.60. Both have a DV01 of USD 100,000 per bp. An analyst sizes the hedge by the reciprocal of the reverse regression slope. The standard deviation of the resulting daily P&L is closest to:

  1. USD 400,000.Why not: This is the correct regression hedge: 5 x sqrt(1 - 0.36) = 4 bp.
  2. USD 1,202,000.Why not: Adds the covariance term instead of subtracting it, treating the hedge as if it were an extra long position.
  3. USD 500,000.Why not: This is the DV01-neutral or unhedged risk of 5 bp; it ignores the extra variance from the over-sized hedge.
  4. USD 667,000. - correct

Reverse-implied ratio h = 5/(0.6 x 6) = 1.389. Variance = 25 + h² x 36 - 2 x h x 0.6 x 5 x 6 = 25 + 69.44 - 50 = 44.44, so 6.67 bp = USD 667,000. The over-hedge is worse than no hedge at all.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Regression Hedging and Principal Component Analysis