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Set out what a regression hedge is and show how it improves on the plain DV01-neutral hedge.

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

The idea

A regression hedge starts from the observed history of yield changes instead of assuming two yields move one for one. Regress the change in the yield of the position being hedged (the dependent variable) on the change in the yield of the hedging instrument (the explanatory variable). Write Target and Hedge for those two yield changes. The fitted line is as follows. Target = α + β × Hedge + ε. The DV01-neutral hedge amount is then scaled by β, so a β of 0.8 means hedging with 80 percent of the DV01-neutral amount. The DV01-neutral hedge is the special case of a β of 1. What remains unhedged is the error term, and R² is the share of variance removed. Comparing the variances of the P&L, the result is as follows. Hedged = (1 − R²) × Unhedged. The exam point is the direction of the regression, and that β adjusts the DV01 hedge rather than replacing it.

Formulas

Variance left after a regression hedge

σhedged² = (1 − R²) × σunhedged²

R²
the R² of the regression of target yield changes on hedge yield changes
σunhedged²
the variance of the unhedged P&L

It applies to variance; for the standard deviation take √(1 − R²). Multiplying the standard deviation by 1 − R² understates the residual risk.

A practice question

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

What is the main way in which a regression hedge improves on a DV01-neutral hedge?

  1. It removes every source of risk in the hedged position, including factors unrelated to the hedging yield.Why not: Overstates what the regression does; the error term, the part not explained by the hedging yield, remains as residual risk.
  2. It removes the need to compute DV01 for either security.Why not: The regression hedge still uses DV01s; beta only multiplies the DV01-neutral amount.
  3. It scales the hedge by the estimated historical sensitivity of the hedged yield to the hedging yield, rather than assuming equal yield moves. - correct
  4. It makes the position immune to curve twists, since a single beta captures slope changes.Why not: One beta links two yields only; twists that are unrelated to the hedging yield stay in the residual.

A DV01-neutral hedge is a regression hedge with beta = 1. Estimating beta from history lets the hedge reflect how much the target yield typically moves per basis point of the hedging yield.

A harder one

A regression of daily changes in the 30-year yield on daily changes in the 10-year yield gives a slope of 0.85. Compared with a DV01-neutral hedge of a 30-year position using 10-year bonds, the regression hedge:

  1. uses a larger offsetting position, because the slope is positive.Why not: Treats a positive slope as a reason to hedge more; only whether beta is below or above 1 sets the size relative to DV01 neutrality.
  2. uses the same offsetting position, because a positive slope confirms the yields move together.Why not: Treats the sign of the slope as the whole story; the size of beta scales the hedge.
  3. uses an offsetting position equal to 15% of the DV01-neutral hedge.Why not: Applies 1 - beta instead of beta; the hedge is beta, here 85%, of the DV01-neutral amount.
  4. uses a smaller offsetting position, because the 30-year yield has historically moved less than one-for-one with the 10-year yield. - correct

Regression hedge amount = beta x DV01-neutral amount = 0.85 x DV01-neutral amount, so the hedge is 15% smaller, not 15% of its size.

The hardest tier

The standard deviations of daily yield changes are 6 bp for the position being hedged and 8 bp for the hedging instrument, with correlation 0.90. Both have a DV01 of USD 100,000 per bp. The standard deviation of daily P&L under a regression hedge is closest to:

  1. USD 369,000.Why not: This is the DV01-neutral hedge: it uses beta = 1 and gives sqrt(36 + 64 - 2 x 0.9 x 6 x 8) = 3.69 bp.
  2. USD 114,000.Why not: Uses 6 x (1 - 0.81) rather than 6 x sqrt(1 - 0.81); 1 - rho-squared is a variance share.
  3. USD 262,000. - correct
  4. USD 600,000.Why not: This is the unhedged risk, 6 bp x USD 100,000; it ignores the hedge altogether.

Beta = 0.9 x 6/8 = 0.675. Residual standard deviation = 6 x sqrt(1 - 0.81) = 2.615 bp, times USD 100,000 = USD 261,500. The DV01-neutral hedge would give 3.69 bp = USD 369,000.

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