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Account for the shortcomings of hedging a bond position with a DV01-neutral hedge.

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

The idea

A DV01-neutral hedge sizes the offsetting position so that its DV01 equals the DV01 of the position being hedged. Let Target be the face value of the position being hedged, and DV01T and DV01H the DV01 per unit of face of the target and of the hedge. The face value of the hedge then follows from this relation. Hedge = Target × (DV01T ÷ DV01H). The position is protected only if the two yields change by the same number of basis points, that is, a parallel shift. In practice yields at different maturities do not move one for one. A 10-year yield might move only 0.7 bp for each 1 bp move in the 2-year yield, and the curve also steepens, flattens and twists. If the hedged yield moves less than the hedging yield, the DV01-neutral hedge is too large and the position takes on the risk of the hedging instrument itself. The exam point: DV01 neutrality is exact only for equal yield changes, and the hedging error comes from yield changes that are not one for one.

Formulas

Face amount of a DV01-neutral hedge

Hedge = Target × (DV01T ÷ DV01H)

Target
the face amount of the position being hedged
DV01T
the DV01 of the position being hedged, per the same face amount as the hedge's
DV01H
the DV01 of the hedging instrument

Exact only if both yields move by the same number of basis points. If the hedged yield moves less than one-for-one, this hedge is too large.

A practice question

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

A trader hedges a bond position with a DV01-neutral offsetting position. Which assumption about yield changes does this hedge rely on?

  1. The yield of the hedged bond and the yield of the hedging instrument change by the same number of basis points. - correct
  2. Yield changes at the two maturities are uncorrelated, so the two risks diversify away.Why not: Confuses hedging with diversification; a DV01-neutral hedge relies on one-for-one co-movement, not zero correlation.
  3. The hedged yield changes by the hedging yield change multiplied by the ratio of the two durations.Why not: Adds a duration-ratio scaling that the DV01 hedge does not make; DV01 already scales dollar sensitivity and the yield moves are assumed equal.
  4. Longer-maturity yields always move by more basis points than shorter-maturity yields.Why not: Assumes a fixed volatility ordering across maturities, which the hedge never uses and which the data do not support.

DV01 gives the price change per basis point for each security. Matching DV01s offsets P&L only if both yields move by the same number of basis points, a parallel shift.

A harder one

A trader is long a 10-year note and short a 2-year note, with equal DV01 so the hedge is DV01-neutral. Which market move produces a loss on this combined position?

  1. Both yields rise by 25 bp.Why not: A parallel shift is the case the hedge is built for; the loss on the long leg is offset by the gain on the short leg, so net P&L is about zero.
  2. The 10-year yield falls 10 bp while the 2-year yield falls 4 bp.Why not: Reverses which leg gains most: the long 10-year gains 10 units and the short 2-year loses only 4, a net gain.
  3. The 2-year yield falls 10 bp while the 10-year yield falls 4 bp. - correct
  4. The 2-year yield rises 10 bp while the 10-year yield rises 4 bp.Why not: Overlooks that the short 2-year leg gains when its yield rises; it gains 10 units against a 4 unit loss on the long 10-year, a net gain.

Net P&L per unit of DV01 = (change in 10-year yield with sign flipped) minus (change in 2-year yield with sign flipped) for a long 10-year and short 2-year position. With yields falling 4 and 10 bp, the long gains 4 and the short loses 10, so the combined position loses.

The hardest tier

A position long a 10-year note and short a 2-year note is DV01-neutral, with each leg carrying a DV01 of USD 90,000 per bp. Past data show the 10-year yield moves 0.60 bp per 1.00 bp move in the 2-year yield. If the 2-year yield now falls 25 bp, the expected P&L of the hedged position, using that relationship, is closest to:

  1. a gain of USD 900,000.Why not: Flips the sign: the short 2-year leg loses 25 bp of DV01, which more than offsets the gain on the long 10-year leg.
  2. a loss of USD 900,000. - correct
  3. a loss of USD 1,350,000.Why not: Takes the 10-year leg move (15 bp x 90,000) as the net result and shows it as a loss, ignoring the 2-year hedge leg altogether.
  4. a gain of USD 1,500,000.Why not: Inverts beta, expecting the 10-year yield to fall 25/0.6 = 41.7 bp, and so shows a net gain.

Expected 10-year yield fall = 0.60 x 25 = 15 bp. Long 10-year gains 15 x 90,000 = 1,350,000; short 2-year loses 25 x 90,000 = 2,250,000; net = -900,000. The DV01-neutral hedge is too big because the 10-year yield moves less than one-for-one.

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