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Compute how much face value each of the multiple swaps must carry when a regression hedge uses two explanatory variables.

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

The idea

A regression hedge can use two explanatory variables, for example hedging a 10-year bond with a 2-year and a 30-year swap. Let Y be the target yield change, and S2 and S30 the changes in the 2-year and 30-year swap rates. Regress the target yield change on both swap-rate changes, as in the following equation. Y = α + β2 × S2 + β30 × S30 + ε. Each swap is then sized on its own. Let F be the face of the target, and DV01T and DV01S the DV01 per unit of face of the target and of that swap. Then use the following for each swap, scaled by that swap's own β. Face = F × (DV01T ÷ DV01S) × β. The DV01 of each swap therefore equals its β times the DV01 of the target. The betas are partial slopes, holding the other swap rate fixed, so they differ from simple one-variable slopes whenever the swap rates are correlated, and they need not sum to one. The usual errors are using a simple slope for each swap or applying the sum of the betas to each swap.

Formulas

Face amount of swap i in a multi-variable regression hedge

Faceᵢ = Target × (DV01T ÷ DV01ᵢ) × βᵢ

Target
the face amount of the position being hedged
DV01T
the DV01 of the position being hedged, per the same face amount as the swap's
DV01ᵢ
the DV01 of swap i
βᵢ
the partial slope on swap i's rate change in the multiple regression

Each swap uses its own partial beta, not a one-variable slope and not the sum of the betas. The betas need not sum to one.

A practice question

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

A bond is hedged with a 2-year swap and a 30-year swap, using a regression of target yield changes on both swap-rate changes with coefficients beta2 and beta30. The face amount of the 30-year swap is:

  1. Face of target x (DV01 of target / DV01 of 30-year swap) x beta30. - correct
  2. Face of target x (DV01 of target / DV01 of 30-year swap) x (beta2 + beta30).Why not: Applies the sum of the betas to one swap; each swap is sized with its own beta.
  3. Face of target x (DV01 of 30-year swap / DV01 of target) x beta30.Why not: Inverts the DV01 ratio, so a swap with a small DV01 would get too little face.
  4. Face of target x (DV01 of target / DV01 of 30-year swap) / beta30.Why not: Divides by beta30 instead of multiplying.

Each swap is sized so that its DV01 equals its own beta times the target DV01; that gives face_i = F_T x DV01_T / DV01_i x beta_i.

A harder one

A desk holds USD 100 million of a 10-year bond with DV01 0.082 per 100 face and hedges with a 2-year swap (DV01 0.019 per 100) and a 30-year swap (DV01 0.162 per 100). The regression of 10-year yield changes on the two swap-rate changes has beta2 = 0.32 and beta30 = 0.48. The face amount of the 30-year swap is closest to:

  1. USD 50.6 million.Why not: The face needed for a DV01 match with the 30-year swap alone; it applies no beta.
  2. USD 16.2 million.Why not: Uses beta2 = 0.32 on the 30-year swap by mixing the coefficients.
  3. USD 24.3 million. - correct
  4. USD 40.5 million.Why not: Applies the sum of the betas (0.80) to the 30-year swap.

30-year swap face = 100 x 0.082/0.162 x 0.48 = 100 x 0.5062 x 0.48 = USD 24.3 million.

The hardest tier

A position has a DV01 of USD 60,000 per bp. A regression hedge uses a 2-year swap and a 30-year swap with beta2 = 0.30 and beta30 = 0.55. The 30-year swap has a DV01 of USD 1,620 per bp per USD 1 million face. The combined DV01 of the two swap positions and the face amount of the 30-year swap are:

  1. USD 51,000 and USD 20.4 million. - correct
  2. USD 60,000 and USD 20.4 million.Why not: Forces the total hedge DV01 to equal the target DV01, as if the betas had to sum to one; here they sum to 0.85.
  3. USD 51,000 and USD 37.0 million.Why not: Sizes the 30-year swap to the whole target DV01 (60,000/1,620) and ignores beta30.
  4. USD 51,000 and USD 31.5 million.Why not: Puts the whole combined hedge DV01 into the 30-year swap (51,000/1,620) and ignores that part is carried by the 2-year swap.

Combined hedge DV01 = (0.30 + 0.55) x 60,000 = 51,000. The 30-year swap carries 0.55 x 60,000 = 33,000 of DV01, so its face is 33,000/1,620 = USD 20.4 million.

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