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Calculate the hedging instrument positions that offset the key rate risks held in a portfolio.

FRM® Part I · Valuation and Risk Models · Modeling Non-Parallel Term Structure Shifts and Hedging · FRM1.VR.13.d

The idea

To hedge key rate risk, choose one instrument per key rate and make the hedged portfolio's KR01 at every key rate equal zero. Each instrument usually has KR01s at several key rates, not only its own, so the positions come from solving a system of equations. At each key rate, the portfolio KR01 plus the sum over instruments of position × instrument KR01 at that key rate must equal zero. If the longest instrument touches all key rates and the shortest touches one, solve from the longest maturity down, using each solved position to adjust the exposures at shorter key rates. Positions come out as face amounts, found by dividing by the instrument's KR01 per unit of face. A positive portfolio KR01 needs an offsetting short position in instruments with positive KR01s. Common errors are ignoring cross exposures, getting the sign wrong, and mixing up the unit of face.

Formulas

Face amount of a hedge for one key rate

N = −KR01p ÷ KR01h

KR01p
the portfolio's KR01 at the key rate, after any cross exposures from hedges already sized
KR01h
the hedge instrument's KR01 at that key rate per unit of face

A negative N is a sale. When hedges touch several key rates, solve from the longest down; dividing the other way round, or forgetting the cross exposures, gives the wrong size.

A practice question

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

A portfolio is exposed to three key rates (2, 5 and 10 years) and each hedging bond has KR01s at several of them. To neutralise the KR01 at all three key rates, what is generally required?

  1. One instrument, with its maturity chosen to match the portfolio's duration.Why not: Matches only the total DV01; the key rate exposures at the other maturities remain.
  2. Two instruments, since the KR01s are constrained to add up to the DV01.Why not: The sum constraint does not remove a key rate; each key rate needs its own condition and an instrument to meet it.
  3. Three instruments, with positions found by solving three equations together. - correct
  4. Six instruments, one for every pair of key rates.Why not: Confuses the number of hedges with the number of key rate pairs; one instrument per key rate is enough.

Zero KR01 at each of N key rates is N conditions, so N instruments with enough variation in maturity are needed. Because each bond touches several key rates, the positions come from solving the equations together.

A harder one

A portfolio has KR01s (gain from a one basis point fall, USD) of 2,100 at the 5-year key rate and 3,500 at the 10-year key rate. The hedging instruments are a 10-year bond with KR01s of 150 at 5 years and 700 at 10 years per USD 1 million face, and a 5-year bond with a KR01 of 420 at 5 years per USD 1 million face and none at 10 years. The 10-year bond has been sized to hedge the 10-year key rate. What position in the 5-year bond completes the hedge?

  1. Short USD 5.00 million faceWhy not: Ignores the 10-year bond's own KR01 at 5 years, hedging the full 2,100 with the 5-year bond.
  2. Short USD 3.21 million face - correct
  3. Short USD 6.79 million faceWhy not: Adds the cross exposure instead of subtracting it: (2,100 + 750) / 420 should be (2,100 - 750) / 420.
  4. Short USD 1.93 million faceWhy not: Divides the remaining 5-year exposure by the 10-year bond's 10-year KR01 (700) instead of the 5-year bond's own KR01 (420).

The 10-year bond position is -3,500 / 700 = -5 million face. It contributes 150 x (-5) = -750 at 5 years, leaving 2,100 - 750 = 1,350. The 5-year bond must offset that: -1,350 / 420 = -3.21 million face.

The hardest tier

A portfolio has KR01s (gain from a one basis point fall, USD) of 800 at the 2-year key rate, -300 at the 5-year and 1,200 at the 10-year. The hedging bonds have these KR01s per USD 1 million face. The 2-year bond: 190 at 2 years. The 5-year bond: 150 at 2 years and 450 at 5 years. The 10-year bond: 20 at 2 years, 110 at 5 years and 760 at 10 years. What position in the 2-year bond makes the hedged portfolio's KR01 zero at all three key rates?

  1. Short USD 4.21 million faceWhy not: Hedges the 2-year KR01 of 800 with the 2-year bond alone, ignoring the cross exposures of the 5-year and 10-year hedge positions at 2 years.
  2. Long USD 4.88 million faceWhy not: Gets the sign wrong: the 2-year exposure after the other hedges is positive, so the hedge must be a sale.
  3. Short USD 4.88 million face - correct
  4. Short USD 4.16 million faceWhy not: Flips the sign on every cross exposure when solving for the 5-year and 2-year bond positions.

Solve from the longest bond down. 10-year: -1,200 / 760 = -1.579. 5-year: -(-300 + 110 x -1.579) / 450 = +1.053. At 2 years the exposure is 800 + 150 x 1.053 + 20 x -1.579 = 926.3, so the 2-year bond position is -4.88 million face.

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