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Compare forward bucket 01s with KR01s, after defining forward bucket 01 and outlining an interest rate bucketing approach.

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

The idea

A forward bucket 01 measures the change in portfolio value when the forward rates within one maturity bucket fall by one basis point and all other forward rates stay unchanged. The bucketing approach splits the curve into adjacent, non-overlapping maturity ranges, such as 0 to 2, 2 to 5 and 5 to 10 years. It shifts the forward rates in one bucket at a time, producing a step shaped shift. The forward bucket 01s sum to the DV01, just as KR01s do. For a zero coupon bond with continuous compounding, the forward bucket 01 of a bucket is FB01 = PV × 0.0001 × the number of years of the bucket that fall before the bond's maturity. Unlike KR01s, which are spread across neighbouring key rates, forward bucket 01s identify exposure to specific forward rates, so a forward start position shows risk only in its own bucket.

Formulas

Forward bucket 01 of a zero coupon bond

FB01 = PV × 0.0001 × Y

PV
the bond's present value
Y
the years of the bucket that fall before the bond's maturity

Continuous compounding. Y is capped by the bucket's width and is zero for buckets beyond maturity; using the full bucket width for a bond that matures inside it overstates the figure.

A practice question

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

Which definition of a forward bucket 01 is correct?

  1. The change in portfolio value for a one basis point decrease in one spot rate, with neighbouring spot rates adjusted by interpolation.Why not: Describes KR01, which shifts a spot (key) rate with interpolation rather than a block of forward rates.
  2. The change in portfolio value for a one basis point decrease in all forward rates at once.Why not: This is the DV01 of a parallel shift, which equals the sum of all the forward bucket 01s.
  3. The change in value for a one basis point decrease in forward rates within one maturity bucket, other forward rates unchanged. - correct
  4. The change in a forward contract's price for a one basis point change in the spot price of its underlying.Why not: Confuses interest rate buckets with the delta of a forward contract.

A forward bucket 01 shifts only the forward rates inside one bucket (for example the 5 to 10 year forward rates), producing a step in the forward curve, and measures the value change. The bucket values sum to the DV01.

A harder one

Forward rates are continuously compounded. A position is a single zero coupon bond maturing in 8 years with a present value of USD 3,000,000. Buckets are 0 to 2, 2 to 5, 5 to 10 and 10 to 30 years. Using PV x 0.0001 x (years of the bucket before maturity), the forward bucket 01 for the 5 to 10 year bucket is closest to:

  1. USD 1,500Why not: Uses the full 5 year width of the bucket although the bond matures after 3 of those years.
  2. USD 900 - correct
  3. USD 2,400Why not: Uses the whole DV01 (3,000,000 x 8 x 0.0001) instead of the part due to this bucket.
  4. USD 1,440Why not: Reports the KR01 at the 10-year key rate (weight 0.6 of the DV01), not a forward bucket 01.

Only the years of the bucket before the bond's maturity matter: from 5 to 8 years is 3 years. Forward bucket 01 = 3,000,000 x 0.0001 x 3 = USD 900.

The hardest tier

Forward rates are continuously compounded. A portfolio is long a 3-year zero coupon bond with a present value of USD 1,000,000, long a 6-year zero coupon bond with a present value of USD 2,000,000 and short a 12-year zero coupon bond with a present value of USD 1,500,000. Using PV x 0.0001 x (years of the bucket before maturity) for each bond, the forward bucket 01 for the 5 to 10 year bucket is closest to:

  1. -USD 850Why not: Measures the 12-year bond's exposure from 5 years to its maturity (7 years) instead of capping at the end of the bucket (5 years).
  2. USD 950Why not: Ignores the sign of the short 12-year position, adding its exposure instead of subtracting it.
  3. USD 250Why not: Reports the 2 to 5 year bucket result, where the 3-year and 6-year bonds contribute and the 12-year bond is short.
  4. -USD 550 - correct

3-year bond: no exposure beyond 5 years. 6-year bond: 1 year in the bucket, 2,000,000 x 0.0001 x 1 = 200. 12-year short bond: 5 years in the bucket, -1,500,000 x 0.0001 x 5 = -750. Net = -550.

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