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Construct a barbell that has the same cost and duration as a particular bullet investment, then explain what each of the bullet and barbell structures does well and badly.

FRM® Part I · Valuation and Risk Models · Applying Duration, Convexity, and DV01 · FRM1.VR.12.i

The idea

A bullet concentrates cash flows around one maturity, while a barbell splits the same money between a short bond and a long bond and holds nothing in the middle. To match the bullet's cost and duration, the weights must sum to 1 and the weighted duration must equal the bullet duration. Solving those two conditions gives the weight in the long bond as wlong = (Dbullet − Dshort) ÷ (Dlong − Dshort). The rest goes in the short bond. Because the barbell's cash flows are more spread out, it has higher convexity, so it does better in large parallel yield moves in either direction. In a market that prices convexity it usually gives up yield. A twist in the curve can make the two perform differently. A common error is reversing the weights.

Formulas

Weight in the long bond of a cost- and duration-matched barbell

wL = (DB − DS) ÷ (DL − DS)

DB
the bullet's duration
DS
the short bond's duration
DL
the long bond's duration

The short bond takes 1 − wL, so the barbell has the bullet's cost and duration. Swapping the two weights is the common slip.

Convexity of a barbell

CBar = wS × CS + wL × CL

wS
the weight in the short bond
CS
the short bond's convexity
wL
the weight in the long bond
CL
the long bond's convexity

Convexity is a value-weighted average of the parts. Taking a simple average, or swapping the weights, gives the wrong figure.

A practice question

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

Which description of bullet and barbell structures is correct?

  1. A bullet concentrates cash flows around one maturity, while a barbell holds short and long maturities with nothing in the middle. - correct
  2. A bullet holds short and long maturities and a barbell concentrates cash flows around one maturity.Why not: Reverses the two definitions.
  3. Both hold bonds across all maturities but in different proportions.Why not: Neither structure holds a spread across all maturities; a barbell has a gap in the middle.
  4. A bullet holds only zero-coupon bonds and a barbell holds only coupon bonds.Why not: Coupon type does not define the structure; the distribution of maturities does.

A bullet clusters holdings at one point on the curve. A barbell splits them between the short and long ends, leaving the intermediate maturities empty.

A harder one

A bullet has a market value of USD 100 million and a duration of 7.5. A barbell of the same cost uses a short bond with duration 1.9 and a long bond with duration 16.0. The investment in the long bond is closest to:

  1. USD 60.3 millionWhy not: Swaps the weights, putting the short bond's share in the long bond.
  2. USD 39.7 million - correct
  3. USD 50.0 millionWhy not: Splits the investment equally, which gives a duration of 8.95 not 7.5.
  4. USD 46.9 millionWhy not: Uses the ratio of the bullet duration to the long duration (7.5 / 16.0).

Weight in long = (7.5 - 1.9) / (16.0 - 1.9) = 5.6 / 14.1 = 0.3972. Long bond investment = USD 39.7 million and the short bond receives USD 60.3 million.

The hardest tier

A USD 100 million bullet has convexity 70.0 and a cost- and duration-matched barbell has convexity 153.9. Ignoring yield differences, the estimated value difference between the barbell and the bullet after a parallel move of 150bp in either direction is closest to:

  1. The bullet is ahead by USD 944,000Why not: Sign error: higher convexity helps the barbell in both directions.
  2. The barbell is ahead by USD 1,888,000Why not: Omits the factor of 0.5 in the convexity term.
  3. The barbell is ahead by USD 944,000 - correct
  4. There is no difference, because the durations are equalWhy not: Ignores the second-order term, which is the source of the difference.

Convexity difference = 83.9. Value difference = 0.5 x 83.9 x 0.015² x 100 million = USD 943,875 in favor of the barbell, in either direction.

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