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  1. 069Portfolio A holds only 5-year zero-coupon bonds. Portfolio B holds 2-year and 8-year zeros, weighted so its duration is also 5. Which portfolio does better after a large parallel move in yields, and which one loses if the curve steepens?Duration and ratesHardTreasury and ALMBank market risk

    Try it first

    After a 200 basis point parallel move in either direction, which portfolio is ahead?

    Show the worked solution

    The barbell wins a large parallel move either way; it loses when the curve steepens. Both portfolios have duration 5, but the barbell's convexity is 34 against the bullet's 25. For a 200 basis point rise the barbell loses 9.35% against 9.52%, and for a fall it gains 10.72% against 10.52%. If the curve steepens 10 basis points per year around 5 years, the bullet is flat and the barbell loses 0.88%.

    If the durations match, why do the portfolios behave differently?

    Two families each spend an average of Rs 50,000 a month. One spends it evenly; the other spends Rs 20,000 some months and Rs 80,000 others. A small price rise hits them alike, but a big shock hits the second family's heavy months much harder. Averages match; the spread does not. Duration is the average maturity, so it only describes small, parallel moves; the barbell spreads its money further from 5 years, which gives it more convexity and exposes it to changes in the curve's shape.

    Same duration of 5: the barbell wins big parallel moves and loses a steepener0y2y5y8y10yMaturityafter: steeperbeforebullet50%50%yieldBarbell: 0.5 x 2y + 0.5 x 8y, duration 5, convexity 34 against 25ScenarioBulletBarbell+200 bp parallel-9.52%-9.35%barbell better-200 bp parallel+10.52%+10.72%barbell betterSteepener+0.00%-0.88%barbell worseEqual duration is not equal risk:convexity bought, curve risk sold
    The bullet sits at 5 years and the barbell splits evenly between 2 and 8 years, both at duration 5. The barbell does better in a 200 basis point parallel move either way because its convexity is 34 against 25, and it loses 0.88% when the curve steepens around the 5-year point while the bullet is unaffected.

    Why does the steepener hurt only the barbell?

    In this steepener the 5-year yield does not move, so the bullet does not move. The 2-year yield falls 30 basis points and the 8-year yield rises 30. The barbell's long leg carries four times the duration of its short leg, so the loss on the 8-year bond swamps the gain on the 2-year. Roughly: half the money times 8 years times 0.30% is a 1.2% loss, and half times 2 years times 0.30% is a 0.3% gain, a net loss of about 0.9%.

    The relationship
    w⋅2+(1−w)⋅8=5⇒w=0.5Cbarbell=0.5(22)+0.5(82)=34>25=52w\cdot 2 + (1-w)\cdot 8 = 5 \Rightarrow w = 0.5 \qquad C_{\text{barbell}} = 0.5(2^2) + 0.5(8^2) = 34 > 25 = 5^2
    wthe share of value in the 2-year zero
    Cconvexity; for a zero with continuous compounding it is maturity squared
    What it says in wordsMatching duration fixes the weights at half and half, and the barbell ends up with more convexity.

    The limitation: the gains from convexity are small for small moves and are usually priced, since a barbell typically yields less than a bullet of the same duration. In a quiet market that yield give-up can outweigh the convexity benefit, so the choice is a view on volatility and curve shape, not a free lunch.

    Where candidates lose it

    The trap is saying the two are the same risk because duration matches. Duration is one number describing one kind of move, and the interviewer built the question to see if you know the two other risks it misses: convexity and curve shape.

    The second slip is getting convexity right and missing the steepener. Give both halves: the barbell wins a big parallel move and loses when the long end sells off relative to the short end.

    What the interviewer asks next

    • Which portfolio is hurt by a flattening instead?
    • How would you hedge the barbell's steepener exposure?
    • Why does a barbell usually yield less than a bullet of the same duration?