Portfolio Management puzzles, solved step by step
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100You can hold a 7-year bullet yielding 7.0%, or a barbell of 2-year bonds at 6.5% and 12-year bonds at 7.2% with the same duration. Which does better if yields shift 100 basis points in parallel, either way? What does the barbell give up?Fixed incomeInstitutional asset management
Try it first
On an immediate 100 basis point parallel move, up or down, which wins?
Show the worked solution
The barbell wins both ways on an immediate parallel move, by about 0.12 points if yields fall and 0.10 if they rise, but it yields about 0.15 points less a year. With duration matched, convexity is the only difference: 71 for the barbell against 49 for the bullet. Over a year the barbell needs a move of roughly 113 to 123 basis points to earn back its lower yield, and it loses if the curve steepens.
Why does spreading money to the two ends add convexity?
Balancing a plank on a see-saw with one heavy child in the middle is steady; balancing it with a child at each end makes it swing more for the same push. Price sensitivity grows faster than linearly with maturity, so a mix of short and long bonds with the same average duration as a middle bond has more curvature than the middle bond: here about 71 against 49. More convexity means gaining a little more when yields fall and losing a little less when they rise.
The bullet and barbell price changes almost coincide across parallel shifts, but the barbell's advantage grows with the square of the move, from about 0.10 to 0.12 points at 100 basis points to over 0.7 at 300; it only clears the 0.15 points of yield it gives up each year beyond moves of roughly 113 to 123 basis points. What does the barbell give up, and when does it lose?
First, yield. Weighting the 2-year at about 50% to match duration gives a blended yield near 6.85%, about 0.15 points below the bullet's 7.0%. That is the price of convexity: if yields sit still, the bullet simply earns more, and over a year the barbell's edge only pays for itself on a parallel move of more than about 1.2 points. Second, curve shape. If the curve steepens, with short yields falling and long yields rising, the barbell's long leg loses and the bullet wins; if the 7-year sector rallies against the two ends, the bullet wins too.
Bullet Barbell Yield, blended 7.00% 6.85% Modified duration 6.54 6.54 Convexity 48.9 70.6 Yields fall 100 bp +6.79% +6.91% Yields rise 100 bp -6.30% -6.20% Priced as zero-coupon bonds with modified duration matched at 6.54, the barbell gains 6.91% against 6.79% on a 100 basis point fall and loses 6.20% against 6.30% on a rise, while yielding 0.15 points less. State the assumptions: zero-coupon bonds, a blended yield taken as a weighted average, which is an approximation, and an instantaneous shift. The conclusion survives all three: the barbell buys protection against large parallel moves with yield, and takes on curve-shape risk.
Where candidates lose it
The common slip is saying same duration, same result. Duration matching removes only the first-order difference, and the question is about what is left over.
The second loss is praising the barbell without the cost. It gives up yield and carries curve risk, and a candidate who names both sounds like someone who has run a bond book.
What the interviewer asks next
- How would you build the barbell to match both duration and yield? What would you give up?
- Which does better if the curve flattens with no change in the 7-year yield?
- Why do liability-driven investors often prefer bullets matched to their payment dates?