Performance of Duration-Neutral Bullets and Barbells Given a Change in the Yield Curve
Imagine two very simple portfolios of bonds. Both have a modified duration of exactly 10. One is a bullet portfolio of zero-coupon bonds of maturity 10.2 years, while the other is evenly weighted between coupon bonds with a mix of near-zero maturity (A) and 20.5 years (B).
| Portfolio | Maturity | Yield | Duration |
|---|---|---|---|
| Bullet | 10.2 | 4% | 10 |
| Barbell (A) | 0 | 2% | 0 |
| Barbell (B) | 20.5 | 5% | 20 |
The duration here is the modified duration, so it's the maturity divided by the gross semiannual yield in each case. Which portfolio will have a higher yield?
Exactly.
You can see that this portfolio has a 4% yield, while the barbell will have a 3.5% yield. How would you relate this result to convexity?
No.
The barbell yield will be 3.5%, and that's no higher than that of the bullet.
No.
Consider the expected yield of the barbell, and compare it to the 4.0% yield of the bullet.
Right.
No.
Just the opposite—lower yields for higher convexity portfolios.
The barbell will have more convexity, and that general benefit requires that the bullet is compensated with higher yield. It's not a rule, but it reflects the most common yield curve shape.
Now try some moves. Maybe the curve shifts in parallel: all rates drop by 50 bps. Given the match in duration and difference in convexity, which portfolio will outperform?
Absolutely. Same duration, more convexity.
No, the barbell portfolio will.
It has the same duration but will also benefit from a higher convexity adjustment.
Would the "winner" be different if the parallel yield curve shift was an increase?
You're right!
Actually, it won't.
The price adjustment for positive convexity is positive either way, so the barbell wins.
Now consider some twists. Maybe the 20-year yield stays the same, while the 10-year yield rises, and the near-zero yield rises more. The bullet is clearly going to lose value, since that 10-year yield is rising. What do you think will happen to the barbell?
Not quite.
This is tricky, but consider that the 20-year yield isn't changing, and the near-zero rate has a duration of zero.
No.
While "a rising tide lifts all boats," as they say, rising yields don't help bond prices.
Correct.
The 20-year yield isn't changing, and the near-zero rate has a duration of zero. So there's no market value change for the barbell, and the barbell again outperforms.
Maybe that was a trick of how the curve flattened, so try flattening with a twist instead. Suppose the near-zero yield rises 100 bps while the 20-year yield falls 100 bps, and the 10-year yield is unchanged. The curve just rotates a little around that 10-year point, flattening that way. How would you expect these portfolios to perform?
No.
Yields are changing here, and only the near-zero yield can be assumed to have zero effect when it moves.
No.
The bullet is not a winner here. Consider that there is really only one yield causing a price change.
Yes!
The barbell benefits from that lower 20-year yield, so the price is going up. The near-zero yield doesn't have a price effect, and the 10-year yield isn't changing.
So the general rule is that a flattening of the curve helps barbells more than bullets. If you play this logic in reverse, you'll see that a steepening of the curve generally helps bullets more than barbells.
In the real world, nothing is quite this simple. But you can still place these two portfolios in the back of your mind to digest situations like this:
| Maturity | Partial BPV, Portfolio X | Partial BPV, Portfolio Y |
|----|-----|-----|
| 1 Year | 0.0021 | 0.0043 |
| 3 Year | 0.0068 | 0.0058 |
| 5 Year | 0.0107 | 0.0095 |
| 10 Year | 0.0114 | 0.0102 |
| 20 Year | 0.0041 | 0.0040 |
| 30 Year | 0.0185 | 0.0251 |
If you expected the yield curve to flatten in the very near future, which portfolio would you probably rather be holding?
No, Portfolio Y would be a better choice.
Look back at the sensitivities here. What does Portfolio Y look like, compared to portfolio X—more like a bullet or a barbell?
Excellent.
And why?
No.
Portfolio X is more like a bullet. It has greater sensitivities away from the extremes.
There you go!
Right again!
Portfolio Y has greater sensitivities at the extremes, making it more like a barbell. Barbells do better when the curve flattens. And those BPV estimates can even give you an estimate of the price changes.
To summarize:
[[summary]]
The bullet portfolio
The barbell portfolio
They should be the same
Lower yields for higher-convexity portfolios
Higher yields for higher-convexity portfolios
The bullet portfolio
The barbell portfolio
No
Yes
No change
It will gain value
It will lose value as well
They would both be unchanged
The bullet would outperform the barbell
The barbell would outperform the bullet
Portfolio X
Portfolio Y
A bullet
A barbell
It's more like a bullet
It's more like a barbell
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