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Bonds with a longer duration will have a greater impact on price change, generally speaking. This also depends on how it is weighted, though I think the barbell is typically 50/50.
When rates rise, the value of the long term bonds will decrease and the value of short term bonds increase. Given a $100 million portfolio:
$50m in ST bonds with a duration of 1
$50m in LT bonds with a duration of 10
Short term rates decrease by 5%
Long term rates increase by 5%
The value of the ST bonds will increase by $2.5 million ($50*-.05*-1)
The value of LT bonds will decrease by $25 million ($50*.05*-10)
Let's also say you have a bullet with a duration of 5. In this case it would be insensitive to changes in the steepening of the curve. Because the cash flows are centered around maturity, the bullet will be most sensitive to whatever happens to the 5 year rate.
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