Convexity in Barbell and Bullet

CFA Level III
#1

I have a question.

For bonds convexity is supposed to be a positive thing as when the Yields fall the bond with higher convexity rises more in price than a bond with lower convexity and likewise falls less when there is an increase in yield.

Barbell structure has higher convexity (dispersion) than a bullet structure portfolio.

now my question is when the slope of the yield curve increase the price decrease of a barbell portfolio is more than that of a bullet portfolio because the longer term bonds fall more in value than the increase in value of short term bonds in a barbell (due to the fact that Duration of Long term bonds in Barbell is greater than the short term bond’s duration in a barbell). Which makes sense.

But now in terms of convexity, shouldn’t the Barbell portfolio’s value decrease less than the bullet because of its higher convexity when the yield curve’s slope increase?

I hope i am able to explain my question well.

Thanks.

#2

Barbell portfolio value always decreases less than bullet portfolio value when rate rises due to its higher convexity.

#3

To be more specific, in rising interest rate scenario, the more extreme barbell the portfolio is, the better performer the portfolio is.