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Convexity in Barbell and Bullet

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.

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

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

Jahid Hassan