if the yield curve parallel shifts upward, a portfolio with higher convexity decrease less than a portfolio with lower convexity. thus, a higher convexity portfolio should be used instead of lower convexity portfolio. ( cfa level 2)
if the yield curve parallel shifts upward, we should use bullet portfolio rather than barbell pofolio ( cfa level 3)
question: which one is correct? because the barbell has higher convexity, should be used in situation 1 but not in situation 2?
Higher convexity portfolios will outperform in parallel shifts of yield curve.
Convexity is valuable because bond will suffer less when rates rise and gain more when rates go down.
I am not sure where you got point 2 but in CFAI reading 20 it clearly states, for example:
In an instantaneous downward parallel shift, as illustrated in Exhibit 22, the higher-convexity barbell portfolio AC will outperform bullet portfolio B slightly because of Portfolio AC’s greater sensitivity to declining yields and rising prices.
#2 is not correct. Under the assumption that the durations are the same for the bullet and barbell, the barbell has higher convexity and therefore outperforms in a parallel upward shift.
Perhaps you took this from an example where the durations were not equal?