Roll Down Return a small clarifcation

CFA Level III Fixed Income
#1

can we say the following sentence is true for bonds that we plan on holding till maturity meaning that the roll down is price will go toward the par regardless of any yield curve changes?

“The roll down return demonstrates how the price of a bond typically moves closer to par regardless of yield curve changes over the strategy horizon.”

Whereas if we are not planning to hold the bond until maturity, there wont be any such as thing pull to par because depending on how the interest rate moves, we may seller it prior to maturity

#2

Roll down applies regardless of the intention of the holder.

If a bond is one day nearer to maturity, then its duration is pretty much one day less and the holder wears slightly less risk. This is what roll down is.

There is also another, smaller effect which is that this change in duration reduces slightly the bond convexity.

These two effects, together with accrued coupon, is carry.