The PHB (page 115) defines the maximum hit points of the Arcane Ward but not the initial hit points.
Arcane Ward
The ward has a hit point maximum equal to twice your wizard level + your intelligence modifier
Does the ward begin at maximum hit points or does it start with hit points equal to twice the spell level as with recharging the ward?
Whenever you cast an abjuration spell of 1st level or higher, the ward regains a number of hit points equal to twice the level of the spell
Best Answer
The ward begins at maximum hit-points. The first version of the PHB stated the same passage as:
I believe the change in text was made to address questions about the maximum, but in doing so I guess it introduced this question.