Why old battleships escorted high-value convoys

Why were battleships assigned to convoy escort during the Second World War?

Their presence made an attack too costly and slow for enemy surface raiders, protecting the convoy even when the escort was older or weaker.

A surface raider attacking a protected convoy had to fight the escort while the convoy escaped and its position was reported. Even a favorable duel could leave the raider damaged far from repair and vulnerable to pursuing forces. The escort did not have to sink Bismarck, Scharnhorst, or a similar attacker to defeat the raid's purpose. The battleship's value was therefore deterrence and time, not merely the probability of winning a one-on-one engagement.