The Collaborative International Dictionary of English v.0.48:

Presidentship \Pres"i*dent*ship\, n.
   The office and dignity of president; presidency. --Hooker.
   [1913 Webster]

WordNet (r) 3.0 (2006):

presidentship
    n 1: the office and function of president; "Andrew Jackson
         expanded the power of the presidency beyond what was
         customary before his time" [syn: presidency,
         presidentship]