Shrewd, sagacious, perspicacious, astute can all mean acute in perception and sound in judgment, especially in reference to practical affairs.
Shrewd implies native cleverness, acumen, and an exceptional ability to see below the surface; it often also connotes hardheadedness.
Sagacious is usually applied to persons or their decisions, their judgments, and their methods of pursuing their ends; it stresses penetration, discernment, judiciousness, and often, farsightedness.
Perspicacious is applied chiefly to mental sight or insight and suggests unusual power to see through and to understand what is dark, hidden, mysterious, or puzzling.
Astute implies a combination of shrewdness and perspicacity and often, in addition, connotes an ability to keep one’s counsel or an incapacity for being fooled, especially where one’s own interests are concerned.
Astute, opprobriously used, heightens the suggestion, sometimes present in perspicacious, of artfulness, diplomacy, or craft. It may connote merely shrewd discernment and sagacity.