It’s the dead of winter, not the beginning of a new fiscal or calendar year, so why January 20th? 
Blame it on the 20th amendment to the U.S. Constitution which was ratified in 1933. It states in Section 1:
The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3rd day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
The above taken from a good explanation found at Wikipedia,
