https://www.sefaria.org/Mishnah_Rosh_Hashanah.1?ven=The_Mishna_with_Obadiah_Bartenura_by_Rabbi_Shraga_Silverstein&lang=bi

There are four New Years: The first of Nissan is the New Year for kings and for festivals. [The kings of Israel count from Nissan. If a king began to reign in Shevat or Adar, when Nissan arrives, that year ends and they begin to count a second year for him. This is derived from (I Kings 6:1): "And it came to pass in the four hundred and eightieth year after the children of Israel went out of the land of Egypt, in the fourth year, in the month of Ziv, which is the second month, of the reign of Solomon over Israel, etc." The reign of Solomon is being likened to the exodus from Egypt. Just as the exodus from Egypt is counted from Nissan, so the reign of Solomon is counted from Nissan. And the reign of the gentile kings is counted from Tishrei. As stated below: "On the first of Tishrei is the New Year for years"; that is, for the years of the gentile kings." A (practical) difference: They used to indicate the date on bills (of payment) according to the years of the king's reign, for the sake of shalom malchuth ("the peace of the kingdom"). And if a king began reigning on Adar, and a scribe wrote (on a bill) in Nissan thirteen months later: "Nissan of the second year," since Nissan is the New Year for kings, this is a shtar mukdam ("a pre-dated bill") and is invalid; for that Nissan was the Nissan of the third year.

(http://www.autoadmit.com/thread.php?thread_id=3921801&forum_id=2#35629623)