Ray Dalio sold his last remaining stake in Bridgewater Associates and stepped off its board, capping a tumultuous transition at the hedge-fund firm he founded. 

Bridgewater bought the remaining shares held by its charismatic billionaire founder, the firm told clients in a letter last week. People familiar with the matter said Bridgewater subsequently issued new shares to the sovereign-wealth fund of Brunei in a multibillion-dollar deal that gave the Southeast Asian fund a nearly 20% stake in the Connecticut firm.