Billionaire hedge-fund manager George Soros lost nearly $1 billion as a result of the stock-market rally spurred by Donald Trump’s surprise presidential election.

But Stanley Druckenmiller, Mr. Soros’s former deputy who helped Mr. Soros score $1 billion of profits betting against the British pound in 1992, anticipated the market’s recent climb and racked up sizable gains, according to people close to the matter.

The...