Pepsi soft drinks are displayed at a convenience store in San Francisco, California.Justin Sullivan | Getty ImagesPepsiCo shares popped Tuesday after Elliott Investment Management took a significant stake as the activist investor sees a “rare” and “historic” opportunity for a turnaround in the iconic soft drink giant.Shares of PepsiCo climbed more than 5% in premarket trading. The stock is down about 2% this year, significantly lagging the broader market.Loading chart…The Paul Singer-founded Elliott’s bet in Pepsi is worth $4 billion, becoming the consumer giant’s top five active investors excluding index funds, according to FactSet. The activist investor sent a presentation and letter to Pepsi’s board of directors Tuesday, detailing a clear agenda focused on restoring business momentum.”While unfortunate, this disappointing trajectory has created a historic opportunity: With the right mindset and an appropriately ambitious turnaround plan, PepsiCo today represents a rare chance to revitalize a leading global enterprise and unlock significant shareholder value,” Elliott wrote in its letter.The Wall Street Journal first reported Elliott’s new stake earlier Tuesday.Elliott said it hopes to work together to help Pepsi build on its legacy of success and achieve its full potential.Pepsi has been cutting costs and trying to improve its profit margins. The company closed two manufacturing plants for its North American food business during the quarter. Pepsi said it is trying to make its transportation and logistics more efficient. The company is also evaluating how it spends its marketing dollars to make sure it is getting the best return on its investment.In July, Pepsi reported quarterly earnings and revenue that topped analysts’ expectations, as the company projected that weak North American demand will rebound as strategy changes take hold.Elliott has a history of activism that has at times yielded strong returns for investors. It’s a large holder of Phillips 66 and Southwest Airlines and has been driving changes at those two companies. …