Investors are paying perhaps more than $2 billion a year because of changes in the portfolios that make up the Standard & Poor’s 500 and Russell 2000 indices, according a new study. The research, conducted by Vijay Singal of Virginia Tech’s Pamplin College of Business, Honhui Chen of the University of Central Florida, Orlando and Gregory Nornha of the Milgard School of Business at the University of Washington, Tacoma, found that investors are paying the price for losing stocks that are leaving the index and the higher cost of adding rising stocks. The paper concludes, according to CFA Institute, that the losses “result largely from index fund managers possessing mandates that require low tracking error and from the predictability and/or preannouncement of index changes.”