Elan has been a partner with Biogen Idec Inc. on the drug since 2000, but announced earlier this month that it was ending that arrangement. As part of that deal, which is expected to close soon, Biogen will get full ownership of Tysabri and Elan will get the $3.25 billion payment, as well as royalties on all sales of the drug.
Elan has said that it plans to use part of the payment to buy back $1 billion of its stock, boosting value for the company's shareholders.
Royalty Pharma said Monday it plans to finance the acquisition through a combination of cash and debt.
Over the past 52 weeks, Elan's U.S. shares have traded between $9.37 and $15.27. Over the past year, the shares have lost about 15 percent of their value.