Mortgage Settlement Monitoring Act of 2013 - Establishes the Office of the Independent Monitor to determine the compliance with its terms of all parties to the amended consent orders (settlement) finalized on February 28, 2013, between the Board of Governors of the Federal Reserve System and the Office of the Comptroller of the Currency and 11 mortgage servicing companies subject to enforcement actions for unsafe and unsound practices related to residential mortgage loan servicing and foreclosure processing, as well as any future agreement between the Board and the Office and a mortgage servicing company.