If MoneyGram money transfer, for some reason has not been received by the addressee, the sender will receive a refund. The company continues to automate and enhance MoneyGram MoneyGram launched a new in-person bill payment service which will be delivered through agents in their domestic network, giving consumers an easy, low-cost method to pay their bills. Until 2004, the owners of MoneyGram were Viad Corp and Travelers Express Company Inc.