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 At the heart of MoneyGram is trust and reliability of customers. You can quickly send and receive money, as money transfers usually come to the recipient within 10 minutes. Until 2004, the owners of MoneyGram were Viad Corp and Travelers Express Company Inc. MoneyGram financial results were driven by money transfer business, which delivered more than 40 percent growth in transaction volume.