Arna was born in Reykjavík to Ellert Schram and Anna Guðlaug Ásgeirsdóttir. She grew up in Reykjavík where she attended Vesturbæjarskóli and Hagaskóli. In 1988, she graduated from Menntaskólinn í Reykjavík. Arna graduated with a BA degree in political science and philosophy from the University of Iceland and the University of Copenhagen, as well as an MBA degree with an emphasis on management, operations and marketing from Reykjavík University.
