GA and Oak Hill first picked up 60 per cent of the India-headquartered company when it was still known as GE Capital International Services in 2004.
The pair paid $500m and later divested some of their stake during the company’s 2007 IPO.
GIC and Bain’s deal values the business at about $3.3bn, and represents 68 million common shares bought at $14.76 each.
Bain Capital employees Amit Chandra, David Humphrey, Mark Nunnelly and Mark Verdi will all join the business to replace current directors from GA and Oak Hill Capital following the deal.
The firm first announced it had agreed a deal for the business at the beginning of August.
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