Both firms are listed on the Johannesburg Stock Exchange as well as the London market.
Anglo was up more than three per cent at the time of writing, while Old Mutual had risen more than five per cent.
Zuma stood down last night, after the African National Congress (ANC) told him on Monday night that he had 48 hours to resign, or face being voted out by them.
During his nine-year reign, Zuma survived many personal and professional controversies such fathering a child out of wedlock, comments about showering after sex to reduce the likelihood of HIV contraction, a “midnight political reshuffle” which caused markets to panic, as well as an expenses scandal for home renovations.
He is still fighting 783 counts of corruption over a 30bn rand (£1.8bn) arms deal in the late 1990s when he was deputy president.
ANC chief whip Jackson Mthembu said last night that newly elected ANC leader Cyril Ramaphosa could be sworn in as South Africa's new head of state as early as today.