Mr Goeltz, 53, joined NatWest in February 1992 from Seaman, the entertainment conglomerate, in New York.
He arrived at a time when the bank was suffering badly from the effects of recession in the UK. Its American retail arm, NatWest Bancorp, lost over pounds 150m in the two years to 1992 while the UK high street operation lost over pounds 100m in that year alone.
Mr Goeltz's tenure has seen the American arm move back to profit and sold off for pounds 2.3bn, while the UK side has also recovered strongly.
NatWest Markets, the investment banking arm formerly known as County NatWest, was threatened with closure by the bank about the time Mr Goeltz arrived. Under new chief executive Martin Owen it now forms a main plank of the group's expansion plans.
Lord Alexander, chairman of NatWest, paid tribute to Mr Goeltz's role in rebuilding the group's fortunes in a difficult period: "I would like to thank Richard for his valued contribution to the group over the past four years."
At American Express Mr Goeltz will replace Michael Monaco, who quit in April following a dispute with his boss, executive chairman Harvey Golub, over the assignments of several senior people.
Mr Goeltz will report to Mr Golub, who said the company had broadened the finance director's role to include serving as a member of the chief executive's office.
That means Mr Goeltz will share responsibility with vice-chairmen Ken Chenault, Jon Linen and Chuck Farr for reviewing business strategies and handling issues that affect more than one business unit, Mr. Golub said.
Diary, page 18