Dobson will stay on until April when he will be replaced by Peter Harrison, head of investment, who was widely tipped as his successor. Dobson will then become non-executive chairman.
Schroders' share price rose 2.8 per cent as the asset manager also revealed a strong set of full year results, with profit before tax and exceptional items up eight per cent to £609.7m while pre-tax profits rose 14 per cent to £589m.
Earnings per share rose 12 per cent to 171.1p. Schroders' recommended dividend is up 12 per cent to 87p per share.
However net inflows were almost half that of last year, down to £13bn from £24.8bn.
"2015 was another good year for Schroders," said Dobson.
“Schroders is well placed for a more challenging environment with a highly diversified business and a strong financial position. We see many interesting long-term growth opportunities and we will again be taking advantage of a period of dislocation to invest behind them.”