The move comes in the wake of Philip Howell stepping down from the post of chief executive of Williams de Broe.
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Meanwhile, the integration of Investec Wealth & Investment with Williams de Broe is expected to conclude by March 2013.
Though Howell has stepped down with immediate effect, Wragg will take the position of chief executive of the combined business only in September 2012, said the company.
Howell had joined Williams de Broe in February 2010 and oversaw the acquisition of parent Evolution Group by Investec in September last year.
He also had overseen the acquisition of BNP Paribas Wealth Management by Williams de Broe last year, which had added GBP1.8 billion in assets.
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By GlobalDataIn further developments, the COO of Williams de Broe, David Howard who is set to step down by the end of 2012, will be succeeded by COO of Investec Wealth & Investment, Judy Price.
Investec Wealth & Investment confirmed in April 2012 that it would drop the Williams de Broe name and merge the wealth manager into its parent.
