Roberts was the chairman of Nationwide Building Society before he joined the London-based insurance firm as a non-executive director in 2017.
Christine LaSala, senior independent director will chair the company's board on an interim basis, Beazley said.
LaSala, who has about 45 years of experience in the insurance industry, joined Beazley in 2016.
The company said it would soon start the process of identifying Roberts' successor.
