"It has been fantastic to have been on the journey with Admiral since the management buy-out, a journey which I am confident will continue to deliver sustained success," said Lyons.
The way the business has developed has been astonishing and a tribute to a great team that has created a culture that breeds innovation, the drive and ability to succeed, openness and transparency. I shall miss it, and them, greatly.
During Lyons' tenure he oversaw car insurance specialist's float on the London Stock Exchange in 2004.
Admiral, which has said it could move its European business to Ireland or another country if British insurers lose their right to sell their products across Europe as a consequence of leaving the EU, said a new chairman will be elected at the firm's AGM this Spring.
Chief executive David Stevens said: "There are very few people who deserve as much credit for Admiral's success over the last sixteen years as Alastair. He's been everything our shareholders, and indeed our staff
Admiral is the only FTSE 100 firm which has its headquarters in Wales. Local business champions hailed Lyons' impact and the boost his actions had made to the local economy.
“Under the chairmanship of Alastair Lyons, the Admiral Group, which is Wales’ largest private business, has played an integral part in the transformation of Cardiff’s services sector," said Chris Sutton, lead director at JLL Cardiff and Chair of Central Cardiff Enterprise Zone.
"When he took up this post in in 2000, Cardiff was transitioning from a maritime trade city to the financial and professional services hub which now employs 40,000 people and has an economic output in excess of £2 billion."