Phoenix is also referring clients to free guidance provider The Pension Advisory Service (Tpas) should they require any further help with their decision.
The provider said it had taken it a while to work out how to design the scheme to ensure it was clear to customers what they were giving up in return for the cash.
The provider’s books are well suited to making such an offer as they include a large number of clients with small annuities, whereas other providers may have more larger pot clients, Phoenix said.
A spokeswoman said: “It’s quite a big piece of work for an insurer to do and we have quite a large book of customers with smaller payments.
“[Whether others should follow suit] very much depends on the nature of their client book.”
Aviva said it was not planning to offer the option to its clients due to client protection issues.
A spokesperson said: “We remain of the view that cashing in annuities would not represent a good outcome for customers for a number of reasons. We do not therefore offer customers the option to cash in their annuity.”
But Hunter Wealth Management director William Hunter said although he was not in favour of introducing a secondary annuity market because of the risks involved he did not think the same applied to very small annuities.
He said: “How much is a client really at risk with that sum of money?”
“It’s a good idea, less than a pound a day is not a life-changing amount of money but a few thousand pounds today could help someone.”
Standard Life (now Standard Life Aberdeen) will also not be offering the option.
A spokesperson said: "Prior to the pension freedoms introduced in April 2015, customers typically had an option to take cash benefits if their pension fund met either the 'triviality' or 'small pot' requirements. Many people chose to do this rather than purchase an annuity.
"Companies may offer to buy back very small annuities under some circumstances – but annuities are generally written and priced on the understanding that they provide a customer with an income for life without the option to surrender.”
carmen.reichman@ft.com