Filed pursuant to Rule 497(e)

Registration No. 333-184077

ALLIANZGI CONVERTIBLE & INCOME FUND (formerly, AGIC CONVERTIBLE & INCOME FUND)

Supplement dated May 1, 2013 to the

Prospectus Supplement, Prospectus and Statement of Additional Information, each dated

December 4, 2012, of AllianzGI Convertible & Income Fund

(formerly, AGIC Convertible & Income Fund) (the “Fund”)

This information is important and relates to the name of the Fund and the Sub-Adviser of the Fund. You should read it carefully.

Effective January 1, 2013, the Fund’s sub-adviser, formerly known as Allianz Global Investors Capital LLC, changed its name to Allianz Global Investors U.S. LLC (the “Sub-Adviser”) in connection with reorganization transactions within the Allianz Global Investors U.S. business. In addition, the Fund’s name was changed to AllianzGI Convertible & Income Fund, effective as of January 28, 2013.

The reorganizations involved the combination of four affiliated investment advisory firms with and into the Sub-Adviser, including Allianz Global Investors Managed Accounts LLC and Allianz Global Investors Solutions LLC, each effective January 1, 2013, and RCM Capital Management LLC and Caywood-Scholl Capital Management LLC, each effective April 1, 2013. The reorganizations did not result in any changes to the Fund’s portfolio management team at the Sub-Adviser or in the Fund’s day-to-day investment management.

All references to Allianz Global Investors Capital LLC in the Prospectus Supplement, Prospectus and Statement of Additional Information are hereby deleted and replaced with Allianz Global Investors U.S. LLC. All references to AGIC Convertible & Income Fund in the Prospectus Supplement, Prospectus and Statement of Additional Information are hereby deleted and replaced with AllianzGI Convertible & Income Fund.