Exhibit 32(b)

CERTIFICATION PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

In connection with the Annual Report on Form 10-K of City Holding Company (the “Company”) for the period ending December 31, 2012 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, David L. Bumgarner, Senior Vice President, Chief Financial Officer and Principal Accounting Officer of the Company, certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

1. The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

2. The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

Dated: March 15, 2013

/s/ David L. Bumgarner

David L. Bumgarner

Senior Vice President, Chief Financial Officer and Principal Accounting Officer