Fostering Accountability In Remuneration Fund Act of 2023 or the FAIR Fund Act of 2023 This bill requires large financial institutions to defer part of senior employees' compensation and deposit the deferred compensation in a fund established by the institution. This fund must be used to pay civil or criminal fines levied against the institution or, if the institution is a bank or credit union, to make depositors whole in the event of a failure. If sufficient funds exist at the end of a specified period of time, senior employees must receive their deferred compensation.