You need to change your controller code as below:


    public function indexAction() {
        $this->loadLayout();
        $this->renderLayout();
    }

    public function ajaxAction() {
        $isAjax = Mage::app()->getRequest()->isAjax();
        if ( $isAjax ) {
            $output = $this->getLayout()->createBlock('finance/finance')->setTemplate('finance/ajax.phtml')->toHtml();
            $this->getResponse()->setHeader( 'Content-type', 'application/json' );
            $this->getResponse()->setBody(json_encode(['outputHtml' => $output]));
        }
    }

}