Banks must change the way employees are compensated and take other steps to fix a corporate culture that encourages misdeeds or face being broken up, said William C. Dudley, president of the Federal Reserve Bank of New York.

If bad behavior persists, “the inevitable conclusion will be reached that your firms are too big and complex to manage effectively,” Dudley told industry leaders in a speech Monday at the New York Fed. “In that case, financial stability concerns would dictate that your firms need to be dramatically downsized and simplified so they can be managed effectively.”