Family and Medical Insurance Leave Act or the FAMILY Act This bill establishes the Office of Paid Family and Medical Leave within the Social Security Administration. The bill entitles every individual to a family and medical leave insurance (FMLI) benefit payment for a specified benefit period and prescribes a formula for determining the individual's monthly benefit amount. An FMLI benefit payment shall be coordinated with any periodic benefits received under a state or local temporary disability insurance or family leave program. The bill amends the Internal Revenue Code to impose a tax on employers, employees, and self-employed individuals to fund FMLI benefits. It also establishes the Federal Family and Medical Leave Insurance Trust Fund to hold tax revenues.