S Corporation Modernization Act of 2013 - Amends the Internal Revenue Code to revise the tax treatment of S corporations by: (1) permanently reducing from 10 to 5 years the period during which S corporation built-in gains are subject to tax; (2) repealing mandatory termination of S corporation elections for excessive passive investment income; (3) allowing S corporations to increase passive investment income from 25 to 60% without incurring additional tax; (4) allowing nonresident aliens to be potential current beneficiaries of an electing small business trust (ESBT); (5) allowing individual retirement accounts (IRAs) to be S corporation shareholders; (6) allowing ESBTs to claim expanded charitable tax deductions; and (7) making permanent the rule requiring a basis adjustment to stock of an S corporation making charitable contributions of property.