Firearm Risk Protection Act of 2013 - Amends the Brady Handgun Violence Prevention Act to: (1) prohibit the purchase or sale of a firearm unless the purchaser presents proof to the seller and the seller verifies that the purchaser is covered by a qualified liability insurance policy, and (2) require any person who purchases a firearm on or after this Act's effective date to be covered by such a policy. Exempts the purchase or sale of a firearm for use by a federal, state, or local agency.

Defines "qualified liability insurance policy" to mean a policy that: (1) provides liability insurance covering the purchaser specifically for losses resulting from use of the firearm while it is owned by the purchaser, and (2) is issued by an insurer licensed or authorized to provide the coverage by the state in which the purchaser resides.