I've got a player that has an idea for a barbarian variant that doesn't wear armor. At first glance it doesn't appear to be all that unbalanced, but I would appreciate other opinions.

Unarmored Barbarian Variant
A barbarian who chooses to not wear armor might choose to exchange his proficiency with armor for a natural armor bonus.

Gain
Natural Armor bonus – Gain a natural armor bonus to AC equal to the barbarian’s Constitution modifier, the barbarian loses this bonus if he wears any type of armor. A barbarian who wears armor doesn't gain the benefits of Fast Movement and Damage Reduction.

Lose
Proficiency with light, and medium armor.

Let me know what you think. Thanks.