This is a very common sign of ringworm infection or some kind of skin fungus. It is quite common in chinchillas. It usually starts on the nose or around the tail and spreads outward resulting in fur loss in a pattern that is easy to recognize. Having it checked by a vet is not a bad idea so you can get a positive diagnosis.
Ringworm is curable but is also highly contagious including to people so be careful. It usually gets worse before it gets better during treatment. A common cure for this is to put two tablespoons of desenex in your pet's dust bath and bathe it 10 minutes every day until the fur starts growing back. Then go back to a regular bath and normal dust bath schedule. It helps to scrub the cage about every other day to prevent re-infection. If you have other chinchillas in the same room, and it is ringworm, you should treat them as well because it is so contagious.
