Help needed with renter moving out!

[Kate Davies]

My roommate rents a room in my house. We have no formal agreement.
He told me today that he is giving me his 30 days notice to move out.
He stated that as the 30 days is up on the 26th of March, he will pay me the rent for those 26 days, but that if he finds a place sooner, that he will be entitled to a refund for the days that he is not there.
Do I have to give him a refund?
I live in San Diego, California.
Thanks!