If I rent a car from Avis and do not opt for additional coverage, my own personal auto insurance covers me in the event of an accident.

What happens if I add my adult daughter as an additional driver on this rental?(paying the additional driver daily fee) She is in grad school and does not have own a car. (although she is still listed as an additional driver on my personal auto insurance policy, since she drives my cars when she comes home) Will my personal auto policy cover her for damage to the rental car or liability to other cars? I asked the GEICO agents and they couldn't be more vague about their answers......

Thanks1