I didn't particularly want to live either
It's actually my parents' house that is insured with A&L, not the computer itself. It was the contents part of the insurance that covered it, which covers items that family members living there own even if they take them out of the premises and something happens to them elsewhere.
Best idea would be to phone them up and have a chat.
My parents are looong term customers so it might be an old version of the contract that they have to keep up but don't offer to new customers, but it's worth asking about their policies. Contents insurance certainly isn't anything new
Oh, and keep proof of purchase for
* Your computer
* Any RAM or other expansions
With the latter I also got my 512MB expansion replaced, which I didn't initially think about.