How do I find out when my annuity is finished?

First, what do you mean by "finished"?

Is your annuity currently providing income to you, or is it in the deferral period before income payments begin?

In either case, an annuity is a contract between you and an insurance company.

The contract will state--very clearly--when the deferral period ends or when annuity payments end (in the event that there is a limit on the amount of time or "term" that payments are made).

You should first contact the financial advisor or insurance agent who sold the annuity to you.  Ask him or her to point-out the contract provision that addresses this issue.

If the advisor is not available for some reason, go directly to the insurance company that provides the product with the same question.