If I purchase a fixed rate annuity and die after 15 or so years, what happens to the initial principal or purchase amount?

There are many different features that would affect the payout (or lack thereof) in light of your question.

In the most extreme case, a pure single premium income annuity (SPIA) would only provide payments while the contract owner is alive.  With this pure life annuity, payments would stop when the owner passes away--regardless of the amount of premium that was used to purchase the fixed annuity.

SPIAs are designed to provide income payments for the rest of your life--period.

However, this pure income annuity scenario can be affected in a number of ways.

For example, an income annuity contract can be purchased with several types of guarantee periods.  A guarantee period will make the annuity more expensive (rates or payouts will be a bit lower than the pure life annuity described above), but payments from the annuity will not terminate upon death of the contract owner.

There are a handful of guarantee types:

• Certain Period: payments are guaranteed for a pre-defined or "certain" number of years or months.
• Cash Refund: upon death, repays the outstanding or remaining portion of the premium to beneficiaries as a lump sum.
• Installment Refund: upon death, repays the outstanding or remaining portion of the premium to beneficiaries as a series of periodic payments (rather than lump sum).

So the short answer is that it all depends on how the fixed annuity contract is structured before/when purchasing.  As described above, there are some features (that come at a cost) that provide for a "return of premium" upon the death of the policyholder(s).