Individuals can insure themselves perfectly against uncertainty about the length of life by purchasing deferred annuities early in life. In the absence of other uninsurable uncertainties (e.g., income), there will be no residual purchases or sales of annuities later in life, thereby avoiding any adverse-selection. In contrast, the presence of such uncertainties creates an active residual annuity market based on the arrival of new information. We characterize the equilibrium in the residual annuity market and propose a new financial instrument, refundable annuities with a guaranteed refund price, which enables individuals who hold a portfolio of such annuities to better adjust their optimum consumption plan to different realizations. Refundable annuities are shown to be equivalent to annuity options, that is, options that, if exercised, enable the purchase of annuities later in life at a predetermined price. Holding a variety of refundable annuities is (ex ante) welfare enhancing.