This paper investigates the effect of adverse selection and price competition on the private annuity market in a model with two retirement periods. In this framework annuity companies can offer contracts with different payoffs over the periods of retirement. Varying the time structure of the payoffs affects annuity demand and welfare of individuals with low and high life expectancy in different ways. By this, annuity purchasers can be separated according to their survival probabilities. Our main finding is that a Nash-Cournot equilibrium may not exist; if one exists, it will be a separating equilibrium. On the other hand, even if a separating equilibrium does not exist, a Wilson pooling equilibrium exists.