We study mean–variance hedging of a pure endowment, a term insurance and general annuities by trading in a longevity bond with continuous rate payments proportional to the survival probability. In particular, we discuss the introduction of a gratification annuity as an interesting insurance product for the life insurance market. The optimal hedging strategies are determined via their Galtchouk–Kunita–Watanabe decompositions under specific, yet sufficiently general model assumptions. The results are then further illustrated by assuming a general affine structure of the mortality intensity process. The optimal hedging strategies as well as the residual hedging error of a gratification annuity and a simple life annuity are finally investigated with numerical simulations, which illustrate the nice features of the gratification annuity for the insurance industry.