  This paper proposes a paradigm shift in the valuation of long term annuities,
away from classical no-arbitrage valuation towards valuation under the real
world probability measure. Furthermore, we apply this valuation method to two
examples of annuity products, one having annual payments linked to a mortality
index and the savings account and the other having annual payments linked to a
mortality index and an equity index with a guarantee that is linked to the same
mortality index and the savings account. Out-of-sample hedge simulations
demonstrate the effectiveness of real world valuation.
  In contrast to risk neutral valuation, which is a form of relative valuation,
the long term average excess return of the equity market comes into play.
Instead of the savings account, the num\'eraire portfolio is employed as the
fundamental unit of value in the analysis. The num\'eraire portfolio is the
strictly positive, tradable portfolio that when used as benchmark makes all
benchmarked nonnegative portfolios supermartingales. The benchmarked real world
value of a benchmarked contingent claim equals its real world conditional
expectation. This yields the minimal possible value for its hedgeable part and
minimizes the fluctuations for its benchmarked hedge error. Under classical
assumptions, actuarial and risk neutral valuation emerge as special cases of
the proposed real world valuation. In long term liability and asset valuation,
the proposed real world valuation can lead to significantly lower values than
suggested by classical approaches when an equivalent risk neutral probability
measure does not exist.
