Less-Expensive Valuation of Long Term Annuities Linked to Mortality, Cash and Equity

Abstract

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.