Real World Pricing of Long Term Cash-Linked Annuities and Equity-Linked Annuities with Cash-Linked Guarantees

This paper proposes a paradigm shift in the valuation of long term cash-linked annuities and equity-linked annuities with cash-linked guarantees, away from classical no-arbitrage pricing towards pricing under the real world probability measure. In contrast to risk neutral pricing, which is a form of relative pricing, the long term average excess return of the equity market comes into play. Instead of the savings account, the numeraire portfolio is employed as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales. Intuitively, benchmarked portfolios are in the mean downward trending or trendless. The benchmarked real world price of a benchmarked contingent claim equals its real world conditional expectation. This yields the minimal possible price for its hedgeable part and minimizes the variance for its hedge error. Classical actuarial and risk neutral pricing emerge as special cases of the proposed real world pricing. In long term liability and asset valuation, the proposed real world pricing can lead to signicantly lower prices than suggested by classical approaches. The existence of an equivalent risk neutral probability measure is not required.