Less-expensive long-term annuities linked to mortality, cash and equity

Abstract

This paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others 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 the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.

Long-term option pricing with a lower reflecting barrier

“…(iii) The benchmark approach originated by Eckhard Platen (e.g. Platen, 2002;Buhlmann and Platen, 2003;Platen and Heath, 2010;Fergusson and Platen, 2022). This involves a change of numeraire to the "benchmark portfolio" which maximises expected log-utility of terminal wealth (in other literature, the "growth optimal portfolio").…”