Does the St. Petersburg Paradox Cast Doubt on the Use of Expected Value Theory to Approximate the Random Present Value of a Life Annuity?

Abstract

The classic approximation of the random present value (PV) of a life annuity is its expected PV (EPV) over its term. EPV minimizes mean squared error per expected value theory (EVT). Here, based on the experimental work of Hayden and Platt (2009), lessons are drawn from the St. Petersburg paradox (SPP) which violates EVT, to improve random PV approximations. Although much work over the last 300 years has been done to try and resolve the SPP, the question of whether it offers lessons for approximating any random PV has not been considered in the literature. An additional approximation, the median PV (PV*), which has minimal mean absolute error, is introduced herein. PV* can yield a probability (k) greater than 50% that its absolute error is less than the absolute error in EPV. If k is revealed, would agents value the annuity in terms of its EPV and, thus, willingly allow crosssubsidization to dominate their self-interest to only pay amounts that are reasonably commensurate with the PV of the payouts they hope to get in return? It is proposed that the value of the annuity when approximated as the weighted average of PV* and EPV, with k and 1 – k as weights, respectively, makes cross-subsidization of annuitants fairer in the sense of Rawls (1971) and Sen (1992).