Ruined Moments in Your Life: How Good are the Approximations?

Abstract

In this paper we implement numerical PDE solution techniques to compute the probability of lifetime ruin which is the probability that a fixed retirement consumption strategy will lead to financial insolvency under stochastic investment returns and lifetime distribution. Using equity market parameters derived from US-based financial data we conclude that a 65-year-old retiree requires 30 times their desired annual (real) consumption to generate a 95% probability of sustainability, which is equivalent to a 5% probability of lifetime ruin, if the funds are invested in a well-diversified equity portfolio. The 30-to-1 margin of safety can be contrasted with the relevant annuity factor for an inflation-linked income which would generate a zero probability of lifetime ruin. Our paper then goes on to compare the numerical PDE values with various moment matching and other approximations that have been proposed in the literature to compute the lifetime probability of ruin. Our results indicate that the Reciprocal Gamma approximation provides an accurate fit as long as the volatility of the underlying investment return does not exceed three-fourths = 30% per annum, which is consistent with capital market history. At higher levels of volatility the moment matching approximations break down and we provide some theoretical reasons for this phenomena. Our numerical and methodological results should be of interest to both academics and practitioners who are interested in methods of approximating stochastic present values as well as methods for computing sustainable consumption and withdrawal rates towards the end of the human life cycle.