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. This problem is a variant of the classical and illustrious ruin problem in insurance, but adapted to individual circumstances. 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 portfolio. The 30-to-1 margin of safety contrasts with the relevant annuity factor for an inflation-linked pension which would generate a zero probability of lifetime ruin. Our paper then goes on to compare the PDE-based values with moment matching and comonotonic-based approximations that have been proposed in the literature. 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 σ=30% per annum, which is consistent with capital market history. At higher levels of volatility the moment matching approximations break down. We also confirm that the comonotonic-based lower bound approximation provides remarkably accurate results when the time steps are small though. Our results should be of interest to academics, practitioners and software developers who are interested in computing sustainable consumption and withdrawal rates towards the end of the human life-cycle, but without resorting to crude simulations.