Comonotonic approximations for a general pension problem

In this paper, we search for such an investment strategy that minimises the probability of default (or lifetime ruin probability) given a fixed investment amount during the accumulation phase and a fixed withdrawal rate during the annuitisation part. In solving the above-mentioned problem, we introduce a methodology that helps us to obtain a solution in analytical fashion without resorting to time consuming Monte Carlo (MC) simulation. More precisely, the existing methodology of conditioning Taylor approximation is used to find a solution by approximating the original sum via a comonotonic sum. More specifically, we searched for the optimal multi-period investment strategy of an investor whose accumulation phase (lasting M years) is followed by an annuitisation period (lasting N years). When choosing the optimal asset mix, we restricted our analysis to the class of constant mix dynamically rebalanced strategies, with optimisation criterion set to the probability of default. As it was shown by means of a numerical illustration, the solutions of our approximate procedure closely relate to the results of MC simulation.