A Numerical Solution to the Partial Differential Equation of the Risk Based Capital for Guaranteed Minimum Withdrawal Benefit

This research project is dedicated to implement an algorithm for the partial differential equation solution to a risk management problem of the GMWB variable annuity developed in Feng & Vecer [1]. In the first part of the report, we give an introduction of the insurance product and the mathematical dynamics behind it. In the second part of the report, we introduce the finite difference method in solving partial differential equation with the illustration of an example. Finally, we discuss the implementation of the algorithm in Matlab. 1 Problem Introduction Before introducing the concept of guaranteed minimum withdrawal benefit (GMWB), we first give a brief explanation of variable annuity, which is the base contract of GMWBs. From policyholders’ perspective, variable annuity is a kind of insurance contract which allows the insured to gain financial returns from investing their insurance premium. Usually policyholders have a variety of options to invest their premium payments. Then with the capital that policyholders invest, the insurers transfer the funds to third-party vendors who manage and service the accounts of the policyholders’ choice. The relatiohship between the insurer, the policyholder and the vendor can be represented in such a graph.[Figure 1] Figure 1: Variable Annuity Explanation Then, what is the guaranteed minimum withdrawal benefit (GMWBs)? The GMWBs are usually sold as add-ons to variable annuity base contracts, which allow annual withdrawals of a certain percentage of the benefit base until the base is exhausted, even if the policyholder’s account value itself had already fallen to zero before the ending date. Hence with GMWBs, policyholders receive two parts of return: one is a certain percent of the benefit base of which rate is predetermined and the other is partial financial return from capital gains of investment