Risk based capital for guaranteed minimum withdrawal benefit

The guaranteed minimum withdrawal benefit (GMWB) is a recent innovation in the insurance market. It is sold as a rider to variable annuity contracts, which guarantees the return of total purchase payment regardless of the performance of the underlying investment funds. The valuation of GMWB has been extensively covered in the previous literature, but a more challenging problem is the computation of the risk based capital (RBC) which is required by the risk management and regulatory reasons. For the risk management purposes, one needs to find the whole distribution of the profit– loss function, not only its expected payoff which is required for pricing the GMWB contract. GMWB has embedded two option features – the fund fees are proportional to the current value of the fund which results in an average price of the accumulated fund. Thus the contract resembles an Asian option. However, the fees are charged only up to the time of the fund hitting zero which resembles a barrier option payoff. Thus the GMWB is mathematically more complicated than Asian or barrier options traded on the financial markets. We can find the RBC as a solution to a partial differential equation in two spatial dimensions that can be solved numerically.