Risk based capital for guaranteed minimum withdrawal benefit

The guaranteed minimum withdrawal benefit (GMWB), which is sold as a rider to variable annuity contracts, 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 for risk management and regulatory reasons. One needs to find the tail distribution of the profit–loss function, which differs from its expected payoff required for pricing the GMWB contract. GMWB has embedded two option-like features: Management fees are proportional to the current value of the policyholder’s account which results in an average price of the account. Thus the contract resembles an Asian option. However, the fees are charged only up to the time of the account 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. To the authors’ best knowledge, this is the first paper in the literature to formulate and analyse profit–loss distribution using PDE methods of such a product with intricate option-like features. Our approach is much more efficient than the current market practice of rather intensive and expensive Monte Carlo simulations due to the lack of samples for extreme cases.