In this paper we present a numerical valuation of variable annuities with
combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum
Death Benefit (GMDB) under optimal policyholder behaviour solved as an optimal
stochastic control problem. This product simultaneously deals with financial
risk, mortality risk and human behaviour. We assume that market is complete in
financial risk and mortality risk is completely diversified by selling enough
policies and thus the annuity price can be expressed as appropriate
expectation. The computing engine employed to solve the optimal stochastic
control problem is based on a robust and efficient Gauss-Hermite quadrature
method with cubic spline. We present results for three different types of death
benefit and show that, under the optimal policyholder behaviour, adding the
premium for the death benefit on top of the GMWB can be problematic for
contracts with long maturities if the continuous fee structure is kept, which
is ordinarily assumed for a GMWB contract. In fact for some long maturities it
can be shown that the fee cannot be charged as any proportion of the account
value -- there is no solution to match the initial premium with the fair
annuity price. On the other hand, the extra fee due to adding the death benefit
can be charged upfront or in periodic instalment of fixed amount, and it is
cheaper than buying a separate life insurance.Comment: arXiv admin note: substantial text overlap with arXiv:1410.860