Interest rate is one of the main risks for the liability of the variable annuity (VA) due to its long maturity. However, most existing studies on the risk measures of the VA assume a constant interest rate. In this paper, we propose an efficient two-dimensional willow tree method to compute the liability distribution of the VA with the joint dynamics of the mutual fund and interest rate. The risk measures can then be computed by the backward induction on the tree structure. We also analyze the sensitivity and impact on the risk measures with regard to the market model parameters, contract attributes, and monetary policy changes. It illustrates that the liability of the VA is determined by the long-term interest rate whose increment leads to a decrease in the liability. The positive correlation between the interest rate and mutual fund generates a fat-tailed liability distribution. Moreover, the monetary policy change has a bigger impact on the long-term VAs than the short-term contracts.

This paper provides a detailed overview of variable annuities. Consideration is given first to the definition of the term variable annuity. Common terminology used in the variable annuity market is introduced. The current state of the United Kingdom and other international markets is described. Then, by reference to a simplified product, an analysis of customer outcomes, pricing, reserving, risk management and hedging is carried out. The paper ends with a description of current U.K. pensions legislation and how it potentially constrains product development.