Valuation of EIAs with the Threshold GARCH model

In recent years, due to the keen competition in the insurance industries, we witness the prolication and innovation in insurance products in the global insurance market. One of the popular insurance products is an equity-indexed annuity (EIA). This product is particularly popular in the North America. Basically, an EIA is an annuity which earns interest depending on the performance of an equity index, or a certain share. A typical index is the S&P 500. One of the key issues in trading EIAs is to provide an objective or scientic way to value the liabilities. In this paper, we develop a fair valuation method for valuing liabilities under EIAs under a nonlinear time series model, namely the Threshold-GARCH model. This model can incorporate two important stylised empirical features of equity indices, namely, the conditional heteroscedasticity and regime-switching characteristics; the latter is of particular importance in describing the long-term behaviour of investment returns. The market in the Threshold-GARCH model is incomplete. We shall adopt a time-honored tool in actuarial science, namely, the Esscher transform, in specifying a pricing kernel. This method can be justied by the maximisation of an expected power utility. Monte Carlo simulation is used to compute fair EIA values for five most common designs of EIAs in the insurance industries. Empirical studies are provided using the S&P 500 index.