This paper evaluates the long-term risk for equity-linked insurance products. We consider a specific type of equity-linked insurance product with guaranteed minimum maturity benefits (GMMBs), and assume that the underlying equity follows the stochastic volatility model which allows the return's latent volatility component to be short- or long-memory. The explicit form of the quantile reserve or the Value at Risk and its confidence intervals are derived for both the long-memory and short-memory stochastic volatility models. To illustrate the effect of long-memory volatility, we use the S&P 500 index as an example of linked equity. Simulation studies are performed to examine the accuracy of the quantile reserve and to demonstrate the consequence of low coverage probability if model misspecification takes place. The empirical results show that the confidence interval of quantile reserve could be severely underestimated if the long-memory effect in equity volatility is ignored.

Let {X n , n ≥ 0} be a stationary Gaussian sequence of standard normal random variables with covariance function r(n) = E X 0 X n . Let Under some mild regularity conditions on r(n) and the condition that r(n)lnn = o(1) or (r(n)lnn)−1 = O(1), the asymptotic distribution of is obtained. Continuous-time results are also presented as well as a tube formula tail area approximation to the joint distribution of the sum and maximum.

Let X 1, X 2, ·· ·be stationary normal random variables with ρn = cov(X 0, Xn ). The asymptotic joint distribution of and is derived under the condition ρn log n → γ [0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.