On the calculation of risk measures for variable annuities with guaranteed benefits

With the development of the life insurance industry, different types of life insurance products, in addition to the traditional ones, are being developed. A common and well-known life insurance product is the variable annuity with different types of guaranteed benefit riders, which provides policyholders a high rate of investment return with downside risk protections. Two typical distortion risk measures, VaR (value at risk) and CTE (conditional tail expectation), are widely used to manage insurers' future liabilities to avoid the potential of insolvency. In this project, we consider variable annuities with certain types of guaranteed benefits and various asset price processes, and focus on the calculation of the two risk measures of insurers' net and gross liabilities at the maturity date. Specifically, we consider two types of guaranteed benefit riders, the guaranteed minimum death benefit (GMDB) and the guaranteed minimum maturity benefit (GMMB), and assume that the logarithm of underlying asset returns follows a Cauchy or a skew-normal distribution. Analytical expressions of VaR and CTE for insurers' future liabilities are obtained, and numerical calculation algorithms are proposed. Comparisons of the calculated risk measure results with that under the normal distribution are also presented.