Pricing and Hedging Equity-Linked Products Using Risk Measures

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The main objective of this project is to develop pricing and hedging techniques for equity-indexed annuities based on risk measures. An equity-indexed annuity (EIA) is an insurance product with benefits linked to the performance of an equity market. It provides limited participation in the performance of an equity index (e.g. S&P 500) while guaranteeing a minimum rate of return. First introduced by Keyport Life Insurance Co in 1995, EIAs have been the most innovative annuity product over the last 20 years. They have become increasingly popular since their debut; the sales of EIAs have steadily increased and hit a record high of $33.9 billion in 2012, as indicated in the fourth quarter 2012 U.S. Individual Annuities Sales survey from Life Insurance and Market Research Association (LIMRA).Equity-indexed annuities are usually wrapped with some death, surrender, withdrawal, and accumulation guarantees that protect policyholders. The costs of these guarantees are evaluated based on financial and actuarial models. Research on evaluating these products has been carried using various approaches; see Hardy ("Investment Guarantees: Modeling and Risk Management for Equity-Linked life Insurance", 2003) and the references therein. I am primarily interested in the application of optimization techniques to reduce the risk of equity-indexed annuity portfolios. In a complete market, all financial claims can be hedged perfectly. However, in an incomplete financial market, the number of securities is less than the number of possible outcomes. In this case, the perfect hedge is unattainable. This is the appropriate framework when complex underlying asset models, mortality risks, and withdrawals are considered. There are different pricing and hedging techniques that could be used in such cases; we shall focus on risk measures. Two different objectives could be achieved while defining the hedging strategy using risk measures. On the one hand, the insurance company could minimize the hedging cost for a given risk level. On the other hand, the issuer could minimize the risk associated to a hedging strategy chosen among self-financing ones. Gaillardetz and Moghtadai (Equity-Linked Product Evaluation Using Iterated Risk Measures, 2013) focus on the latter where the hedging strategy is formed of the underlying risky and risk-free assets. Using the result of Rockafeller and Uryasev (Conditional value-at-risk for general loss distributions, 2002), which converts the evaluation of the tail value-at-risk into a linear programming problem, we propose to generalize the hedging strategies to multiple risky assets. We will also restrict our hedging strategies to self-financing strategies and try to minimize a given risk measure. This could be solved using the linear programming approach with state variables. Surrender and withdrawal options are usually priced assuming that policyholders are rational. It requires that the investors be fully informed of the financial market and thus can always take optimal decisions. This strong assumption leads to an upper bound for prices since it considers the worse case scenario for the issuer. Under the proposed risk measure approach, surrender and withdrawal processes will be modeled under the physical probability measures, which means that some endogenous and exogenous effects will be used.