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Figure 9: Impact of the foreign risk free rate
5. CONCLUSIONS
Equity-indexed annuities are one innovative product brought into the insurance market recently and the sales have been growing rapidly. Among several product designs of EIAs, ratchet EIAs are the most popular probably because returns are credited periodically with a guaranteed minimum and the account value never decreases once the return is credited. Pricing ratchet EIAs is, however, challenging
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due to the complex contract features and payoff structures. For instance, Hardy (2004) claimed that the value of the simple version of ratchet EIAs is not analytically tractable. Kijima and Wong (2007) could not obtain closed-form solutions for the compound version with a return cap either.
Our major contribution in this dissertation is that we derive the pricing formulas for various ratchet EIA contracts under the Black-Scholes assumptions. Our formulas cover both simple and compound versions of ratchet EIAs. They may have a return cap and can adopt either no return averaging or two types of averaging schemes. The broader coverage of our closed-form solutions make the analyses of various contract features easier than the numerical methods provided by the literature.
Our pricing formulas will be a useful tool for actuaries to design ratchet EIA contracts in terms of controlling guarantee costs and market variable risks such as interest rate level and linked-index’s volatility. The numerical analyses using these formulas can further assist actuaries to evaluate how contract features such as return cap, return averaging, and return accumulation affect the contract value. Our numerical results show that the value of the contract increases with the return cap, decreases with the frequency of averaging, and is higher for the compound version. Furthermore, the results demonstrate that the impacts of contract features are affected by each other.
The impact of return cap is the most significant when returns are accumulated
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compoundedly and when there is no return averaging. The impact of return averaging is reduced significantly by return cap, and the impact of return accumulation is reduced by both return cap and return averaging. Actuaries therefore should always take into account contract features simultaneously when designing and managing ratchet EIA products.
Our formulas will also be useful in hedging the risks of the ratchet EIA products.
Insurers can hedge the risks introduced by embedded options using a passive approach or the dynamic-hedging approach (Boyle and Hardy, 1997). Under the passive method the insurance company offsets the liability associated with the embedded option by purchasing appropriate options in an exchange and/or from another financial institution. For instance, the insurer may purchases call options with the same underlying stock indexes in an exchange to hedge the embedded call options in the ratchet EIA products. These exchange-traded options have short maturities only, but the insurer may roll them over to provide longer-term protections.
If the insurer is concerned with the basis risk resulted from the complex contract features of the ratchet EIA products (e.g., return averaging), it may purchase average rate options in an over-the-counter market. It may even arrange an equity swap with an investment bank. Our formulas will help insurers to assess the due prices/costs of the above hedging arrangements.
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Insurers can employ our formulas in the dynamic hedging as well. Under the dynamic-hedging approach, the assets of the portfolio are adjusted on an ongoing basis so that the fund at maturity provides the minimum guaranteed amount when the guarantee is operative and the value of the assets otherwise. The insurer can employ our formulas to derive the compositions of the replicating portfolios that will be adjusted dynamically to reflect the changing indexes and time to maturity. Due to the existence of transactions costs, the insurer has to adjust the replicating portfolios discretely rather than continuously and will incur hedging errors. It therefore faces the tradeoff between discrete hedging errors and transaction costs. Hardy (2003;
chapter 8) provides detailed descriptions and assessments on this dynamic-hedging approach. Her results, in general, showed that the pricing formulas derived under simple Black-Scholes assumptions can have good hedging capacity for more general assumptions about linked-index and interest rate, which provide another justification for using the B-S framework.
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