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Practical Considerations in Managing Variable Annuities Stochastic Modeling Symposium Toronto

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## Practical Considerations in Managing Variable Annuities Stochastic Modeling Symposium Toronto

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**Practical Considerations in Managing Variable**AnnuitiesStochastic ModelingSymposiumToronto By Thomas S.Y. Ho Sang Bin Lee Yoon Seok Choi April 3-4 2006**Problem Statement**• Variable annuities (GMIB) have embedded capital market options • Equity options (equity put option) • Interest rate options (annuity value) • Pricing of the options • Capital market method: Cost of replications of the options • Identifying the embedded options • Practical considerations**Outline of the Presentation**• The valuation model • Pathwise immunization and LPS • Practical considerations • Computational efficiency • Equity/rate correlations and cost of guarantees • Product risks • Cost benefit analysis • Suboptimal exercise of options • Alternative to equity funds**A Model of GMIB**• Invest in equity/bond funds • End of the accumulation period, a choice: the account value or a fixed annuity • The guarantee is a put option to the insurer • If the fund is an equity fund, it is an equity put option, where the “strike price” is a bond, subject to interest rate risks**Capital Market Approach: Equity/Interest Rate Model**• 2-factor interest rate model • Recombining lattice, orthogonal yield curve movements • Fit the term structure of interest rates and volatilities (arbitrage-free) • Combining lognormal and normal behavior • Equity returns are lognormal with the instantaneous rate of return = short rate**Variable Annuity and GMIB Models**• Policyholders annuitize when the annuity value exceeds the account value • GMIB is the policyholder’s equity put option with a stochastic strike price based on the annuity value. • Can we represent the GMIB as a portfolio of equity options and bond options?**GMIB model: Capital Market Valuation Approach**• dV = dS – fVdt • Max[V(T), B(T, T*)] • GMIB value = average of the pathwise values**Decomposition using Linear Path Space (LPS)**• The path space is a representation of all the possible scenarios • The recombining lattice offers a “co-ordinate system” to represent the possible scenarios • Structured sampling of the path space provides equivalent classes of the possible scenarios, and the lattice framework enables us to measure the size of the classes**GMIB Value and the Correlations**• X-axis: correlation of equity returns with steepening movement • Y-axis: correlation with the parallel movement • Negative correlation implies natural hedging**GMIB Value and Correlations**• The correlation with the steepening effect has greater impact on the GMIB value • The worst scenario: yield curve steepens, rates and equity value fall • Negative correlation with the steepening movement lowers the probability of the worst scenario**GMIB & Account Value**• Increase account value: GMIB value falls • Impact of the correlation on GMIB is significant, particularly when the account value is high**Effectiveness of the Pathwise Immunization**• GMIB pathwise values are plotted against the hedging portfolio pathwise values • The fit is 99% R- square**A Solution in Pricing**• Separate pricing from valuation • Use the decomposition to identify the pricing strategy • The solution is a combination of hedging and directional investment • Product design to seek the optimal solution**Fixed Annuities**• Based in interest rate models • Investment portfolio can be fixed income • The embedded options would be interest rate options • Decomposition based on yield curve scenarios**Computational Efficiency**• Extended LPS model • Structured sampling approach • Convergence**Equity/rate correlations and cost of guarantees**• Correlations of equity returns to the yield curve movement can be incorporated • This approach is important to GMIB valuation because it focuses on the worst scenario: steep yield curve, falling rates, and underperforming equity market**Product risks**• Lapse risk, mortality risk, partial withdrawal can be simulated as risk sources • Distribution of the pathwise values would be higher • Hedging would not less perfect resulting in lower hedged positions**Cost benefit analysis**• Pathwise values provide the present value and risk adjusted values by scenarios • The distribution of the pathwise values post partial hedging provides a measure of the downside risks • The market view can be incorporated in the risk and return tradeoff in the “optimal hedging”**Suboptimal exercise of options**• The exercise rule can be adjusted for inefficiency in exercising the options • Valuation and hedging can be adjusted by the model results**Alternative to equity funds**• Fixed income fund can be evaluated in the same framework • Policyholders’ behavior can be simulated by the model resulting in the appropriate tradeoff of risk and returns**Implications on the Product Design**• Depends on the insurer’s business model • Market directions and balance sheet positions are relevant • Correlations of the fund returns and the guarantees are important • Deciding on the risk and return tradeoff**Conclusions**• Pricing of a product begins the determination of the building blocks of value of the guarantees • Then determine the optimal combination of hedge cost and directional investment • Business model and the portfolio structure must be taken into consideration**References**• Ho, Lee and Choi: “Practical Considerations in Managing Variable Annuities” Working Paper 2006 • Ho and Mudavanhu: “Decomposing and Managing Multivariate Risks: the Case of Variable Annuities” Journal of Investment Management 2005 • Ho and Mudavanhu “Managing Stochastic Volatility Risk of Interest Rate Options: Key Rate Vega” working paper 2006 • Papers available at www.thomasho.com