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Practical Considerations in Managing Variable Annuities Stochastic Modeling Symposium Toronto By Thomas S.Y. Ho Sang Bin Lee Yoon Seok Choi April 3-4 2006 Slide 2 2 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 Slide 3 3 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 Slide 4 4 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 Slide 5 5 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 Slide 6 6 Interest Rate Model Slide 7 7 Variable Annuity and GMIB Models Policyholders annuitize when the annuity value exceeds the account value GMIB is the policyholders 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? Slide 8 8 GMIB model: Capital Market Valuation Approach dV = dS fVdt Max[V(T), B(T, T*)] GMIB value = average of the pathwise values Slide 9 9 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 Slide 10 10 Linear Path Space Slide 11 11 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 -0.50.00.5 -0.57.4911.9814.90 0.010.8714.6917.46 0.513.1316.7919.59 Slide 12 12 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 Slide 13 13 GMIB & Account Value Increase account value: GMIB value falls Impact of the correlation on GMIB is significant, particularly when the account value is high Slide 14 14 Effectiveness of the Pathwise Immunization GMIB pathwise values are plotted against the hedging portfolio pathwise values The fit is 99% R- square Slide 15 15 Decomposition Solution: Pathwise Immunization Portfolio ABCDEFGHI Hedging Instrument StrikeFair Value Regression Coeffs. t-statisticsDollar valueDeltaDuration Dollar Duration Cash-1.003.324.973.320.00 Bond Call1057.20-0.19-10.20-10.810.0021.41-231.43 Bond Call2050.90-0.16-10.25-8.110.0022.78-184.87 Bond Call3044.61-0.11-10.33-4.820.0024.54-118.25 Bond Call4038.31-0.02-6.51-0.730.0026.87-19.57 Bond Call5032.010.149.204.410.0030.13132.72 Bond Call6025.710.429.7110.840.0034.98379.22 Bond Call8013.940.3312.354.620.0050.85234.92 Bond Call1005.420.2812.951.540.0069.42107.12 Bond Call1201.530.178.690.260.0087.9622.57 Equity Put400.141.564.040.22-0.0153.5411.71 Equity Put1006.980.375.222.55-0.1632.8683.86 Equity Put15022.550.5114.8811.40-0.3526.07297.28 Replicating Portfolio 14.69-0.2448.68715.26 GMIB 14.69-0.2248.34710.09 Slide 16 16 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 Slide 17 17 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 Slide 18 18 Computational Efficiency Extended LPS model Structured sampling approach Convergence Slide 19 19 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 Slide 20 20 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 Slide 21 21 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 Slide 22 22 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 Slide 23 23 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 Slide 24 24 Implications on the Product Design Depends on the insurers 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 Slide 25 25 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 Slide 26 26 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.comwww.thomasho.com