Economic Representative Scenarios for Variable Annuity Dynamic Hedging of GMMB and GMDB

Abstract

Variable annuities introduce significant risk for the insurers. To control this risk, insurers generally use dynamic hedging models. In practice, calculations for dynamic hedging models for variable annuities are computationally intensive since they require many nested stochastic scenario projections with outer and inner loops. In this article, we study the relationship between the hedging errors and the descriptive statistics of economic scenarios. Then, using the relationships between the hedging errors and the descriptive statistics of the economics scenarios, we present two novel scenario selection algorithms to determine economic representative scenarios (ERS) to run for dynamic risk hedging models. We do this for guaranteed minimum maturity benefit (GMMB) and guaranteed minimum death benefit (GMDB). We also benchmark these new algorithms against a control variate algorithm and Monte Carlo simulations. One of the two proposed novel scenario selection algorithms, the adapted Clara algorithm gives a lower mean absolute deviation of the conditional tail expectation (CTE) than the control variate method and Monte Carlo simulations for the same number of scenarios. Also, 500 representative scenarios selected by the adapted Clara algorithm generate a mean absolute deviation of the CTE95 of the hedging errors approximately equivalent to 1000 (or more in some cases) Monte Carlo simulations for 5 years GMMB and GMDB.

ACKNOWLEDGMENTS

The authors bear full responsibility for the information contained therein. The views and opinions expressed in this article are those of the authors and do not necessarily reflect the official policy or position of the Autorité des marchés financiers (AMF Québec). All errors and omissions are the sole responsibility of the authors. We thank Frédéric Godin for his helpful comments.

FUNDING

Financial support from the Autorité des marchés financiers is gratefully acknowledged (AMF Québec).

Notes

1 Also, a different value of $\sigma$ (e.g., 0.16 or 0.15) would not have a significant impact on the results of this article. A sensitivity analysis was performed and is presented in Appendix 1 in Table 26.

2 We standardize the vectors $\psi ,$ $\beta ,$ and $\lambda$ in order to give them equal weight in the scenario selection algorithm by using their standardized versions $\tilde{\psi },$ $\tilde{\beta },$ and $\tilde{\lambda },$ which are calculated using Equation (42)(42) $${\tilde{x}}_j=\left(x_j-\underset{1\le j\le N}{\mathrm{\text{min}}}{x}_j\right){\left(\underset{1\le j\le N}{\mathrm{\text{max}}}{x}_j-\underset{1\le j\le N}{\mathrm{\text{min}}}{x}_j\right)}^{-1}$$(42) .

3 We consider the Clara algorithm in the R package cluster. The Clara algorithm is defined in Kaufman and Rousseeuw (1990).

4 We standardize the vectors $\psi ,$ $\beta ,$ and $\lambda$ in order to give them equal weight in the scenario selection algorithm by using their standardized versions $\tilde{\psi },$ $\tilde{\beta },$ and $\tilde{\lambda },$ which are calculated using Equation (42)(42) $${\tilde{x}}_j=\left(x_j-\underset{1\le j\le N}{\mathrm{\text{min}}}{x}_j\right){\left(\underset{1\le j\le N}{\mathrm{\text{max}}}{x}_j-\underset{1\le j\le N}{\mathrm{\text{min}}}{x}_j\right)}^{-1}$$(42) .

5 See Glasserman (2003) for more details on the control variate method.

6 OL means outer loop and IL means inner loop.

7 MAD means “mean absolute deviation.”

8 NJ means “no Poisson jumps.”

9 WJ means “with Poisson jumps.”