Replicated Stratified Sampling

1. Replicated Stratified Sampling A Practical Approach to Financial Modeling 2010 IABA Annual Meeting August 6 - 7, 2010 Jay Vadiveloo, PhD, FSA, MAAA,CFA

2. Notice This presentation has been prepared solely for informational purposes and Towers Watson does not make any representation or warranty, either express or implied, as to the accuracy, completeness or reliability of the information contained in this presentation. Your organization should consult its own counsel, tax, actuarial and financing advisors as to legal and other matters concerning any of the material presented herein. Towers Watson expressly disclaims any and all liability relating or resulting from the use of this presentation.

3. Background • Actuarial valuation of insurance liabilities typically involves production–based, seriatim calculations. • Today’s insurance products include complex features with investment oriented characteristics that require stochastic modeling of market and interest rate performance. • Commercial actuarial software has been designed to handle large, complex stochastic modeling of insurance liabilities. • In most actuarial analyses, for both regulatory and management purposes, the focus is on the risk exposure at the tails (typically the 90th percentile and beyond). • Long run times and lack of a management tool for what-if, actionable analysis.

4. Solutions from several sources have been explored Actuarial Methods • Scenario reduction • Modeling/compression Advantages • Familiar and well-understood Disadvantages • Reduces accuracy • Exposed to model risk • Run-time savings not sufficient Technological Methods • Grid processing Advantages • Brute force method so easy to understand • Can always buy more computers Disadvantages • Costly • Battle for grid time • Still long run times Market-driven Methods • Replicating portfolios Advantages • Closed-form solutions so extremely fast • Allows processing of many scenarios Disadvantages • Only works for market-based parameters – can’t analyze mortality or lapse scenarios • Fit to insurance liabilities

5. Statistical Sampling Approaches • Availability • Entire population • Detailed policy information • Leads to seriatim calculations or grouping methods • Perception that more detail is always better • Analysis of entire population gives more precise information than analysis of a random sample • Sampling error difficult to quantify • Lacking a bridge between academia and industry • UConn Actuarial Center is that bridge Non-existent in actuarial modeling techniques! Why?

6. Towers Watson Replicated Stratified Sampling (RSS) Our patent-pending approach rapidly accelerates run times for many actuarial models, and has the following characteristics: • Based on sound fundamentals of statistical inference • Combination of stratified sampling and sample replication • Reduces run time for any complex stochastic model • with large underlying population • with easy access to underlying population • Produces stable results • Produces robust results with measurable, pre-determined sampling error • Simple to understand, implement and maintain

7. Uniqueness of the RSS Approach • Does not attempt to “simplify” or “approximate” the underlying population characteristics. • Builds on the existing company actuarial models. • Allows for detailed analysis of cash flows under both economic scenarios (equity and interest rate changes) and changes in actuarial assumptions (mortality, lapses, policyholder behavior, etc). • The entire underlying population distribution is approximated under RSS at a prescribed level of accuracy for each quantile. • Convergence time is independent of the size of the population. • Convergence speed and accuracy of RSS technique are based on well-established and tested statistical inference theory.

8. RSS Pilot Study • RSS technique applied to a variable annuity block of a major life insurance company. • Analyzed impact of - an immediate15% drop in equity funds on VACARVM reserves - an immediate 35% drop in equity funds on VACARVM reserves • Analysis done for 3 legal entities both beforeandafter reinsurance. • Analysis compared the change in the VACARVM reserve in the population versus using the RSS technique on 50, 100, 150 and 200 samples of 30 policies each. • Error rate defined as: where A = change in VACARVM reserve using the RSS technique B = change in VACARVM reserve in the population

9. PopulationSummary After Reinsurance Before Reinsurance

10. RSS Results – Sensitivity 1 After Reinsurance Before Reinsurance

11. RSS Results – Sensitivity 2 After Reinsurance Before Reinsurance

12. Advantages of the RSS Approach Significantly reduces run time, allowing more flexibility and transparency in trade-offs between speed and accuracy: • Increase accuracy • Run more stochastic scenarios, improving tail risk analysis • Reduce use of grouping techniques, improving risk analysis in general • Reduce use of shortcuts in modeling approach, decreasing model risk • Map complex investment funds directly, eliminating basis risk • Minimize sampling bias • Increase speed • Maintain model and population complexity but decrease run time • Broad Applicability • Can be used across a range of models and calculations

13. Potential Applications of the RSS Approach VACARVM Production, impact testing, attribution analysis Economic Capital Hedge Programs Hedge effectiveness testing, Explanation of breakage U.S. GAAP SFAS 133, SOP 03-1 Production, impact testing, forecasting Strategic Planning PBA for life insurance products Analysis of Inforce Profitability Any type of analysis that relies on complex, stochastic calculations is a candidate for the RSS approach

14. RSS as a Strategic Management Tool • RSS is ideally suited for any type of “what if” management analysis • Instead of 1,000 scenarios, run 10,000 or 100,000 • Instead of 5 or 10 sensitivities, run 50, 100, or 500 • Results are more robust, more accurate, more timely and therefore more actionable Using RSS, management can be prepared for so called 4th quadrant (low probability, high severity) events that threaten the long-term sustainability of the insurance industry.

15. Open Analytical Research Topics • Mathematical proof, using existing convergence theorems in statistics, that the RSS algorithm generates unbiased and efficient estimates of the change in the population risk measure and is independent of the underlying risk measure being analyzed. • Analytical justification, using numerical analysis and asymptotic techniques, on the number of replications required to achieve a prescribed accuracy level of the RSS estimate of the change in the population risk measure. • Use of clustering analysis techniques to determine the optimal set of risk classes in order to minimize processing time subject to a prescribed level of accuracy of the RSS estimates.

16. Conclusions • Complex actuarial modeling in response to increasingly complex insurance products has led to run times that are prohibitive. • To cope, management has been forced to make trade-offs that are costly either in speed, accuracy or dollar costs. • Towers Watson’s Replicated Stratified Sampling (RSS) approach offers a paradigm shift in measuring and managing risk using actuarial modeling, by dramatically reducing run time for: • Stochastic models, including hedging tools, • Models with large databases, • Models with easy access to underlying population. • The RSS approach allows management more flexibility to proactively participate in the risk management process and better understand the impact of current and potential market, economic, actuarial and customer behavior changes.

17. Contact details Jay Vadiveloo, PhD, FSA, MAAA, CFA Towers Watson Consulting Actuary Towers Watson Professor, University of Connecticut Work: (860) 843-7073 Cell: (860) 916-1010 Email: Jay.Vadiveloo@towerswatson.com