© 2010 Towers Watson. All rights reserved.
Replicated Stratified SamplingA Practical Approach to Financial Modeling
2010 IABA Annual Meeting
August 6 - 7, 2010
Jay Vadiveloo, PhD, FSA, MAAA,CFA
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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.
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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.
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Disadvantages Reduces accuracy Exposed to model risk Run-time savings not
sufficient
Disadvantages Reduces accuracy Exposed to model risk Run-time savings not
sufficient
Solutions from several sources have been explored
ActuarialActuarial
Methods Grid processingMethods Grid processing
TechnologicalTechnological
Advantages Brute force method so easy to
understand Can always buy more
computers
Advantages Brute force method so easy to
understand Can always buy more
computers
Disadvantages Costly Battle for grid time Still long run times
Disadvantages Costly Battle for grid time Still long run times
Methods Replicating portfoliosMethods Replicating portfolios
Market-drivenMarket-driven
Advantages Closed-form solutions so
extremely fast Allows processing of many
scenarios
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
Disadvantages Only works for market-based
parameters – can’t analyze mortality or lapse scenarios
Fit to insurance liabilities
Advantages Familiar and well-understoodAdvantages Familiar and well-understood
Methods Scenario reduction Modeling/compression
Methods Scenario reduction Modeling/compression
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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?
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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
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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.
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RSS Pilot Study
RSS technique applied to a variable annuity block of a major life insurance company.
Analyzed impact of - an immediate 15% drop in equity funds on VACARVM reserves - an immediate 35% drop in equity funds on VACARVM reserves
Analysis done for 3 legal entities both before and after 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
A B
B
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Population Summary
LegalEntity
Number of policies
BaselineReserves
Sensitivity 1Reserves
Sensitivity 2Reserves
POP Ratio 1
POPRatio 2
1 1,027,572 -2,765,845 -4,605,188 -11,714,900 1.665 4.236
2 397,781 -13,298,401 -19,171,991 -20,009,138 1.442 1.505
3 547,883 -3,470,570 -12,610,502 -87,276,482 3.634 25.148
After ReinsuranceAfter Reinsurance
Before ReinsuranceBefore ReinsuranceLegalEntity
Number of policies
BaselineReserves
Sensitivity 1Reserves
Sensitivity 2Reserves
POP Ratio 1
POPRatio 2
1 1,027,572 -79,535,358 -168,055,144 -582,368,081 2.113 7.322
2 397,781 -890,538,871 -1,621,720,896 -3,807,237,541 1.821 4.275
3 547,883 -8,777,155 -27,902,521 -178,654,004 3.179 20.354
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RSS Results – Sensitivity 1
LegalEntity
# Of samples
RSSRatio
POPRatio
ErrorRate
1
50 1.513 1.665 9.14%
100 1.615 3.00%
150 1.640 1.52%
200 1.667 0.12%
2
50 1.611 1.442 11.73%
100 1.486 3.10%
150 1.424 1.25%
200 1.450 0.59%
3
50 3.015 3.634 17.02%
100 3.482 4.18%
150 3.735 2.78%
200 3.633 0.02%
LegalEntity
# Of samples
RSSRatio
POPRatio
ErrorRate
1
50 2.067 2.113 2.17%
100 2.124 0.51%
150 2.106 0.32%
200 2.110 0.15%
2
50 1.826 1.821 0.25%
100 1.812 0.49%
150 1.818 0.17%
200 1.822 0.03%
3
50 2.853 3.179 10.24%
100 3.028 4.74%
150 3.153 0.82%
200 3.177 0.07%
After ReinsuranceAfter Reinsurance Before ReinsuranceBefore Reinsurance
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RSS Results – Sensitivity 2
LegalEntity
# Of samples
RSSRatio
POPRatio
ErrorRate
1
50 2.771 4.236 34.58%
100 3.783 10.68%
150 4.071 3.88%
200 4.216 0.45%
2
50 1.117 1.505 25.75%
100 1.339 11.00%
150 1.386 7.89%
200 1.506 0.10%
3
50 17.574 25.148 30.12%
100 23.239 7.59%
150 26.601 5.78%
200 25.225 0.31%
LegalEntity
# Of samples
RSSRatio
POPRatio
ErrorRate
1
50 5.575 7.322 23.86%
100 7.043 3.81%
150 7.099 3.04%
200 7.330 0.10%
2
50 4.884 4.275 14.24%
100 5.117 19.70%
150 4.652 8.81%
200 4.282 0.15%
3
50 13.845 20.354 31.98%
100 19.059 6.37%
150 21.203 4.17%
200 20.336 0.09%
After ReinsuranceAfter Reinsurance Before ReinsuranceBefore Reinsurance
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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
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PBA for life insurance products
U.S. GAAPSFAS 133, SOP 03-1
Production, impact testing, forecasting
VACARVMProduction, impact
testing, attribution analysis
Analysis of Inforce Profitability
Hedge ProgramsHedge effectiveness
testing, Explanation of breakage
Economic Capital
Any type of analysis that relies on complex, stochasticcalculations is a candidate for the RSS approach
Any type of analysis that relies on complex, stochasticcalculations is a candidate for the RSS approach
Potential Applications of the RSS Approach
Strategic Planning
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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 threatenthe long-term sustainability of the insurance industry.
Using RSS, management can be prepared for so called 4th quadrant (low probability, high severity) events that threatenthe long-term sustainability of the insurance industry.
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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.
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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.
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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: [email protected]
Contact details
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