replicated stratified sampling

© 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

2towerswatson.com© 2010 Towers Watson. All rights reserved. Proprietary and Confidential. For Towers Watson and Towers Watson client use only.

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.


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.


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


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?


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


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.


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


Population Summary

LegalEntity

Number of policies

BaselineReserves

Sensitivity 1Reserves


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



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


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


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


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


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


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.


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.


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.


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

replicated stratified sampling

Documents