Exploring Policyholder Behavior in the Extreme Tail Yuhong (Jason) Xue, FSA MAAA

Exploring Policyholder Behavior in the Extreme TailYuhong (Jason) Xue, FSA MAAA

Session C-19 Yuhong (Jason) Xue

AgendaIntroductionsPolicyholder behavior risk as a strategic riskCopulas and Extreme Value Theory (EVT)Applying EVT to behavior studyThe methodologyThe example: data, model fitting and simulationSummary and Implications

Session C-19 Yuhong (Jason) Xue*


Introduction - Policyholder Behavior RiskWhy its important to manage both short term and long term risksRisk functions tend to focus more on short term risksWhen it comes to long term strategic risks which are sometimes unknown or slow emerging, few are good at itYet the root cause of companies failure is often failing to recognize a emerging trend



Introduction - Policyholder Behavior RiskPolicyholder behavior risk is a strategic risk for insurersHow will policyholders behave in the tail is largely unknownYet assumption of this behavior is embedded in pricing, reserving, hedging and capital determinationIt is of strategic importance to the whole industry



Introduction - CopulasCopula C is a joint distribution function of uniform random variables:

Sklar (1959) showed that a multivarite distribution function can be written in the form of a copula and their marginal distribution functions:

The dependence structure of F can be fully captured by the copula C independent of the marginal distributions Session C-19 Yuhong (Jason) Xue*


Introduction - EVTPickands (1975) used Generalized Pareto (GP) distribution to approximate the conditional distribution of excesses above a sufficiently large threshold The distribution of Pr(X > u + y | X > u), where y > 0 and u is sufficiently large, can be modeled by

In the multivariate case, joint excesses can be approximated by a combination of marginal GP distributions and a copula that belongs to certain copula families such as Gumbel, Frank, and Clayton



Introduction - EVTPredictive power of EVTQuestion: how are random variables relate to each other in the extremesIf enough data beyond a large threshold is available so that a multivariate EVT model can be reasonably fitted, the relationship of the variables in the extreme can be analyzedEVT has lots of applications in insurance



Applying EVT to Behavior Study - MethodologyPolicyholder behavior in extreme economic conditions in math terms is essentially how two or more random variables correlate in the tailMethodologyMarginal distributionAnalyze marginal empirical data and define thresholdFit GP to data that exceeds the thresholdCopula fittingGiven the GP marginal distribution and the thresholds for each variable, find a copula that provides a good fit for the excessesSimulationSimulate the extreme tail using the fitted multivariate distribution model Session C-19 Yuhong (Jason) Xue*


Applying EVT to Behavior Study Variable Annuity ExampleThe VA blockHypothetical VA block with Guaranteed Lifetime Withdrawal BenefitsResembles common patterns of lapse experience observed in the market placeMostly L share business with 4 years of surrender charge Session C-19 Yuhong (Jason) Xue*


Applying EVT to Behavior Study Variable Annuity ExampleDataVariable annuity (VA) shock lapse: lapse rate of 1st year surrender charge is zeroIn-The-Moneyness = PV of future payment / Account value - 1

Raw data: Strong dependenceData exceeding 90th percentile: weak dependenceScatter plot of ITM and 1/Lapse Session C-19 Yuhong (Jason) Xue*


Applying EVT to Behavior Study Variable Annuity ExampleModel fittingWe chose 3 thresholds: 55th, 85th and 90th percentile and 3 copula families: Gumbel, Frank and Clayton to fit the dataThe results for GP marginals:

The results for Copulas:



Applying EVT to Behavior Study Variable Annuity ExampleSimulationSimulated ITM and lapse rates in the extreme tail using the modelImplied dynamic lapse functiondynamic lapse factor is applied to the base lapse assumption to arrive at actual lapse rate when policies are deep in the moneyDynamic lapse curves on the right are developed using regressionBecause lack of data in the region, the curve based on raw data extrapolates strong dependence from the less extreme area Combined raw data with simulated data, the curves show less dependence in the tail Session C-19 Yuhong (Jason) Xue*


Summary and ImplicationsEVT can reveal insightful information about policyholder behavior in the extreme tail compared to traditional methodsThis insight can lead to strategic advantage in better managing the behavior risk: more informed pricing, better reserving and more adequate capitalThe result from the VA example should not be generalized as it can be data dependentThreshold selection in applying EVT is often a tradeoff between having a close approximation and allowing enough data for fitting. There can be situations where finding the tradeoff is difficult



QuestionsJason [email protected] Session C-19 Yuhong (Jason) Xue*


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Exploring Policyholder Behavior in the Extreme Tail Yuhong (Jason) Xue, FSA MAAA