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Exploring Policyholder Behavior in the Extreme Tail Yuhong (Jason) Xue, FSA MAAA. Agenda. Introductions Policyholder behavior risk as a strategic risk Copulas and Extreme Value Theory (EVT) Applying EVT to behavior study The methodology The example: data, model fitting and simulation
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Exploring Policyholder Behavior in the Extreme Tail Yuhong (Jason) Xue, FSA MAAA
Agenda • Introductions • Policyholder behavior risk as a strategic risk • Copulas and Extreme Value Theory (EVT) • Applying EVT to behavior study • The methodology • The example: data, model fitting and simulation • Summary and Implications Session C-19 Yuhong (Jason) Xue
Introduction - Policyholder Behavior Risk • Why it’s important to manage both short term and long term risks • Risk functions tend to focus more on short term risks • When it comes to long term strategic risks which are sometimes unknown or slow emerging, few are good at it • Yet the root cause of companies’ failure is often failing to recognize a emerging trend Session C-19 Yuhong (Jason) Xue
Introduction - Policyholder Behavior Risk • Policyholder behavior risk is a strategic risk for insurers • How will policyholders behave in the tail is largely unknown • Yet assumption of this behavior is embedded in pricing, reserving, hedging and capital determination • It is of strategic importance to the whole industry Session C-19 Yuhong (Jason) Xue
Introduction - Copulas Copula 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 - EVT • Pickands (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 Session C-19 Yuhong (Jason) Xue
Introduction - EVT • Predictive power of EVT • Question: how are random variables relate to each other in the extremes • If 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 analyzed • EVT has lots of applications in insurance Session C-19 Yuhong (Jason) Xue
Applying EVT to Behavior Study - Methodology • Policyholder behavior in extreme economic conditions in math terms is essentially how two or more random variables correlate in the tail • Methodology • Marginal distribution • Analyze marginal empirical data and define threshold • Fit GP to data that exceeds the threshold • Copula fitting • Given the GP marginal distribution and the thresholds for each variable, find a copula that provides a good fit for the excesses • Simulation • Simulate the extreme tail using the fitted multivariate distribution model Session C-19 Yuhong (Jason) Xue
Applying EVT to Behavior Study – Variable Annuity Example • The VA block • Hypothetical VA block with Guaranteed Lifetime Withdrawal Benefits • Resembles common patterns of lapse experience observed in the market place • Mostly L share business with 4 years of surrender charge Session C-19 Yuhong (Jason) Xue
Applying EVT to Behavior Study – Variable Annuity Example Scatter plot of ITM and 1/Lapse Data exceeding 90th percentile: weak dependence Raw data: Strong dependence • Data • Variable annuity (VA) shock lapse: lapse rate of 1st year surrender charge is zero • In-The-Moneyness = PV of future payment / Account value - 1 Session C-19 Yuhong (Jason) Xue
Applying EVT to Behavior Study – Variable Annuity Example • Model fitting • We chose 3 thresholds: 55th, 85th and 90th percentile and 3 copula families: Gumbel, Frank and Clayton to fit the data • The results for GP marginals: • The results for Copulas: Session C-19 Yuhong (Jason) Xue
Applying EVT to Behavior Study – Variable Annuity Example • Simulation • Simulated ITM and lapse rates in the extreme tail using the model • Implied dynamic lapse function • dynamic lapse factor is applied to the base lapse assumption to arrive at actual lapse rate when policies are deep in the money • Dynamic lapse curves on the right are developed using regression • Because 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 Implications EVT can reveal insightful information about policyholder behavior in the extreme tail compared to traditional methods This insight can lead to strategic advantage in better managing the behavior risk: more informed pricing, better reserving and more adequate capital The result from the VA example should not be generalized as it can be data dependent Threshold 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 Session C-19 Yuhong (Jason) Xue
Questions Jason Xue Yuhong_xue@glic.com 212-598-1621 Session C-19 Yuhong (Jason) Xue