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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

Applying EVT to Behavior Study – Variable Annuity Example 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 Raw data: Strong dependence Data exceeding 90 th percentile: weak dependence Scatter plot of ITM and 1/Lapse Session C-19 Yuhong (Jason) Xue 10

Applying EVT to Behavior Study – Variable Annuity Example Model fitting – We chose 3 thresholds: 55 th, 85 th and 90 th 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 11

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 12

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 13

Questions Jason Xue Session C-19 Yuhong (Jason) Xue 14