Xue 2012 erm ppt

Exploring Policyholder Behavior in the Extreme Tail

Yuhong (Jason) Xue, FSA MAAA

2 Session C-19 Yuhong (Jason) Xue

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


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


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


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


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


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


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


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



• 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 90th percentile: weak dependence

Scatter plot of ITM and 1/Lapse



• 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:

Threshold Variable location Scale shape 55th ITM -0.005 0.197 -0.193

1/lapse 3.448 0.282 1.387 85th ITM 0.161 0.259 -0.446

1/lapse 4.545 1.986 -0.156 90th ITM 0.223 0.245 -0.476

1/lapse 5.000 2.222 -0.217

Threshold 55th 85th 90th Number data pairs

560 145 95

Copula Parameter Pseudo Max Loglikelihood

Parameter Pseudo Max Loglikelihood

Parameter Pseudo Max Loglikelihood

Gumbel 1.715 140.869 1.278 8.893 1.106 1.236 Frank 4.736 134.379 2.420 10.678 0.912 1.043 Clayton 0.801 69.952 0.601 10.881 0.148 0.531



• 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

67%59%

53%48%

43%40%

37%34%

32%30%

29%27%

26%24%

23%22%

21%20%

20% -

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Dynamic Lapse Factor as a function of AV/Guar- Calculated from GLM Regression

Combined w Gumbel Combined w Clayton Raw Data


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


Questions

Jason [email protected]

212-598-1621

mailto:[email protected]

Xue 2012 erm ppt

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