ERM Symposium 2012

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

2. 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
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3. 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
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4. 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
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5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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
Scatter plot of
ITM and 1/Lapse
Raw data: Strong dependence Data exceeding 90th percentile:
weak dependence
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11. 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:
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
– The results for Copulas:
Threshold 55th 85th 90th
Number 560 145 95
data pairs
Copula Parameter Pseudo Max Parameter Pseudo Max Parameter Pseudo Max
Loglikelihood Loglikelihood 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
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12. Applying EVT to Behavior Study –
Variable Annuity Example
• Simulation
– Simulated ITM and lapse rates in the
extreme tail using the model Dynamic Lapse Factor as a function of AV/Guar
• Implied dynamic lapse function - Calculated from GLM Regression
– dynamic lapse factor is applied to 0.80
the base lapse assumption to 0.70
arrive at actual lapse rate when 0.60
policies are deep in the money 0.50
0.40
– Dynamic lapse curves on the right 0.30
are developed using regression 0.20
0.10
– Because lack of data in the region, -
the curve based on raw data
67%
61%
56%
51%
48%
44%
42%
39%
37%
35%
33%
32%
30%
29%
28%
27%
26%
25%
extrapolates strong dependence
Combined w Gumbel Combined w Clayton Raw Data
from the less extreme area
– Combined raw data with simulated
data, the curves show less
dependence in the tail
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13. 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
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14. Questions
Jason Xue
Yuhong_xue@glic.com
212-598-1621
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