Presented on 2008 SOA Annual Meeting

1. watsonwyatt.com
2008 SOA Annual Meeting
Predictive Modeling in Life Insurance
Yuhong (Jason) Xue, FSA, MAAA
October 21, 2008

2. Agenda
Theoretical Background of Predictive Modeling
– Generalized Linear Modeling (GLM)
Applications of GLM in Life Insurance
– Mortality analysis
– Policy holder behavior study
– Stochastic modeling
1

3. Theoretical Background
2

4. Predictive Modeling
 Statistical model that relates an event (death) with a number of
risk factors (age, sex, YOB, amount, marital status, etc.)
Age
Sex
Y.o.B. Expected
Model mortality
Married
Amount
etc.
3

5. Generalized Linear Models (GLMs)
 Special type of predictive modelling
 A method that can model
– a number
as a function of
– some factors
 For instance, a GLM can model
– Motor claim amounts as a function of driver age, car type, no
claims discount, etc …
– Motor claim frequency (as a function of similar factors)
 Historically associated with P&C pricing (where there was a
pressing need for multivariate analysis)
4

6. Understanding GLM Results
 A GLM will model the ‘observed amount’ (eg motor claims
frequency, mortality rate, economic capital results from a life
model) as
Amount = Base level × Factor 1 × Factor 2 …
 For example, if ‘observed amount’ is mortality, Factor 1 is
gender, and Factor 2 is annuity payment band, then
Base GLM Payment GLM
Level Gender Factor Band Factor
0.005 M 1.0 100-500 1.5
F 0.8 500-1000 1.1
1000-2000 1.0
>2000 0.9
 Mortality for Female with Payment in band 100-500 =
0.005 x 0.8 x 1.5 = 0.006
5

7. Mathematical Form of GLM
E[Y] =  = g ( X -1
)
Some function Parameters to be
(user defined) estimated
(the answer!)
Some matrix based on data
Observed thing (user defined)
(data) as per linear models
6

8. Bedtime Reading
 Copies available at
www.watsonwyatt.com/glm
7

9. Applications of GLM in Mortality Analysis
8

10. Mortality Analysis of Annuitant
 The traditional approach: experience study
– Focus on limited risk factors, such as Age, Sex, may extend to
other factors (i.e. amount)
– Calculate A/E ratio with slicing and dicing techniques to come
up with a set of weights (or multipliers)
– Limitation: Ignore interaction
 For example, a simple tabulation of mortality by annuity amount
ignores impact of other risk factors such as marital status
 Advantages of GLM
– A multivariate analysis including all risk factors simultaneously
– Isolate impact of a single risk factor
– Unique ability of using calendar year as a risk factor, making it
possible to study many years of data
9

11. Examples of Mortality Analysis
Examples Using GLM to Analyze Annuitant Mortality
Based on dataset representing a life company’s
typical portfolio of retirees currently receiving benefits
10

12. Example 1: Effect of Annuity Amount
Generalized Linear Modeling Illustration
Income Effect
0.06
0% 1600000
0
1400000
-6%
-0.06
1200000
Exposure (years)
Log of multiplier
-0.12 -15% 1000000
800000
-0.18 -18%
600000
-0.24
400000
-0.3
200000
-29%
-0.36 0
<= 30K <= 50K <= 75K <= 100K > 100K
Income
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
Results show evidence of reduced mortality with increased benefits
11

13. Example 2: Calendar Year Trend
Generalized Linear Modeling Illustration
Run 1 Model 2 - GLM - Significant
0.1
700000
0.08
600000
0.06 5%
500000
Exposure (years)
4%
Log of multiplier
4%
0.04
400000
2%
0.02 300000
1%
0%
0 200000
-0.02 100000
-0.04 0
2002 2003 2004 2005 2006 2007
Calendar year
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
Mortality improvements 1% per annum over previous six years
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14. Example 3: The Selection Effect
Generalized Linear Modeling Illustration
Run 1 Model 2 - GLM - Significant
0.01
3000000
0%
0
2500000
-0.01
-0.02
Exposure (years)
2000000
Log of multiplier
-3%
-0.03
1500000
-0.04
-0.05 1000000
-0.06
500000
-0.07
-0.08 0
<=5 5+
Duration
Approx 95% confidence interval Smoothed estimate
Selection effect is inconclusive
13

15. Example 4: Birth Cohort Effect
Generalized Linear Modeling Illustration
Birth Cohort
0.15
500000
0.1
7%
5% 400000
5% 5%
4% 5%
0.05 4%
3%
Exposure (years)
Log of multiplier
2%
1%
0% 0% 300000
-1% -1% -1% -1%
0 -2% -1%
-2%
-4%
200000
-0.05
100000
-0.1
-0.15 0
<= 1915 <= 1918 <= 1921 <= 1924 <= 1926 <= 1928 <= 1931 <= 1933 <= 1936 <= 1940
Smoothed estimate, Sex: M Smoothed estimate, Sex: F
No Cohort Effect for male and Female
14

16. Example 5: Effect of Joint Life Status
Generalized Linear Modeling Illustration
Joint Survivor Status
0.08
2500000
0.06
0.04 3% 2000000
Exposure (years)
0.02
Log of multiplier
1500000
0%
0
-0.02 1000000
-4%
-0.04
500000
-0.06
-0.08 0
Single Life Joint Life Primary Joint Life Surviving Spouse
Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
Evidence of “broken heart syndrome” which may influence pricing
15

17. Mortality Varies by Postcode
 Map shows age-
standardised mortality
rates in England &
Wales
 From red = high to
blue = low
16

18. Why Use GLM in Analyzing Mortality
Valuation
– More accurate mortality rates can impact the present
value of cash flow by 1 – 2% which is significant in bulk
buyout situations
Pricing
– Characteristics identified by GLM that influence
mortality can be used for pricing purposes
Understanding Risks
– Certain characteristics identified by GLM, such as
geographical location, can be used to focus marketing
efforts
17

19. Use GLM to Study Policy Holder Behavior
18

20. Example of Lapse Study
 Advantages of GLM in studying policy holder behavior
– Better quantify effects of factors: age/sex, duration,
calendar year of exposure, benefit amount, geographical
location, distribution channel, …
– Can Include standard economic measures such as GDP
and equity market returns to study dynamic lapses
– Can also study correlations of guarantee utilization rate with
factors like In-The-Moneyness and value of liability
 The following examples are based on a portfolio of single
premium deferred annuities
19

21. The Effect of Duration
GLM life surrender analysis - duration
0.6
0.3 3000000
0 2500000
Exposure (years)
Log of multiplier
-0.3 2000000
-0.6 1500000
-0.9 1000000
-1.2 500000
-1.5 0
0 1 2 3 4 5 6 7 8 9 10 >=11
Oneway relativities Unsmoothed estimate
20

22. Application of GLM in Stochastic Modeling
21

23. Example of Economic Capital (EC) Modeling
 Economic Capital (EC) is the end of year one capital
requirement at 99.95% confidence level
 Treat result of every scenario in the stochastic run as
one observation
 Treat the parameters in the ESG as risk factors
 Advantages
– Quick independent check of the model as stochastic
results are difficult to validate
– Provides a closed-form solution of EC which can be used
as approximations to avoid nested stochastic loops in
certain applications
22

24. Economic Capital Modeling
Change in credit spread
Equity Property
Simulation return return Pc1 Pc2 Pc3 AAA AA A Capital yr 1
1 -18.1% -11.3% 1.1728 -0.0694 -0.0764 0.14% 0.19% 0.19% 351,956,232
2 37.5% 28.4% -0.8093 0.1426 0.0376 0.15% 0.19% 0.20% 182,869,264
3 -16.0% 12.9% -1.1597 -0.2165 0.0151 0.10% 0.08% 0.09% 295,234,182
4 34.0% 0.9% 1.5612 0.3284 -0.0514 0.40% 0.57% 0.60% 273,541,440
5 -7.6% 42.1% 5.7572 0.1840 -0.2618 0.00% -0.06% -0.08% 132,504,095
6 21.9% -19.8% -4.3497 -0.1075 0.1720 0.23% 0.32% 0.34% 401,335,715
7 -28.9% 0.8% 2.4245 0.0486 -0.1218 0.11% 0.14% 0.15% 310,364,028
8 58.4% 13.0% -1.9035 0.0247 0.0747 0.08% 0.05% 0.01% 192,173,551
9 11.5% 45.0% -2.9855 -0.6720 0.0549 0.00% -0.07% -0.10% 188,914,076
10 1.0% -21.5% 3.8398 0.9810 -0.0792 0.22% 0.30% 0.32% 303,942,207
11 -8.5% 3.5% 3.5653 0.7616 -0.0941 0.02% -0.05% -0.03% 221,069,505
12 22.9% 10.0% -2.7940 -0.5229 0.0594 0.07% 0.06% 0.05% 276,782,151
13 4.2% -12.3% -2.3709 0.1354 0.1092 0.17% 0.22% 0.24% 355,975,365
14 8.1% 29.9% 3.0075 -0.0947 -0.1628 0.35% 0.52% 0.57% 223,679,045
15 -9.1% -9.2% 0.7996 -0.5391 -0.1028 0.02% -0.04% -0.06% 327,166,411
16 23.8% 25.5% -0.4267 0.6100 0.0772 -0.02% -0.12% -0.14% 151,541,793
17 14.8% 12.6% -4.6239 -0.1730 0.1771 0.36% 0.41% 0.51% 338,591,627
18 -2.0% 1.2% 6.1837 0.2795 -0.2719 0.20% 0.40% 0.45% 245,649,584
19 -28.8% -4.4% 1.2037 0.2253 -0.0464 0.01% -0.06% -0.09% 303,185,900
20 58.2% 19.2% -0.8391 -0.1285 0.0094 0.14% 0.32% 0.31% 188,498,682
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25. Initial Results (good)
Preliminary analysis of ICA results
The higher- All factors,return the less the
the normal identity, no interactions (Genmod used)
Run 1 Model 3 - Initial runs
0.3
capital requirement
24%
0.2 18% 10000
15%
11%
7%
0.1
4%
0% 8000
0 -4%
Number of claims
-7%
Log of multiplier
-11%
-0.1 -14% 6000
-18%
-0.2 -21%
-24%
-29% 4000
-0.3
-0.4
2000
-40%
-0.5
-0.6 0
Property return
Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
P value = 0.0%
Rank 5/8
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26. Initial Results (bad)
Spike- Initial runs - All factors, normalproblemresults used)
Preliminary analysis of ICA in the
Run 1 Model 3
indicated a identity, no interactions (Genmod
0.012
model
1%
12000
0.008
0%
0% 10000
0.004 0%
0% 0%
0% 0%
Number of claims
0%
Log of multiplier
0%
8000
0
0%
6000
-0.004
-1%
-0.008 4000
-0.012 2000
-1%
-0.016 0
Change credit spread A
P value = 0.0%
Approx 95% confidence interval Unsmoothed estimate Smoothed estimate
Rank 1/8
25