watsonwyatt.com

2008 SOA Annual Meeting

Predictive Modeling in Life Insurance

Yuhong (Jason) Xue, FSA, MAAAOctober 21, 2008

1

Agenda

Theoretical Background of Predictive Modeling– Generalized Linear Modeling (GLM)

Applications of GLM in Life Insurance– Mortality analysis– Policy holder behavior study– Stochastic modeling

2

Theoretical Background

3

Predictive Modeling

Statistical model that relates an event (death) with a number ofrisk factors (age, sex, YOB, amount, marital status, etc.)

Amount

Y.o.B.

Age

etc.

Sex

Married

ExpectedmortalityModel

4

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)

5

Understanding GLM Results

Base Level0.005

GenderGLMFactor

M 1.0 F 0.8

A GLM will model the ‘observed amount’ (eg motor claims frequency, mortality rate, economic capital results from a life model) asAmount = 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

Mortality for Female with Payment in band 100-500 =0.005 x 0.8 x 1.5 = 0.006

Payment Band

GLMFactor

100-500 1.5 500-1000 1.1 1000-2000 1.0 >2000 0.9

6

E[Y] = = g ( X )-1

Observed thing(data)

Some function(user defined)

Some matrix based on data(user defined)

as per linear models

Parameters to beestimated

(the answer!)

Mathematical Form of GLM

7

Bedtime Reading

Copies available atwww.watsonwyatt.com/glm

8

Applications of GLM in Mortality Analysis

9

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

10

Examples of Mortality Analysis

Examples Using GLM to Analyze Annuitant MortalityBased on dataset representing a life company’s

typical portfolio of retirees currently receiving benefits

11

Example 1: Effect of Annuity Amount

Results show evidence of reduced mortality with increased benefits

Generalized Linear Modeling IllustrationIncome Effect

-29%

-18%

-15%

-6%

0%

-0.36

-0.3

-0.24

-0.18

-0.12

-0.06

0

0.06

Income

Log

of m

ultip

lier

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

<= 30K <= 50K <= 75K <= 100K > 100K

Expo

sure

(yea

rs)

Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

12

Example 2: Calendar Year Trend

Mortality improvements 1% per annum over previous six years

Generalized Linear Modeling IllustrationRun 1 Model 2 - GLM - Significant

0%1%

2%

4%4%

5%

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

Calendar year

Log

of m

ultip

lier

0

100000

200000

300000

400000

500000

600000

700000

2002 2003 2004 2005 2006 2007

Expo

sure

(yea

rs)

Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

13

Example 3: The Selection Effect

Selection effect is inconclusive

Generalized Linear Modeling IllustrationRun 1 Model 2 - GLM - Significant

0%

-3%

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

Duration

Log

of m

ultip

lier

0

500000

1000000

1500000

2000000

2500000

3000000

<=5 5+

Expo

sure

(yea

rs)

Approx 95% confidence interval Smoothed estimate

14

Example 4: Birth Cohort EffectGeneralized Linear Modeling Illustration

Birth Cohort

0%

4%

-1%

5%5%

7%

5%5%4%

3%

-1%

2%

-4%

0%-1%

-2%-1%

1%

-2%-1%

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Log

of m

ultip

lier

0

100000

200000

300000

400000

500000

<= 1915 <= 1918 <= 1921 <= 1924 <= 1926 <= 1928 <= 1931 <= 1933 <= 1936 <= 1940

Expo

sure

(yea

rs)

Smoothed estimate, Sex: M Smoothed estimate, Sex: F

No Cohort Effect for male and Female

15

Example 5: Effect of Joint Life Status

Evidence of “broken heart syndrome” which may influence pricing

Generalized Linear Modeling IllustrationJoint Survivor Status

3%

-4%

0%

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Log

of m

ultip

lier

0

500000

1000000

1500000

2000000

2500000

Single Life Joint Life Primary Joint Life Surviving Spouse

Expo

sure

(yea

rs)

Oneway relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

16

Mortality Varies by Postcode

Map shows age-standardised mortality rates in England & Wales

From red = high to blue = low

17

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

18

Use GLM to Study Policy Holder Behavior

19

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

20

The Effect of DurationGLM life surrender analysis - duration

-1.5

-1.2

-0.9

-0.6

-0.3

0

0.3

0.6

Log

of m

ultip

lier

0

500000

1000000

1500000

2000000

2500000

3000000

0 1 2 3 4 5 6 7 8 9 10 >=11

Exp

osur

e (y

ears

)

Oneway relativities Unsmoothed estimate

21

Application of GLM in Stochastic Modeling

22

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

23

Economic Capital Modeling Change in credit spread

SimulationEquity return

Property 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% 351,956,2322 37.5% 28.4% -0.8093 0.1426 0.0376 0.15% 0.19% 0.20% 182,869,2643 -16.0% 12.9% -1.1597 -0.2165 0.0151 0.10% 0.08% 0.09% 295,234,1824 34.0% 0.9% 1.5612 0.3284 -0.0514 0.40% 0.57% 0.60% 273,541,4405 -7.6% 42.1% 5.7572 0.1840 -0.2618 0.00% -0.06% -0.08% 132,504,0956 21.9% -19.8% -4.3497 -0.1075 0.1720 0.23% 0.32% 0.34% 401,335,7157 -28.9% 0.8% 2.4245 0.0486 -0.1218 0.11% 0.14% 0.15% 310,364,0288 58.4% 13.0% -1.9035 0.0247 0.0747 0.08% 0.05% 0.01% 192,173,5519 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% 303,942,20711 -8.5% 3.5% 3.5653 0.7616 -0.0941 0.02% -0.05% -0.03% 221,069,50512 22.9% 10.0% -2.7940 -0.5229 0.0594 0.07% 0.06% 0.05% 276,782,15113 4.2% -12.3% -2.3709 0.1354 0.1092 0.17% 0.22% 0.24% 355,975,36514 8.1% 29.9% 3.0075 -0.0947 -0.1628 0.35% 0.52% 0.57% 223,679,04515 -9.1% -9.2% 0.7996 -0.5391 -0.1028 0.02% -0.04% -0.06% 327,166,41116 23.8% 25.5% -0.4267 0.6100 0.0772 -0.02% -0.12% -0.14% 151,541,79317 14.8% 12.6% -4.6239 -0.1730 0.1771 0.36% 0.41% 0.51% 338,591,62718 -2.0% 1.2% 6.1837 0.2795 -0.2719 0.20% 0.40% 0.45% 245,649,58419 -28.8% -4.4% 1.2037 0.2253 -0.0464 0.01% -0.06% -0.09% 303,185,90020 58.2% 19.2% -0.8391 -0.1285 0.0094 0.14% 0.32% 0.31% 188,498,682

24

Initial Results (good)

Preliminary analysis of ICA resultsRun 1 Model 3 - Initial runs - All factors, normal identity, no interactions (Genmod used)

-40%

-29%

-24%-21%

-18%-14%

-11%-7%

-4%0%

4%7%

11%15%

18%24%

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Property return

Log

of m

ultip

lier

0

2000

4000

6000

8000

10000

Num

ber o

f cla

ims

Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value = 0.0%Rank 5/8

The higher the return the less the capital requirement

25

Initial Results (bad)

Preliminary analysis of ICA resultsRun 1 Model 3 - Initial runs - All factors, normal identity, no interactions (Genmod used)

0%

1%

0%

0%0%

0%0%0%0%

0%

0%

-1%

-1%

-0.016

-0.012

-0.008

-0.004

0

0.004

0.008

0.012

Change credit spread A

Log

of m

ultip

lier

0

2000

4000

6000

8000

10000

12000

Num

ber o

f cla

ims

Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value = 0.0%Rank 1/8

Spike indicated a problem in the model