Southwest Actuarial Forum

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Interactions and SaddlesSouthwest Actuarial Forum

Serhat Guven, FCAS, MAAA

June 10, 2011

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Agenda

Background Interactions Saddles Validation


Background

SIMPLE MODEL Relationship between rating levels of one factor is constant for all levels

of other rating variables Assume two rating variables

Age: Youthful, Adult (Base), Mature, Senior Gender: Male (Base), Female

Simple Model: Age + Gender

5 Parameters

β0+ βM + βF

β0+ βMMatureβ0+ βFβ0Adult

β0+ βS

β0+ βY

Male

β0+ βS + βF

Seniors

β0+ βY + βF

YouthfulFemale


Background

0.5

0.7

0.9

1.1

1.3

1.5

1.7

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3435-39

40-4445-49

50-5455-59

60-6465-69

70+Policyholder Age


5 Parameters

β0+ βM + βF


β0+ βS

β0+ βY

Male

β0+ βS + βF

Seniors

β0+ βY + βF

YouthfulFemale

SIMPLE MODEL

Difference between male and female is the same regardless of age


Interactions

INTERACTION MODEL Relationship between rating levels of one rating factor is different for all

levels of another rating variable Assume two rating variables

Age: Youthful, Adult (Base), Mature, Senior Gender: Male (Base), Female


5 Parameters

β0+ βM + βF


β0+ βS

β0+ βY

Male

β0+ βS + βF

Seniors

β0+ βY + βF

YouthfulFemale Male Female

Youthful β0+ βYβ0+ βY + βF+

βYF

Adult β0 β0+ βF

Mature β0+ βMβ0+ βM + βF+

βMF

Seniors β0+ βSβ0+ βS + βF+

βSF

Interaction: Age + Gender+Age.Gender

8 Parameters


Background

INTERACTION MODEL

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3435-39

40-4445-49

50-5455-59

60-6465-69

70+Policyholder Age

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3435-39

40-4445-49

50-5455-59

60-6465-69

70+Policyholder Age

Youthful males are significantly higher than youthful females – resulting structure is still volatile

Male Female

Youthful β0+ βYβ0+ βY + βF+

βYF

Adult β0 β0+ βF

Mature β0+ βMβ0+ βM + βF+

βMF

Seniors β0+ βSβ0+ βS + βF+

βSF

Interaction: Age + Gender+Age.Gender

8 Parameters


Why are interactions present Because that's how the factors behave Because the multiplicative model can go wrong at the edges

— 1.5 * 1.4 * 1.7 * 1.5 * 1.8 * 1.5 * 1.8 = 26!

Interaction challenges Identification

— Given n factors there are n choose 2 possible interactions to study Simplification

— Once an interaction has been identified the structure needs to be simplified to avoid overfitting

7

Background


Interactions

Age x Gender Main effects construction

Gender

b b - b b b b b b b

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -

- - - - - - - - - - -

Age b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -b - - - - - - - - - -b - - - - - - - - - -


Interactions

Imbalances within the cells exists – suggesting some form of interaction


Interactions

Age x Gender Full Interaction

Gender

b b - b b b b b b b

b b b - b b b b b b b



- - - - - - - - - - -

Age b b b - b b b b b b b



b b b - b b b b b b bb b b - b b b b b b bb b b - b b b b b b b


Interactions

Rating Area x Vehicle Value Interaction needs to be simplified

Vehicle Value

b b - b b b b b b b

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -

- - - - - - - - - - -

Rat

ing

Are

a

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -b - - - - - - - - - -b - - - - - - - - - -

Vehicle Value

b b - b b b b b b b




- - - - - - - - - - -

Rat

ing

Are

a




b b b - b b b b b b bb b b - b b b b b b bb b b - b b b b b b b

Vehicle group

b b - b b b b b b b

b - - - - - - - - - -

b - - - - - - - - - -

b - - - - - - - - - -

- - - - - - - - - - -

Age b - - - - - - - - - -

b - - - - -

b - - - - -

b - - - - -b - - - - -b - - - - -

b

Rat

ing

Are

a

Vehicle Value


Rating Area x Vehicle Value Simplification emphasizes interaction strength

Interactions


Saddles

Quadrant saddle: revisiting an simple main effect model

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39S1

S5

S9

S13

S17

S21S25S29S33S37

-400

-200

0

200

400

600

800

1000

1200


Saddles

Quadrant saddle: interaction terms twist the paper

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39S1

S5

S9

S13

S17

S21S25S29S33S37

-800

-600

-400

-200

0

200

400

600

800


Saddles


Res

pons

e

Factor Levels

Saddles

Transforming predictors into single parameter variates


Res

pons

e

Factor Levels

b

Saddles

Transforming predictors into single parameter variates


Saddles

Parameterization


5 Parameters

β0+ βM + βF


β0+ βS

β0+ βY

Male

β0+ βS + βF

Seniors

β0+ βY + βF

YouthfulFemale

Variate Model: a1*bAge + a2*bGender

3 Parameters

β0+ a1*βM + a2*βF

β0+ a1*βMMatureβ0+ a2* βFβ0Adult

β0+ a1*βS

β0+ a1*βY

Male

β0+ a1*βS + a2* βF

Seniors

β0+ a1*βY + a2*βF

YouthfulFemale

Fitted values from both structures are the same Variates

X-Values are the beta parameters from the simple model Coefficient should be 1.000


Saddles

Parameterization Nonlinear terms introduced via interaction among variates

Male FemaleYouthful b0 + a1*βY β0+ a1*βY + a2*βF + a3*bAge * bGender

Adult β0 β0+ a2* βF + a3*bAge * bGender

Mature β0+ a1*βM β0+ a1*βM + a2*βF + a3*bAge * bGender

Seniors β0+ a1*βS β0+ a1*βS + a2* βF + a3*bAge * bGender

Full Saddle: a1*bAge + a2*bGender + a3*bAge * bGender


Saddles

Advantages Non linear variate parameter

— Easier to validate— Simpler shape

Useful in identifying interactions in more volatile structures— High dimensional factors— N-way interaction structures— Severity models

Disadvantages Difficult to detect interactions when a factor is not a main effect


Saddles

Rating Area x Vehicle Value Revisited


Saddles

Driver Age x Vehicle Age


Validation

Model Comparison Auto frequency: Out of sample

4%

6%

8%

10%

12%

14%

16%

18%

20%

50% 70% 90% 94% 98% 102% 106% 110% 130% 150%0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

Exposure Observed Saddle Original


Validation

Model Comparison Auto frequency: Out of time

0.006

0.011

0.016

0.021

0.026

< 80% 80% -84%

82% -86%

84% -88%

86% -90%

88% -92%

90% -94%

92% -96%

94% -98%

96% -100%

98% -102%

100% -104%

102% -106%

104% -108%

106% -110%

108% -112%

110% -114%

112% -116%

114% -118%

116% -120%

118% -122%

> 120%0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

Exposure Observed Saddles Original


Validation

Model Comparison Renewals: Out of sample

0.15

0.2

0.25

0.3

0.35

0.4

< 80% 80% - 86% 83% - 89% 86% - 92% 89% - 95% 92% - 98% 95% - 101% 98% - 104% 101% - 107% 104% - 110% 107% - 113% 110% - 116% 113% - 119% 116% - 122%0

2000

4000

6000

8000

10000

12000

14000

Exposure Observed Saddles Original


Conclusion

Interactions are an important part of the model creation process

Interactions have their own challenges Identification Simplification Dimensionality

Effort needed in simplifying and validating identified interaction Saddles use the framework of variate vectors in the design matrix to quickly

simplify and validate new interaction terms

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