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Actuarial Interactions and Saddles: Simplified Validation Models

Explore the relationships between age, gender, and rating variables in actuarial analysis. Learn about simple and interaction models, as well as utilizing saddles for parameterization. Discover the advantages and challenges in identifying interactions in complex structures.

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Actuarial Interactions and Saddles: Simplified Validation Models

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  1. Southwest Actuarial Forum Interactions and Saddles Serhat Guven, FCAS, MAAA June 10, 2011

  2. Agenda • Background • Interactions • Saddles • Validation

  3. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters 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

  4. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters Background SIMPLE MODEL Difference between male and female is the same regardless of age

  5. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters 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 Interaction: Age + Gender+Age.Gender 8 Parameters

  6. Background INTERACTION MODEL Interaction: Age + Gender+Age.Gender Youthful males are significantly higher than youthful females – resulting structure is still volatile 8 Parameters

  7. Background • 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

  8. Gender b b - b b b b b b b b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - - - - - - - - - - - - Age b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - Interactions • Age x Gender • Main effects construction

  9. Interactions • Imbalances within the cells exists – suggesting some form of interaction

  10. Gender b b - b b b b b b b b b b - b b b b b b b b 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 b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b Interactions • Age x Gender • Full Interaction

  11. Interactions • Rating Area x Vehicle Value • Interaction needs to be simplified Vehicle Value Vehicle Value Vehicle Value b b - b b b b b b b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b Rating Area - - - - - - - - - - - - - - - - - - - - - - Rating Area b - - - - - - - - - - Rating Area b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b

  12. Interactions • Rating Area x Vehicle Value • Simplification emphasizes interaction strength

  13. Saddles • Quadrant saddle: revisiting an simple main effect model

  14. Saddles • Quadrant saddle: interaction terms twist the paper

  15. Saddles

  16. Saddles

  17. Saddles

  18. Saddles

  19. Saddles • Transforming predictors into single parameter variates Response Factor Levels

  20. b Saddles • Transforming predictors into single parameter variates Response Factor Levels

  21. Simple Model: Age + Gender Variate Model: a1*bAge+ a2*bGender Male Male Female Female Youthful Youthful β0+ a1*βY β0+ βY β0+ βY + βF β0+ a1*βY + a2*βF Adult Adult β0 β0 β0+ a2* βF β0+ βF Mature Mature β0+ a1*βM β0+ βM β0+ βM + βF β0+ a1*βM + a2*βF Seniors Seniors β0+ βS β0+ a1*βS β0+ βS + βF β0+ a1*βS + a2* βF 3 Parameters 5 Parameters Saddles • Parameterization • Fitted values from both structures are the same • Variates • X-Values are the beta parameters from the simple model • Coefficient should be 1.000

  22. Saddles • Parameterization • Nonlinear terms introduced via interaction among variates Full Saddle: a1*bAge + a2*bGender + a3*bAge * bGender

  23. 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

  24. Saddles • Rating Area x Vehicle ValueRevisited

  25. Saddles • Rating Area x Vehicle ValueRevisited

  26. Saddles • Rating Area x Vehicle ValueRevisited

  27. Saddles • Driver Age x Vehicle Age

  28. Validation • Model Comparison • Auto frequency: Out of sample

  29. Validation • Model Comparison • Auto frequency: Out of time

  30. Validation • Model Comparison • Renewals: Out of sample

  31. 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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