How To Build A Predictive Model In Insurance

Transcription

1 The Do s & Don ts of Building A Predictive Model in Insurance University of Minnesota November 9 th, 2012 Nathan Hubbell, FCAS Katy Micek, Ph.D.

2 Agenda Travelers Broad Overview Actuarial & Analytics Career Opportunities Actuarial & Analytics Leadership Development Program (AALDP) Do s and Don ts of Generalized Linear Models (GLMs) Insurance Background and Motivation Failings of Multiple Linear Regression Basics of GLMs Over-fitting Questions 2

3 About Travelers Offers property and casualty solutions to individuals and companies Second-largest commercial insurer in the U.S. Second-largest personal insurer through the independent agency channel Representatives in every U.S. state, Canada, Ireland and the United Kingdom A member of the Dow Jones Industrial Average the only insurance company on the list 3

4 Analytics at Travelers Who are we? Across Travelers, we form a large (300+) and diverse community of Ph.D., Masters and Bachelors holders in the following disciplines: mathematics statistics physics actuarial science computer science business and more! 4

5 Actuarial & Analytics Leadership Development Program (AALDP) 5-yr program for actuarial students and advanced analytics Actuarial track offers exam support Analytics learn insurance on the job through work projects and seminars Exams not required but support is provided for those interested Leadership development opportunities Career exploration opportunities through rotations Networking opportunities (mentor program, committees) 2012 Pilot Class for Advanced Analytics Offers a flexible career path 5

6 Generalized Linear Models 6

7 Basics of Insurance Insurance companies sell the product of insurance policies, which are the promise to pay in the event that a customer experiences a loss. The unique challenge in insurance is that we don t know what the cost of insuring a customer is when we sell the policy. Example: The cost to insure an auto customer It s impossible to predict if someone is going to Get into an accident The type of accident (hit a telephone pole, hit another vehicle, bodily injury) How bad (cost) the accident will be 7

8 Business Impact of Loss Experience To estimate the cost of insuring policyholders, we must predict losses Two fundamental questions we must answer are: 1. Ratemaking: looking to the future Setting rates for policies How much do we need to charge customers for a policy in order to reach our target profit? Basic idea: price = cost + profit 2. Reserving: looking at the impact of past experience Setting aside reserve money How much money do we need to set aside to pay for claims? Note: We cannot predict losses for each individual. However, if we group our customers together, we can build statistical models to predict average loss over a group. 8

9 First Model Attempt: Multiple Linear Regression / Ordinary Least Squares E[Y] = a 0 + a 1 X a n X n Goal: Fit a linear relationship between the predictors (X 1,, X n ) and the response variable Y. Y, the response Assumptions: 1. Y is normally distributed. 2. The variance of Y is constant. X, the predictor Approach: The parameters (a 0, a 1,, a n ) can be estimated by minimizing the sum of squared errors. 9

10 Oops - DON T assume Y is normally distributed In insurance, we study loss experience in terms of claims. Two aspects of claims must be considered. 1. Frequency: what is the rate that claims are being made? 2. Severity: what is the average size of claim? The underlying distribution in the model depends on what aspect of the loss experience we re investigating. 10

11 Double Oops - DON T assume the Variance of Y is constant Varies by expected loss. High frequency losses have less variance. High severity losses have more variance. 11

12 DO assume an exponential family distribution for Y Note: Non-normal distributions are more suited to highly skewed claim data Poisson - claim frequency Discrete distribution Time-invariant Variance equals mean (m = E[Y]) Gamma - severity Continuous distribution Variance equals mean squared (m 2 = E[Y] 2 ) Gamma Distribution Source: Gamma Distribution, Wikipedia 12

13 Suitable Model Framework: Generalized Linear Models (GLMs) E[Y] = g -1 (a 0 + a 1 X a n X n ) where g(x) is the link function. Goal: fit a non-linear relationship between the predictors (X 1,, X n ) and the response variable Y. Assumptions: 1. Y can be from any exponential family of distributions. 2. Variance depends on expected mean. Approach: The parameters (a 0, a 1,, a n ) can be estimated using maximum likelihood when underlying distribution is fixed.

14 Exponential Family ABCs Source: A Practitioner's Guide to Generalized Linear Models 14

15 Exponential Family Mean and Variance Tweedie V(m) = m p ; 1 < p < 2 Source: A Practitioner's Guide to Generalized Linear Models 15

16 Frequency & Severity All-in-One: Uncle Tweedie Source: A Practitioner's Guide to Generalized Linear Models 16

17 Over-fitting 17

18 DON T Underfit First let s start with under-fitting Expected Auto Claim Frequency: 5% Expected Auto Loss: $10,000 Expected Premium = 5% * $10,000 = $500 Competitors use a segmented rate plan What happens next? We will win all of the business where competitors charge more than $500 We lose all of the business where competitors charge less than $500 Now why would a competitor want to charge more than the average? Hmmm perhaps we need a better approach 18

19 Occam s Razor / Principle of Parsimony 19

20 However, DON T Overfit A noisy model = similar problems to under-fitting Some things you might use to look at fit: Summary Statistics (R 2, AIC, BIC) Deviance Tests / Type III P-value / Parameter Estimate Standard Errors ROC / Lift / Gain Charts However, let s keep our eyes on the prize: We want our model to perform well on future data! 20

21 DO Fit on Non-Training (Future!) Data Cross Validation Validation and Test / Hold out Bias / Variance Tradeoff Regularization 21

22 Cross Validation Training Validation Testing (Holdout) Model Fit Model Stucture Final Model Testing 22

23 Variance / Bias Tradeoff Source: Elements of Statistical Learning 23

24 Price Bias/variance: How would you fit this model? Size 24

25 Price Price Price Bias vs. Variance Size Size Size High bias (under-fit) Just right High variance (over-fit) Source: Coursera Machine Learning 25

26 Price Regularization Size High variance (overfit) Source: Coursera Machine Learning 26

27 References and Resources Actuarial Exams and Career Information Travelers Careers Actuarial and Analytics Research Internship and Full Time A Practitioner's Guide to Generalized Linear Models tion_3.pdf Statistical Modeling: The Two Cultures (Leo Breiman) Elements of Statistical Learning (Hastie, Tibshirani, Friedman) Coursera Machine Learning: 27