1 General Iteration Algorithms by Luyang Fu, Ph. D., State Auto Insurance Company Cheng-sheng Peter Wu, FCAS, ASA, MAAA, Deloitte Consulting LLP 2007 CAS Predictive Modeling Seminar Las Vegas, Oct 11-12, 2007

2 Agenda  History and Overview of Minimum Bias Method  General Iteration Algorithms (GIA)  Conclusions  Demonstration of a GIA Tool  Q&A

3 History on Minimum Bias  A technique with long history for actuaries: Bailey and Simon (1960) Bailey (1963) Brown (1988) Feldblum and Brosius (2002) A topic in CAS Exam 9  Concepts: Derive multivariate class plan parameters by minimizing a specified “bias” function Use an “iterative” method in finding the parameters

4 History on Minimum Bias  Various bias functions proposed in the past for minimization  Examples of multiplicative bias functions proposed in the past:

5 History on Minimum Bias  Then, how to determine the class plan parameters by minimizing the bias function?  One simple way is the commonly used an “iterative” methodology for root finding: Start with a random guess for the values of x i and y j Calculate the next set of values for x i and y j using the root finding formula for the bias function Repeat the steps until the values converge  Easy to understand and can be programmed in almost any tool

6 History on Minimum Bias  For example, using the balanced bias functions for the multiplicative model:

7 History on Minimum Bias  Past minimum bias models with the iterative method:

8 Iteration Algorithm for Minimum Bias  Theoretically, is not “bias”.  Bias is defined as the difference between an estimator and the true value. For example, is bias. If, then xhat is an unbiased estimator of x.  To be consistent with statistical terminology, we name our approach as General Iteration Algorithm.

9 Issues with the Iterative Method  Two questions regarding the “iterative” method: How do we know that it will converge? How fast/efficient that it will converge?  Answers: Numerical Analysis or Optimization textbooks Mildenhall (1999)  Efficiency is a less important issue due to the modern computation power

10 Other Issues with Minimum Bias  What is the statistical meaning behind these models?  More models to try?  Which models to choose?

11 Summary on Historical Minimum Bias  A numerical method, not a statistical approach  Best answers when bias functions are minimized  Use of an “iterative” methodology for root finding in determining parameters  Easy to understand and can be programmed in many tools

12 Connection Between Minimum Bias and Statistical Models  Brown (1988) Show that some minimum bias functions can be derived by maximizing the likelihood functions of corresponding distributions Propose several more minimum bias models  Mildenhall (1999) Prove that minimum bias models with linear bias functions are essentially the same as those from Generalized Linear Models (GLM) Propose two more minimum bias models

13 Connection Between Minimum Bias and Statistical Models  Past minimum bias models and their corresponding statistical models

14 Statistical Models - GLM  Advantages include : Commercial software and built-in procedures available Characteristics well determined, such as confidence level Computation efficiency compared to the iterative procedure

15 Statistical Models - GLM  Issues include: Requires more advanced knowledge of statistics for GLM models Lack of flexibility:  Reliance on commercial software / built-in procedures.  Cannot do the mixed model.  Assumes a pre-determined distribution of exponential families.  Limited distribution selections in popular statistical software.  Difficult to program from scratch.

16 Motivations for GIA  Can we unify all the past minimum bias models?  Can we completely represent the wide range of GLM and statistical models using Minimum Bias Models?  Can we expand the model selection options that go beyond all the currently used GLM and minimum bias models?  Can we fit mixed models or constraint models?

17 General Iteration Algorithm  Starting with the basic multiplicative formula  The alternative estimates of x and y:  The next question is – how to roll up x i,j to x i, and y j,i to y j ?

18 Possible Weighting Functions  First and the obvious option - straight average to roll up  Using the straight average results in the Exponential model by Brown (1988)

19 Possible Weighting Functions  Another option is to use the relativity-adjusted exposure as weight function  This is Bailey (1963) model, or Poisson model by Brown (1988).

20 Possible Weighting Functions  Another option: using the square of relativity-adjusted exposure  This is the normal model by Brown (1988).

21 Possible Weighting Functions  Another option: using relativity-square-adjusted exposure  This is the least-square model by Brown (1988).

22 General Iteration Algorithms  So, the key for generalization is to apply different “weighting functions” to roll up x i,j to x i and y j,i to y j  Propose a general weighting function of two factors, exposure and relativity: W p X q and W p Y q  Almost all published to date minimum bias models are special cases of GMBM(p,q)  Also, there are more modeling options to choose since there is no limitation, in theory, on (p,q) values to try in fitting data – comprehensive and flexible

23 2-parameter GIA  2-parameter GIA with exposure and relativity adjusted weighting function are:

24 2-parameter GIA vs. GLM pqGLM 1Inverse Gaussian 10Gamma 11Poisson 12Normal

25 2-parameter GIA and GLM  GMBM with p=1 is the same as GLM model with the variance function of  Additional special models: 0<q<1, the distribution is Tweedie, for pure premium models 1<q<2, not exponential family -1<q<0, the distribution is between gamma and inverse Gaussian  After years of technical development in GLM and minimum bias, at the end of day, all of these models are connected through the game of “weighted average”.

26 3-parameter GIA  One model published to date not covered by the 2- parameter GMBM: Chi-squared model by Bailey and Simon (1960)  Further generalization using a similar concept of link function in GLM, f(x) and f(y)  Estimate f(x) and f(y) through the iterative method  Calculate x and y by inverting f(x) and f(y)

27 3-parameter GIA

28 3-parameter GIA  Propose 3-parameter GMBM by using the power link function f(x)=x k

29 3-parameter GIA  When k=2, p=1 and q=1  This is the Chi-Square model by Bailey and Simon (1960)  The underlying assumption of Chi-Square model is that r 2 follows a Tweedie distribution with a variance function

30 Additive GIA

31 Mixed GIA  For commonly used personal line rating structures, the formula is typically a mixed multiplicative and additive model: Price = Base*(X + Y) * Z

32 Constraint GIA  In real world, for most of the pricing factors, the range of their values are capped due to market and regulatory constraints

33 Numerical Methodology for GIA  For all algorithms: Use the mean of the response variable as the base Starting points:1 for multiplicative factors; 0 for additive factors Use the latest relativities in the iterations All the reported GIAs converge within 8 steps for our test examples  For mixed models: In each step, adjust multiplicative factors from one rating variable proportionally so that its weighted average is one. For the last multiplicative variable, adjust its factors so that the weighted average of the product of all multiplicative variables is one.

34 Conclusions  2 and 3 Parameter GIA can completely represent GLM and minimum bias models  Can fit mixed models and models with constraints  Provide additional model options for data fitting  Easy to understand and does not require advanced statistical knowledge  Can program in many different tools  Calculation efficiency is not an issue because of modern computer power.

35 Demonstration of a GIA Tool  Written in VB.NET and runs on Windows PCs  Approximately 200 hours for tool development  Efficiency statistics: Efficiency for different test cases Excel DataCSV Data # of Records# of VariablesLoading TimeModel TimeLoading TimeModel Time sec0.5 sec0.1 sec0.5 sec 25, sec6 sec1 sec5 sec 48, sec50 sec1.5 sec50 sec 642,12849N/A 40 sec

36 Q & A