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
1

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

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
3

History on Minimum Bias


Various bias functions proposed in the past for minimization
Examples of multiplicative bias functions proposed in the past:
Balanced Bias
Squared Bias

 i
 j , i
 j , w i , w i , j j
( r i ,
( r i , j j

 x i x i y j
) y j
) 2
Chi

Squared Bias
 i
 j , w i , j
( r i , j
 x i w i , j x i y j y j
)
2
4

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 finding formula for the bias function and y j using the root
Repeat the steps until the values converge
Easy to understand and can be programmed in almost any tool
5

History on Minimum Bias
 For example, using the balanced bias functions for the multiplicative model:
Balanced Bias
 i
 j , w i , j
( r i , j
 x i y j
)
Then ,
ˆ i , t j , t


 j
 j w i w i
, j
,
 i
 i w i w i
, j
, j r i , j j , t

1 j r i , j
ˆ i , t

1

0
6

History on Minimum Bias
 Past minimum bias models with the iterative method:
ˆ i , t
ˆ i , t
ˆ i , t

 j
 j w i w i
, j
, j r i , j j , t

1





 j
 j w i , w i j
, r i ,
2 j j

1 j , t

1 j , t

1




1 / 2

1 n
 j r i , j j , t

1
ˆ i , t
ˆ i , t


 j
 j w i
2
, w i
2
, j r i j
,
 j
 j w i , w i j
, r i , j j j j , t

1
2 j , t

1 j , t

1
2 j , t

1
7

Iteration Algorithm for Minimum Bias


 i

, j
,
 j
Bias is defined as the difference between an estimator and the x i
 i i
 x
ˆ  i
0 then xhat is an unbiased estimator of x.
To be consistent with statistical terminology, we name our approach as General Iteration Algorithm.
8

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
9

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

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
11

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
12

Connection Between Minimum Bias and
Statistical Models
 Past minimum bias models and their corresponding statistical models
ˆ i , t


 j j w i , w i , j y
ˆ j r i , j j , t

1

Poisson
ˆ i , t





 j
 j w i , w i , j r i ,
2 j j

1 j , t

1 j , t

1




1 / 2
 
2
ˆ i , t

1 n
 j r i , j j , t

1

Exponentia l
ˆ i , t
ˆ i , t


 j
 j w i
2
, w i
2
, j r i j
,
 j
 j w i w i
, j
, r i , j j j j , t

1
2 j , t

1 j , t

1
2 j , t

1

Normal

Least Squared
13

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
14

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.
15

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?
16

General Iteration Algorithm



Starting with the basic multiplicative formula r
 i , j x i y j
The alternative estimates of x and y: i , j
 r i , j
/ y j
, j

1 , 2 , to n j , i
 r i , j
/ x i
, i

1 , 2 , to m ,
The next question is – how to roll up x i,j to x i
, and y j,i to y j
?
17

Possible Weighting Functions

ˆ i
First and the obvious option - straight average to roll up
  j j
  i
1 n
1 m
ˆ i , j j , i

1 n
 j r i , j j

1 m
 i r i ,
ˆ i j
 Using the straight average results in the Exponential model by Brown (1988)
18

Possible Weighting Functions

 i
Another option is to use the relativity-adjusted exposure as weight function j
  
 j i , w i j
, j j
   i i , w i j
, j i i j





ˆ i , j j , i
  
 j i , w i j
, j j
   i i , w i j
, j i i j


 r i , j j

 r i ,
ˆ i j 

 j
 j w i , w i , j j r i , j j
 i
 i w i w i
,
, j j r i
ˆ i
, j
This is Bailey (1963) model, or Poisson model by Brown
(1988).
19

Possible Weighting Functions

ˆ i
Another option: using the square of relativity-adjusted exposure
  
 j w i
2
, j w i
2
, j
2 j
2 j


ˆ i , j

 j
 j w i
2
, w i
2
, j r i j
, j
2 j j j
   i w i
2
, j w i
2
,
ˆ i
2 j
ˆ i
2

 j , i

 i
 i w i
2
, w i
2
, j r i j
, j
ˆ i
2
ˆ i
 This is the normal model by Brown (1988).
20

Possible Weighting Functions


Another option: using relativity-square-adjusted exposure
ˆ i
  
 j i , j w i , j
2 j
2 j


ˆ i , j

 w i j
 j
, w i j
, r i , j j
2 j j j
   i i , j w i ,
ˆ i
2 j
ˆ i
2

 j , i

 w i i
 i
, w i j
, r i , j j
ˆ i
2
ˆ i
This is the least-square model by Brown (1988).
21

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
22

2-parameter GIA
 2-parameter GIA with exposure and relativity adjusted weighting function are:
ˆ i j
  
 j i p
, j w i p
, j q j q j


ˆ i , j
   i i p
, j w i p
,
ˆ i q j
ˆ i q

 j , i


 j w i p
,
 j j r i w i p
,
, j j q j q

1 j
 i
 i w i p
, j w i p
, r i j
, j
ˆ i q

1
ˆ i q

1
23

2-parameter GIA vs. GLM
1
1
1 p
1
0
1
2 q
-1
GLM
Inverse Gaussian
Gamma
Poisson
Normal
24

2-parameter GIA and GLM



GMBM with p=1 is the same as GLM model with the variance function of V (

)
 
2
 q
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”.
25

3-parameter GIA




One model published to date not covered by the 2parameter 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)
26

3-parameter GIA f ( x
ˆ i
)
  
 j w i p
, j w i p
, j q j q j


 f ( x
ˆ i , j
)

 j w i p
, j q j
 j w i p
, j f ( r i , j q j j
) f ( j
)
   i w i p
, j w i p
, x
ˆ i q j
ˆ i q

 f ( j , i
)

 i w i p
, j
 i
ˆ i q w i p
, j f ( r i , j x
ˆ i
ˆ i q
)
27

3-parameter GIA
 Propose 3-parameter GMBM by using the power link function f(x)=x k
ˆ i j
 

 j w i p
,
 j j r i w i p
, k
, j j q j q
 k j
 

 i w i p
,
 i j r i w i p
, k
, j j
ˆ i q
ˆ i q
 k



1 / k



1 / k
28

3-parameter GIA


 When k=2, p=1 and q=1 i j


 j w i
 j
, j w i r i
, j
2
, j




 i w i
 i
, j w i r i
, j
2
, j i j

1 j

1 i

1


/ 2

1


/ 2
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
V (

)
 
1 .
5
29

Additive GIA i j


 j w i p
, j
 j
( r i , j w i p
, j

 i w i p
, j
( r i , j
 i w i p
, j
 y j
) x i
)
30

Mixed GIA
 For commonly used personal line rating structures, the formula is typically a mixed multiplicative and additive model:
 Price = Base*(X + Y) * Z i , j , h i , j , h i , j , h


 r i , j , h  z h r i , j , h  z h x i r i , j , h
 y j y j x i
31

Constraint GIA
 In real world, for most of the pricing factors, the range of their values are capped due to market and regulatory constraints x
ˆ
1 x
ˆ
2
 


 j w
1 , p
 j j r
1 , k w
1 , p j j y y q j q
 k j





1 / k max( 0 .
75 x
ˆ
1
, min( 0 .
95 x
ˆ
1
,



 j w
2 p
,
 j j r
2 k
, w
2 p
, j j y y q j j q
 k




1 / k
))
32

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.
33

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.
34

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
# of Records
508
25,904
48,517
642,128
# of Variables
3
6
Excel Data
Loading Time Model Time
0.7 sec
2.8 sec
0.5 sec
6 sec
CSV Data
Loading Time Model Time
0.1 sec
1 sec
0.5 sec
5 sec
13
49
9 sec
N/A
50 sec 1.5 sec
40 sec
50 sec
35

Q & A
36