Transcript Slides – Math .umn.edu

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University of Minnesota
December 3rd, 2010
Nathan Hubbell, FCAS
John Renze, PhD, FCAS
Agenda
• Travelers
– Broad Overview
– Analytics Career Opportunities
• Predictive Modeling
– Generalized Linear Models (GLM’s)
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About Us
• Offers property and casualty solutions to individuals and companies of all
sizes
• Second-largest commercial insurer in
the U.S.
Revenue of
• Second-largest personal insurer
$25 billion and total
through the independent agency channel
assets of $110 billion
• No. 98 on the Fortune 500 list of largest
in fiscal year 2009
U.S. companies
• 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
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Company History
The Saint Paul – 1853
Travelers – 1864
Seeing the need for a local insurance
company, Alexander Wilkin and 16
fellow Saint Paul businessmen founded
St. Paul Fire and Marine Insurance.
J.G. Batterson and nine others formed Travelers
Insurance Company for the purpose of insuring
travelers against death or injury while
journeying by railway or steamboat.
St. Paul Travelers – 2004
The St. Paul and Travelers merged on April 1, 2004 forming
The St. Paul Travelers Companies, Inc.
Travelers – 2007
The company name changed to The Travelers Companies,
Inc. and began trading on the New York Stock Exchange
under the symbol TRV. The 137-year-old insurance icon, the
red umbrella, was reinstated.
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Our Organization
• Business Insurance: Offers a broad array of property casualty, specialty
insurance and related services to businesses of all sizes.
• Financial, Professional & International Insurance: Includes international
products and surety, crime and financial liability products that use creditbased underwriting processes.
• Personal Insurance: Offers products including automobile, homeowners,
renters and condominium policies to individual consumers.
• Claim: Includes 13,000 trained Claim professionals in four countries and
all 50 states who respond to customers 24 hours a day, seven days a
week, 365 days a year.
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What is insurance and what good does it serve?
Insurance restores individuals to the financial state they were in prior to a loss
(e.g. car accident; tree fell on house)
For this benefit, customers pay a premium to the insurance company
If a customer doesn’t have a loss, then their premium
A. Helps the insurance company cover the loss another customer did have
B. Keeps the insurance company functioning so it can continue providing this service
If the customer does have a loss, it’s the other insureds that are helping them!
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Analytics at Travelers – Who are we?
Across the four business units, we form a large (100+) and diverse community of
Ph.D., Masters and Bachelors holders in the following disciplines:
mathematics
statistics
physics
actuarial science
computer science
business
… and more!
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Analytics at Travelers – Who are we?
What makes Travelers special?
Teamwork isn’t a buzzword here – it’s real and we live it
• We share information & technology openly with each other
• Learning something new can be as simple as asking a local expert; they
make the time, despite busy schedules
• We each have a unique combination of strengths; we are valued for them
and our managers help us grow into the careers we desire
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Analytics at Travelers – What do we do?
We ask and answer questions requiring sophisticated analyses
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How much will it cost to insure a customer?
How expensive will this claim be?
How likely is it that this customer will purchase our product?
How many claims adjusters will we need in two years?
What new statistical methods will help move our business into the future?
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Analytics at Travelers – What do we do?
Why are these questions hard?
Example: How much it costs to insure an auto customer
It’s impossible to predict if someone is going to
A. Get into an accident
B. The type of accident (telephone pole, another vehicle)
C. How bad the accident will be
But if we have enough customers, we can start to group them…
in group A) we expect 1 / 10 to get into an accident, costing on average $1000
in group B) etc.
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Analytics at Travelers – What do we do?
Why it matters…
The more finely we group, the more accurate the price. In that case:
1. If our competitors charge more, the customer will choose us and we will grow
profitably
2. If our competitors charge less, the customer will choose them and they’ll
grow unprofitably
Either way, we win!
The trick is finding the right groups, and getting the right price for them…
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Analytics at Travelers - Methodologies
To stay ahead of our competitors, we sift through the literature searching for the
most applicable techniques. Some examples:
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–
–
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GAMs
Elastic net & adaptive LASSO
MARS
Gradient boosted trees
We pick the methods that have real-world value to our business and give us the
competitive edge
We do a significant amount of proprietary methodology development internally
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Analytics at Travelers - Technology
We have millions of customers. To support this volume of data and facilitate
the use of the latest methodologies, we rely on cutting-edge technology
Teradata data warehouses – hundreds of TB at our fingertips!
Multi-processor linux servers for analytic software
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SAS
R
Salford Systems
Custom software (C++, FORTRAN)
…and more!
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Analytics at Travelers – Why us?
We are at the cutting-edge
We grow our people; we give them the training and opportunities they need
to move their careers ahead
We are a team – it’s our combined focus that makes Travelers the leader of
the industry that we are today.
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Predictive Modeling
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Predictive Modeling
• Using Generalized Linear Models (GLM’s) and other statistical methods to
predict exposure to loss at detailed level.
– Recently, property-casualty insurance companies have embraced
predictive modeling as a strategic tool for competing in the marketplace.
• Originally introduced as a method of increasing precision for personal auto
insurance pricing
• Extended to homeowners and commercial lines
• Today, it is applied in areas such as marketing, underwriting, pricing, and
claims management
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How do you differentiate your rates?
Automobile
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Age
Gender
Marital status
# vehicles
# drivers
Home policy
Driving record
Years Licensed
Limits
Prior Insurance
Student/Nonstudent
Location (Garage/driven)
Annual Mileage
Homeowners
–
–
–
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Age of home
# occupants
Primary / Secondary
Prior claim experience
Construction
Protection
Roof Type
Location (CAT?)
Amount of Insurance
Auto Policy
Responsibility of owner
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How do you set the prices?
• Old Way:
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–
–
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Group data by class  class relativities
Sort data into age groups  age relativities
Group data by territory  territorial relativities
Rating factor = class x age x territory
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What is wrong with the old way?
Example
Size of car
Class
Claim
Count
Large cars
Medium
cars
Small cars
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Age
Exposures Frequency
Relativity
(number of (probability of
cars)
a claim)
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400
0.038
1
110
1700
0.065
1.725
143
900
0.159
4.237
Class
Old
Young
Claim
Count
Exposures Frequency
Relativity
(number of (probability of
cars)
a claim)
80
188
1800
1200
0.044
0.157
1
3.325
•
A young driver of a small car would be charged
• 4.237 x 3.325 = 14.088
times what an old driver of a large car would be charged.
•
Important point: Some of this effect is double-counted, as size of car is correlated with age.
•
(numbers are illustrative only)
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Possible Solution: Multiple Linear Regression
• E[Y] = a0 + a1X1 + …+ anXn
• Two Key Assumptions:
– Y is Normally distributed random variable.
– Variance of Y is constant (homoscedastic).
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Problems With Regression – Part I
• Y is NOT normally distributed.
– Number of claims is discrete
– Claim sizes are skewed to the right
– Probability of an event is in [0,1]
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Problems With Regression – Part II
• Variance of Y is NOT constant.
– Varies by expected loss.
• High frequency losses have less variance.
• High severity losses have more variance.
– Varies by exposure.
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Problems With Regression – Part III
2.5
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1.5
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0.5
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Loss Relativity
3.5
Nonlinear
relationship between X’s and Y’s.
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Example:
Age of driver
Driver Age
(numbers are illustrative only)
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Generalized Linear Models (GLMs)
• E[Y] = g-1(a0 + a1X1 + …+ anXn)
• Fewer restrictions:
– Non-linear relationships.
• g(x) = x  Additive model
• g(x) = exp(x)  Multiplicative model
• g(x) = 1 / (1+exp(x))  Logistic model
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Generalized Linear Models (GLMs)
• E[Y] = g-1(a0 + a1X1 + …+ anXn)
• Fewer restrictions:
– Y can be from any exponential family of distributions.
• Poisson (number of claims)
• Binomial (probability of renewing)
• Gamma (loss severity)
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Generalized Linear Models (GLMs)
• E[Y] = g-1(a0 + a1X1 + …+ anXn)
• Fewer restrictions:
– Variance depends on the expected mean.
• Normal: Variance is constant.
• Poisson: Variance equals mean.
• Gamma: Variance equals mean squared.
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Generalized Linear Models (GLMs)
• What’s the catch?
– No closed form solution.
– Use maximum likelihood estimation.
• Iterative process.
• Make a guess and linearize.
• Solve the linear problem to find next guess.
– Increased computational complexity.
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Key Steps in Model Building – Part I
• What are you modeling?
• How will you implement?
• Gather and clean internal data.
• Link other sources: internal and external.
• Create training and validation sets
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Key Steps in Model Building – Part II
• Build Model on Training Set
• Univariate analysis – statistically test each predictor
• Build multivariate models using significant predictors
• Select best multivariate predictive model
 Keep most relevant predictors
 Principle of parsimony – simplicity is good
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Key Steps in Model Building – Part III
• Measure predictive power on validation set
• Was training set over fit?
• Peer review
• Implement
• Post-implementation monitoring
• Adjust with new knowledge
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