University of Minnesota
November 9th, 2012Nathan Hubbell, FCAS Katy Micek, Ph.D.
The “Do’s & Don’ts” of Building
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
• 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
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!
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
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
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.
First Model Attempt:
Multiple Linear Regression / Ordinary Least Squares
E[Y] = a0 + a1X1 + …+ anXn• Goal: Fit a linear relationship between the predictors (X1, …, Xn) and the response variable Y.
• Assumptions:
1. Y is normally distributed.
2. The variance of Y is constant.
• Approach: The parameters (a0, a1, …, an) can be estimated by minimizing the sum of
squared errors. X, the predictor Y, t h e re sp o n se
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.
Double Oops - DON’T assume the Variance of Y is constant
– Varies by expected loss.
• High frequency losses have less variance.
DO assume an exponential family distribution for Y
• Poisson - claim frequency
• Discrete distribution • Time-invariant
• Variance equals mean (m = E[Y]) • Gamma - severity
• Continuous distribution
• Variance equals mean squared (m2 = E[Y]2)
Note: Non-normal distributions are more suited to highly skewed claim data
Gamma Distribution
Suitable Model Framework:
Generalized Linear Models (GLMs)
E[Y] = g-1(a0 + a1X1 + …+ anXn)
where g(x) is the link function.
• Goal: fit a non-linear relationship between the
predictors (X1, … , Xn) 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 (a0, a1, …, an) can be estimated using maximum likelihood when
Exponential Family ABCs
Exponential Family – Mean and Variance
Frequency & Severity All-in-One: Uncle Tweedie
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
However, DON’T Overfit
• A noisy model = similar problems to under-fitting
• Some things you might use to look at fit:
– Summary Statistics (R2, 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:
DO Fit on Non-Training (Future!) Data
• Cross Validation
– Validation and Test / Hold out – Bias / Variance Tradeoff
