Institute for Mathematics and Its Applications Estimating Losses with Predictive Modeling: Analytics Careers at Travelers Institute for Mathematics and Its Applications Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. March 3rd, 2013 Catherine (Katy) A. Micek, Ph.D.

Agenda My Career Path Insurance 101 About Travelers Business Impact of Loss Experience Use of Data GLMs: an Example of Estimating Losses with Predictive Modeling Analytics at Travelers AALDP Questions Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 2

My Career Path B.A. in Mathematics (2004) Minors in Physics & QMCS M.S. in Mathematics (2008) Ph.D. in Applied Mathematics (2010) Visiting Assistant Professor Taught undergraduate math courses Visiting Professor Error analysis in FEM method Sr Consultant, Research & Analytics Predictive Modeling for Business Insurance University of St. Thomas, 2000-2004 University of Minnesota, 2004-2010 Augsburg College, 2010-2012 Adventium Labs, May – August 2011 Travelers , June 2012 - present Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved.

Insurance 101: Travelers Who are we? Travelers is a property and casualty insurance company that sells a wide variety of insurance and surety products and risk management services to businesses, organizations, and individuals. What do we sell? Our principal products are insurance policies and surety bonds, which are, in essence, promises to pay in the event that customers experience certain types of losses. How do we use mathematics? The unique challenge in insurance is that we don’t know what the cost of insuring a customer is when we sell the policy, so we use mathematics to predict the expected losses for groups of customers. 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another vehicle, bodily injury) How bad (cost) the accident will be Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 4 4

Insurance 101: Business Impact of Loss Experience When looking at the business impact of loss experience, there are two fundamental questions that need to be answered. 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 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? To answer these questions, we must do data analysis. Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 5

Insurance 101: Data at Travelers Traditional Actuarial Usage Univariate analysis Includes external data Multivariate analysis Future development Continued use of sophisticated statistical methods Loss, Premium, and Financial Data Research & Development Unstructured Example: Estimating Losses With Predictive Modeling Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 6

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. Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 7

Oops - DON’T assume Y is normally distributed Oops! Why This Approach Doesn’t Work … 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. Frequency: what is the rate that claims are being made? 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 High frequency losses have less variance. High severity losses have more variance. Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 8

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 (m2 = E[Y]2) Generalized linear models are one example of a suitable framework for our modeling goals. Gamma Distribution Source: “Gamma Distribution,” Wikipedia Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 9

Generalized Linear Models For generalized linear models, we have that 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: Y can be from any exponential family of distributions. Variance depends on expected mean. Approach: The parameters (a0, a1, …, an) can be estimated using maximum likelihood when underlying distribution is fixed. Maximum Likelihood Source: “A Practitioner's Guide to Generalized Linear Models” Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 10

Analytics at Travelers Our problems are challenging in their scope and interdisciplinary nature. Written and verbal communication skills are important Lifelong learners The analytics group at Travelers is a large (300+) and diverse community. Bachelors, Masters, Ph.D.s Mathematics, statistics, physics, actuarial science, computer science, business Ph.D.s are a valuable asset within the Travelers analytic community! Creative and experienced problem-solvers Ability to see the “big picture” and stream-line processes Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 11

Actuarial & Analytics Leadership Development Program (AALDP) 5-yr program for actuarial students and analytics participants Actuarial students offered exam support Analytics participants learn insurance on the job through work projects and seminars (exams are optional) Leadership development opportunities Career exploration opportunities through rotations Networking opportunities (mentor program, committees) 2012 – Pilot Class for Analytics Participants Offers a flexible career path Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 12

References and Resources Travelers Careers http://www.travelers.com/careers Analytics Research Full Time Opportunities A Practitioner's Guide to Generalized Linear Models http://www.towerswatson.com/assets/pdf/2380/Anderson_et_al_Edition_3.pdf Copyright © 2013 The Travelers Indemnity Company, Unpublished Work. All rights reserved. 13