Modeling the Loss Process for Medical Malpractice Bill Faltas GE Insurance Solutions CAS Special Interest Seminar … Predictive Modeling “GLM and the Medical Malpractice Crisis” Session October 4, 2004 Chicago, IL © Employers Reinsurance Corporation Patent Pending

2 © Employers Reinsurance Corporation Patent Pending "The work of science is to substitute facts for appearances and demonstrations for impressions.“ John Ruskin

3 © Employers Reinsurance Corporation Patent Pending Regression Modeling  Simply: A functional relationship between one unknown (Y) and one or more knowns (X’s) Y = f (X 1, X 2,..., X n )  error  Statistically: A distribution for Y with parameters that vary with X Example: Ordinary Least Squares (OLS) (“linear regression”) Y ~ Normal (  i,  2 ) Estimate both  i and  2  have a measure of variability (  2 ) E(Y) is a linear combination of X’s  i = a + b 1 X 1i + b 2 X 2i +…+ b n X ni Estimate parameters a, b 1, b 2, …, b n

4 © Employers Reinsurance Corporation Patent Pending Terminology X’s (explanatory / covariate / predictor / independent variables) could be: (1) Numerical:(a) Continuous [e.g., years of practice, square feet] (b) Discrete [e.g., # past claims] (2) Categorical: (a) Ordinal [e.g., income or state group (H / M / L)] (b) Nominal [e.g., gender (M/F), state] Y (response / dependent variable) could be: (1) Continuous [e.g., total $ losses from an insurance policy] (2) Discrete [e.g., # of insurance claims] (3) Binary [e.g., whether an insurance policy is likely to have a claim (Y/N)]

5 © Employers Reinsurance Corporation Patent Pending Popular Regression Modeling Choices Y Continuous Ordinary Least Squares Model (OLS) Y Binary (0,1) Logistic Model Y Positive (Y>0) Exponential Model Y Discrete {0,1,2,3, …} Poisson Model Y

6 © Employers Reinsurance Corporation Patent Pending Model Item GLMOLSLogistic Form of Y AnyContinuousBinary (0,1) Distribution of Y Y ~ Exponential Family Y ~ Normal ( ,  2 ) (in exponential family) Y (=1/0) ~ Bernoulli (P) (in exponential family) Model [E(Y)] Mean(Y) = h(X  ) Mean(Y i ) = f(a + b 1 X 1i + … + b n X ni ) f(linear combination of X’s)  = X   i = a + b 1 X 1i + …+ b n X ni (linear combination of X’s) P = e X  / (1 + e X  ) P i = P(Y i =1) = e L i / (1+ e Li ) where L i =a + b 1 X 1i + … + b n X ni Method of Estimating a, b 1, …, b n M.L.E. Method of Least Squares (same as M.L.E. for Normal) M.L.E. GLM, OLS, and Logistic

7 © Employers Reinsurance Corporation Patent Pending Loss Process Model for Medical Malpractice  Line Characteristic: low frequency / high severity  Objective: Build models to forecast emergence and ultimate values for (Y’s) # notices (a.k.a. incidents) # notices that turn into claims with indemnity payment $ losses  Based on Four Types of X’s Policyholder attributes … state, specialty, years of practice, etc. Policy attributes … form type, limit, etc. Environmental attributes … lawyers per 1000, births per 1000, etc. Time … e.g., policy age measures time since effective date

8 © Employers Reinsurance Corporation Patent Pending pdf Dependence of Likelihood on X 1 X1X1 Not significantly different Likelihood of Notice Claim Likelihood for doctor rises and falls with Age Likelihood at policy age 2.5 years (mode), rises and falls with X 2 Likelihood is a function of many (X) variables, including policy age Likelihood changes with X 1 and X 2  include both in model Y is binary (1/0), “whether there is a notice or not” Likelihood at policy age 2.5 years (mode) increases with X 1

9 © Employers Reinsurance Corporation Patent Pending To model: P = Likelihood of Notice = Pr(Y=1) Likelihood of Notice A Logistic Model (a GLM application) Transform some of the X variables, including policy age Develop model based on 70% data Validate model on remaining 30% of data Compare actual vs. modeled triangles of ‘# policies with notices’ Finalize parameters on 100% of data P = P(Y=1) = e L / (1+ e L ) where L =a + b 1 X 1 + … + b n X n

10 © Employers Reinsurance Corporation Patent Pending Model Validation Approaches Set aside sample Develop parameters using remaining data Verify model works against sample Finalize model using all data Set aside 1 st sample Develop parameters using remaining data Verify model works against 1 st sample Resample and redo … n times Finalize model using all data Divide data into n partitions (often 4-6) Set aside 1 st partition Develop parameters using other partitions Verify model works against 1 st partition Repeat process for all other partitions Finalize model using all data SamplingResamplingPartitioning Uses all data

11 © Employers Reinsurance Corporation Patent Pending Notice to Claim … Waiting Time Approach Waiting time defined as time from notice to claim Waiting time approach enables lack of claim data to be used as information # Claims = (# notices) x (prob. of notice turning into a claim) Area represents probability of turning into a claim years after receiving notice (no actual data prior to 1.0 year). “Waiting time” varies by different values of attribute X 2  include X 2 in notice-to-claim model

12 © Employers Reinsurance Corporation Patent Pending Estimate Claim Closing Values (Claim Sizes) Model trended claim sizes using standard actuarial approaches –Closed claims, without regard to closing lag –Closed claims by closing lag –Closed claims by policyholder attributes Compare company data and models with external benchmarks Select model(s) Test modeled severities against actual severities –Actual severities in development triangles –Modeled severities: f(policyholder, policy, closing year)

13 © Employers Reinsurance Corporation Patent Pending P.D.F of Log of claim sizes by 8 groups of X 1 LN(Claim Size) Density Claim Size Distribution Claim size distribution varies by different values of attribute X 1  include X 1 in claim size modeling modes Claim size model parameters are a function of significant attributes Model location and shape varies w/attributes A way to introduce distributional variation

14 © Employers Reinsurance Corporation Patent Pending Modeling Summary # Notices (Logistic Model) Claim Size Distribution Policyholder Attributes Policy Attributes Environmental Attributes CLAIM COUNTS Notices Becoming Claims (Waiting Time) # Claims = # Notices x Prob of Notice to Claim CLAIM SIZES $ LOSSES $ Losses = # Claims x Claim Size

15 © Employers Reinsurance Corporation Patent Pending GLM Application Advantages  Useful for all lines, including low freq / high sev  Identifies and uses significant variables simultaneously  Effective in dealing with interacting variables  Can use time element to model emergence and ultimates  Variability of modeled estimates can be byproduct and useful for measurements of risk/uncertainty  Multiple applications  Underwriting  Pricing  Reserving  Risk