A Claim Counts Model for Discerning the Rate of Inflation from Raw Claims Data

1. A Claim Counts Model for Discerning the Rate of Inflation from Raw Claims Data Spring 2006 CASE Meeting Frank A. SchmidSenior Economist Thanks to John Robertson and Jon Evans for comments Slide 1 of 17

2. The Problem • What is the rate of inflation in a set of raw claims? • I define the rate of inflation as the common trend • Assume you have a set of claims from 1999, and another set of claims from 2003, and you would like to know by which rate the losses have inflated? • In workers’ comp, do indemnity claims inflate by more than a given, published wage index, or by less? • Similarly, do workers’ comp medical claims inflate by more than the medical price index, or by less? • The answers to these questions are of import for rate-making Slide 2 of 17

3. Outline • The concepts of severity, utilization, and inflation • The impact on severity of changes in the shape of the claims distribution • A claims migration model • Estimation technique • Empirical findings • Conclusion Slide 3 of 17

4. The Concepts of Severity, Utilization, and Inflation • Severity, utilization, and inflation are abstract concepts • In keeping with common usage, I employ operational definitions for these concepts—that is, these concepts are defined by the way they are measured • Inflation (change in price level) • The common trend in the dollar amounts of claims for a given set of claims • Severity • The average loss per claim in a given set of claims • Utilization (residual) • The amount of services consumed for a set of claims, as measured by the ratio of severity to price level Slide 4 of 17

5. The Impact on Severity of Changes in the Shape of the Claims Distribution (1) • Traditionally (and as used above), we define the rate of change in utilization as the difference between the rate of change in severity and the rate of inflation • Although this definition of utilization (or rate of utilization change) can be useful, it can also be misleading—as shown below • Utilization, when defined as a residual (as above) is an aggregate concept and, hence, conveys little information about individual claims Slide 5 of 17

6. The Impact on Severity of Changes in the Shape of the Claims Distribution (2) • Example • For simplicity, assume that there is no inflation • In 1999, there were 2 minor back injuries at $1,000, and 1 severe back injury at $10,000 • In 2003, there were 1 minor back injury at $1,000, and 1 severe back injury at $10,000 • Taken together, we observe a utilization increase of $1,500, although there has been no increase in consumption of medical services for any of the claims • On the other hand, if we (correctly) assume that there was no increase in utilization for a given claim, we may (erroneously) conclude (according to the aforementioned traditional definition) that the rate of inflation equals 9.1 percent Slide 6 of 17

7. The Impact on Severity of Changes in the Shape of the Claims Distribution (3) • The aforementioned example illustrates the import of gauging the rate of inflation that applies to a given set of claims • Further, the example shows that the change in the shape of the distribution of claims count by size can account entirely for changes in what is commonly labeled “utilization” • Finally, the example demonstrates that, although the traditional definition of utilization change (as presented above) warrants great caution when it is used for judging the rate of inflation or, conversely, the rate of change of consumption of medical (or indemnity) services Slide 7 of 17

8. A Claims Migration Model (1) • Claims are binned by dollar size of claim • All bins are of equal width on the logarithmic scale • The bin width is comfortably larger than (e.g. twice as large as) my prior belief about the (compounded) rate of inflation • Inflation moves (some) claims up, into higher bins • The large bin width ensures that inflation does not cause claims to leap over bins • From the fraction of claims moving up, we can factor out the rate of inflation (the common trend) Slide 8 of 17

9. A Claims Migration Model (2) • The change in claim count in any given bin is not only affected by inflation, but also by a possible change in the shape of the distribution of claims count • The change in claims count is assumed linear (but not necessarily proportional) in the size of claim (as measured by the respective lower bin break point) • The resulting statistical claims migration model is a system of equations, with one equation for each bin • The claims count in any given bin is a function of • The initial claims count in this bin • The claims count in the neighboring lower-claims-size bin • The rate of inflation • The change in claims count distribution by claims size Slide 9 of 17

10. Estimation Technique (1) • The claims count is modeled as a gamma mixture of Poisson distributions • In a Poisson distribution, mean and variance are identical • This property of the Poisson is not always desirable • The Poisson distribution may be replaced by a Poisson-gamma mixture, among with are the two negative binomial models (NB1 and NB2) • I chose a gamma mixture suggested by Jim Albert (1999, “Criticism of a hierarchical model using Bayes factors,” Statistics in Medicine18, pp. 287-305) Slide 10 of 17

11. Estimation Technique (2) • The (Bayesian) statistical model reads: Slide 11 of 17

12. Estimation Technique (3) • The model is estimated using WinBUGS • WinBUGS is the MS Windows application of BUGS • BUGS: Bayesian inference Using Gibbs Sampling • BUGS is a software platform for Bayesian analysis of complex statistical models using Markov Chain Monte Carlo (MCMC) techniques • There are several incarnations of BUGS, among them an open source version (OpenBUGS) • WinBUGS is the version for use with MS Windows • WinBUGS was developed and is maintained by the MRC Biostatistics Unit of the University of Cambridge, UK • http://www.mrc-bsu.cam.ac.uk/bugs/welcome.shtml Slide 12 of 17

13. Empirical Findings (1) • Observed claims count of medical claims • (Same bin break points for both years) • Did the shape of the distribution change? • The migration of claims to higher bins could all be due to inflation, but is it? Slide 13 of 17

14. Empirical Findings (2) • Observed versus fitted claims count • (Again, same bin break points for both years) Slide 14 of 17

15. Empirical Findings (3) • The re-binning test (for goodness of fit) • Take the estimated rate of inflation and move up the 1999 claims count accordingly • Take the estimated rate of inflation, scale up the dollar numbers of the 1999 claims, and then re-bin Slide 15 of 17

16. Empirical Findings (4) • The change of the claims count distribution • The 1999 claims are migrated up according to the estimated rate of inflation (common trend) Slide 16 of 17

17. Conclusion • What is the driving force behind an increase (or decrease) in severity? • There may be a common trend in the dollar amounts of all claims • This across-the-board increase, we call inflation • In part, this common trend may include across-the-board price increases as caused by improvements in the quality of services • There may be a change in the shape of the claims count distribution by claims size • For instance, some small claims may turn into large claims (e.g., same minor injury now induces more expensive treatment) • At the same time, some small claims may disappear (e.g., some type of minor injury has become less common) Slide 17 of 17