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

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

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

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 conceptsthat
  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

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 misleadingas shown below
• Utilization, when defined as a residual (as
  above) is an aggregate concept and, hence,
  conveys little information about individual claims

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

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

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)

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

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 Medicine 18, pp.
  287-305)

11
Estimation Technique (2)

• The (Bayesian) statistical model reads

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

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?

14
Empirical Findings (2)

• Observed versus fitted claims count
• (Again, same bin break points for both years)

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

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)

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)