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Transition Matrix Theory and Loss Development John B. Mahon CARe Meeting June 6, 2005

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Transition Matrix Theory and Loss DevelopmentJohn B. MahonCARe MeetingJune 6, 2005

- Transition Matrix Theory
- TMT Applied to GC data
- Distributional Model of Loss Development
- Independence and variation
- Effects on Expected Values and Increased Limits Factors

- Andrey Markov 1856 – 1922
- Best known for work on stochastic process theory
- Markov Chains
- Two essential parts
- Initial distribution
- Transition matrix

- Need to define the array of available states
- Transition matrix has two dimensions
- Both are defined as the available states

- Use standard matrix multiplication
- Markov Property
- Any information dating from before the last step for which the state of the process is known is irrelevant in predicting its state at a later step.

- Claims database of large reinsurance intermediary
- DB prepared manually from submission for recovery
- All values entered ground-up at 100%
- Isolated claims coded GL
- Eliminated claims based on loss cause description
- In order to eliminate “non-standard” losses

- Losses trended with Bests/Masterson’s GL BI trend

- mu= 1.005 * ln(x),
- Ingnoring the class 001 data

- Sigma = 1/(maturity *0.001205+ln(loss size)*0.078874-0.34447

- Transition Matrix method assumes independence
- Real Life claims are not independent
- More mature transitions depend on earlier transitions

- TM has no memory as to where it came from, real claims do
- The TM method may introduce excessive variation

- Create initial to “final” transitions to compare with TM results
- Initial is the first evaluation of a claim
- “Final” is the latest evaluation of a claim
- Assumed to be closed

- Prepared a series of transition matrices representing various initial maturities.
- Data initially as counts
- Division by the initial total produces probability

- Test the relationship between the TM sigmas and the emperical sigmas
- Take ratio emperical sigma / TM sigma
- Select the average of the ratio as an adjustment to TM sigmas

- Final Test – Compare fitted adjusted sigma surface to emperical sigma surface
- Difference is TM minus the emperical
- Take difference and plot

- Open claims are a distribution at ultimate
- The lognormal distribution is a good model for this distribution
- It is possible to model ultimate losses with four parameters and two variables
- Transition matrix introduces additional variation due to its independent nature
- This additional variation can be removed by simple adjustment

- Devise a method to measure the effect of distributional loss development
- Synthesize a set of losses
- Lognormal distribution
- Mu =13, sigma =1
- Used stratified sampling

- Can enter into fitting routines to get parametized distributions
- Can use directly for calculating limited expected values and ILF’s
- Limited to the largest size of loss

- Can use directly for simulation
- Lacks ability to estimate parameter uncertainty

- LDF is higher at first
- Distributional method rises to meet LDF values
- Distributional and unadjusted data are similar for 0 - $1M range

- The results vary as the basic limit is changed
- For a $100,000 basic limit, the developed losses resemble each other and are 50% higher than unadjusted
- For a $1,000,000 basic limit the distributional adjusted is much higher than the LDF adjusted

- Distributional Loss Development shifts the ultimate severity distribution differently than application of loss development factors
- This can result in differences in increased limits factors up to 30%
- Increase limits factors are sensitive to the selection of basic limits
- The differences shown here are upper limits
- This study assumed all losses to be developed
- In a real life collection of claims, the majority would be closed, and the minority would be developed

- Development errors could result in 10 – 15% errors