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Instrat. Transition Matrix Theory and Loss Development John B. Mahon CARe Meeting June 6, 2005. Ovewview. Transition Matrix Theory TMT Applied to GC data Distributional Model of Loss Development Independence and variation Effects on Expected Values and Increased Limits Factors. Instrat.

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


Ovewview
Ovewview

  • Transition Matrix Theory

  • TMT Applied to GC data

  • Distributional Model of Loss Development

  • Independence and variation

  • Effects on Expected Values and Increased Limits Factors




Markov chains
Markov Chains

  • 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.










Data source
Data Source

  • 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
















Formula for estimating mu
Formula for estimating mu

  • mu= 1.005 * ln(x),

  • Ingnoring the class 001 data







Function to forecast sigma values
Function to forecast sigma values

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







Comparison of transition matrix results to direct transitions
Comparison of Transition Matrix results to Direct Transitions

  • 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


Comparison of transition matrix results to direct transitions1
Comparison of Transition Matrix results to Direct Transitions

  • 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


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  • 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


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  • Final Test – Compare fitted adjusted sigma surface to emperical sigma surface

  • Difference is TM minus the emperical

  • Take difference and plot


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Instrat formulation of a working model
Instrat TransitionsFormulation of a working model

  • 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


Effect of distributional loss development
Effect of Distributional Loss Development Transitions

  • 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


Simulated losses
Simulated losses Transitions


Simulated losses1
Simulated losses Transitions



Original losses emperical cumulative probability
Original Losses TransitionsEmperical Cumulative Probability


Ldf losses emperical cumulative probability
LDF Losses TransitionsEmperical Cumulative Probability


Distributional loss development emperical cumulative probability
Distributional Loss Development TransitionsEmperical Cumulative Probability




Cumulative distributions
Cumulative Distributions Transitions

  • 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





Effects on limited average severity
Effects on limited average severity Transitions

  • LDF is higher at first

  • Distributional method rises to meet LDF values

  • Distributional and unadjusted data are similar for 0 - $1M range


Increased limits factors basic limits 100 000
Increased Limits Factors TransitionsBasic Limits =$100,000


Increased limits factors basic limits 100 0001
Increased Limits Factors TransitionsBasic Limits =$100,000


Increased limits factors basic limits 100 0002
Increased Limits Factors TransitionsBasic Limits =$100,000


Increased limits factors basic limits 1 000 000
Increased Limits Factors TransitionsBasic Limits =$1,000,000


Increased limits factors basic limits 1 000 0001
Increased Limits Factors TransitionsBasic Limits =$1,000,000


Effects on increased limits factors
Effects on increased limits factors Transitions

  • 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


Conclusions
Conclusions Transitions

  • 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


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