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Correlation and Aggregation in Stochastic Loss Reserve Models

Correlation and Aggregation in Stochastic Loss Reserve Models. Glenn Meyers ISO Innovative Analytics CLRS – September 2007. Drivers of Correlation A Simple Example. Select X 1 and X 2 independently at random Select B 1 , B 2 and B at random B 1 X 1 and B 2 X 2 are not correlated

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Correlation and Aggregation in Stochastic Loss Reserve Models

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  1. Correlation and Aggregation inStochastic Loss Reserve Models Glenn Meyers ISO Innovative Analytics CLRS – September 2007

  2. Drivers of CorrelationA Simple Example • Select X1 and X2 independently at random • Select B1, B2 and B at random • B1X1 and B2X2 are not correlated • BX1 and BX2 are correlated • Bad things (high B) happen at the same time

  3. Drivers of CorrelationB Causes Correlation

  4. Drivers of CorrelationB Increases Volatility of Sum

  5. A Stochastic Loss Reserve Model • XAY,Lag is the loss paid in (AY,Lag) • E[XAY,Lag] = f(PremiumAY,ELR,a,b) • PremiumAY is given • ELR = Expected Loss Ratio • a and b are parameters of the b distribution • XAY,Lag ~ Collective Risk Model • Count distribution is • Negative binomial or • Negative multinomial • Claim severity increases with settlement lag

  6. Expected Loss Model b is the cdf of the beta distribution

  7. Negative Binomial Modelfor Claim Count • Given expected claim count lAY,Lag • Select CAY,Lag from a gamma distribution with mean 1 and variance c • NAY,Lag~ Poisson with mean CAY,LaglAY,Lag • NAY,Lag’s are: • independent for all (AY,Lag) pairs

  8. Negative Multinomial Modelfor Claim Count • Given expected claim count lAY,Lag • Select CAY from a gamma distribution with mean 1 and variance c • NAY,Lag~ Poisson with mean CAYlAY,Lag • NAY,Lag’s are: • Correlated within a given AY • Independent between different AY’s

  9. Using Model to Predict the Distribution of Outcomes • Model predicts the distribution of each XAY,Lag • We need to find sum over future payments: • To evaluate the distribution of the sum we need to consider correlations • More precisely we need to consider the structure of the drivers of correlation

  10. Using FFT’s to Calculate Distribution of Sums Independent Dependence Driven by q

  11. Parameters Used in Example • Realistic - Derived from fitting real data • My call paper from last year’ CLRS “Estimating Predictive Distributions for Loss Reserve Models” • Parameters • Premium = $50 million • Claim severity distributions given • c = 0.01 • ELR = 0.807 • a = 1.654 • b = 4.780

  12. Negative Binomial/Multinomial Thicker tails for the negative multinomial model Correlation makes a difference!

  13. Correlation Driven by Parameter/Model Risk • The choice of ELR, a and b parameters affects all XAY,Lag’s • A good estimation procedure provides a distribution of possible estimates • Used Gibbs sampler (a Bayesian method) • To be described in future paper • Random parameters drives correlation

  14. Parameters Generated by Gibbs Sampler

  15. Negative Multinomial Model

  16. Parameter Risk Across Two Lines • For demonstration purposes, treat the above as two separate lines. • Parameters sets can be selected: • Simultaneously • Independently

  17. Combining Two Identical Lines

  18. Summary Statistics Relevant to Capital Management • Standard Deviation Capital @ 99.5% = 2.576 × Standard Deviation • Tail Value at Risk @ 99% = Average Loss over the 99th percentile • TVaR Capital = TVaR @ 99% – Expected Value

  19. Summary Statistics for Each Model • Model dependencies (i.e. correlation) can have a significant effect in capital management

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