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Insurance Applications of Bivariate Distributions

This presentation discusses the application of bivariate distributions in insurance, using mathematical techniques to address pricing and estimation problems. Includes numerical examples.

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Insurance Applications of Bivariate Distributions

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  1. Insurance Applications of Bivariate Distributions David L. Homer & David R. Clark CAS Annual Meeting November 2003

  2. AGENDA: • Explain the Insurance problem being addressed • Show the mathematical “machinery” used to address the problem • Provide a numerical example

  3. AGENDA: • Explain the Insurance problem being addressed • Show the mathematical “machinery” used to address the problem • Provide a numerical example

  4. The Players: Insured: Dietrichson Drilling A large account with predictable annual losses Insurer: Pacific All Risk Insurance Co. Actuary: You

  5. The Pricing Problem: Pacific All-Risk Insurance Company sells a product that provides coverage on both • Specific Excess – individual losses above 600,000 • Aggregate Excess – above the sum of all retained losses capped at 3,000,000 in the aggregate

  6. Policy Structure proposed for our insured Dietrichson Drilling 1,000,000 Per-Occurrence Layer 600,000 Per-occurrence Retained by the insured Stop Loss Layer 3,000,000 8,000,000 Aggregate Losses

  7. How do we price this product? • Expected Losses are straight-forward • Expected Losses for the two coverages are additive • Separate distributions are straight-forward • A combined distribution is NOT

  8. AGENDA: • Explain the Insurance problem being addressed • Show the mathematical “machinery” used to address the problem • Provide a numerical example

  9. How do we estimate a single distribution? Define frequency and severity distributions, then: • Heckman-Meyers • Recursive methods (Panjer) • Simulation • Fast Fourier Transform (FFT)

  10. Key Elements of FFT Technique: • Discretized severity vector x=(x0 ,…,xn-1) • FFT formula • IFFT formula

  11. Convolution Theorem: The transform of the sum is equal to the product of the transforms. To sum up j independent identical variables:

  12. Probability Generating Function: The PGFN is a short-cut method for combining the distributions for each possible number of claims. It does all of the convolutions for us!

  13. Putting it all together we obtain the aggregate probability vector z from the severity probability vector x and the claim count PGFN :

  14. Bivariate case is the same, but using a MATRIX instead of a VECTOR. becomes…

  15. AGENDA: • Explain the Insurance problem being addressed • Show the mathematical “machinery” used to address the problem • Provide a numerical example

  16. Pacific All-Risk: Severity Distribution Bivariate Severity Mx Single Claim Severity x Primary Excess Marginal 0 200,000 400,000 600,000 0 0.00% 0.00% 0.00% 0.00% 0.00% 200,000 37.80% 0.00% 0.00% 0.00% 37.80% 400,000 23.50% 0.00% 0.00% 0.00% 23.50% 600,000 14.60%9.10%15.00% 0.00% 38.70% 800,000 0.00% 0.00% 0.00% 0.00% 0.00% 1,000,000 0.00% 0.00% 0.00% 0.00% 0.00% Excess Marginal 75.90% 9.10% 15.00% 0.00% 100.00% 0 0.00% 200,000 37.80% 400,000 23.50% 600,000 14.60% 800,000 9.10% 1,000,000 15.00% Primary

  17. Negative Binomial PGF for claim counts with Mean=5 and Variance=6:

  18. Bivariate Aggregate Matrix Mz Excess 0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 0 1.05% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 200,000 1.65% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 400,000 2.38% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 600,000 3.09% 0.40% 0.66% 0.00% 0.00% 0.00% 0.00% 800,000 3.34% 0.65% 1.07% 0.00% 0.00% 0.00% 0.00% 1,000,000 3.39% 0.96% 1.58% 0.00% 0.00% 0.00% 0.00% 1,200,000 3.22% 1.27% 2.16% 0.26% 0.21% 0.00% 0.00% 1,400,000 2.86% 1.40% 2.44% 0.44% 0.36% 0.00% 0.00% 1,600,000 2.43% 1.45% 2.59% 0.66% 0.54% 0.00% 0.00% 1,800,000 1.97% 1.40% 2.57% 0.90% 0.78% 0.09% 0.05% 2,000,000 1.54% 1.26% 2.38% 1.02% 0.92% 0.15% 0.08% 2,200,000 1.17% 1.09% 2.12% 1.08% 1.01% 0.24% 0.13% 2,400,000 0.86% 0.90% 1.80% 1.08% 1.05% 0.33% 0.20% 2,600,000 0.62% 0.72% 1.47% 1.00% 1.01% 0.38% 0.24% 2,800,000 0.43% 0.55% 1.16% 0.88% 0.93% 0.42% 0.27% 3,000,000 0.29% 0.41% 0.89% 0.75% 0.82% 0.43% 0.29% Primary

  19. Probability Distribution for Per-Occurrence Excess Losses

  20. Expected Per-Occurrence Loss = 391,000 overall = 830,334 in scenarios where stop loss is hit Both coverages go bad at the same time!

  21. Other Applications: • Generation of Large & Small losses for DFA • Loss and ALAE with separate limits • Any other bivariate phenomenon (e.g., WC medical and indemnity)

  22. Questions or Comments?

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