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The COTOR Challenge

The COTOR Challenge. Committee on the Theory of Risk November 2004 Annual Meeting. History of the Challenge. Last spring a COTOR member challenged actuarial geeks to estimate 500k xs 500k layer based on list of 250 claims Emails flew back and forth furiously

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The COTOR Challenge

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  1. The COTOR Challenge Committee on the Theory of Risk November 2004 Annual Meeting

  2. History of the Challenge • Last spring a COTOR member challenged actuarial geeks to estimate 500k xs 500k layer based on list of 250 claims • Emails flew back and forth furiously • A number of different approaches were used • Literature about heavy tailed distributions was recommended • Winner was Phil Heckman using mixture of 2 lognormals

  3. History cont. • Criticism existed around the sample since some sample statistics were too far from the real distribution • COTOR feels that the solution of this problem is of interest ot the actuarial community • Our data is almost never normal/lognormal • Our data is typically heavy tailed • It is likely that in many real situations, a sample of 250 claims would not represent a random draw from any distribution

  4. History cont. • Another challenge was issued under well defined conditions • Stuart Klugman picked the sample • 250 claims randomly generated from an inverse transformed gamma • Challenge was to estimate severity in the $5M xs $5M layer (mean and 95% confidence intervals)

  5. The Sample 250 claims randomly selected from an inverse transformed gamma

  6. Purpose of Session • Raise awareness of audience of how frequently extreme values need to be dealt with • Present relatively easy to use approaches • Make audience aware of how difficult this problem is to solve

  7. Normal Distribution Assumption • The normal or lognormal assumption is common in finance application • Option pricing theory • Value at risk • CAPM • Evidence that asset return data does not follow the normal distribution is widely available • 1968 Fama paper in Journal of the American Statistical Association

  8. 1.15 T h e o r e t i c a l V a l u e 1.10 1.05 1.00 0.95 0.90 0.85 0.8 0.9 1.0 1.1 1.2 1.3 Observed Value Test of Normal Distribution Assumption Normal Q-Q Plot of Monthly Return on S&P

  9. Test of Normal Distribution Assumption

  10. Consequences of Assuming Normality • The frequency of extreme events is underestimated – often by a lot • Example: Long Term Capital • “Theoretically, the odds against a loss such as August’s had been prohibitive, such a debacle was, according to mathematicians, an event so freakish as to be unlikely to occur even once over the entire life of the universe and even over numerous repetitions of the universe” • When Genius Failed by Roger Lowenstein, p. 159

  11. Criteria for Judging • New and creative way to solve the problem • Methodology that practicing actuaries can use • Clarity of exposition • Accuracy of known answer • Estimates of confidence interval

  12. Table of Results

  13. Observations Regarding Results • These estimations are not easy • Nearly 13 to 1 spread between lowest and highest mean • Only 10% of answers came within 10% of right result • All responders recognized tremendous uncertainty in results (range from upper to lower CL went from 8 to infinity) • Our statistical expert could not understand the description of the method of 30% of the respondents

  14. Observations • All but 2 of the methods relied on approaches commonly found in the literature on heavy tailed distributions and extreme values • It is clear that it is very difficult to get accurate estimates from a small sample • The real world is even more challenging than this • 250 claims probably don’t follow any known distribution • Trend • Development • Unforeseen changes in environment • Consulting with claims adjusters and underwriters should provide valuable additional insights

  15. Observations • The closest answer was 5% below the true mean • Half of the responses below the true mean, Half were above • Average response was 40% higher than the mean • Average response (ex outlyer) was within 2% of the mean • Read: “The Wisdom of Crowds: Why the Many are Smarter than the Few and How Collective Wisdom Shapes Business, Economics, Societies and Nations” by: James Surowiecki • Implications for Insurance Companies?

  16. Speakers • Meyers • Evans • Flynn • Woolstenhulme • Venter • Heckman

  17. Announcement of Winners • Louise Francis – COTOR Chair

  18. Possible Next Steps • Make the results of the challenge available to the membership • COTOR subcommittee to evaluate how to make techniques readily available • Another round making the challenge more real world • Include trend and development • Give multiple random samples

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