Correlation Estimation for Property and Casualty Underwriting Losses

1. Correlation Estimation for Property and Casualty Underwriting Losses Fred Klinker Insurance Services Office, Inc.

2. Mathematical vs. Physical Models for Correlation • Mathematical models/ treatments: convenient and parsimonious ways of encoding what we know about correlation: simulation, Fast Fourier Transforms, copulas, etc. • Physical models for the drivers of correlation that therefore capture the structure: parameter uncertainty, natural and man-made catastrophes, mass torts.

3. Estimation of Correlation • For a number of lines of business, companies, and years, estimate expected losses or loss ratios • Measure deviations of actual ultimates from these expectations • Estimate correlations among these deviations as the correlations relevant to required capital

4. Issues • Deviations about long-term means not the most relevant, because they probably include a predictable component driven by known rate and price indices, trends, knowledge of current industry competitiveness, losses emerged to date, etc. • What is relevant are unpredictable deviations from expectations varying predictably over time.

5. Thought Experiment 1 • Rose Colored Glasses Insurance Company—will probably estimate larger correlations than a company that estimates its expected losses more accurately. • A cautionary conclusion—the correlations we estimate to some extent depend on how we estimate the expectations.

6. Thought Experiment 2: How We Might Like to Estimate Correlations • Mimic P&C industry real-time forecasting: rolling one-year-ahead forecasts based on what industry would have known compared to estimated actual ultimates • What we need: Multiple decade time series of loss ratios and predictors, one decade to calibrate the time series, plus more to check for time varying correlations • We lack the requisite data

7. An Alternative Calculation • One decade of data, no predictors • By LOB, a generalized additive model with main effects for company and year • Year effect captured by a non-parametric smoother • Fitted values respond to both earlier and later years, as opposed to one-year-ahead forecasts

8. A Question • Could the year smoother “forecast” even better than the best true one-year-ahead forecast, thereby understating deviations and covariances? • Perhaps, but probably not vastly better.

9. A Correlation Model Based on Parameter Uncertainty • From recent papers by Glenn Meyers, assuming frequency parameter uncertainty only:

10. where: • Lijk is annual aggregate ultimate loss for LOB i, company j, and year k. • δii´ is 1 if and only if i = i´ and 0 otherwise. Likewise for δjj´. • δGiGi´is 1 if and only if first and second LOBs are in the same covariance group, otherwise 0. • μi and σi are the mean and standard deviation of the severity distribution associated with LOB i. • Eijk = E[Lijk] • gi is the covariance generator associated with LOB i.

11. Recall the definition of covariance: Define the normalized deviation: Divide the original equation by EijkEi’j’k to find: .

12. Model for Expected Losses • Model loss ratios, then multiply by denominators. • By LOB, a generalized additive model with main effects for company and year • Year smoothing parameters chosen so that model responds to long term trends without responding much to individual year effects. • Loss ratio volatility declines significantly with increasing company size; a weighted model strongly recommended.

15. Appearance of roughly parallel lines supports main effects model. • At least for LOB 1, considerable correlated ups and downs from year to year. • After visual inspection of these graphs, would not be surprised to find greater correlation for LOB 1 than for LOB 2.

17. Variance Model

19. Other Pairwise Products of Deviations

21. In each pairwise product, first and second deviations share common year and LOB 1, but different companies: cross-company, within-LOB correlation. • Pairwise products are not independent; many share a common first or second factor. • Regression line indicates modest positive correlation between first and second deviations, plus considerable noise. • A visual aid only; actual inference not based on this line.

23. For illustrative purposes only, ignores year effects; measures deviations against decade average, separately by company. • Ignoring long-term trends and patterns, probably predictable, inflates apparent correlations.

24. Bootstrap Estimates of Standard Errors • Pairwise products of deviations not independent; can’t use the usual sqrt(n) rule. • Don’t bootstrap on pairwise products directly; this destroys two-way structure of data on company and year. • Bootstrap on year, take all companies. Then bootstrap on company, take all years. • Combined standard error is square root of sum of squared standard errors due to year and company separately.

25. Representative Results

26. Correlation Parameter Estimates: LOB 1 Between companies: g Estimate: 0.0026 Standard error due to years: 0.0008 Standard error due to companies: 0.0009 Full standard error: 0.0012 Within company: c + g Estimate: 0.0226 Standard error due to years: 0.0048 Standard error due to companies: 0.0078 Full standard error: 0.0092

27. With respect to g, standard errors due to years and companies are comparable. • Estimate is more than twice the full standard error, so significant. • g is the variance of a frequency multiplier acting in common across companies within LOB 1. Square root of about .05: common underlying effects have the potential to drive frequencies across companies within LOB 1up or down by 5 or 10%. • Contagion is 0.02.

28. Correlation Parameter Estimates: LOB 2 Between companies: g Estimate: 0.0007 Standard error due to years: 0.0002 Standard error due to companies: 0.0003 Full standard error: 0.0004 Within company: c + g Estimate: 0.0090 Standard error due to years: 0.0007 Standard error due to companies: 0.0022 Full standard error: 0.0023

29. g just barely significant at two standard errors. • Both g and c smaller than for LOB 1, as expected from graphical evidence.

30. Correlation Parameter Estimates:LOB 1 vs. LOB 2 Between and within companies: g Estimate: 0.0005 Standard error due to years: 0.0005 Standard error due to companies: 0.0003 Full standard error: 0.0006

31. What is here labeled g is actually geometric average of gs for LOBs 1 and 2, if in the same covariance group, or 0 otherwise. • Parameter estimate not significantly different from 0: no statistical evidence that LOBs 1 and 2 are in the same covariance group.

32. Additional Observations • Parameter estimates are pooled across companies, not separate by company size, stock/ mutual, etc. • Correlation in the body of a multivariate distribution vs. “correlation in the tails”: correlation due to parameter uncertainty vs. correlation due to catastrophes.

33. What Else is in Appendix? • Expected losses derived from expected loss ratio models. We tested several denominators: premium, PPR, exposures. • Adjusted normalized deviations for degrees of freedom. • More thorough treatment of weights in all models: loss ratio, variance, other pairwise products of deviations. • Tested correlation model parameters for dependence on size of company: none found.

34. Bibliography • Glenn Meyers, “Estimating Between Line Correlations Generated by Parameter Uncertainty,” CAS Forum, Summer 1999. http://www.casact.org/pubs/forum/99sforum/99sf197.pdf • Glenn Meyers, Fred Klinker, and David Lalonde, “The Aggregation and Correlation of Insurance Exposure,” CAS Forum, Summer 2003. http://www.casact.org/pubs/forum/03sforum/03sf015.pdf