The Aggregation and Correlation of Reinsurance Exposure Glenn Meyers Insurance Services Office, Inc. Fredrick Klinker Insurance Services Office, Inc. David Lalonde AIR Worldwide
Themes of Paper • Dependency models derived from data • Aggregate data from insurers • Catastrophe models driven by engineering analyses and geographic distribution of insured properties • Using aggregate loss models to derive capital costs (i.e. capacity charges) for reinsurance contracts
The Negative Binomial Distribution • Select a at random from a gamma distribution with mean 1 and variance c. • Select the claim count K at random from a Poisson distribution with mean al • K has a negative binomial distribution with:
Multiple Line Parameter Uncertainty • Select b from a distribution with E[b] = 1 and Var[b] = b. • For each line h, multiply each loss by b.
Multiple Line Parameter UncertaintyA simple, but nontrivial example E[b] = 1 and Var[b] = b
About Correlation • There is no direct connection between r and b. • For the same value of b: • Small insurers have large process risk and hence smaller correlation • Large insurers have smaller process risk and hence larger correlations. • Pay attention to the process that generates correlations.
Dependency Analyses are Directed Toward Goal of Evaluating Reinsurer Capital Costs • If bad things happen at the same time, your need more capital.
Summary – Correlation and Capital • At least conceptually, we have established that correlation is important in determining overall capital needs. • Now let’s examine the evidence for correlation. • Fred – Standard insurer losses • David – Catastrophe models
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
Issues • Deviations about long-term means not the most relevant, because they probably include a predictable component, given known rate and price indices, trends, knowledge of current industry competitiveness, losses emerged to date, etc. • What is relevant are unpredictable deviations from possibly time varying expectations, which may vary predictably from long-term averages.
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 from which we calculate deviations.
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, at least one decade to calibrate the time series model, plus a couple more to check for changing correlations over time • We lack the requisite data
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 (loess: quadratic local regression) • Fitted values respond to both earlier and later years, as opposed to one-year-ahead forecasts
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.
A Correlation Model Based on Parameter Uncertainty • From recent papers by Glenn Meyers, assuming frequency parameter uncertainty only:
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.
Recall the definition of covariance: Define the normalized deviation: Divide the original equation by EijkEi’j’k to find: .
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.
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.
In each pairwise product, first and second deviations share common year and LOB 1, but different companies. This is a measure of cross-company, within-LOB correlation. • Pairwise products are not independent; many share a common first or second factor (deviation). • 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.
Just for illustrative purposes, ignores year effects; measures deviations against decade average, separately by company. • Ignoring long-term trends and patterns, probably predictable, inflates apparent correlations.
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.
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
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. Its square root is about .05, indicating that common underlying effects have the potential to drive frequencies across companies within LOB 1up or down by 5 or 10%. • Contagion is 0.02.
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
g just barely significant at two standard errors. • Both g and c smaller than for LOB 1, as expected from graphical evidence.
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
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. There is no statistical evidence that LOBs 1 and 2 are in the same covariance group. A priori, we did not expect them to be.
Additional Observations • Parameter estimates are pooled across companies, not separate by company size, stock/ mutual, etc.
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.
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