Correlation Estimation for Correlation Estimation for Property and Casualty Property and Casualty Underwriting LossesUnderwriting Losses

Fred KlinkerFred KlinkerInsurance Services Office, Inc.Insurance Services Office, Inc.

Mathematical vs. Physical Models Mathematical vs. Physical Models for Correlationfor Correlation

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

Estimation of CorrelationEstimation of Correlation

For a number of lines of business, For a number of lines of business, companies, and years, estimate expected companies, and years, estimate expected losses or loss ratioslosses or loss ratios

Measure deviations of actual ultimates Measure deviations of actual ultimates from these expectationsfrom these expectations

Estimate correlations among these Estimate correlations among these deviations as the correlations relevant to deviations as the correlations relevant to required capitalrequired capital

IssuesIssues

Deviations about long-term means not the Deviations about long-term means not the most relevant, because they probably most relevant, because they probably include a predictable component driven by include a predictable component driven by known rate and price indices, trends, known rate and price indices, trends, knowledge of current industry knowledge of current industry competitiveness, losses emerged to date, competitiveness, losses emerged to date, etc.etc.

What is relevant are unpredictable What is relevant are unpredictable deviations from expectations varying deviations from expectations varying predictably over time.predictably over time.

Thought Experiment 1Thought Experiment 1

Rose Colored Glasses Insurance Rose Colored Glasses Insurance Company—will probably estimate larger Company—will probably estimate larger correlations than a company that correlations than a company that estimates its expected losses more estimates its expected losses more accurately.accurately.

A cautionary conclusion—the correlations A cautionary conclusion—the correlations we estimate to some extent depend on we estimate to some extent depend on how we estimate the expectations.how we estimate the expectations.

Thought Experiment 2: How We Thought Experiment 2: How We Might Like to Estimate CorrelationsMight Like to Estimate Correlations

Mimic P&C industry real-time forecasting: Mimic P&C industry real-time forecasting: rolling one-year-ahead forecasts based on rolling one-year-ahead forecasts based on what industry would have known what industry would have known compared to estimated actual ultimatescompared to estimated actual ultimates

What we need: Multiple decade time What we need: Multiple decade time series of loss ratios and predictors, one series of loss ratios and predictors, one decade to calibrate the time series, plus decade to calibrate the time series, plus more to check for time varying correlationsmore to check for time varying correlations

We lack the requisite dataWe lack the requisite data

An Alternative CalculationAn Alternative Calculation

One decade of data, no predictorsOne decade of data, no predictors

By LOB, a generalized additive model with By LOB, a generalized additive model with main effects for company and yearmain effects for company and year

Year effect captured by a non-parametric Year effect captured by a non-parametric smoothersmoother

Fitted values respond to both earlier and Fitted values respond to both earlier and later years, as opposed to one-year-ahead later years, as opposed to one-year-ahead forecastsforecasts

A QuestionA Question

Could the year smoother “forecast” even Could the year smoother “forecast” even better than the best true one-year-ahead better than the best true one-year-ahead forecast, thereby understating deviations forecast, thereby understating deviations and covariances?and covariances?

Perhaps, but probably not vastly better.Perhaps, but probably not vastly better.

A Correlation Model Based on A Correlation Model Based on Parameter UncertaintyParameter Uncertainty

From recent papers by Glenn Meyers, assuming From recent papers by Glenn Meyers, assuming frequency parameter uncertainty only:frequency parameter uncertainty only:

i'j'kijki'iGG

ijkiiijkii

ijj'ii'i'j'kijk

EEggδ

E)cg(Eμμ

σδδ],LCov [L

i'i

2

2

1

where:where:

LLijkijk is annual aggregate ultimate loss for LOB is annual aggregate ultimate loss for LOB ii, ,

company company jj, and year , and year k.k.

δδii´ii´ is 1 if and only if is 1 if and only if i i = = i´i´ and 0 otherwise. and 0 otherwise.

Likewise for Likewise for δδjj´jj´..

δδGiGi´GiGi´ is 1 if and only if first and second LOBs are is 1 if and only if first and second LOBs are

in the same covariance group, otherwise 0.in the same covariance group, otherwise 0.

μμii and and σσii are the mean and standard deviation of are the mean and standard deviation of

the severity distribution associated with LOB the severity distribution associated with LOB ii..

EEijkijk = = EE[[LLijkijk] ]

ggii is the covariance generator associated with is the covariance generator associated with

LOB LOB ii. .

Recall the definition of covariance:Recall the definition of covariance:

Define the normalized deviation:Define the normalized deviation:

Divide the original equation by EijkEi’j’k to find:

[ , ] [( [ ])( [ ])]ijk i j k ijk ijk i j k i j kCov L L E L E L L E L

[ ]

[ ]ijk ijk

ijkijk

L E L

E L

.

2

'

' '[ ] (1 )

iii jj i

iijk i j k ii jj i i GiGi i i

ijk

E g c g gE

Model for Expected LossesModel for Expected Losses

Model loss ratios, then multiply by denominators.Model loss ratios, then multiply by denominators.

By LOB, a generalized additive model with main By LOB, a generalized additive model with main effects for company and yeareffects for company and year

Year smoothing parameters chosen so that Year smoothing parameters chosen so that model responds to long term trends without model responds to long term trends without responding much to individual year effects.responding much to individual year effects.

Loss ratio volatility declines significantly with Loss ratio volatility declines significantly with increasing company size; a weighted model increasing company size; a weighted model strongly recommended.strongly recommended.

1990 1992 1994 1996 1998

Year

0.5

1.0

1.5

Lo

ss R

atio

Loss Ratios by Company (LOB 1)

1990 1992 1994 1996 1998

Year

0.6

0.8

1.0

1.2

1.4

Lo

ss R

atio

Loss Ratios by Company (LOB 2)

Appearance of roughly parallel lines Appearance of roughly parallel lines supports main effects model.supports main effects model.

At least for LOB 1, considerable correlated At least for LOB 1, considerable correlated ups and downs from year to year.ups and downs from year to year.

After visual inspection of these graphs, After visual inspection of these graphs, would not be surprised to find greater would not be surprised to find greater correlation for LOB 1 than for LOB 2.correlation for LOB 1 than for LOB 2.

90 92 94 96 98

Year

-0.1

5-0

.10

-0.0

5-0

.00

0.0

50

.10

Ye

ar

Eff

ect

Loss Ratio Year Effect (LOB 1)

Variance ModelVariance Model

5000 10000 50000 100000 500000 1000000

Expected Loss (E) (thousands)

0.0

0.2

0.4

0.6

0.8

1.0

Sq

ua

red

De

via

tion

Squared Deviation vs. Expected Loss (LOB 1)

Other Pairwise Products of Other Pairwise Products of DeviationsDeviations

-4 -2 0 2 4 6

First Standardized Deviation

-5

-3

-1

1

3

5

Se

co

nd

Sta

nd

ard

ize

d D

evia

tion

Deviation vs. Deviation (LOB 1, Full Trend Model)

In each pairwise product, first and second In each pairwise product, first and second deviations share common year and LOB 1, but deviations share common year and LOB 1, but different companies: cross-company, within-LOB different companies: cross-company, within-LOB correlation.correlation.

Pairwise products are not independent; many Pairwise products are not independent; many share a common first or second factor.share a common first or second factor.

Regression line indicates modest positive Regression line indicates modest positive correlation between first and second deviations, correlation between first and second deviations, plus considerable noise.plus considerable noise.

A visual aid only; actual inference not based on A visual aid only; actual inference not based on this line.this line.

-3 -1 1 3 5

First Standardized Deviation

-4

-2

0

2

4

Se

co

nd

Sta

nd

ard

ize

d D

evia

tio

nDeviation vs. Deviation (LOB 1, No Trend Model)

For illustrative purposes only, ignores year For illustrative purposes only, ignores year effects; measures deviations against effects; measures deviations against decade average, separately by company.decade average, separately by company.

Ignoring long-term trends and patterns, Ignoring long-term trends and patterns, probably predictable, inflates apparent probably predictable, inflates apparent correlations.correlations.

Bootstrap Estimates of Standard Bootstrap Estimates of Standard ErrorsErrors

Pairwise products of deviations not independent; Pairwise products of deviations not independent; can’t use the usual sqrt(n) rule.can’t use the usual sqrt(n) rule.

Don’t bootstrap on pairwise products directly; Don’t bootstrap on pairwise products directly; this destroys two-way structure of data on this destroys two-way structure of data on company and year.company and year.

Bootstrap on year, take all companies. Then Bootstrap on year, take all companies. Then bootstrap on company, take all years.bootstrap on company, take all years.

Combined standard error is square root of sum Combined standard error is square root of sum of squared standard errors due to year and of squared standard errors due to year and company separately.company separately.

Representative ResultsRepresentative Results

Correlation Parameter Estimates: LOB 1Correlation Parameter Estimates: LOB 1

Between companies: Between companies: ggEstimate: 0.0026Estimate: 0.0026Standard error due to years: 0.0008Standard error due to years: 0.0008Standard error due to companies: 0.0009Standard error due to companies: 0.0009Full standard error: 0.0012Full standard error: 0.0012

Within company: Within company: c + gc + gEstimate: 0.0226Estimate: 0.0226Standard error due to years: 0.0048Standard error due to years: 0.0048Standard error due to companies: 0.0078Standard error due to companies: 0.0078Full standard error: 0.0092Full standard error: 0.0092

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

Correlation Parameter Estimates: LOB 2Correlation Parameter Estimates: LOB 2

Between companies: Between companies: ggEstimate: 0.0007Estimate: 0.0007Standard error due to years: 0.0002Standard error due to years: 0.0002Standard error due to companies: 0.0003Standard error due to companies: 0.0003Full standard error: 0.0004Full standard error: 0.0004

Within company: Within company: c + gc + gEstimate: 0.0090Estimate: 0.0090Standard error due to years: 0.0007Standard error due to years: 0.0007Standard error due to companies: 0.0022Standard error due to companies: 0.0022Full standard error: 0.0023Full standard error: 0.0023

g just barely significant at two standard g just barely significant at two standard errors.errors.

Both g and c smaller than for LOB 1, as Both g and c smaller than for LOB 1, as expected from graphical evidence.expected from graphical evidence.

Correlation Parameter Estimates:Correlation Parameter Estimates:LOB 1 vs. LOB 2LOB 1 vs. LOB 2

Between and within companies: Between and within companies: gg

Estimate: 0.0005Estimate: 0.0005

Standard error due to years: 0.0005Standard error due to years: 0.0005

Standard error due to companies: 0.0003Standard error due to companies: 0.0003

Full standard error: 0.0006Full standard error: 0.0006

What is here labeled g is actually What is here labeled g is actually geometric average of gs for LOBs 1 and 2, geometric average of gs for LOBs 1 and 2, if in the same covariance group, or 0 if in the same covariance group, or 0 otherwise.otherwise.

Parameter estimate not significantly Parameter estimate not significantly different from 0: no statistical evidence different from 0: no statistical evidence that LOBs 1 and 2 are in the same that LOBs 1 and 2 are in the same covariance group.covariance group.

Additional ObservationsAdditional Observations

Parameter estimates are pooled across Parameter estimates are pooled across companies, not separate by company size, companies, not separate by company size, stock/ mutual, etc.stock/ mutual, etc.

Correlation in the body of a multivariate Correlation in the body of a multivariate distribution vs. “correlation in the tails”: distribution vs. “correlation in the tails”: correlation due to parameter uncertainty correlation due to parameter uncertainty vs. correlation due to catastrophes.vs. correlation due to catastrophes.

What Else is in Appendix?What Else is in Appendix?

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

BibliographyBibliography

Glenn Meyers, “Estimating Between Line Glenn Meyers, “Estimating Between Line Correlations Generated by Parameter Correlations Generated by Parameter Uncertainty,” Uncertainty,” CAS ForumCAS Forum, Summer 1999., Summer 1999.http://www.casact.org/pubs/forum/99sforum/http://www.casact.org/pubs/forum/99sforum/99sf197.pdf99sf197.pdfGlenn Meyers, Fred Klinker, and David Lalonde, Glenn Meyers, Fred Klinker, and David Lalonde, “The Aggregation and Correlation of Insurance “The Aggregation and Correlation of Insurance Exposure,” Exposure,” CAS ForumCAS Forum, Summer 2003., Summer 2003.http://www.casact.org/pubs/forum/03sforum/http://www.casact.org/pubs/forum/03sforum/03sf015.pdf03sf015.pdf