Correlation Estimation for
Property and Casualty
Underwriting Losses
Fred Klinker
Insurance Services Office, Inc.
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.
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 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.
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.
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
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
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:
 σ i2


2
Cov [Lijk ,Li'j'k ]  δii'δ jj'   μi  Eijk  ( 1  gi )ci Eijk 

 μi

 δGi Gi' gi gi' Eijk Ei'j'k
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:
Cov[ Lijk , Lijk ]  E[( Lijk  E[ Lijk ])( Lij k  E[ Lij k ])]
Define the normalized deviation:
 ijk 
.
Lijk  E[ Lijk ]
E[ Lijk ]
Divide the original equation by EijkEi’j’k to find:
  i2

 ii ' jj   i 
i

    (1  g )c  
E[ijk ijk ] 
ii ' jj '
i
i
GiGi  gi gi 
Eijk
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.
1.0
0.5
Loss Ratio
1.5
Loss Ratios by Company (LOB 1)
1990
1992
1994
Year
1996
1998
1.0
0.8
0.6
Loss Ratio
1.2
1.4
Loss Ratios by Company (LOB 2)
1990
1992
1994
Year
1996
1998
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.
-0.00
-0.05
-0.10
-0.15
Year Effect
0.05
0.10
Loss Ratio Year Effect (LOB 1)
90
92
94
96
Year
98
Variance Model
0.6
0.4
0.2
0.0
Squared Deviation
0.8
1.0
Squared Deviation vs. Expected Loss (LOB 1)
5000
10000
50000 100000
Expected Loss (E) (thousands)
500000 1000000
Other Pairwise Products of
Deviations
Deviation vs. Deviation (LOB 1, Full Trend Model)
Second Standardized Deviation
5
3
1
-1
-3
-5
-4
-2
0
2
First Standardized Deviation
4
6
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.
Deviation vs. Deviation (LOB 1, No Trend Model)
Second Standardized Deviation
4
2
0
-2
-4
-3
-1
1
First Standardized Deviation
3
5
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.
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.
Representative Results
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. 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.
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: no statistical evidence
that LOBs 1 and 2 are in the same
covariance group.
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.
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.
Bibliography
Glenn Meyers, “Estimating Between Line
Correlations Generated by Parameter
Uncertainty,” CAS Forum, Summer 1999.
http://www.casact.org/pubs/forum/99sforum/99sf
197.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/03sf
015.pdf