Insurance Risk Study - Aon

Insurance Risk Study
Modeling Dependence
Dependence is a core component of economic capital
modeling. Risk managers frequently discuss correlation,
and this Study includes numerous correlation matrices.
But correlation alone does not fully describe
dependence. There are many ways to combine two
variables to have the same linear correlation coefficient.
For example, the familiar symmetric, elliptical contours
of the normal copula can have the same linear
correlation as a more pinched distribution, and
pinching can occur either on the left, the right or both
sides. The impact of dependence is most clearly seen in
the distribution of the sum (or portfolio return) of the
two variables, with extreme tail correlation producing
an aggregate distribution with much fatter tails.
Variables in financial markets often exhibit such
extreme tail correlation, as seen in the left chart
below. In this plot, the outliers at the 10.0 percent
and 1.0 percent significance levels assuming a
multivariate normal distribution comprise 11.9 percent
and 2.1 percent of the observations. This kind of
behavior has led many analysts to reject the normal
distribution as a model for dependence.
Academics and risk managers have introduced many
different copulas as means of modeling dependence
with flexible tail behavior. But the appropriateness of
different copulas for insurance losses has been less
well tested.
Daily Stock Returns of Two Financial
Stocks Through the Crisis
16
3.0
2.0
1.0
-1.0
-2.0
-3.0
Normal Transformed Data
0
-3.0 -2.0 -1.0 0 1.0 2.0 3.0
Our study of U.S. data shows that apart from
correlation driven by property catastrophe events there
is little evidence of multi-line extreme correlation. The
right chart below compares results for other liability
occurrence and workers compensation. In this case, the
outliers at the 10.0 percent and 1.0 percent significance
levels assuming multivariate normal distribution
represent 10.9 percent and 1.0 percent of the
observations — well within expectations. Analysis of
other U.S. lines shows similar results.
There is still the possibility that events with long return
periods are not shown in our 18-year data sample — for
example, the impact of asbestos on other liability
occurrence and products liability. We may yet observe
extreme tail correlation in insurance results. But its
absence during the past 18 years suggests that it is not
nearly as commonplace as in financial markets.
We conclude that while the traditional approach to
modeling dependence using the normal copula has
known limitations, it is not rejected by the data as a
model for correlation between non-catastrophe
insurance lines. Catastrophe simulation models address
this issue for catastrophe lines by including correlation
as a model output.
Products Liability Occurrence vs.
Workers Compensation
3.0
2.0
1.0
-1.0
-2.0
-3.0
Normal Transformed Data
0
-3.0 -2.0 -1.0 0 1.0 2.0 3.0