Integrating Reserve Risk Models into
Economic Capital Models
Stuart White, Corporate Actuary
Casualty Loss Reserve Seminar, Washington D.C.
18-19 September 2008
Disclaimer
—
The views and opinions expressed in this presentation are solely
those of the Speaker and not those of XL Capital Ltd or any of its
subsidiary companies
© 2008, XL Capital Ltd. All rights reserved.
Page 2 _26-Jul-16
Integrating Reserve Risk into Economic
Capital Models
1)
Selecting Reserve Risk Distributions
•
•
2)
Evaluate model fit
Interpreting model output
Integrating Reserve Risk
•
•
•
Fitting distributions to held reserves
One year vs ultimate
Correlation or dependence structure
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1) Selecting Reserve Risk Distributions
— Consider
multiple methods
— Bootstrap,
Mack, other
— Paid vs Incurred
— Historical reserve data
— Industry data
— Consistency
with current year pricing
distributions / RDS / scenario tests
— How does the output compare with regulator and
rating agency capital benchmarks?
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Evaluate model fit
— Practical
methods
— How
well does the reserve predicted by the model
match the held reserve? If no, then can you trust the
predicted standard deviation as well?
— Do relationships across lines make sense?
— Back-testing – Compare historical prior year
development to model predictions
— Changes to the mix of business or terms and
conditions. May need to adjust the data or bear in
mind when selecting results.
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Back-test reserve ranges
80% confidence interval
Line A
1-in-4-years strengthening
Line B
1-in-3-years release
Line C
1-in-2-years strengthening
Line D
1-in-7-years release
Line E
1-in-2-years release
Line F
1-in-2-years strengthening
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Percentiles
—
—
A back-test compares reserve development over a one-year period
against the original estimate of a reserve range
In this exhibit red indicates a strengthening of reserves, green a release
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Evaluate model fit
—
Statistical methods
—
—
“Need to check the reasonableness of model assumptions”
• Plot residuals – do they match assumed distributions?
• Variance or Heteroskedasticity (can be a problem across
development periods or underwriting years)
• Outliers
• Leverage – identify points with excessive influence on model
results
• Significance tests of model parameters
• Compare goodness of fit with historical data
On following two slides example output of good and bad fits from
a standard statistical package
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(Good Fit)
Points are along 45
deg line
Relatively flat trend in variance of
residuals
May want to
investigate point
#72, but not an
extreme outlier
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(Poor Fit)
Points deviate from
45 deg line
Variance increasing with size of
predicted values
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Interpreting model output
— Plot
the output of model iterations
— Benefit of fitting a distribution to simulated model
output
•
More stable tail values
— Model
Output may be closer to a “shifted”
lognormal distribution
•
•
How do you measure the coefficient of variation?
Comparability across methods
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Interpreting model output:
Plot distributions
35
50
65
80
95
110
125
140
155
170
Graphing distributions scaled to a common mean gives a good basis
for comparison and sense checking results across different methods or
lines of business
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Interpreting model output:
Shifted lognormal example
—
Mean of lognormal distribution is:
—
—
Shifted Lognorm al Distribution
Overall mean
70%
Shift
30%
Mean of un-shifted distribution
40%
Standard deviation
30%
Coefficient of variation is:
—
—
30%+40% = 70%
30% / 70% = 42.9%
CV for un-shifted distribution is:
—
30% / 40% = 75%
—
This can be a real issue when
comparing distributions from
different models that are of
different shapes. The shift is
important!
—
continued …
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0%
Page 12 _26-Jul-16
50%
100%
150%
200%
250%
Interpreting model output:
Shifted lognormal example continued
—
—
—
Suppose your model produced the
distribution described on the
previous slide
But you only measured the mean
and CV (70% and 42.9%), and
then applied this to a lognormal
distribution
Actual Distribution
Mean and CV applied to LogN
Distribution
This widens a typical reserve
range interval (e.g. 10-90
percentile), but reduces capital
percentiles (e.g. 99 or 99.5):
Percentile
Actual
LogN Distribution
10%
43.6%
38.0%
90%
105.3%
108.9%
99%
181.4%
167.2%
99.5%
208.8%
185.3%
0%
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50%
100%
150%
200%
250%
2) Integrating Reserve Risk
Fitting distributions to held reserves
—
—
—
—
What if modeled reserves don’t match held reserves?
Decision on whether to scale or shift output distributions
in order to match model means (exactly the same issues
shown in the previous example)
Incurred claims models predict the variability in IBNR.
Therefore need to make an assumption about case
reserves (fixed or stochastic?)
Exclusions from modeled data:
•
•
•
Large losses
Expense reserves, unrecoverables
Ceded amounts
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One Year vs Ultimate
—
—
Does it makes sense to aggregate ultimate reserve risk distributions
with different tail lengths?
Consider time horizon of other risk categories, which are often
looked at over a single year
•
•
•
•
Asset
Credit
FX
Operational
—
How would you combine these with an ultimate reserve risk
distribution?
—
Implication would be to use a one-year reserve distribution, which
makes aggregation with other risks more straightforward
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One Year vs Ultimate
—
Therefore either need a methodology to produce oneyear reserve risk volatilities or a way to transform
ultimate distributions
—
Run-off ultimate using loss development pattern
Wacek – “The path of the ultimate loss ratio” (CAS
Forum Winter 2007)
Adjust confidence level used to derive capital
—
—
•
•
For example the FSA approach that 97.5% over a 5 year time
horizon is equivalent to 99.5% over 1 year
May use the multi-year default curves published by rating
agencies combined with the expected duration of capital held
to estimate this
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Correlation or dependence structure
—
Possibilities:
•
•
•
—
An example of a multi-stage approach would be to model reserve and
current year risk together within a model
•
•
•
—
Large correlation/copula matrix
Multiple stages of aggregation
Dependence
A stochastic claims model (such as bootstrap) can be adjusted to produce
loss projections for the current year
Model a rate level index using time series to capture inter-year dependence
Combine the two for a model that considers both current and prior year risk
Dependence structures would include:
•
•
Linking reserve or underwriting distributions to common factors. These may
be items such as shock losses or economic variables such as inflation or
interest rates
Such approaches offer a more detailed link with other risks such as asset
and credit
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