The Cotor Challenge, Round 2
Matthew Flynn
(860) 633-4119 x8764
Matt.Flynn@sas.com
Copyright © 2003, SAS Institute Inc. All rights reserved.
A little EDA … Proc GCHART; The data are dominated by single large claim, dashed
horizontal lines are at the 95% and 99% percentiles
$10M
$5M
99th pctile
95th pctile
A little EDA … Proc BOXPLOT The data are dominated by single large claim
A little EDA continued… Proc UNIVARIATE;
A little EDA … Proc UNIVARIATE; Loss Histogram – very, very long tail
A little EDA … Proc UNIVARIATE; Losses verses Exponential distribution Large loss
(upper right)
A little EDA continued … Proc UNIVARIATE - logLoss; overall fits are unlikely to fit tails well.
A little EDA … Proc GCHART; The data are dominated by single large claim, vertical lines
are at $5m and $10m
A little EDA continued … Proc UNIVARIATE - logLoss;
A little EDA continued… - logLoss; Top loss = 60% of total dollars, 90% of all dollars are in
the top 25 (or 1%) losses
Sample Mean Excess Distribution
en
n

(
x


)
i k

( ) 
n  k 1
The sample mean excess distribution is the sum of the excesses over
the threshold u divided by the number of data points, n − k + 1, which
exceed the threshold u.
The sample mean excess function describes the expected excess of a
threshold given that exceedance occurs and is an empirical estimate of
the mean excess function; e(u) = E [x − u|x > u].
If a graph of the sample mean excess function is horizontal, the tail is
exponential. An upward sloping graph is said to be ‘fat-tailed’, relative to
an exponential.
Extreme Value Theory – “Peaks Over Threshold” and the
Generalized Pareto distribution
G ,
1  (1    x /  )1 /  if ε  0

1  exp( x /  ) if ε  0
{
Next, fitting A GPD fit the tail of the loss distribution via SAS Proc NLMIXED.
proc nlmixed data=Cotor(where=(logLoss>11.9));
parms sigma=1 xi=0.3;
bounds sigma >= 0;
if (1 + xi * ((logLoss – 11.9) / sigma)) <= 0 then lnlike = 11.9 ** 6;
else
lnlike = -log(sigma) - (1 + (1 / xi))*log(1 + xi * ((logLoss – 11.9) / sigma));
model logLoss ~ general(lnlike);
run;
Quantile or Tail Estimator – VaR (Value at Risk)


ˆ



ˆ   n

1  p   1
xˆ p   

ˆ   Nu





See: McNeil, Alexander J. The Peaks over Thresholds Method for Estimating High Quantiles of Loss
Distributions, ASTIN Colloquium, 1997, equation 5, page 10.
Expected Shortfall – Tail VaR – Conditional Tail Expectation
If things go bad, how bad is bad?




ES p  VaRp  E  x  VaRp | x  VaRp 




y


p
Expected value of a layer from r to R
E y | x     r ( x  r ) f x dx  ( R  r )(1  Fx ( R))
R


GPD Model Fit – Parameter estimates
GPD Model Fit – additional estimates – estimated percentiles, expected shortfall
Sensitivity analysis – expected shortfall, varying size of single largest loss
$5M xs $5M layer price estimate $2,364
The entire analysis can be run directly from Excel
The entire analysis can be run directly from Excel
The entire analysis can be run directly from Excel
The entire analysis can be run directly from Excel
The entire analysis can be run directly from Excel
Bibliography/Resources
Beirlant, Jan and Gunther Matthys, G., Heavy Tailed Distributions and Rating, ASTIN
Bulletin, 2001, v.31, n.1, p.37-58, http://www.casact.org/library/astin/vol31no1/37.pdf
Cebrian, Ana C. , Michel Denuit, and Philippe Lambert, Generalized Pareto Fit to the
Society Of Actuaries Large Claims Database, North American Actuarial Journal, 2003,
v.7, n.3, p.18-36, http://www.soa.org/bookstore/naaj03_07.html#generalized
Chavez-Demoulin, Valerie and Paul Embrechts, Smooth Extremal Models in Finance
and Insurance, Journal of Risk and Insurance, 2001, v. 71, n. 2, p. 183-199,
http://statwww.epfl.ch/people/chavez/
Coles, Stuart, An Introduction to Statistical Modeling of Extreme Values , Springer,
2001, http://www.maths.bris.ac.uk/~masgc/ismev/summary.html
Corradin, Stefano, Economic Risk Capital and Reinsurance: an Application to Fire
Claims of an Insurance Company, WP, 2001,
http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/StefanoCorradin.pdf
Bibliography/Resources, cont.
Cummins, J. David, Christopher M. Lewis and Richard D. Phillips, Pricing
excess-of-loss reinsurance contracts against catastrophic loss, Wharton WP,
1998, n. 98-9, http://fic.wharton.upenn.edu/fic/papers/98/9809.pdf
Joossens, Elisabeth and Johan Segers, Modeling large 3rd party claims in car
insurance with an extended GPD, WP, June 2004,
http://dad.ulyssis.org/~bettie/motorfleet/tekst.pdf
McNeil, Alexander J., Estimating the Tails of Loss Severity Distributions using
Extreme Value Theory, ASTIN Bulletin, 1997, v. 27, n. 1, p. 117-137,
http://www.casact.org/library/astin/vol27no1/117.pdf
McNeil, Alexander J., The Peaks over Thresholds Method for Estimating High
Quantiles of Loss Distributions, ASTIN Colloquium, 1997
Bibliography/Resources, cont.
Reiss, Rolf-Dieter and Michael Thomas , Statistical analysis of extreme values,
extended 2nd edition with applications to insurance, finance, hydrology and other
fields, Birkhauser, 2001, http://www.xtremes.math.uni-siegen.de/
Smith, Richard L., Statistics of extremes, with applications in environmental
science, insurance and finance, U. North Carolina, Statistics WP, July 2002,
http://www.stat.unc.edu/postscript/rs/semstatrls.ps
See also:
SAS Online Docs –
Proc NLMIXED
Matt Flynn
(860) 633-4119 x8764
Matt.Flynn@sas.com