Probability Models of Natural Catastrophe Losses
PAVLA JINDROVÁ, VIERA PACÁKOVÁ
Institute of Mathematics and Quantitative Methods
Faculty of Economics and Administration
University of Pardubice
Studentská 95, 532 10 Pardubice 2
CZECH REPUBLIC
Abstract: - In this article we describe parametric curve-fitting methods for modeling historical natural
catastrophe losses. We summarize relevant theoretical results above Excess over Threshold Method (EOT) and
provide its application to the data about total catastrophe losses in the world in period 1970-2014, published in
No2/2015 Swiss Re study Sigma.
Key-Words: - Excess over Threshold Method, Generalized Pareto Distribution, losses, natural catastrophe
.
1 Introduction
Catastrophic events affect various regions of the
world with increasing frequency and intensity
(Fig. 1). Many regions are threatened by
catastrophic risks large range, where extensive
disruptions are frequently, sometimes more than
once a year. Large catastrophic events can be caused
by natural phenomena or are caused by man. It
should be noted that many of the events and natural
character are to a large extent influenced by human
activity. This mainly concerns climate change, but
also, for example, the influence of the mining
industry. Serious events in recent years are often the
result of terrorist acts.
0
50
100
150
200
250
300
1970 1975 1980 1985 1990 1995 2000 2005 2010
Man-made disasters Natural catastrophes
Fig.1 Number of catastrophic events 1970-2014
According to the latest sigma study, global
insured losses from natural catastrophes and man-
made disasters were USD 35 billion in 2014, down
from USD 44 bi llion in 2013 a nd well below the
USD 64 billion-average of the previous 10 years.
There were 189 natural catastrophe events in 2014,
the highest ever on sigma records, causing global
economic losses of USD 110 b illion. (Swiss Re
sigma No 2/2015, p. 4)
Catastrophe modelling is a risk management
tool that uses computer technology to help insurers,
reinsurers and risk managers better assess the
potential losses caused by natural and man-made
catastrophes. The models use historical disaster
information to simulate the characteristics of
potential catastrophes and to determine the potential
losses cost.
We are interested in probability modelling
catastrophe losses, specifically the tails of loss
severity distributions. Thus is of particular relevance
in reinsurance if we are required to choose or price a
high-excess layer. In this situation it is essential to
find a good statistical model for the largest observed
historical losses.
2 Problem Formulation
Suppose losses are the independent, identically
distributed (iid) random variables
12
, , ...XX
, with
common distribution function
( ) ( )
xXPxF
X
≤=
, where
0>x
(1)
The generalized Pareto Distribution (GPD) is
the limit distribution of scaled excess of high
thresholds.
PROOF
DOI: 10.37394/232020.2022.2.1
Pavla Jindrová, Viera Pacáková
E-ISSN: 2732-9941
1
2.1 GPD Theorem
Suppose
...,,
21
XX
are iid with distribution F.
Than
( )
−
+−=
−
ξ
σ
µ
ξ
1
1exp x
xG
(2)
where
01 >
−
+
σ
µ
ξ
x
(3)
is the limit distribution of the maxima Mn = X1,n =
max(X1, …, Xn). Then for a large enough threshold u,
the conditional distribution function of Y = (X – u /
X > u) is approximately
( )
ξ
σ
ξ
1
~
11
−
+−= x
xH
(4)
defined on
( ){ }
0 >
~
x/1 and 0 > x :x
σξ
+
, where
( )
µξσσ
−+= u
~
.
The family of distributions defined by equation
(4) is called the generalized Pareto family (GPD).
For a fixed high threshold u, the two parameters are
the shape parameter ξ and the scale parameter
σ
.
For simpler notation, we may just use σ for the scale
parameter if there is no confusion.
2.2 Excess over Threshold Method
The modelling using the excess over threshold
method follows the assumptions and conclusions in
GPD Theorem. Suppose
n
xxx ...,,,
21
are raw
observations independently from a common
distribution F(x). Given a high threshold u, assume
( ) ( ) ( )
k
xxx ...,,,
21
are an observation that exceeds u.
Here we define the ascendances as
( )
uxx
ii
−=
for
ki ,...,2,1=
.
By GPD Theorem i
x
may be regarded as
realization of independently random variable which
follows a generalized Pareto family with unknown
parameters
ξ
and
σ
. In case
0≠
ξ
, the likelihood
function can be obtained directly from (4):
( )( )
∏
=
−−
+=
k
i
i
x
L
1
11
1
1
,
ξ
σ
ξ
σ
σξ
x
(5)
3 Problem Solution
The analysis focus on 2 64 amounts of damage (in
USD millions) of total natural catastrophes in time
period from January 2010 to December 2014,
published in Swiss Re sigma 2011-2015.
Fig.2 Chronologically arranged the total losses
of natural catastrophes 2010-2014
The time series plot (Fig.2) allows us to identify
the most extreme losses and their approximate times
of occurrences.
We have fitted a generalized Pareto distribution
using the maximum likelihood method for
parameters estimation to the data above threshold of
3000 (Fig.3), above threshold of 5000 (Fig.5) and
above threshold 8000 (Fig. 7).
These plots are useful for examining the
distribution based on sample data. We have overlaid
a theoretical distribution function on the same plot
with empirical distribution of the sample to compare
them.
The black stair lines on Fig.3, Fig.5 and Fig.7
show the empirical distribution functions of
empirical sample data and the blue curves present
the theoretical distribution function of the estimated
generalized Pareto distributions for different
thresholds. The red lines are the lower and upper
bounds of the 95% confidence interval estimates of
the distribution function. It can be seen that the
estimated parametric distribution function falls
inside the bands.
In Fig.3, Fig.5 and Fig. 8 we see the good fit of
all three generalized Pareto distributions of total
losses on natural catastrophes.
The QQ-plots (Fig.4, Fig.6, and Fig.8) against
the generalized Pareto distributions is another way
to examine visually the hypothesis that the losses
which exceed a very high threshold come from
estimated distributions.
Table 1 presents the parameters of the fitted
generalized Pareto distributions on the data above
three different thresholds. By p-values in this table
we can state the best fit in the case of threshold
u = 5 000.
PROOF
DOI: 10.37394/232020.2022.2.1
Pavla Jindrová, Viera Pacáková
E-ISSN: 2732-9941
2
Empirical distribution function for u = 3000
Mean = 10092,380952, Std. dev. = 13225,860073, N = 42
Empirical distribution function
Generalized Pareto
95% lower confidence interval
95% upper confidence interval
010000 20000 30000 40000 50000 60000 70000 80000
Total loses
0
10
20
30
40
50
60
70
80
90
100
Percentage (relative freaquency)
Fig.3 GPD fitted to 42 exceedances
of the threshold 3000
Q-Q plot against the GPD of the threshold 3000
010000 20000 30000 40000 50000 60000 70000 80000
Theoretical quantile
0,01 0,75 0,9 0,95
0
10000
20000
30000
40000
50000
60000
70000
80000
Observed quantile
Fig.4 QQ-plot against the GPD fitted
to 42 exceedances of the threshold 3000
Empirical distribution function for u = 5000
Mean = 15585,454545, Std. dev. = 16573,054070, N = 22
Empirical distribution function
Generalized Pareto
95% lower confidence interval
95% upper confidence interval
010000
20000
30000
40000
50000
60000
70000
80000
Total loses
0
10
20
30
40
50
60
70
80
90
100
Percentage (relative freaquency)
Fig.5 GPD fitted to 22 exceedances
of the threshold 5000
Q-Q plot against the GPD of the threshold 5000
010000 20000 30000 40000 50000 60000 70000 80000 90000
Theoretical quantile
0,01 0,75 0,9 0,95
0
10000
20000
30000
40000
50000
60000
70000
80000
Observed quantile
Fig.6 QQ-plot against the GPD fitted
to 22 exceedances of the threshold 5000
Empirical distribution function for u = 8000
Mean = 24800,000000, Std. dev. = 19721,105446, N = 11
Empirical distribution function
Generalized Pareto
95% lower confidence interval
95% upper confidence interval
010000 20000 30000 40000 50000 60000 70000 80000
Total loses
0
10
20
30
40
50
60
70
80
90
100
Percentage (relative feaquency)
Fig.7 GPD fitted to 11 exceedances
of the threshold 8000
Q-Q plot against the GPD of the threshold 8000
010000 20000 30000 40000 50000 60000 70000
Theoretical quantile
0,01 0,5 0,75 0,9 0,95
0
10000
20000
30000
40000
50000
60000
70000
80000
Observed quatile
Fig.8 QQ-plot against the GPD fitted
to 11 exceedances of the threshold 8000
PROOF
DOI: 10.37394/232020.2022.2.1
Pavla Jindrová, Viera Pacáková
E-ISSN: 2732-9941
3
Table 1 Comparisons of estimated GPD for different
thresholds
u = 3 000 u = 5 000 u = 8 000
parametr ξ 2842.322 4195.862 13706.81
parametr σ
-0.67771
-0.75417
-0.19003
p-value 0.850026 0.959389 0.760575
4 Conclusion
We have shown that fitting the generalized Pareto
distribution to natural catastrophic losses which
exceed high thresholds is a useful method for
estimating the tails of loss severity distributions. In
our experience with several insurance datasets we
have found consistently that the generalized Pareto
distribution is a good approximation in the tail.
This is not altogether surprising. As we have
explained, the method has solid foundations in the
mathematical theory of the behaviour of extremes; it
is not simply a question of ad hoc curve fitting.
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PROOF
DOI: 10.37394/232020.2022.2.1
Pavla Jindrová, Viera Pacáková
E-ISSN: 2732-9941
4