Civil Engineering Reference
In-Depth Information
catastrophe risk exposure is popular and inevitable in practice owing to the
lack of empirical loss/claim data; and (ii) information regarding seismic loss
dependence for rare events is critically missing. In Section 28.2, brief sum-
maries of the peak-over-threshold (POT) method and the copula method
are given. Subsequently, salient features of the earthquake-engineering-
based seismic loss model are discussed in Section 28.3. In Section 28.4, the
developed statistical models are applied to calculate several risk measures
for a reinsurance portfolio, such as value-at-risk (VaR) and tail value-at-risk
(TVaR). The results are compared with the estimates based on the earth-
quake-engineering-based model to investigate the accuracy of the proposed
approach.
28.2 Statistical modelling of extreme data
28.2.1 Extreme value theory and marginal
distribution modelling
The occurrence of extremely large losses is of particular concern for insur-
ers and reinsurers who undertake catastrophe earthquake risk. Insurers and
reinsurers need to deal with very rare events, estimate their potential loss,
and design a portfolio to meet regulatory as well as business requirements.
Therefore, accurate assessment of risk exposure due to extreme events is
the primary focus for portfolio analysts and managers. Moreover, it is desir-
able to have a tool that enables quick assessment for sensitivity analysis.
Extreme value theory offers a rational theoretical framework for analys-
ing large seismic loss data. A standard method in extreme value theory is
the block maxima model, which focuses on the maximum data during a
specifi ed period only (e.g. annual maximum fl ood volume). Based on
Fisher-Tippett-Gnedenko theorem, such data can be modelled as the gen-
eralised extreme value (GEV) distribution. The distribution function of the
GEV model is given by:
(
)
1
ξμβ
(
)
−
ξ
⎧
exp
exp
−+ −
⎣
1
x
⎦
ξ
≠
0
0
⎪
⎪
()
=
Fx
,
[28.1]
(
[
(
)
]
)
−−−
exp
x
μβ
ξ
=
where
ξ
,
μ
, and
β
are the GEV model parameters. The shape parameter
ξ
identifi es a suitable type of the three limiting extreme value distributions:
ξ
< 0 indicate the Frechet distribution, Gumbel distribution,
and Weibull distribution, respectively. Given data, the model parameters
can be estimated based on the maximum likelihood method.
Another useful class of extreme value models is the POT model; it con-
centrates on data exceeding a high threshold value and thus uses data more
effi ciently than the block maxima model. Under fairly general conditions
that are applicable to the majority of common probability distributions,
> 0,
ξ
=
0, and
ξ
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