Seismic risk management of insurance losses using extreme value theory and copula - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 775

simulated samples from the fi tted statistical models (both normal and HRT
copulas) match fairly well with the input samples; qualitatively, fi tting of the
HRT copula is slightly superior to the normal copula (however, visually, for
the considered case, both models perform very well). Note that signifi cant
deviation around the annual probability level of 0.003 is due to the adopted
threshold value for determining the POT data. Therefore, for right tail data,
accuracy of the proposed model using the GP distribution and copula in
approximating the results from the earthquake-engineering-based model is
judged to be good. To further demonstrate the applicability of the proposed
method, the same exercise (i.e. marginal and copula fi tting and simulation
from the fi tted statistical model) is carried out by using insurance seismic
loss samples by applying Equation [28.16] with D 1.0.

The results are also included in Fig. 28.10; the proposed statistical method
performs well for the insurance input data. Generally, the above conclusions
are applicable to other cases (e.g. various combinations of groups with dif-
ferent separation distances as well as different insurance policy arrange-
ments). It is noteworthy that the simulation from the fi tted statistical model
is very quick (less than one minute), in comparison with the earthquake-
engineering-based model (several days). Therefore, once the accuracy of
the method is verifi ed, the proposed method can serve as a viable alterna-
tive to conduct various sensitivity analyses.

= 0.1, C = 0.5, and γ =

28.4.3 Assessment of reinsurance risk exposure

The developed statistical models of bivariate insurance seismic loss data
can be useful for evaluating risk measures related to reinsurance. For illus-
tration, consider that a reinsurer covers insurer's seismic loss layer between
20 and 70 million CAD from the combined portfolio of Groups 2 and 3.
The insurance arrangement is set to: D 1.0 (i.e. insur-

ance seismic loss case in Fig. 28.10a). The annual probabilities of exceedance
for the loss threshold values are 0.00135 and 0.00007, respectively (Fig.
28.10a). The annual expected payment for the reinsurance layer is calcu-
lated as 15.474, 12.854, and 13.367 million CAD for the original data, normal
copula case, and HRT copula case, respectively. Similarly, the standard
deviation of the annual reinsurance payment is calculated as 11.360, 11.251,
and 11.743 million CAD for the original data, normal copula case, and HRT
copula case, respectively. These quantities are the fundamental input in
determining reinsurance premium. Overall, the calculated statistics for the
normal and HRT copulas are comparable to those for the original data, and
the statistical seismic loss models can approximate the original data well.
VaR and TVaR are popular and practical risk measures for insurance
portfolio management. VaR is the fractile corresponding to small probabil-
ity of exceedance, whereas TVaR is the conditional expected value above

= 0.1, C = 0.5, and γ =