Seismic risk management of insurance losses using extreme value theory and copula - Seismic Risk Analysis and Management of Civil Infrastructure Systems - page 755

portfolios probabilistically, by taking seismic loss dependence of the port-
folios into account. They showed that seismic losses from geographically
close portfolios can be nonlinearly correlated in the upper tail. A failure to
account for this correlation might result in biased seismic loss estimation
of the combined portfolio. It was also highlighted that a single parametric
model may not be suffi cient to approximate the entire marginal probability
distribution of seismic loss. In particular, a model that fi ts the major part of
small loss data well may not be suitable for describing the upper tail of large
loss data, which is of greater importance for insurers and reinsurers. A
recent study by Goda and Yoshikawa (2012) demonstrated that physical
effects of spatially correlated ground motions affect the insurer's ruin prob-
ability; adequate portfolio management strategies must be sought to meet
regulatory requirements for stable operation, depending on reserve fund,
characteristics of business related to non-catastrophic risks, and insurance
arrangements.
In this chapter, a statistical approach based on extreme value theory and
copula is proposed to model earthquake risk exposures of an insurance
portfolio. Extreme value theory provides a rational theoretical framework
for analysing large seismic loss data. The issue addressed herein is related
to portfolio aggregation of two separate insurance portfolios in a seismic
region. Special attention is given to the characterisation of the upper right
tail of the combined seismic loss distribution. The approach is based on the
generalised Pareto (GP) model to approximate seismic loss data that exceed
some large threshold value for marginal distribution modelling, and employs
the copula technique to combine two random loss datasets. The GP model
for describing the upper tail of data has well-established statistical founda-
tion based on Pickands-Balkema-de Haan theorem (Coles, 2001). More-
over, the copula technique effectively separates dependence modelling
from marginal distribution modelling based on Sklar theorem (McNeil
et al. , 2005). The above method is especially useful for assessing earthquake
risk exposure for insurers and reinsurers who underwrite high loss coverage.
To demonstrate the method, 2400 hypothetical wood-frame houses located
in Vancouver, BC, which refl ect realistic features of existing wood-frame
houses in Vancouver, are considered. The 2400 houses are grouped into six
portfolios (each consisting of 400 houses); the six portfolios are positioned
at different locations with various inter-portfolio distances to investigate
the effects of geographical separation distance on portfolio aggregation. An
earthquake-engineering-based seismic loss model developed by Goda et al.
(2011) is employed to generate seismic loss samples for the 2400 wood-
frame houses; the seismic loss data are used to develop a statistical model
based on the GP distribution and copula function. It is worth mentioning
that the above set-up is motivated by two facts: (i) the use of numerical
computer models for determining insurance coverage rates and assessing