We conducted empirical studies for the PCS data obtained from Property Claim Services. ISO's Property Claim Services unit is the internationally
recognized authority on insured property losses from catastrophes in the United States, Puerto Rico and the U.S. Virgin Islands. PCS investigates
reported disasters and determines the extent and type of damage, dates of occurrence and geographic areas affected (Burnecki, Kukla, and Weron; 2000). The data,
see Figure 4.1, concerns the US market's loss amounts in USD, which occurred between and
and adjusted for inflation using
the Consumer Price Index provided by the U.S. Department of Labor. Only natural perils like hurricane, tropical storm, wind, flooding, hail,
tornado, snow, freezing, fire, ice and earthquake were taken into consideration. We note that peaks in
Figure 4.1 mark the occurrence of Hurricane Andrew (the 24th August 1992) and the Northridge Earthquake (the 17th January 1994).
In order to calibrate the pricing model we have to fit both the distribution function of the incurred losses and the process
governing
the flow of natural events.
The claim size distributions, especially describing property losses, are usually heavy-tailed. In the actuarial literature for describing such
claims, continuous distributions are often proposed (with the domain
), see Chapter 13. The choice of the distribution is
very important because it influences the bond price. In Chapter 14 the claim amount distributions were fitted to the PCS data depicted in
Figure 4.1. The log-normal, exponential, gamma, Weibull, mixture of two exponentials, Pareto and Burr distributions were analysed. The
parameters were estimated via the Anderson-Darling statistical minimisation procedure. The goodness-of-fit was checked with the help of
Kolmogorov-Smirnov, Kuiper, Cramér-von Mises and Anderson-Darling non-parametric tests. The test statistics were compared with the critical values
obtained through Monte Carlo simulations. The Burr distribution with parameters
,
and
passed all tests. The log-normal distribution with parameters
and
was the next best fit.
A doubly stochastic Poisson process governing the occurrence times of the losses was fitted by Burnecki and Kukla (2003). The simplest case with the
intensity equal to a nonnegative constant
was considered. Studies of the quarterly number of losses and the inter-occurence times of the
catastrophes led to the conclusion that the flow of the events may be described by a Poisson process with an annual intensity of
.
The claim arrival process is also analysed in Chapter 14. The statistical tests applied to the annual waiting times led to a
renewal process.
Finally, the rate function
was fitted and the claim arrival process was
treated as a non-homogeneous Poisson process. Such a choice of the intensity function allows modelling of an annual seasonality present in the
natural catastrophe data.
Baryshnikov, Mayo, and Taylor (1998) proposed an intensity function of the form
. Using the least squares procedure (Ross; 2001), we fitted the cumulative intensity function (mean value function) given by
to the accumulated quarterly
number of PCS losses. We concluded that
,
, and
. This choice of the rate function allows the incorporation of both an
annual cyclic component and a trend which is sometimes observed in natural catastrophe data.
It appears that both the mean squared error (MSE) and the mean absolute error (MAE) favour the rate function . In this case MSE =
and MAE =
, whereas
yields MSE =
and MAE =
. Finally the homogeneous Poisson process with the constant intensity gives MSE =
and MAE =
. All three choices of the intensity function
are illustrated in Figure 4.3, where the accumulated quarterly number of
PCS losses and the mean value functions on the interval
years are depicted. This interval was chosen to best illustrate the differences.
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