In this section we illustrate some of the methods described earlier in the chapter. We conduct the analysis for two datasets. The first is the PCS (Property Claim Services, see Insurance Services Office Inc. (ISO) web site: www.iso.com/products/2800/prod2801.html)/A> dataset covering losses resulting from natural catastrophic events in USA that occurred between 1990 and 1999. The second is the Danish fire losses dataset, which concerns major fire losses in Danish Krone (DKK) that occurred between 1980 and 1990 and were recorded by Copenhagen Re. Here we consider only losses in profits. The overall fire losses were analyzed by Embrechts, Klüppelberg, and Mikosch (1997).
The Danish fire losses dataset has been already adjusted for inflation. However, the PCS dataset consists of raw data. Since the data have been collected over a considerable period of time, it is important to bring the values onto a common basis by means of a suitably chosen index. The choice of the index depends on the line of insurance. For example, an index of the cost of construction prices may be suitable for fire and other property insurance, an earnings index for life and accident insurance, and a general price index may be appropriate when a single index is required for several lines or for the whole portfolio. Here we adjust the PCS dataset using the Consumer Price Index provided by the U.S. Department of Labor. Note, that the same raw catastrophe data, however, adjusted using the discount window borrowing rate that refers to the simple interest rate at which depository institutions borrow from the Federal Reserve Bank of New York was analyzed by Burnecki, Härdle, and Weron (2004). A related dataset containing the national and regional PCS indices for losses resulting from catastrophic events in USA was studied by Burnecki, Kukla, and Weron (2000).
As suggested in the proceeding section we first look for the appropriate shape of the distribution. To this end we plot the empirical mean excess functions for the analyzed data sets, see Figure 13.7. Both in the case of PCS natural catastrophe losses and Danish fire losses the data show a super-exponential pattern suggesting a log-normal, Pareto or Burr distribution as most adequate for modeling. Hence, in the sequel we calibrate these three distributions.
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We apply two estimation schemes: maximum likelihood and statistic minimization. Out of the three fitted distributions only the log-normal has closed form expressions for the maximum likelihood estimators. Parameter calibration for the remaining distributions and the
minimization scheme is carried out via a simplex numerical optimization routine. A limited simulation study suggests that the
minimization scheme tends to return lower values of all edf test statistics than maximum likelihood estimation. Hence, it is exclusively used for further analysis.
The results of parameter estimation and hypothesis testing for the PCS loss amounts are presented in Table 13.1. The Burr
distribution with parameters
,
, and
yields the best results and passes all tests at the
2.5% level. The log-normal distribution with parameters
and
comes in second, however, with an unacceptable fit as
tested by the Anderson-Darling statistic. As expected, the remaining distributions presented in Section 13.3 return even worse fits.
Thus we suggest to choose the Burr distribution as a model for the PCS loss amounts. In the left panel of Figure 13.8 we present the empirical and
analytical limited expected value functions for the three fitted distributions. The plot justifies the choice of the Burr distribution.
The results of parameter estimation and hypothesis testing for the Danish fire loss amounts are presented in Table 13.2. The
log-normal distribution with parameters
and
returns the best results. It is the only distribution that passes any of
the four applied tests (
,
,
, and
) at a reasonable level. The Burr and Pareto laws yield worse fits as the tails of the edf are
lighter than power-law tails. As expected, the remaining distributions presented in Section 13.3 return even worse fits. In the right panel of Figure
13.8 we depict the empirical and analytical limited expected value functions for the three fitted distributions. Unfortunately, no definitive conclusions can be drawn regarding the choice of the distribution. Hence, we
suggest to use the log-normal distribution as a model for the Danish fire loss amounts.
Distributions: | log-normal | Pareto | Burr | |
Parameters: | ![]() |
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|
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||
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Tests: | ![]() |
0.0440 | 0.1049 | 0.0366 |
(0.033) | (![]() |
(0.077) | ||
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0.0786 | 0.1692 | 0.0703 | |
(0.022) | (![]() |
(0.038) | ||
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0.1353 | 0.7042 | 0.0626 | |
(0.006) | (![]() |
(0.059) | ||
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1.8606 | 6.1160 | 0.5097 | |
(![]() |
(![]() |
(0.027) |
Distributions: | log-normal | Pareto | Burr | |
Parameters: | ![]() |
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|
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||
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Tests: | ![]() |
0.0381 | 0.0471 | 0.0387 |
(0.008) | (![]() |
(![]() |
||
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0.0676 | 0.0779 | 0.0724 | |
(0.005) | (![]() |
(![]() |
||
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0.0921 | 0.2119 | 0.1117 | |
(0.049) | (![]() |
(0.007) | ||
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0.7567 | 1.9097 | 0.6999 | |
(0.024) | (![]() |
(0.005) |