In this section we will illustrate all approximations presented in Section 15.3. To this end we consider three claim amount distributions which were fitted to the PCS catastrophe data in Chapter 13, namely the mixture of two exponential (a running example in Section 15.3) with
,
and
, log-normal with
and
, and Pareto with
and
distributions.
The logarithm of the ruin probability as a function of the initial capital ranging from USD 0 to
billion for the three distributions is depicted in Figure 15.2. In the case of log-normal and Pareto distributions the reference Pollaczek-Khinchin approximation is used. We see that the ruin probability values for the mixture of exponential distributions are much higher than for the log-normal and Pareto distributions. It stems from the fact that the estimated parameters of the mixture result in the mean equal to
, whereas the mean of the fitted log-normal distribution amounts to
and of Pareto distribution to
.
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Figures 15.3-15.5 describe the relative error of the approximations from Sections 15.3.1-15.3.11 with respect to exact ruin probability values in the mixture of two exponentials case and obtained via the Pollaczek-Khinchin approximation in the log-normal and Pareto cases. The relative safety loading is set to
. We
note that for the Monte Carlo method purposes in the Pollaczek-Khinchin approximation we generate 500 blocks of 100000 simulations. First, we consider the mixture of two exponentials case already analysed in Section 15.3. Only the subexponential approximation can not be used for such a claim amount distribution, see Table 15.16. As we can clearly see in Figure 15.3 the Cramér-Lundberg, De Vylder
and 4-moment gamma De Vylder
approximations work extremely well. Furthermore, the heavy traffic, light traffic, Renyi, and Lundberg approximations show a total lack of accuracy and the rest of the methods are only acceptable.
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In the case of log-normally distributed claims, the situation is different, see Figure 15.4. Only results obtained via Beekman-Bowers, De Vylder and 4-moment gamma De Vylder approximations are acceptable. The rest of the approximations are well off target. We also note that all 11 approximations can be employed in the log-normal case except the Cramér-Lundberg one.
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Finally, we take into consideration the Pareto claim size distribution. Figure 15.5 depicts the relative error for approximations. Only the Cramér-Lundberg and 4-moment gamma De Vylder approximations have to excluded as the moment generating function does not exist and the fourth moment is infinite for the Pareto distribution with
. As we see in Figure 15.5 the relative errors for all approximations can not be neglected. There is no unanimous winner among the approximations but we may claim that the exponential approximation gives most accurate results.
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