15.4 Numerical Comparison of the Infinite Time Approximations

In this section we will illustrate all $ 12$ 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 $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$ and $ a=0.0584$, log-normal with $ \mu= 18.3806$ and $ \sigma= 1.1052$, and Pareto with $ \alpha =3.4081$ and $ \lambda=4.4767\cdot 10^8$ distributions.

The logarithm of the ruin probability as a function of the initial capital $ u$ ranging from USD 0 to $ 50$ 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 $ 2.88\cdot 10^8$, whereas the mean of the fitted log-normal distribution amounts to $ 1.77\cdot 10^8$ and of Pareto distribution to $ 1.86\cdot 10^8$.

Figure 15.2: The logarithm of the exact value of the ruin probability. The mixture of two exponentials (dashed blue line), log-normal (dotted red line), and Pareto (solid black line) clam size distribution.

\includegraphics[width=1.04\defpicwidth]{STFruin16.ps}

Figures 15.3-15.5 describe the relative error of the $ 11$ 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 $ 30\%$. 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.

Figure 15.3: The relative error of the approximations. More effective methods (left panel): the Cramér-Lundberg (solid blue line), exponential (short-dashed brown line), Beekman-Bowers (dotted red line), De Vylder (medium-dashed black line) and 4-moment gamma De Vylder (long-dashed green line) approximations. Less effective methods (right panel): Lundberg (short-dashed red line), Renyi (dotted blue line), heavy traffic (solid magenta line), light traffic (long-dashed green line) and heavy-light traffic (medium-dashed brown line) approximations. The mixture of two exponentials case.

\includegraphics[width=0.7\defpicwidth]{STFruin17a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin17b.ps}

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.

Figure 15.4: The relative error of the approximations. More effective methods (left panel): the exponential (dotted blue line), Beekman-Bowers (short-dashed brown line), heavy-light traffic (solid red line), De Vylder (medium-dashed black line) and 4-moment gamma De Vylder (long-dashed green line) approximations. Less effective methods (right panel): Lundberg (short-dashed red line), heavy traffic (solid magenta line), light traffic (long-dashed green line), Renyi (medium-dashed brown line) and subexponential (dotted blue line) approximations. The log-normal case.
\includegraphics[width=0.7\defpicwidth]{STFruin18a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin18b.ps}

Finally, we take into consideration the Pareto claim size distribution. Figure 15.5 depicts the relative error for $ 9$ 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 $ \alpha =3.4081$. 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.

Figure 15.5: The relative error of the approximations. More effective methods (left panel): the exponential (dotted blue line), Beekman-Bowers (short-dashed brown line), heavy-light traffic (solid red line) and De Vylder (medium-dashed black line) approximations. Less effective methods (right panel): Lundberg (short-dashed red line), heavy traffic (solid magenta line), light traffic (long-dashed green line), Renyi (medium-dashed brown line) and subexponential (dotted blue line) approximations. The Pareto case.
\includegraphics[width=0.7\defpicwidth]{STFruin19a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin19b.ps}