15.7 Numerical Comparison of the Finite Time Approximations

Now, we illustrate all $ 5$ approximations presented in Section 15.6. As in the infinite time case we consider three claim amount distributions which were best fitted to the catastrophe data in Chapter 13, namely the mixture of two exponentials (a running example in Sections 15.3 and 15.6), log-normal and Pareto distributions. The parameters of the distributions are: $ \beta _1=3.5900\cdot 10^{-10}$, $ \beta _2=7.5088\cdot 10^{-9}$, $ a=0.0584$ (mixture), $ \mu= 18.3806$, $ \sigma= 1.1052$ (log-normal), and $ \alpha =3.4081$, $ \lambda=4.4767\cdot 10^8$ (Pareto). The ruin probability will be depicted as a function of $ u$, ranging from USD 0 to $ 30$ billion, with fixed $ T=10$ or with fixed value of $ u=20$ billion USD and varying $ T$ from 0 to $ 20$ years. The relative safety loading is set to $ 30\%$. Figures has the same form of output. In the left panel, the exact ruin probability values obtained via Monte Carlo simulations are presented. The right panel describes the relative error with respect to the exact values. We also note that for the Monte Carlo method purposes we generated $ 50$ x $ 10000$ simulations.

Figure 15.6: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). The Segerdahl (short-dashed blue line), diffusion (dotted red line), corrected diffusion (solid black line) and finite time De Vylder (long-dashed green line) approximations. The mixture of two exponentials case with $ T$ fixed and $ u$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin26a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin26b.ps}

First, we consider the mixture of two exponentials case. As we can see in Figures 15.6 and 15.7 the diffusion approximation almost for all values of $ u$ and $ T$ gives highly incorrect results. Segerdahl and corrected diffusion approximations yield similar error, which visibly decreases when the time horizon gets bigger. The finite time De Vylder method is a unanimous winner and always gives the error below $ 10\%$.

Figure 15.7: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). The Segerdahl (short-dashed blue line), diffusion (dotted red line), corrected diffusion (solid black line) and finite time De Vylder (long-dashed green line) approximations. The mixture of two exponentials case with $ u$ fixed and $ T$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin27a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin27b.ps}

In the case of log-normally distributed claims, we can only apply two approximations: diffusion and finite time De Vylder ones, cf. Table 15.23. Figures 15.8 and 15.9 depict the exact ruin probability values obtained via Monte Carlo simulations and the relative error with respect to the exact values. Again, the finite time De Vylder approximation works much better than the diffusion one.

Figure 15.8: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). Diffusion (dotted red line) and finite time De Vylder (long-dashed green line) approximations. The log-normal case with $ T$ fixed and $ u$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin28a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin28b.ps}

Figure 15.9: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). Diffusion (dotted red line) and finite time De Vylder (long-dashed green line) approximations. The log-normal case with $ u$ fixed and $ T$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin29a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin29b.ps}

Finally, we take into consideration the Pareto claim size distribution. Figures 15.10 and 15.11 depict the exact ruin probability values and the relative error with respect to the exact values for the diffusion and finite time De Vylder approximations. We see that now we cannot claim which approximation is better. The error strongly depends on the values of $ u$ and $ T$. We may only suspect that a combination of the two methods could give interesting results.

Figure 15.10: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). Diffusion (dotted red line) and finite time De Vylder (long-dashed green line) approximations. The Pareto case with $ T$ fixed and $ u$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin30a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin30b.ps}

Figure 15.11: The exact ruin probability obtained via Monte Carlo simulations (left panel), the relative error of the approximations (right panel). Diffusion (dotted red line) and finite time De Vylder (long-dashed green line) approximations. The Pareto case with $ u$ fixed and $ T$ varying.
\includegraphics[width=0.7\defpicwidth]{STFruin31a.ps}\includegraphics[width=0.7\defpicwidth]{STFruin31b.ps}