# Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem

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## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Main Results

#### 3.1. Both Parameters Unknown

#### 3.2. Mean Claim-Size Parameter Unknown

#### 3.3. Safety Loading Parameter Unknown

## 4. Specific Results

## 5. Numerical Experiments

## 6. Final Comments

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Continuous Distributions

- Bivariate distribution proposed by Gómez-Déniz et al. (2014).$$\begin{array}{c}\hfill f(x,y)=\frac{{\sigma}^{\gamma}}{B(\alpha -\gamma ,\beta )\Gamma (\gamma )}{x}^{\alpha -1}{(1-x)}^{\beta -1}{y}^{\gamma -1}exp(-\sigma xy),\end{array}$$$$\begin{array}{c}\hfill \Gamma (z)={\int}_{0}^{\infty}{t}^{z-1}{e}^{-t}\phantom{\rule{0.166667em}{0ex}}dt\end{array}$$$$\begin{array}{c}\hfill B({z}_{1},{z}_{2})={\int}_{0}^{1}{t}^{{z}_{1}-1}{(1-t)}^{{z}_{2}-1}\phantom{\rule{0.166667em}{0ex}}dt.\end{array}$$$$\begin{array}{c}\hfill {f}_{X}(x)=\frac{1}{B(\alpha -\gamma ,\beta )}{x}^{\alpha -\gamma -1}{(1-x)}^{\beta -1}.\end{array}$$$$\begin{array}{c}\hfill {f}_{Y}(y)=\frac{{\sigma}^{\gamma}\Gamma (\alpha +\beta -\gamma )}{\Gamma (\alpha +\beta )B(\alpha -\gamma ,\gamma )}{y}^{\gamma -1}{\phantom{\rule{0.166667em}{0ex}}}_{1}{F}_{1}(\alpha ,\alpha +\beta ,-\sigma y),\end{array}$$$$\begin{array}{c}\hfill {}_{1}{F}_{1}(m,n,z)={\displaystyle \sum _{k=0}^{\infty}}\frac{{(m)}_{k}{z}^{k}}{{(n)}_{k}k!},\end{array}$$Using Kummer’s first theorem we have that (A3) can be rewritten as$$\begin{array}{c}\hfill {f}_{Y}(y)=\frac{{\sigma}^{\gamma}\Gamma (\alpha +\beta -\gamma )}{\Gamma (\alpha +\beta )B(\alpha -\gamma ,\gamma )}{y}^{\gamma -1}{e}^{-\sigma y}{\phantom{\rule{0.166667em}{0ex}}}_{1}{F}_{1}(\beta ,\alpha +\beta ,\sigma y).\end{array}$$$$\begin{array}{c}\hfill cov(X,Y)=\frac{\beta \gamma}{\sigma (\alpha +\beta -\gamma )(\gamma -\alpha +1)},\end{array}$$$$cov(X,Y)\left(\right)open="\{"\; close>\begin{array}{cc}0& \mathrm{if}\phantom{\rule{0.54em}{0ex}}0\alpha -\gamma 1,\\ 0& \mathrm{if}\phantom{\rule{0.54em}{0ex}}\alpha -\gamma 1.\end{array}$$
- Lindley distribution.$$\begin{array}{ccc}\hfill p(y)& =& \frac{{\beta}^{2}}{1+\beta}(1+y)exp\left(\right)open="("\; close=")">-\beta y,\phantom{\rule{1.em}{0ex}}y0,\phantom{\rule{0.166667em}{0ex}}\beta 0,\hfill \end{array}$$
- Inverse Gaussian distribution.$$\begin{array}{ccc}\hfill p(y)& =& \sqrt{\frac{\beta}{2\pi {y}^{3}}}exp\left(\right)open="["\; close="]">-\frac{\beta}{2{\alpha}^{2}y}{(y-\alpha )}^{2},\phantom{\rule{1.em}{0ex}}y0,\phantom{\rule{0.277778em}{0ex}}\alpha 0,\phantom{\rule{0.277778em}{0ex}}\beta 0,\hfill \end{array}$$
- The confluent hypergeometric distribution.$$\begin{array}{ccc}\hfill p(y)& =& \frac{{y}^{\alpha -1}{(1-y)}^{\beta -1}{e}^{-\sigma y}}{B(\alpha ,\beta ){\phantom{\rule{0.277778em}{0ex}}}_{1}{F}_{1}(\alpha ,\alpha +\beta ,-\sigma )},\phantom{\rule{1.em}{0ex}}0<y<1,\phantom{\rule{0.277778em}{0ex}}\alpha >0,\phantom{\rule{0.277778em}{0ex}}\beta >0,\phantom{\rule{0.277778em}{0ex}}\sigma \ge 0,\hfill \\ \hfill M(t)& =& \frac{{}_{1}{F}_{1}(\alpha ,\alpha +\beta ,t-\sigma )}{{}_{1}{F}_{1}(\alpha ,\alpha +\beta ,-\sigma )},\hfill \end{array}$$$$\begin{array}{c}\hfill {}_{1}{F}_{1}(a;c;x)=\frac{\Gamma (c)}{\Gamma (a)\Gamma (c-a)}{\int}_{0}^{1}{z}^{a-1}{(1-z)}^{c-a-1}{e}^{xz}dz,\phantom{\rule{1.em}{0ex}}c>a>x.\end{array}$$

## References

- Albrecher, Hansjörg, Corina Constantinescu, and Stéphane Loisel. 2011. Explicit ruin formulas for models with dependence among risks. Insurance: Mathematics and Economics 48: 265–70. [Google Scholar] [CrossRef]
- Asmussen, Søren, and Hansjörg Albrecher. 2010. Ruin Probabilities. Hackensack: World Scientific. [Google Scholar]
- Dickson, David C. M. 2005. Insurance Risk and Ruin. Cambridge: Cambridge University Press. [Google Scholar]
- Dickson, David C. M., and Howard R. Waters. 1992. The probability and severity of ruin in finite and infinite time. ASTIN Bulletin 22: 178–90. [Google Scholar] [CrossRef]
- Egami, Masahiko, and Kazutoshi Yamazaki. 2014. Phase-type fitting of scale functions for spectrally negative Lévy processes. Journal of Computaional and Applied Mathematics 264: 1–22. [Google Scholar] [CrossRef]
- Garcia, Jorge M. A. 2005. Explicit solutions for survival probabilities in the classical risk model. ASTIN Bulletin 35: 113–30. [Google Scholar] [CrossRef]
- Gerber, Hans U. 1979. An Introduction to Mathematical Risk Theory. Philadelphia: Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania. [Google Scholar]
- Gerber, Hans U., Marc J. Goovaerts, and Rob Kaas. 1987. On the probability and severity of ruin. ASTIN Bulletin 17: 151–63. [Google Scholar] [CrossRef]
- Ghitany, Mohamed E., Barbra Atieh, and Saralees Nadarajah. 2008. Lindley distribution and its application. Mathematics and Computer in Simulation 78: 493–506. [Google Scholar] [CrossRef]
- Gómez-Déniz, Emilio, Mohamed E. Ghitany, and Dhaifalla Al-Mutari. 2016. A note on probability and severity of ruin for generalized Lindley claim size distribution. Anales del Instituto de Actuarios Españoles 3: 25–40. [Google Scholar]
- Gómez-Déniz, Emilio, Agustín Hernández-Bastida, and M. Pilar Fernández-Sánchez. 2014. Computing credibility bonus-malus premiums using the aggregate claims distribution. Hacettepe Journal of Mathematics and Statistics 43: 1047–61. [Google Scholar]
- Gordy, Michael B. 1998. Computationally convenient distributional assumptions for common-value auctions. Computational Economics 12: 61–78. [Google Scholar] [CrossRef]
- Grandell, Jan. 1990. Aspects of Risk Theory. New York: Springer. [Google Scholar]
- Heilmann, Wolf-Rüdiger. 1989. Decision theoretic foundations of credibility theory. Insurance: Mathematics and Economics 8: 75–95. [Google Scholar] [CrossRef]
- Kou, Steven G., and Hui Wang. 2003. First passage times of a jump diffusion process. Advances in Applied Probability 35: 504–31. [Google Scholar] [CrossRef]
- Kuznetzov, Alexey, Andreas E. Kyprianou, and Juan C. Pardo. 2012a. Meromorphic Lévy processes and their fluctuation identities. The Annals of Applied Probability 22: 1101–35. [Google Scholar] [CrossRef]
- Kuznetzov, Alexey, Andreas E. Kyprianou, and Victor Rivero. 2012b. The Theory of Scale Functions for Spectrally Negative Lévy Processes in Lévy Matters II. Berlin/Heidelberg: Springer. [Google Scholar]
- Kyprianou, Andreas E. 2014. Fuluctuations of Lévy Processes with Applications: Introductory Lectures. Berlin/Heidelberg: Springer Science & Business Media. [Google Scholar]
- Lindley, Dennis. 1958. Fiducial distributions and Bayes’s theorem. Journal of the Royal Statistical Society Series B 20: 102–7. [Google Scholar]
- Nadarajah, Saralees. 2005. Exponentiated beta distributions. Computers & Mathematics with Applications 49: 1029–35. [Google Scholar]
- Politis, Konstadinos. 2006. A functional approach for ruin probabilities. Stochastic Models 22: 509–36. [Google Scholar] [CrossRef]
- Ramsay, Colin M. 2003. A solution to the ruin problem for Pareto distributions. Insurance: Mathematics and Economics 33: 109–16. [Google Scholar] [CrossRef]
- Ramsay, Colin M., and Miguel A. Usabel. 1997. Calculating ruin probabilities via product integration. ASTIN Bulletin 27: 263–71. [Google Scholar] [CrossRef]
- Rolski, Tomasz, Hanspeter Schmidli, Volker Schmidt, and Jozef L. Teugels. 1999. Stochastic Processes for Insurance and Finance. Hoboken: John Wiley & Sons. [Google Scholar]
- Sarabia, José María, Emilio Gómez-Déniz, Faustino Prieto, and Vanesa Jordá. 2018. Aggregation of dependent risks in mixtures of exponential distributions and extensions. ASTIN Bulletin 48: 1079–107. [Google Scholar] [CrossRef]
- Schmidli, Hanspeter. 1999. On the distribution of the surplus prior and at ruin. ASTIN Bulletin 29: 227–44. [Google Scholar] [CrossRef]
- Seal, Hilary L. 1980. Survival probabilities based on Pareto claim distributions. ASTIN Bulletin 11: 61–71. [Google Scholar] [CrossRef]
- Tamturk, Muhsin, and Sergey Utev. 2018. Ruin probability via quantum mechanics approach. Insurance: Mathematics and Economics 79: 69–74. [Google Scholar] [CrossRef]
- Wei, Li, and Hai-liang Yang. 2004. Explicit expressions for the ruin probabilities of Erlang risk process with Pareto individual claim. Acta Mathematicae Applicatae Sinica 20: 495–506. [Google Scholar] [CrossRef]
- Yamazaki, Kazutoshi. 2017. Phase-type approximation of the gerber-shiu function. The Operations Research Society of Japan 60: 337–52. [Google Scholar] [CrossRef]

1 |

Distribution on$\Pi $ | Mixture Ruin Function | Mixture Severity of Ruin |

Bivariate | ${\tilde{\psi}}_{BV}(u)\equiv \frac{\beta}{\alpha +\beta -\gamma}{\left(\right)}^{\frac{\sigma}{\sigma +u}}\gamma $ | $\begin{array}{cc}\hfill {\tilde{G}}_{BV}(u,y)=& {\tilde{\psi}}_{BV}(u)\left(\right)open="["\; close>1-{\left(\right)}^{\frac{\sigma +u}{y}}\gamma \hfill & \frac{\Gamma (\alpha +\beta -\gamma +1)\Gamma (\alpha )}{\Gamma (\alpha -\gamma )\Gamma (\alpha +\beta +1)}\end{array}$ |

Distribution on$\Lambda $ | Mixture Ruin Function | Mixture Severity of Ruin |

Gamma | $\tilde{\psi}}_{G}(u)=\frac{{(1+\theta )}^{\alpha -1}}{{\left(\right)}^{1}$ | ${G}_{G}(u,y)={\tilde{\psi}}_{G}(u)\left(\right)open="\{"\; close="\}">1-{\left(\right)}^{\frac{1+\theta (1+\beta u)}{(1+\theta )(1+\beta y)+\theta \beta u}}\alpha$ |

Exponential | $\tilde{\psi}}_{E}(u)=\frac{1}{1+\theta (1+\beta u)$ | $G}_{E}(u,y)={\tilde{\psi}}_{E}(u)\left(\right)open="["\; close="]">1-\frac{1+\theta (1+\beta u)}{(1+\theta )(1+\beta y)+\theta \beta u$ |

Lindley | $\tilde{\psi}}_{L}(u)=\frac{{\beta}^{2}(1+\beta +\theta (1+\beta +u))}{(1+\beta ){(\beta +\theta (\beta +u))}^{2}$ | ${G}_{L}(u,y)={\psi}_{L}(u)\left(\right)open="\{"\; close="\}">1-{\mathcal{G}}_{L}(\beta ,\alpha ,\theta ,u,y)$ |

Inverse Gaussian | $\tilde{\psi}}_{IG}(u)=\frac{1}{1+\theta}exp\left(\right)open="\{"\; close="\}">\frac{1}{\alpha}\left(\right)open="["\; close="]">\beta -\sqrt{\frac{\beta (\beta (1+\theta )+2{\alpha}^{2}\theta u)}{(1+\theta )}$ | ${G}_{IG}(u,y)={\tilde{\psi}}_{IG}(u)\left(\right)open="\{"\; close="\}">1-{\mathcal{G}}_{IG}(\beta ,\alpha ,\theta ,u,y)$ |

Distribution on${\rm Y}$ | Mixture Ruin Function | Mixture Severity of Ruin |

Uniform | ${\tilde{\psi}}_{U}(u)=\frac{\lambda u+{e}^{-\lambda u}-1}{{\lambda}^{2}{u}^{2}},\phantom{\rule{0.54em}{0ex}}u>0$ | ${G}_{U}(u,y)={\tilde{\psi}}_{U}(u)(1-exp(-\lambda y))$ |

CH | $\tilde{\psi}}_{CH}(u)=\frac{\beta}{\alpha +\beta}\frac{{}_{1}{F}_{1}(\alpha ,\alpha +\beta +1,-\sigma -\lambda u)}{{}_{1}{F}_{1}(\alpha ,\alpha +\beta ,-\sigma )$ | ${G}_{CH}(u,y)={\tilde{\psi}}_{CH}(u)(1-exp(-\lambda y))$ |

Beta | ${\tilde{\psi}}_{B}(u)=\frac{\beta}{\alpha +\beta}\phantom{\rule{0.277778em}{0ex}}{}_{1}{F}_{1}(\alpha ;\alpha +\beta +1;-\lambda u)$ | ${G}_{B}(u,y)={\tilde{\psi}}_{B}(u)(1-exp(-\lambda y))$ |

Power | ${\tilde{\psi}}_{P}(u)=\frac{1}{\alpha +1}\phantom{\rule{0.277778em}{0ex}}{}_{1}{F}_{1}(\alpha ;\alpha +2;-\lambda u)$ | ${G}_{B}(u,y)={\tilde{\psi}}_{P}(u)(1-exp(-\lambda y))$ |

arcsin | ${\tilde{\psi}}_{AS}(u)=\frac{1}{2}{\phantom{\rule{0.277778em}{0ex}}}_{1}{F}_{1}(1/2,2,-\lambda u)$ | ${G}_{AS}(u,y)={\tilde{\psi}}_{AS}(u)(1-exp(-\lambda y))$ |

**Table 2.**Ruin probabilities for exponential claims with $\lambda =1$ and different values of the premium loading factor $\theta $.

u | $\mathit{\theta}=0.10$ | $\mathit{\theta}=0.25$ | $\mathit{\theta}=0.50$ | $\mathit{\theta}=0.75$ | $\mathit{\theta}=1.00$ |
---|---|---|---|---|---|

1 | 0.830092 | 0.654985 | 0.477688 | 0.372251 | 0.303265 |

2 | 0.757957 | 0.536256 | 0.342278 | 0.242499 | 0.183940 |

3 | 0.692091 | 0.439049 | 0.245253 | 0.157973 | 0.111565 |

4 | 0.631949 | 0.359463 | 0.175731 | 0.102910 | 0.067667 |

5 | 0.577033 | 0.294304 | 0.125917 | 0.067039 | 0.041042 |

6 | 0.526889 | 0.240955 | 0.090223 | 0.043672 | 0.024893 |

7 | 0.481103 | 0.197278 | 0.064648 | 0.028449 | 0.015098 |

8 | 0.439296 | 0.161517 | 0.046322 | 0.018533 | 0.009157 |

9 | 0.401121 | 0.132239 | 0.033191 | 0.012073 | 0.005554 |

10 | 0.366264 | 0.108268 | 0.023782 | 0.007865 | 0.003368 |

50 | 0.009650 | 0.000036 | 3.850 × 10^{−8} | 2.82 × 10^{−10} | 6.94 × 10^{−12} |

100 | 0.000102 | 1.640 × 10^{−9} | 2.22 × 10^{−15} | 1.39 × 10^{−19} | 9.64 × 10^{−23} |

${\tilde{\mathit{\psi}}}_{\mathit{BV}}(\mathit{u})$ | ${\tilde{\mathit{\psi}}}_{\mathit{G}}(\mathit{u})$,$\mathit{\alpha}=2,\phantom{\rule{0.277778em}{0ex}}=\mathit{\beta}=1/2$ | |||||||||

$\mathit{u}$ | $\mathit{\alpha}=\mathbf{1.21}$ $\mathit{\beta}=\mathbf{11.79}$ | $\mathit{\alpha}=\mathbf{1.09}$ $\mathit{\beta}=\mathbf{4.26}$ | $\mathit{\alpha}=\mathbf{1.06}$ $\mathit{\beta}=\mathbf{2.06}$ | $\mathit{\alpha}=\mathbf{1.05}$ $\mathit{\beta}=\mathbf{1.36}$ | $\mathit{\alpha}=\mathbf{1.04}$ $\mathit{\beta}=\mathbf{1.01}$ | $\mathit{\theta}=\mathbf{0.10}$ | $\mathit{\theta}=\mathbf{0.25}$ | $\mathit{\theta}=\mathbf{0.50}$ | $\mathit{\theta}=\mathbf{0.75}$ | $\mathit{\theta}=\mathbf{1.00}$ |

1 | 0.909091 | 0.800000 | 0.666667 | 0.571429 | 0.500000 | 0.831758 | 0.661157 | 0.489796 | 0.387543 | 0.320000 |

2 | 0.898100 | 0.790328 | 0.658606 | 0.564520 | 0.493955 | 0.763889 | 0.555556 | 0.375000 | 0.280000 | 0.222222 |

3 | 0.890382 | 0.783536 | 0.652947 | 0.559669 | 0.489710 | 0.704000 | 0.473373 | 0.296296 | 0.211720 | 0.163265 |

4 | 0.884442 | 0.778309 | 0.648590 | 0.555935 | 0.486443 | 0.650888 | 0.408163 | 0.240000 | 0.165680 | 0.125000 |

5 | 0.879617 | 0.774063 | 0.645053 | 0.552902 | 0.483789 | 0.603567 | 0.355556 | 0.198347 | 0.133175 | 0.098765 |

6 | 0.526889 | 0.240955 | 0.090223 | 0.043672 | 0.024893 | 0.561224 | 0.312500 | 0.166667 | 0.109375 | 0.080000 |

7 | 0.875559 | 0.770492 | 0.642076 | 0.550351 | 0.481557 | 0.523187 | 0.276817 | 0.142012 | 0.091428 | 0.066116 |

8 | 0.872058 | 0.767411 | 0.639509 | 0.548151 | 0.479632 | 0.488889 | 0.246914 | 0.122449 | 0.077562 | 0.055555 |

9 | 0.868982 | 0.764704 | 0.637254 | 0.546217 | 0.477940 | 0.457856 | 0.221607 | 0.106667 | 0.066627 | 0.047337 |

10 | 0.866240 | 0.762291 | 0.635243 | 0.544494 | 0.476432 | 0.429688 | 0.200000 | 0.093750 | 0.057851 | 0.040816 |

50 | 0.824438 | 0.725506 | 0.604588 | 0.518218 | 0.453441 | 0.084876 | 0.022222 | 0.007653 | 0.004164 | 0.002743 |

100 | 0.807942 | 0.710989 | 0.592491 | 0.507849 | 0.444368 | 0.029562 | 0.006611 | 0.002136 | 0.001136 | 0.000739 |

${\mathit{\psi}}_{\mathit{L}}(\mathit{u})$,$\mathit{\beta}=\sqrt{\mathbf{2}}$ | ${\mathit{\psi}}_{\mathit{IG}}(\mathit{u})$,$\mathit{\alpha}=\mathbf{1},\mathit{\beta}=\mathbf{1}$ | |||||||||

$\mathit{u}$ | $\mathit{\theta}=\mathbf{0.10}$ | $\mathit{\theta}=\mathbf{0.25}$ | $\mathit{\theta}=\mathbf{0.50}$ | $\mathit{\theta}=\mathbf{0.75}$ | $\mathit{\theta}=\mathbf{1.00}$ | $\mathit{\theta}=\mathbf{0.10}$ | $\mathit{\theta}=\mathbf{0.25}$ | $\mathit{\theta}=\mathbf{0.50}$ | $\mathit{\theta}=\mathbf{0.75}$ | $\mathit{\theta}=\mathbf{1.00}$ |

1 | 0.832812 | 0.664911 | 0.496879 | 0.396288 | 0.329431 | 0.833247 | 0.666071 | 0.498347 | 0.397569 | 0.330430 |

2 | 0.767518 | 0.566663 | 0.392956 | 0.300174 | 0.242641 | 0.768697 | 0.568483 | 0.393376 | 0.299056 | 0.240461 |

3 | 0.711083 | 0.492388 | 0.323521 | 0.240273 | 0.190909 | 0.712858 | 0.493426 | 0.320614 | 0.234702 | 0.183940 |

4 | 0.661881 | 0.434490 | 0.274170 | 0.199681 | 0.156854 | 0.663937 | 0.433616 | 0.267050 | 0.189374 | 0.145262 |

5 | 0.618656 | 0.388225 | 0.237447 | 0.170497 | 0.132858 | 0.620637 | 0.384737 | 0.226020 | 0.155878 | 0.117345 |

6 | 0.580416 | 0.350490 | 0.209139 | 0.148573 | 0.115094 | 0.581987 | 0.344021 | 0.193683 | 0.130277 | 0.096433 |

7 | 0.546376 | 0.319180 | 0.186697 | 0.131536 | 0.101441 | 0.547240 | 0.309591 | 0.167639 | 0.110210 | 0.080333 |

8 | 0.515901 | 0.292817 | 0.168498 | 0.117936 | 0.090635 | 0.515814 | 0.280121 | 0.146303 | 0.094168 | 0.067667 |

9 | 0.488477 | 0.270341 | 0.153459 | 0.106840 | 0.081879 | 0.487240 | 0.254640 | 0.128578 | 0.081137 | 0.057531 |

10 | 0.463681 | 0.250967 | 0.140834 | 0.097622 | 0.074645 | 0.461140 | 0.232419 | 0.113683 | 0.070412 | 0.049302 |

50 | 0.147573 | 0.063149 | 0.032234 | 0.021630 | 0.016275 | 0.103113 | 0.022243 | 0.005169 | 0.002066 | 0.001075 |

100 | 0.078514 | 0.032393 | 0.016351 | 0.010934 | 0.008213 | 0.030961 | 0.003601 | 0.000484 | 0.000140 | 0.000058 |

${\mathit{\psi}}_{\mathit{B}}(\mathit{u})$ | ${\mathit{\psi}}_{\mathit{CH}}(\mathit{u})$ | |||||||||

$\mathit{u}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{9}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{3}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{0.33}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{1.1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{9}$ $\mathit{\sigma}=\mathbf{0.1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{3}$ $\mathit{\sigma}=\mathbf{0.1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{1}$ $\mathit{\sigma}=\mathbf{0.1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{0.33}$ $\mathit{\sigma}=\mathbf{0.1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{0.1}$ $\mathit{\sigma}=\mathbf{0.1}$ |

1 | 0.824511 | 0.621830 | 0.367879 | 0.167418 | 0.389129 | 0.826698 | 0.631616 | 0.395369 | 0.224775 | 0.651310 |

2 | 0.759969 | 0.527252 | 0.283834 | 0.120673 | 0.302500 | 0.762419 | 0.536637 | 0.306354 | 0.163002 | 0.456691 |

3 | 0.704272 | 0.455508 | 0.227754 | 0.091973 | 0.244135 | 0.706900 | 0.464402 | 0.246685 | 0.124855 | 0.341109 |

4 | 0.655801 | 0.399725 | 0.188645 | 0.073327 | 0.203097 | 0.658545 | 0.408106 | 0.204899 | 0.099936 | 0.268109 |

5 | 0.613293 | 0.355394 | 0.160270 | 0.060566 | 0.173121 | 0.616107 | 0.363277 | 0.174469 | 0.082798 | 0.219271 |

6 | 0.575751 | 0.319479 | 0.138958 | 0.051421 | 0.150483 | 0.578603 | 0.326893 | 0.151541 | 0.070465 | 0.184886 |

7 | 0.542384 | 0.289886 | 0.122468 | 0.044604 | 0.132887 | 0.545249 | 0.296867 | 0.133752 | 0.061238 | 0.159596 |

8 | 0.512553 | 0.265138 | 0.109380 | 0.039352 | 0.118873 | 0.515414 | 0.271722 | 0.119601 | 0.054108 | 0.140303 |

9 | 0.485740 | 0.244170 | 0.098767 | 0.035191 | 0.107474 | 0.488583 | 0.250391 | 0.108103 | 0.048446 | 0.125136 |

10 | 0.461522 | 0.226200 | 0.090000 | 0.031819 | 0.098035 | 0.464337 | 0.232091 | 0.098589 | 0.043847 | 0.112913 |

50 | 0.152137 | 0.056541 | 0.019600 | 0.006555 | 0.021517 | 0.153571 | 0.058331 | 0.021619 | 0.009102 | 0.022970 |

100 | 0.082505 | 0.029117 | 0.009900 | 0.003289 | 0.010879 | 0.083346 | 0.030067 | 0.010930 | 0.004571 | 0.011508 |

**Table 4.**Classical ruin probabilities for Pareto distribution (above) with $m=1$ and different values of the premium loading factor $\theta $. Probabilities based on expression (17) and Beta mixing distribution (below).

u | $\mathit{\theta}=0.10$ | $\mathit{\theta}=0.25$ | $\mathit{\theta}=0.50$ | $\mathit{\theta}=0.75$ | $\mathit{\theta}=1.00$ |
---|---|---|---|---|---|

10 | 0.627722 | 0.372683 | 0.206648 | 0.138243 | 0.102523 |

20 | 0.498175 | 0.245262 | 0.119275 | 0.075909 | 0.055050 |

30 | 0.411440 | 0.178339 | 0.081426 | 0.051056 | 0.036887 |

40 | 0.347896 | 0.137560 | 0.060856 | 0.038038 | 0.027509 |

50 | 0.299157 | 0.110519 | 0.048164 | 0.030142 | 0.021847 |

60 | 0.260646 | 0.091524 | 0.039650 | 0.024884 | 0.018080 |

70 | 0.229552 | 0.077594 | 0.033588 | 0.021150 | 0.015402 |

80 | 0.204018 | 0.067029 | 0.029075 | 0.018369 | 0.013404 |

90 | 0.182761 | 0.058793 | 0.025596 | 0.016222 | 0.011859 |

100 | 0.164859 | 0.052226 | 0.022838 | 0.014516 | 0.010630 |

$\mathit{u}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{9}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{3}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{1}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{0.33}$ | $\mathit{\alpha}=\mathbf{1}$ $\mathit{\beta}=\mathbf{1.1}$ |

10 | 0.656535 | 0.413391 | 0.206860 | 0.084824 | 0.221515 |

20 | 0.560347 | 0.322256 | 0.150307 | 0.059235 | 0.161739 |

30 | 0.497584 | 0.270725 | 0.121416 | 0.046852 | 0.130993 |

40 | 0.451470 | 0.236169 | 0.103170 | 0.039268 | 0.111501 |

50 | 0.415428 | 0.210890 | 0.090365 | 0.034055 | 0.097784 |

60 | 0.386135 | 0.191370 | 0.080780 | 0.030212 | 0.087497 |

70 | 0.361666 | 0.175726 | 0.073284 | 0.027243 | 0.079438 |

80 | 0.340804 | 0.162838 | 0.067230 | 0.024868 | 0.072923 |

90 | 0.322732 | 0.151995 | 0.062221 | 0.022918 | 0.067527 |

100 | 0.306875 | 0.142719 | 0.057997 | 0.021285 | 0.062971 |

y | Exponential | Mixing Distribution | |||
---|---|---|---|---|---|

Gamma | Lindley | Inverse Gaussian | |||

$u=0$ | 1 | 0.505696 | 0.444444 | 0.411775 | 0.415263 |

5 | 0.794610 | 0.734694 | 0.680567 | 0.721115 | |

10 | 0.799964 | 0.777777 | 0.736850 | 0.777757 | |

∞ | 0.800000 | 0.800000 | 0.800000 | 0.800000 | |

$u=5$ | 1 | 0.186035 | 0.155555 | 0.137258 | 0.152318 |

5 | 0.292321 | 0.305556 | 0.286781 | 0.325647 | |

10 | 0.294290 | 0.336621 | 0.330539 | 0.366766 | |

∞ | 0.294304 | 0.355555 | 0.388225 | 0.384737 | |

$u=10$ | 1 | 0.068438 | 0.072000 | 0.066816 | 0.078125 |

5 | 0.107539 | 0.160494 | 0.162841 | 0.187195 | |

10 | 0.108263 | 0.183673 | 0.197878 | 0.217767 | |

∞ | 0.108268 | 0.200000 | 0.250967 | 0.232419 | |

$u=100$ | 1 | 1.040 × 10^{−9} | 0.000562 | 0.001506 | 0.000515 |

5 | 1.630 × 10^{−9} | 0.002222 | 0.006353 | 0.001880 | |

10 | 1.640 × 10^{−9} | 0.003486 | 0.010625 | 0.002720 | |

∞ | 1.640 × 10^{−9} | 0.006611 | 0.032393 | 0.003601 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gómez-Déniz, E.; Sarabia, J.M.; Calderín-Ojeda, E.
Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem. *Risks* **2019**, *7*, 68.
https://doi.org/10.3390/risks7020068

**AMA Style**

Gómez-Déniz E, Sarabia JM, Calderín-Ojeda E.
Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem. *Risks*. 2019; 7(2):68.
https://doi.org/10.3390/risks7020068

**Chicago/Turabian Style**

Gómez-Déniz, Emilio, José María Sarabia, and Enrique Calderín-Ojeda.
2019. "Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem" *Risks* 7, no. 2: 68.
https://doi.org/10.3390/risks7020068