# Optimal Retention Level for Infinite Time Horizons under MADM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Classical Risk Model

## 3. Ultimate Ruin Probability

#### 3.1. Ultimate Ruin Probability for the Gamma Process

#### 3.2. Ultimate Ruin Probability for the Translated Gamma Process

## 4. Optimal Reinsurance Criteria

#### 4.1. Released Capital

#### 4.2. Expected Profit

#### 4.3. Expected Utility

#### 4.3.1. Exponential Utility

#### 4.3.2. Fractional Power Utility

#### 4.3.3. Logarithmic Utility

## 5. Multi-Criteria Decision Making

#### 5.1. Multiple-Attribute Decision Making

#### 5.2. The Analytical Hierarchy Process

- Step 1:
- The summation of the values in each column of the pairwise comparison matrix is found,
- Step 2:
- Each element is divided by its column total value (the normalized pairwise comparison matrix),
- Step 3:
- The average of elements in each row is calculated (relative priorities-priority index).

- Step 1:
- The pairwise comparison matrix and its relative priorities are multiplied,
- Step 2:
- The weighted sum vector elements is divided by the associated priority value,
- Step 3:
- A consistency index $(CI)$ is calculated by the average of the values of Step 2, that is to say the maximum eigenvalue ${\lambda}_{max}$, such that:$$CI=\frac{{\lambda}_{max}-n}{n-1}.$$
- Step 4:
- A consistency ratio $(CR)$ is calculated as:$$CR=\frac{CI}{RI}.$$

#### 5.3. TOPSIS Method with Euclidean Distance

- Step 1:
- The decision matrix is normalized by using the vector-normalization technique:$${r}_{ij}=\frac{{x}_{ij}}{\sqrt{{\displaystyle \sum _{i=1}^{m}{({x}_{ij})}^{2}}}},$$
- Step 2:
- Weighted-normalized values are calculated by using the weight vector $w=({w}_{1},{w}_{2},\cdots ,{w}_{n})$.$${V}_{ij}(x)={w}_{j}\phantom{\rule{3.33333pt}{0ex}}{r}_{ij},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,\cdots ,n;\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}j=1,\cdots ,m.$$
- Step 3:
- Let the positive ideal points and negative ideal points (anti-ideal) be ${\mathit{S}}^{+}$ and ${\mathit{S}}^{-}$, respectively. The positive ideal points are equivalent to the maximum value under each criterion:$$\begin{array}{ccc}\hfill {\mathit{S}}^{+}& =& \left\{{S}_{1}^{+},{S}_{2}^{+},\cdots ,{S}_{j}^{+},\cdots ,{S}_{n}^{+}\right\}\hfill \\ & =& {\displaystyle \{(\underset{i}{max}\phantom{\rule{3.33333pt}{0ex}}{V}_{ij}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}j\in J),(\underset{i}{min}\phantom{\rule{3.33333pt}{0ex}}{V}_{ij}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}j\in {J}^{\prime})\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}i=1,2\cdots ,m)\}}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathit{S}}^{-}& =& \left\{{S}_{1}^{-},{S}_{2}^{-},\cdots ,{S}_{j}^{-},\cdots ,{S}_{n}^{-}\right\}\hfill \\ & =& {\displaystyle \{(\underset{i}{min}{V}_{ij}\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}j\in J),(\underset{i}{max}{V}_{ij}\phantom{\rule{3.33333pt}{0ex}}|j\in {J}^{\prime})\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}i=1,2\cdots ,m)\},}\hfill \end{array}$$
- Step 4:
- The distance between each alternative is calculated by using n-dimensional Euclidean distance. The distance between the alternative ${A}_{i}$ and the ideal solution is:$${d}_{i}^{+}=\sqrt{{\displaystyle \sum _{j=1}^{n}{({V}_{ij}-{S}_{j}^{+})}^{2}}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,2,\cdots ,m,$$$${d}_{i}^{\phantom{\rule{3.33333pt}{0ex}}-}=\sqrt{{\displaystyle \sum _{j=1}^{n}{({V}_{ij}-{S}_{j}^{-})}^{2}}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,2,\cdots ,m.$$
- Step 5:
- The relative closeness of each alternative to the ideal solution (closeness index) is calculated as:$${C}_{i}=\frac{{d}_{i}^{-}}{{d}_{i}^{+}+{d}_{i}^{-}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}i=1,2,\cdots ,m,$$

#### 5.4. TOPSIS Method with Mahalanobis Distance

## 6. Numerical Analysis

**R**programming. The corresponding retention levels M of the smallest initial surplus are calculated.

- AHP-1
- uses the linear scale transformation normalization technique. The normalized values are obtained by dividing the outcome of a criterion by its maximum value. Then, we assume that the maximum normalized value of the criterion is equal to nine as in Saaty’s matrix (Table 2) (Hwang and Yoon [28]). Thus, the scale of measurement varies precisely from 1–9 for each criterion.
- AHP-2
- uses the vector normalization technique as used in the TOPSIS method in (20).
- AHP-3
- uses the min-max normalization technique, which is given below:$${r}_{ij}=\frac{{x}_{ij}-min({x}_{ij})}{max({x}_{ij})-min({x}_{ij})}.$$
- AHP-4
- uses the automating pairwise comparison technique [32]. In this technique, a pairwise comparison matrix ${\mathbf{B}}^{(j)}$ is comprised of the m criteria, $j=1,2,\cdots ,m$. The matrix ${\mathbf{B}}^{(j)}$ is a $n\times n$ real matrix, where n is the number of alternatives. Each element ${b}_{ih}^{(j)}$ of the matrix ${\mathbf{B}}^{(j)}$ represents the evaluation of the $\mathit{i}$-th alternative compared to the $\mathit{h}$-th alternative with respect to the $\mathit{j}$-th criterion.

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The graph of the optimal initial surplus and optimal retention level when $\theta =0.1$ and $\xi =0.15$ for the exponential claims.

**Figure 3.**The graph of the optimal initial surplus and optimal retention level when $\theta =0.1$ and $\xi =0.15$ for the Pareto claims.

Alternatives (${\mathit{A}}_{\mathit{i}}$) | Attributes (Criteria) $({\mathit{C}}_{\mathit{j}})$ | ||||
---|---|---|---|---|---|

${\mathit{C}}_{\mathbf{1}}$ | ${\mathit{C}}_{\mathbf{2}}$ | ${\mathit{C}}_{\mathbf{3}}$ | ⋯ | ${\mathit{C}}_{\mathit{n}}$ | |

${A}_{1}$ | ${X}_{11}$ | ${X}_{12}$ | ${X}_{13}$ | ⋯ | ${X}_{1n}$ |

${A}_{2}$ | ${X}_{21}$ | ${X}_{22}$ | ${X}_{23}$ | ⋯ | ${X}_{2n}$ |

${A}_{3}$ | ${X}_{31}$ | ${X}_{32}$ | ${X}_{33}$ | ⋯ | ${X}_{3n}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |

${A}_{m}$ | ${X}_{m1}$ | ${X}_{m2}$ | ${X}_{m3}$ | ⋯ | ${X}_{mn}$ |

Definition | Index |
---|---|

Equally important | 1 |

Equally or slightly more important | 2 |

Slightly more important | 3 |

Slightly too much more important | 4 |

Much more important | 5 |

Much too more important | 6 |

Far more important | 7 |

Far more important to extremely more important | 8 |

Extremely more important | 9 |

n | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

RI | 0.58 | 0.90 | 1.12 | 1.24 | 1.32 | 1.41 |

Individual Claims | $\mathit{\theta}=\mathbf{0.1}$ | $\mathit{\theta}=\mathbf{0.2}$ |
---|---|---|

Exponential Distribution | 49.638 | 26.591 |

Pareto Distribution | 79.774 | 45.090 |

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.15}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.2}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.3}$ | $\mathit{\theta}=\mathbf{0.2}$ and $\mathit{\xi}=\mathbf{0.3}$ | |
---|---|---|---|---|

Exponential Claims | 27.798 | 38.307 | 45.758 | 14.367 |

Pareto Claims | 30.382 | 44.510 | 57.616 | 15.692 |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.15}$ | 1 | 27.763 | 0.852 | 21.875 | 18.000 | 0.582 | 1.123 | 0.116 |

2 | 27.863 | 0.917 | 21.775 | 20.022 | 0.598 | 1.124 | 0.117 | |

3 | 27.963 | 0.946 | 21.675 | 20.865 | 0.604 | 1.124 | 0.117 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

217 | 49.363 | 7.300 | 0.275 | 49.949 | 0.832 | 1.136 | 0.128 | |

218 | 49.463 | 7.780 | 0.175 | 49.968 | 0.832 | 1.136 | 0.128 | |

219 | 49.563 | 8.800 | 0.075 | 49.989 | 0.832 | 1.136 | 0.128 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.2}$ | 1 | 38.302 | 1.548 | 11.336 | 28.740 | 0.713 | 1.130 | 0.122 |

2 | 38.402 | 1.6523 | 11.236 | 30.839 | 0.723 | 1.131 | 0.123 | |

3 | 38.502 | 1.705 | 11.136 | 31.822 | 0.728 | 1.131 | 0.123 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

112 | 49.402 | 7.420 | 0.236 | 49.940 | 0.833 | 1.136 | 0.128 | |

113 | 49.502 | 8.158 | 0.136 | 49.971 | 0.833 | 1.136 | 0.128 | |

114 | 49.602 | 9.752 | 0.036 | 49.994 | 0.833 | 1.136 | 0.128 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.3}$ | 1 | 45.758 | 2.669 | 3.880 | 39.602 | 0.789 | 1.134 | 0.126 |

2 | 45.858 | 2.907 | 3.780 | 41.808 | 0.797 | 1.134 | 0.126 | |

3 | 45.958 | 3.017 | 3.680 | 42.661 | 0.800 | 1.135 | 0.127 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

37 | 49.358 | 7.050 | 0.280 | 49.870 | 0.832 | 1.136 | 0.128 | |

38 | 49.458 | 7.57 | 0.180 | 49.923 | 0.832 | 1.136 | 0.128 | |

39 | 49.558 | 8.631 | 0.080 | 49.973 | 0.832 | 1.136 | 0.128 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.2}$ and $\mathit{\xi}=\mathbf{0.3}$ | 1 | 14.367 | 0.835 | 12.224 | 34.941 | 0.611 | 1.123 | 0.116 |

2 | 14.467 | 0.925 | 12.124 | 40.550 | 0.651 | 1.125 | 0.117 | |

3 | 14.567 | 0.967 | 12.024 | 42.949 | 0.667 | 1.125 | 0.118 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

121 | 26.367 | 6.943 | 0.224 | 99.855 | 0.902 | 1.137 | 0.129 | |

122 | 26.467 | 7.65 | 0.124 | 99.928 | 0.902 | 1.137 | 0.129 | |

123 | 26.567 | 9.764 | 0.024 | 99.991 | 0.902 | 1.137 | 0.129 |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |
---|---|---|---|---|---|---|---|---|

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.15}$ | 1 | 30.382 | 0.935 | 49.392 | 16.774 | 0.593 | 1.123 | 0.115 |

2 | 30.482 | 1.003 | 49.292 | 18.432 | 0.606 | 1.123 | 0.116 | |

3 | 30.582 | 1.034 | 49.192 | 19.152 | 0.611 | 1.124 | 0.117 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

492 | 79.482 | 335.481 | 0.292 | 49.999 | 0.897 | 1.091 | 0.087 | |

493 | 79.582 | 504.920 | 0.192 | 49.999 | 0.897 | 1.091 | 0.087 | |

494 | 79.682 | 1041.301 | 0.092 | 50 | 0.897 | 1.091 | 0.087 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.2}$ | 1 | 44.510 | 1.795 | 35.264 | 25.511 | 0.729 | 1.130 | 0.122 |

2 | 44.610 | 1.917 | 35.164 | 27.288 | 0.737 | 1.130 | 0.123 | |

3 | 44.710 | 1.972 | 35.064 | 28.039 | 0.741 | 1.131 | 0.123 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

351 | 79.510 | 369.987 | 0.264 | 50.000 | 0.897 | 1.138 | 0.129 | |

352 | 79.610 | 589.278 | 0.164 | 50.000 | 0.897 | 1.138 | 0.129 | |

353 | 79.710 | 1490.600 | 0.064 | 50.000 | 0.897 | 1.138 | 0.129 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.3}$ | 1 | 57.616 | 3.371 | 22.158 | 34.336 | 0.812 | 1.134 | 0.126 |

2 | 57.716 | 3.615 | 22.058 | 36.011 | 0.817 | 1.134 | 0.126 | |

3 | 57.816 | 3.728 | 21.958 | 36.700 | 0.820 | 1.134 | 0.126 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

220 | 79.516 | 378.826 | 0.258 | 50.000 | 0.900 | 1.138 | 0.129 | |

221 | 79.616 | 612.581 | 0.158 | 50.000 | 0.900 | 1.138 | 0.129 | |

222 | 79.716 | 1653.219 | 0.058 | 50.000 | 0.900 | 1.138 | 0.129 | |

Loading Factors | Number of Alternatives | Initial Surplus (u) | Retention Level (M) | Released Capital (RC) | Expected Profit (EP) | Expected Exponential Utility (EeU) | Expected Fractional Utility (EfU) | Expected Logarithmic Utility (ElU) |

$\mathit{\theta}=\mathbf{0.2}$ and $\mathit{\xi}=\mathbf{0.3}$ | 1 | 15.692 | 0.917 | 29.398 | 32.612 | 0.603 | 1.123 | 0.116 |

2 | 15.792 | 1.010 | 29.298 | 37.179 | 0.636 | 1.124 | 0.117 | |

3 | 15.892 | 1.054 | 29.198 | 39.195 | 0.650 | 1.124 | 0.117 | |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | |

292 | 44.792 | 310.269 | 0.298 | 100.000 | 0.924 | 1.138 | 0.129 | |

293 | 44.892 | 465.985 | 0.198 | 100.000 | 0.924 | 1.138 | 0.129 | |

294 | 44.992 | 938.272 | 0.098 | 100.000 | 0.924 | 1.138 | 0.129 |

Exponential Claims | ||||
---|---|---|---|---|

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.15}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.2}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.3}$ | $\mathit{\theta}=\mathbf{0.2}$ and $\mathit{\xi}=\mathbf{0.3}$ | |

TOPSIS-E | 30.663 and 1.326 | 39.202 and 1.932 | 45.858 and 2.907 | 16.067 and 1.334 |

AHP-1 | 35.363 and 1.883 | 39.802 and 2.086 | 45.858 and 2.907 | 18.367 and 1.829 |

AHP-2 | 31.463 and 1.420 | 38.902 and 1.846 | 45.858 and 2.907 | 16.167 and 1.355 |

AHP-3 | 48.363 and 5.392 | 49.202 and 6.675 | 49.558 and 8.631 | 25.067and 4.445 |

AHP-4 | 31.363 and 1.407 | 38.302 and 1.548 | 45.958 and 3.017 | 16.167 and 1.355 |

TOPSIS-M | 28.263 and 1.010 | 38.502 and 2.010 | 46.858 and 3.699 | 14.567 and 0.967 |

Pareto Claims | ||||

$\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.15}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.2}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.3}$ | $\mathit{\theta}=\mathbf{0.1}$ and $\mathit{\xi}=\mathbf{0.4}$ | |

TOPSIS-E | 37.782 and 1.740 | 47.910 and 2.815 | 58.916 and 4.987 | 20.092 and 2.266 |

AHP-1 | 47.682 and 3.409 | 51.710 and 3.676 | 59.516 and 4.780 | 25.292 and 3.405 |

AHP-2 | 40.882 and 2.378 | 47.910 and 2.815 | 58.416 and 4.728 | 21.792 and 2.435 |

AHP-3 | 62.582 and 7.643 | 74.110 and 21.763 | 79.716 and 1653.219 | 34.992 and 8.805 |

AHP-4 | 33.882 and 1.513 | 47.710 and 2.770 | 58.116 and 3.973 | 15.992 and 1.090 |

TOPSIS-M | 38.382 and 2.182 | 44.510 and 1.795 | 79.616 and 612.581 | 44.792 and 310.269 |

© 2016 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/).

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**MDPI and ACS Style**

Bulut Karageyik, B.; Şahin, Ş. Optimal Retention Level for Infinite Time Horizons under MADM. *Risks* **2017**, *5*, 1.
https://doi.org/10.3390/risks5010001

**AMA Style**

Bulut Karageyik B, Şahin Ş. Optimal Retention Level for Infinite Time Horizons under MADM. *Risks*. 2017; 5(1):1.
https://doi.org/10.3390/risks5010001

**Chicago/Turabian Style**

Bulut Karageyik, Başak, and Şule Şahin. 2017. "Optimal Retention Level for Infinite Time Horizons under MADM" *Risks* 5, no. 1: 1.
https://doi.org/10.3390/risks5010001