# Intelligent Decision Support in Proportional–Stop-Loss Reinsurance Using Multiple Attribute Decision-Making (MADM)

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## Abstract

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## 1. Introduction

#### 1.1. Background

- The optimal reinsurance form, under given criteria;
- Given the reinsurance form, the choice of reinsurance parameters. (e.g., optimal retention portion for proportional reinsurance, optimal retention limit for stop-loss reinsurance, etc.)

#### 1.2. Paper Development

## 2. Literature Review

## 3. Methodology

#### 3.1. Decision Flow

#### 3.2. The Proportional–Stop-Loss Reinsurance Model

#### 3.3. Variable Definition

#### 3.4. Simulating Alternatives Using MODM

#### 3.5. Calculating Decision Criteria

#### 3.5.1. Expected Profit of the Insurance Company (Reinsured Party): ${\mathit{PROFIT}}_{i}$

#### 3.5.2. Expected Shortfall ${\mathrm{ES}}_{\mathsf{\alpha}}$

#### 3.5.3. Ruin Probability

#### 3.5.4. Expected Utility

#### 3.6. Selecting the Best Alternative Using MADM

- Formulate decision matrix $D$ with $m$ alternatives ${A}_{1},{A}_{2},\dots ,{A}_{m}$ and $n$ decision criteria ${C}_{1},{C}_{2},\dots ,{C}_{n}$. The attribute value of ${A}_{i}$ on ${C}_{j}$ for $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$ is represented as ${d}_{ij}$.
- Calculate weight of the criteria using entropy technique as follows:$$\begin{array}{c}{q}_{ig}=\frac{{d}_{ig}}{{d}_{1g}+{d}_{2g}+\dots {d}_{mg}};\forall g\in \left\{1,2,\dots ,c\right\}\\ {\Delta}_{g}=-k{\displaystyle \sum}{q}_{ig}\cdot lo{g}_{2}\left({q}_{ig}\right);\forall g\in \left\{1,2,\dots ,c\right\}\\ {d}_{g}=1-{\Delta}_{g},{w}_{g}=\frac{{d}_{g}}{\left({d}_{1}+\dots +{d}_{g}\right)}\\ {w}_{g}{}^{\prime}=\frac{{\lambda}_{g}\cdot {w}_{g}}{{\lambda}_{1}\cdot {w}_{1}+{\lambda}_{2}\cdot {w}_{2}+\dots +{\lambda}_{c}\cdot {w}_{c}}\end{array}$$
- Normalize the decision matrix using the following formula:$${r}_{ij}=\frac{{d}_{ij}}{\sqrt{{{\displaystyle \sum}}_{i=1}^{m}{d}_{ij}^{2}}}$$One may notice that by scaling the criteria (multiplying a constant to ${d}_{ij}$), the decision will not change. However, it will not necessarily return same decision for different utility functions that generate the same decision under expected utility measurement, as adding a constant to ${d}_{ij}$ in the ${r}_{ij}$ formula will change the resulting ${r}_{ij}$.
- Calculate the weighted normalized decision matrix by using the normalized decision matrix parameter ${r}_{ij}$ and weight vector $\omega =({\omega}_{1},{\omega}_{2},\dots ,{\omega}_{n}$) to return the weighted normalized decision matrix parameter ${V}_{ij}={\omega}_{j}\cdot {r}_{ij}$. If criteria are given same weight, ${\omega}_{1}={\omega}_{2}=\dots ={\omega}_{n}=\frac{1}{n}$.
- Compute the vectors of positive ideal solutions and the negative ideal solutions, denoted by:$$\begin{array}{c}{S}^{+}=\left({S}_{1}^{+},{S}_{2}^{+},{S}_{3}^{+},\dots ,{S}_{n}^{+}\right)\\ {S}^{-}=\left({S}_{1}^{-},{S}_{2}^{-},{S}_{3}^{-},\dots ,{S}_{n}^{-}\right)\end{array}$$
- Calculate the distance between each alternative and the positive and negative ideal points. The distance between alternative ${A}_{i}$ and the positive ideal points is:$${D}_{i}^{+}=\sqrt{{\displaystyle \sum}_{j=1}^{n}{({V}_{ij}-{S}_{j}^{+})}^{2}},\text{}\mathrm{for}\text{}i=1,2,\dots ,m;$$The distance between alternative and the negative ideal solutions are:$${D}_{i}^{-}=\sqrt{{\displaystyle \sum}_{j=1}^{n}{({V}_{ij}-{S}_{j}^{-})}^{2}},\text{}\mathrm{for}\text{}i=1,2,\dots ,m;$$
- Calculate the relative closeness coefficient of each alternative represented as:$${C}_{i}=\frac{{D}_{i}^{-}}{{D}_{i}^{+}+{D}_{i}^{-}},\text{}{C}_{i}\in \left[0,1\right]$$
- Rank the alternatives according to ${C}_{i}$. The alternative with higher ${C}_{i}$ value is preferred over lower ${C}_{i}$ alternatives.

## 4. Case Study

#### 4.1. Loss Distribution Modeling

#### 4.2. Generating Alternatives from the Viewpoints of Reinsurers

#### 4.3. Constructing Decision Matrix

#### 4.4. Selecting Alternatives Using TOPSIS

topsis (decisionMakingMatrix,lambdaWeight,criteriaSign) |

## 5. Managerial Implications

- The best alternative suggested by TOPSIS does not necessarily optimize any one single criterion, rather, it has an overall highest ranking due to its relative weighted closeness to all four criteria. In reality, if reinsurance is chosen merely according to expected profit, the insurance company may suffer from a high probability of financial crisis. On the other hand, if the decision merely considers constraining higher shortfalls, the insurance company may appear to have poor performance based on their profit and loss statement due to the low profits obtained.By increasing the ceding amount from a = 0.6 to a = 0.75, 0.9, one result from Trial 1,2,3,4 (Trial 2–4 are reported in Appendix A) suggests that the ranking of alternatives is different when parameters are changed. When the ceding portions are fixed at relatively lower level (such as a = 0.6 to a = 0.75) the best alternative to choose will have the retention limit equaling to mean value of loss. Thus, if the given reinsurance parameters (either $a$, $\theta $ or $M2$) are altered, it is recommended for the insurance company to evaluate once again the reinsurance plans instead of extrapolating conclusions from previous experiences.
- In addition, Trial 4 with a = 1 models an excess-of-loss reinsurance form where ${X}_{r}=MAX\left(0,X-M\right)=1\ast {(0,X-M)}_{+}$. Accordingly, results from Trial 4 are in accordance with previous knowledge on excess-of-loss reinsurance. Under excess-of-loss reinsurance, the best form is given at $M={M}_{max}$, which is in correspondence with Section 2 in Payandeh-Najafabadi and Panahi-Bazaz (2017).
- In each trial, the Alternative 1 M = 0 simulates the scenario of pure proportional reinsurance. Trial 5 attempts to model different retention level under proportional reinsurance ($M=X$) with fixed reinsurance premium loading factor $\theta $. The result shows that given same premium loading factor, retention level of 0.6 would be most preferable.
- By setting (a, M) to (0, 0), we could also model the scenario of no reinsurance. The results show that with no reinsurance, the expected shortfall of insurance company will be significantly higher than all other alternatives, and the ruin probability will be higher as well. This suggests that insurance company without reinsurance is more likely to become bankrupt if large losses are incurred. As compensation, the expected profit and utility will increase by a small amount for the insurance company due to high profit from insurance premium and low probability of large losses. However, noting the high ruin probability, which suggests a much higher risk of bankruptcy, the insurance company will often seek for reinsurance to keep ruin probability low.
- Furthermore, through the simulation process, the variance and profitability of the reinsurer are also being observed and calculated (as can be seen from Figure 9). The result was in correspondence with our previous argument that by scaling the ceding portion a to larger values, both the variance and the profitability of the reinsurer will increase, suggesting that there is a trade-off between high profit and high risk of large losses. Thus, this supports our previous assumption that the reinsurer is ambiguous towards a design that only differs with respect to parameter a.

## 6. Conclusions

#### 6.1. Contributions

- To the best of our knowledge, this is the first theoretical study using MADM to approach proportional–stop-loss reinsurance model, though there are a few recent studies using MADM in designing either pure proportional or pure stop-loss reinsurance contracts;
- This is one of the few studies taking a non-discriminatory position considering both the insurance and the reinsurance company in designing an optimal reinsurance contract, and the study made significant contribution by incorporating existing MODM models and the promising MADM model into one decision flow process to arrive at a robust decision for reinsurance design;
- This study demonstrates the feasibility of incorporating intelligent decision supporting systems in reinsurance deal-making. As observed by the author through industry experiences, @Risk has grown its popularity recently for actuarial study in modeling risk and claims. The prototype of TOPSIS implemented through Matlab suggests that a software of multi-criteria decision support would be promising.
- As previous research suggested (Bazaz and Najafabadi 2015), MADM is not likely to address finding of optimal type of reinsurance. However, with the generic formulation of proportional–stop-loss reinsurance, we would be able to model proportional reinsurance and stop-loss reinsurance as special cases of proportional–stop-loss, thus the choice between proportional and non-proportional reinsurance using MADM could be possible under this formulation of reinsurance.

#### 6.2. Limitations

- In terms of the scope of study, due to time and resource constraints the study only considers proportional-stop-loss treaty reinsurance, while basing the decision process on ruin probability, CVaR, and expected utility criteria. Other types of reinsurance and decision measurements have not been elaborated and tested.
- In terms of methodology, this study attempts to utilize the simulation software @Risk to model the loss and claim distribution and to use numerical TOPSIS model in modeling decisions from the insurance company, without reaching to a close-form solution. Thus, the conclusions were drawn based on simulation result rather than robust theoretical derivation.
- In terms of model implementation, due to resource constraints, this study only includes a numerical made-up case instead of existing cases to conduct archival research in addressing the decision process in the reinsurance purchase decisions.

#### 6.3. Future Directions

## Author Contributions

## Conflicts of Interest

## Appendix A. TOPSIS Trials #2, #3 and #4

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**Figure 1.**Trade procedure P1 (Liang 2014).

**Figure 2.**Trade procedure P2 (Liang 2014).

**Figure 3.**Graphical representation of technique for order of preference by similarity to ideal solution (TOPSIS) (Chauhan and Vaish 2013).

Variable | Variable Explanation |
---|---|

$t$ | the time period of one contract, in our case study $t=1$; |

$N$ | the number of claims incurred in period t (during one contract); |

$W\left(t\right)$ | the wealth held by insurance company at time t; |

$\zeta $ | the loading factor of the reinsurance premium paid to the reinsurer; |

$\theta $ | the loading factor of the premium paid to the reinsured party; |

$X$ | the claim amount of one single loss; |

${X}_{i}$ | the claim amount payable by the insurance company (the reinsured party); |

${X}_{r}$ | the claim amount payable by the reinsurance company (the reinsurer); |

$S\left(t\right)$ | the aggregate loss of an insurance portfolio; |

${S}_{i}\left(t\right)$ | the aggregate claim (loss) incurred to insurance company (the reinsured party); |

${S}_{r}\left(t\right)$ | the aggregate claim (loss) incurred to the reinsurance company (reinsurer); |

${F}_{S}\left(X\right)$ | the cumulative distribution function of S; |

$\overline{{F}_{S}}=1-{F}_{S}\left(X\right)$ | the survival distribution function of S; |

$\left(a,M\right)$ | the proportional–stop-loss reinsurance parameter,${X}_{r}=a{(X-M)}_{+}$; |

$c$ | the total premium per unit time; |

${c}_{i}$ | the premium gained by the insurance company; |

${c}_{r}$ | the premium payable to the reinsurer; |

$E{S}_{\alpha}$ | the expected shortfall with a confidence level of $\alpha $; |

$PROFI{T}_{i}$ | the expected profit gained by insurance company; |

$\psi \left(i\right)$ | the ruin probability of insurance company’s wealth $U\left(t\right)$; |

${U}_{i}\left(t\right)$ | the utility of insurance company at the end of period t; |

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## Share and Cite

**MDPI and ACS Style**

Wang, S.J.X.; Poh, K.L. Intelligent Decision Support in Proportional–Stop-Loss Reinsurance Using Multiple Attribute Decision-Making (MADM). *J. Risk Financial Manag.* **2017**, *10*, 22.
https://doi.org/10.3390/jrfm10040022

**AMA Style**

Wang SJX, Poh KL. Intelligent Decision Support in Proportional–Stop-Loss Reinsurance Using Multiple Attribute Decision-Making (MADM). *Journal of Risk and Financial Management*. 2017; 10(4):22.
https://doi.org/10.3390/jrfm10040022

**Chicago/Turabian Style**

Wang, Shirley Jie Xuan, and Kim Leng Poh. 2017. "Intelligent Decision Support in Proportional–Stop-Loss Reinsurance Using Multiple Attribute Decision-Making (MADM)" *Journal of Risk and Financial Management* 10, no. 4: 22.
https://doi.org/10.3390/jrfm10040022