Multiple Attribute Decision Making ( MADM ) in Proportional-Stop-Loss Reinsurance

This article addresses reinsurance decision making process, which involves the insurance company and the reinsurance company, and is negotiated through reinsurance intermediaries. The article proposes a decision flow to model the reinsurance design and selection process. In contrast to existing literature on pure proportional reinsurance or stop-loss reinsurance, this article focuses on the combination into Proportional-Stop-loss reinsurance design which better addresses interest of both parties. In terms of methodology, the significant contribution of the study is to incorporate Multiple Attribute Decision Making (MADM) into modelling the reinsurance selection. The MultiObjective Decision Making (MODM) model is applied in designing reinsurance alternatives. Then MADM is applied to aid insurance companies in choosing the most appropriate reinsurance contract. To illustrate the feasibility of incorporating intelligent decision supporting system in reinsurance market, the study includes a numerical case study using simulation software @Risk in modeling insurance claims, and programming in MATLAB to realize MADM. Managerial implications could be drawn from the case study results. More specifically, when choosing the most appropriate reinsurance, insurance companies should base their decision on multiple measurements instead of single-criteria decision making models for their decisions to be more robust.


Background
Reinsurance is generally known as "the insurance for insurance".Following similar concepts and principles as insurance, it provides financial compensation to insurance companies for the risk of large losses.The reinsured (or "insurance companies") buys reinsurance from the reinsurer (or "reinsurance companies") in exchange for loss limitation, revenue protection and free up capital.In recent years, reinsurance has grown in both market value and diversity due to global trends such as the global climate change, increase in insurance mega losses, volatility in equity markets and emerging risks such as terrorism.Regardless of the financial size, an insurance company rarely retains all of their risk.Thus it is of interest to understand the decision making process of reinsurance contract, which in reality is usually done with the negotiation intermediaries, i.e. the reinsurance brokers.Typically, there are two categories of reinsurance decisions, both of which will be addressed in this study:

•
Optimal reinsurance form under given criteria; • Given the reinsurance form of decision, decide on the reinsurance parameters.(e.g.optimal retention portion for proportional reinsurance, optimal retention limit for stop-loss reinsurance, etc.) The common forms of reinsurance are shown in Figure A1 and their complete definitions are included in Appendix A. This study focuses on treaty reinsurance which covers an entire portfolio with multiple single risks.Both facultative and treaty reinsurance could be further broken down into proportional reinsurance and non-proportional reinsurance (Carter 1979).Early research has shown that under variance risk measurement with fixed premium, stop-loss contract is the optimal reinsurance form for reinsurance buyers ( (Borck 1960) and (Hürlimann 2011)), whereas that quotashare best addresses the interest of the reinsurer (Vajda 1962).Clearly, there would be conflicts of choice between two parties.Thus, this study addresses a combinational form of proportional and stop-loss treaty reinsurance (See Section 2.2), following the definition by Samson and Thomas (1985).Quota-share reinsurance and stop-loss reinsurance could be considered as special cases of Proportional-Stop-Loss reinsurance.In deciding the optimal reinsurance parameters, this study attempts to utilize Multiple Attribute Decision Making (MADM) which improves from previous literatures on single criterion.

Papaer Development
The paper is organized as follows.Section 2 recalls recent research that this study is built upon.Section 3 develops the decision flow based on the form of Proportional-Stop-loss Reinsurance and determines the optimal reinsurance parameters using Multi-Objective Decision Making (MODM).Section 4 includes a numerical case study, which models claims using @Risk and implements MADM for buyer's selection using MATLAB.Section 5 discusses the contributions, limitations, further directions and concludes the study.

Literature Review
Decision analysis models on single criterion have been extensively discussed both for the reinsurer (see Appendix B.1) in structuring reinsurance and for the reinsured (see Appendix B.2) for evaluating and selecting the most appropriate reinsurance product (Samson and Thomas 1985).Only recently did researchers begin to look into the cooperative behavior of both parties to reach a joint-party optimality (see Appendix B.3).In addition, recent growth of promising decision analysis based on multiple criteria has ignited sparks in the reinsurance field of study (See Appendix B.2.4).Complete review of existing decision-making methodologies in reinsurance is included in this study in Appendix B. In particular, this study is developed upon three recent researches ((Bazaz and Najafabadi 2015), (Bulut Karageyik and Şahin 2017) and (Payandeh-Najafabadi and Panahi-Bazaz 2017)) which focus on Multi-Attribute Decision Making (MADM) and Proportional-Stop-loss Reinsurance.
Basak and David (2015) first proposes to use MADM to the problem of selecting optimal reinsurance level under competing criteria.In choosing the input alternatives, they use ruin probability as a constraint, i.e. the insurance company should not have a probability of ruin greater than 1%.Loss distribution was modeled as the translated gamma process and the reinsurance forms considered was pure proportional and pure stop-loss reinsurance.The study also includes comparison with single criterion decision making and concludes that MADM is extremely insightful for selecting optimal reinsurance.
Later, Basak and Sule (2017) improves on the research to include Value-at-Risk (VaR) measurement into consideration, specifically targeting at optimal retention level in excess-of-loss reinsurance design.Key measurement criteria are expected profit, expected shortfall, finite time ruin probability and variance of risk.By comparing and contrasting different MADM techniques, the authors safely conclude that under the case of reinsurance where correlation between measurements are low enough, different MADM techniques will generate similar optimal retention level.
However, both studies were focusing either pure proportional reinsurance or pure nonproportional reinsurance, with neither considering the combination of both.The only comprehensive discussion of proportional-stop-loss reinsurance up-to-date is conducted by Hürlimann (2017), which took a viewpoint from both the insurer and the reinsurer.However, the limitation of this research is that it has a narrow focus on only VaR measurement to arrive at a closed-form determination of optimal reinsurance.
To the best of my knowledge, there is no previous research conducting analysis of proportionalstop-loss reinsurance based on multiple measurement schemes and considering decisions from both parties.Thus, this study serves the purpose of filling this gap.More specifically, Section 3 will incorporate reinsurance into the trade procedure of deals done between two parties.Decision making models with single measurement of variance of risk will be used in modeling reinsurer offerings and MADM model will be used for best reinsurance selection by insurance company.

Decision Flow
As reinsurance decision making involves the seller party (the reinsurance company, or the reinsurer) and the buyer party (the insurance company, or the reinsured), it could be safely viewed as a two-sided trade process, which involves negotiation between the selling and buying party.Furthermore, reinsurance deals could adapt to established two-sided trade matching models P1(Figure 1) or P2 (Figure 2) with the existence of a broker (Liang 2014).For the first trade procedure P1, both reinsurer and the reinsured exchange information through the broker.Previous researches have suggested to model P1 using two-sided cooperative game with incomplete information (See (Borch 1960) and (Wang 2003)), while little literature has discussed about the reinsurance deals settled under procedure P2.Under P2, the seller (the reinsurer) will provide several plans for the buyer to choose from.However, noticing the prevalence of procedure P2 in reinsurance trading practice, this study attempts to model reinsurance scenario under P2 by using MODM in providing reinsurance alternatives and by using MADM in selecting appropriate reinsurance design for the reinsured.Thus, the research design develops as follows in Flowchart 3

The Proportional-Stop-Loss Reinsurance Model
The paper discusses Proportional-Stop-Loss reinsurance adopting definitions from (Samson and Thomas 1985), (Hürlimann 2011) and (Payandeh-Najafabadi and Panahi-Bazaz 2017).Under which, given a single loss of X and a reinsurance arrangement with parameters ( , ) the reinsurer is bonded to pay a claim amount of: where a is the fraction ceded to the reinsurer and M is the retention limit.The reinsured will pay the rest of the claim = − .When M=0, the Proportional-Stop-Loss model becomes the classical quota-share reinsurance model, and when a=1, it becomes the classical stop-loss reinsurance model.

Variable Definition
To ensure the consistency of notations in this paper, we define the key variables as Table 3.1.Almost all definitions follow previous literature, and necessary elaborations will be given in later sections.the premium payable to the reinsurer; the expected shortfall with a confidence level of ; the expected profit gained by insurance company; ( ) the ruin probability of insurance company's wealth ( ); ( ) the utility of insurance company at the end of period t;

Providing Alternatives Using MODM
Considering the reinsurance practices and following previous research on joint-party reinsurance problem, this study attempts to model the reinsurer pricing objectives under Value-at-Risk (VaR) measurement.We attempt to formulate a model maximizing the reinsurer expected profit while minimizing the variance of profit.In deciding a reinsurance design, the reinsurer needs to specify the premium and the arrangement of reinsurance claim amount, in other words, the reinsurance premium loading factor and the reinsurance design parameter ( , ).Under the Expected Value Premium Principle, insurance premium must be at least greater than expected individual loss (Bulut Karageyik and Şahin 2017).Thus, the bi-objective model is formulated as: where is defined as the aggregated claim of loss (compounded from individual loss ), is the premium paid to reinsurance company per unit time, defined according to expected value premium principle.(Formulas in Section 2.6.1).
Clearly, there is conflict between two objectives and there is no single design of ( , , ) that could achieve all objectives.The closed-form optimality derivations (Hürlimann 2011) are omitted and the optimal solution would be an efficient frontier analyzed in closed form.The optimal pairs will satisfy: Note that for an increasing ceding level a, the reinsurer risk and expected profit will both increase proportionally, thus the reinsurer preference will be ambiguous for different ceding portion while fixing the pair of ( , ).This is in line with (Payandeh-Najafabadi and Panahi-Bazaz 2017) which suggests that optimal design ( , , ) depend on the loss distribution (in our case, ) but not on the market premium ( ) , and does not depend on the portion retained ( ).Thus, it would be flexible for reinsurance company to select an appropriate ceding portion given their risk appetite and their financial capability (which is often not necessarily known by the broker).In Section 4, we will briefly discuss the resulting effects of choosing different ceding portion , based on numerical case study.Thus, the alternatives provided by the reinsurance firm will be in the form of ( , , ).These are inputing alternatives we will use to apply MADM.For illustration, Figure 6 shows the optimal pairs of ( , ) given other parameters in the case study.Peer-reviewed version available at J. Risk Financial Manag.2017, 10, 22; doi:10.3390/jrfm10040022

Calculating Decision Criteria
Now we need to define the selection criteria for reinsurance design.In this study, we are concerned with expected profit, expected shortfall, ruin probability and expected utility as selection criteria.All of them are calculated taking the viewpoints of reinsurance buyers (the reinsured).

Expected Profit of Insurance Company (the Reinsured)
In general, the expected profit of the reinsured is calculated as the difference between the insurer's income and the claims paid to the policyholders.Net premium gained by the insurance company is calculated under the Expected Value Premium Principle, defined as: * =Total Premium Income -Reinsurance Premium The net insurance profit after considering the reinsurance arrangement is Our objective is to maximize the expected profit of the insurance company.

Expected Shortfall ES
Expected shortfall is calculated under Value at Risk (VaR) measurement.VaR given a confidence level of (0,1) is defined as the smallest such that the probability of loss < * is at least (Bazaz and Najafabadi 2015), i.e.
Expected shortfall is the financial risk measurement to investigate market risk of the portfolio.It is calculated as the expected value of tail distribution of as follows: An increase in retention level M will cause the insurer's liability to insurance policyholders to increase, and thus ES will increase accordingly.In contrast, a larger ceding portion of a will release insurer from burden and thus will decrease the amount of liability held by the reinsured.Our objective is to find the optimal (a,M) pair that could minimize the expected shortfall of insurance company.

Ruin Probability
The ruin probability criterion is based on definitions of finite time ruin probability measurement.The insurer's asset is represented as ( ) and is defined by: In equation ( 8) * is the net premium income per unit time gained by the insurance company, and S(t) is the aggregate claim amount up to time t, which is calculated by: The finite time ruin probability, ( , ), is given as: ( , ) = ( ( ) < 0) for some s, 0 < ≤ (11) In our study, the ruin probability is approximated through simulation study as the closed form ruin probability for compounding exponential loss distribution under Proportional-Stop-Loss reinsurance design is hard to obtain.Our objective is to minimize the ruin probability of ( , ) such that the insurance company would be less likely to bankrupt if there is a large loss incurred.

Expected Utility
To address the utility theory used in vast literature on reinsurance optimization (Samson and Thomas 1983), the utility function of the reinsured is defined as exponential utility function, which assumes constant absolute risk aversion: In reality, utility function may have much more complexity and may be different for different insurance companies.However, as long as the value of utility could be obtained in numeric value, decision could be made through MADM.In deciding the optimal reinsurance alternative, one of our objectives is to maximize the expected utility of the insurance company.

Selecting the Best Alternative Using MADM
In Section 2, we reviewed decision analysis techniques on reinsurance decisions under single measurement.In order to model the decision of the reinsurance purchasing party (the insurance company or the reinsured) under multiple measurement criteria, this study adopts Multi-Criteria Decision Making techniques.In particular, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), reviewed by previous research (Bazaz and Najafabadi 2015) as the most popular MADM technique and most suitable for pure numerical criteria, is applied to the reinsurance selection problem.Furthermore, suggested by (Bulut Karageyik and Şahin 2017), the correlation between criteria in reinsurance problem is small enough to return similar results from different TOPSIS methodologies, thus in this study we choose the classical TOPSIS method to support our analysis.
Following similar definitions of TOPSIS in previous study (Bazaz and Najafabadi 2015), (Ameri Sianaki 2015) and (Bulut Karageyik and Şahin 2017), we briefly describe the steps of applying the method as follows.This study attempts to implement the TOPSIS decision supporting system by storing reinsurance alternatives in Excel and processing the input matrices with MATLAB code.Part of the MATLAB code was developed with reference to previous efforts by Amari (Ameri Sianaki 2015), and was revised accordingly to serve the needs of this study.Below is the complete procedure of conducting TOPSIS.One may notice that by scaling the criteria (multiplying a constant to ), the decision would not change; However, it will not necessarily return same decision for different utility functions that generate same decision under expected utility measurement as adding a constant to in the formula will change the resulting .4. Calculate the weighted normalized decision matrix by using normalized decision matrix parameter and weight vector = ( , , .
6. Calculate the distance between each alternative and the positive and negative ideal points.The distance between alternative and the positive ideal points is The distance between alternative and the negative ideal solutions are: Calculate the relative closeness coefficient of each alternative represented as: 8. Rank the alternatives according to .The alternative with higher value is preferred over lower alternatives.
A graphical representation of TOPSIS is shown in Figure 4.Each blue ball represents one available alternative.The red ball represents the negative ideal solution and the green ball represents the positive ideal solution.The blue ball that is relatively near to the green ball and away from the red ball would be the best alternative amongst all.Given at least 4 selecting criteria, it would be hard to visualize the alternatives in 3-dimensional space, thus the calculation of distance is coded using MATLAB.Peer-reviewed version available at J. Risk Financial Manag.2017, 10, 22; doi:10.3390/jrfm10040022

Case study
Following the decision flow in Section 2 this study models the reinsurance deal procedure as illustrated in Figure 5.We will choose a loss distribution model in Section 3.1, generate reinsurance offering alternatives in Section 3.2, tabulate decision matrix in Section 3.3 and finally apply MADM in selecting from alternatives in Section 3.4.
Thus, S is the compounding distribution of ( ) identical, independently distributed risk each with distribution X.

Generating Alternatives from Viewpoints of Reinsurers
Following Section 4.2, we define the case by setting the portion ceded as a=0.6, a=0.75, a=0.9 and a=1 respectively in assessing the differences in results when retention level is changed.The optimal pair ( , ) (Figure 6 with on x-axis and M on y-axis) are sought by grid search as solving Formula 3.2 in mathematics form may not be succinct.When < 0.1, constraint < will be violated; all points to the right and below of the efficient frontier is deemed as inferior to the points on the efficient frontier.In managerial terms, the reinsurance designs with parameters to the southeast of the efficient frontier will cause the reinsurance company to likely generate less profit while suffering from a larger risk.

Constructing decision matrix
For the first trial, we fixed the ceding portion at a=0.6 and select 35 pairs of the optimal ( , ) as the reinsurance design parameters for TOPSIS alternatives.Profit for insurance company and expected utility after claims are calculated using theoretical mean of the random distributions.However, given the claim process as compounding exponential loss with Poisson occurrence, ruin probability and expected shortfall at 95% confidence level are hard to obtain in analytic terms.Thus, by using Monte Carlo Simulation with 100,000 iterations, loss and claim is modeled as exponential value with Poisson occurrence and .and ruin probability ( , ) are calculated in Excel as follows: By retrieving criteria value and reformatting in the tabulated workbook for processing TOPSIS, the decision matrix is built as shown in Figure 8 and is ready for processing using MATLAB coded developed in Section 2.

Selecting alternative using TOPSIS
With above alternatives as inputting decision matrix, we could observe that if we only consider minimizing expected shortfall and ruin probability, the profit or expected utility will be exceptionally low.The color scale shows intuitively this conflict with red representing smallest value and green representing largest value.We define the weight vector of criteria as all equals (equal values in blue cells Range F39:I39).Calling the Matlab function built earlier in Section 2 by executing the following code: topsis (decisionMakingMatrix,lambdaWeight,criteriaSign) we could get normalized weight matrix, the identified ideal solutions and the distance between each alternative and the ideal optimality.All these results are stored in Excel sheet "TOPSIS OUTPUT Variables" with selection ranking results shown in Figure 9.

Managerial Implications
The result shows several interesting findings: 1.The best alternative suggested by TOPSIS 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 high probability of financial crisis.On the other hand, if the decision merely considers constraining higher shortfalls, insurance company may look bad on their profit and loss statement due to low profit earned.By increasing the ceding amount from a=0.6 to a=0.75,0.9 and 1 result from Trial 1,2,3,4 suggests that the ranking of alternatives is different when parameters are changed.When ceding portion 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 , or ) are altered, it is recommended for insurance company to evaluate again the reinsurance plans instead of extrapolating conclusions from previous experiences.2. In addition, Trial 4 with a=1 is modeling excess-of-loss reinsurance form where = (0, − ) = 1 * (0, − ) .Accordingly, result from Trial 4 are in correspondence with previous knowledge on excess-of-loss reinsurance.Under excess-of-loss reinsurance, the best form is given at = , which is in correspondence with Section 2 in Payandeh-Najafabadi & Panahi-Bazaz (Payandeh-Najafabadi and Panahi-Bazaz 2017).3.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 ( = ) with fixed reinsurance premium loading factor .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 bankrupt if large losses are incurred.As a 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, note the high ruin probability which suggests a much higher risk of bankrupting, the insurance company will often seek for reinsurance to keep ruin probability low. 5. Furthermore, through the simulation process, the variance and profitability of the reinsurer are also being observed and calculated (as could be seen from Figure 10.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 design that only differs in parameter a.

Contributions
The research has the following contributions: 1. To the best of our knowledge, this is the first theoretical study using MADM to approach Proportional-Stop-loss reinsurance model, though there is small amount of recent studies using MADM in designing either pure proportional or pure stop-loss reinsurance contract; 2. It is one of the few studies taking a non-discriminatory position considering both the insurance and the reinsurance company in designing optimal reinsurance contract, and the study made significant contribution by incorporating existing MODM models and promising MADM model in one decision flow to arrive at robust decision for reinsurance design; 3.This study demonstrates the feasibility to incorporate intelligent decision supporting system 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.4. 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 Exposure rating, however, rely more on current exposure in predicting future losses.This technique relates premium to likely losses.To achieve this, underwriter needs to set up risk profiles of the portfolio in bands.A "first loss curve" (WillisRe 2013) could be used to estimate the split of premium required between the deductible and cover of any insured or reinsured risk.The first loss curves are tailored for each portfolio under examination and assumes a common form of Gamma, Lognormal, etc.According to WillisRe (2013), Pareto model is the most widely used model in reinsurance and could be used in conjunction with experience and exposure rating to estimate risk premiums for excess of loss treaties with high deductibles, where loss experience is insufficient and could be sometimes misleading.However, some criticize the classical Pareto model of its deficiency in modelling loss frequency (Fackler n.d.).The classic Pareto model either fits the small-to-medium risk or large risk well but not both at the same time.Fackler (Fackler n.d.) discusses the generalization and extension of Pareto Model to improve its fitness of loss distribution.

B.1.4 Stochastic pricing
In traditional actuarial pricing models, models used to approximate expected losses from specific occurrences are usually deterministic based on specific loss occurrences (WillisRe 2013).Recent research has been approaching the model using stochastic (probabilistic) approach taking account of the probability attached to various outcomes.Stochastic modeling is considered to sit inbetween experience and exposure methods in that assumptions could depend on either loss history or current exposure or a combination of both.Stochastic models generally provide a better approximation than simple deterministic calculations.

B.2 Decision making for ceding companies
Insurance company usually choose to purchase appropriate reinsurance in the hope of achieving: (1) Increased diversity and financial flexibility in balance sheet; (2) Increased sustainability of operating performance, especially in accident years; and (3) Diversified business profile in enterprise risk management.For an insurer to choose the most appropriate reinsurance design regarding type (as in Appendix A) and parameters, quantitative theories have been developed and applied to aid the decision making process.

B.2.1 Operations research.
Since the seminal work of Borch (1960) (Borck 1960), optimal insurance/reinsurance problems have been studied extensively (Cheung et al. 2014).Recently, because of the promising risk measures such as value at risk (VaR) and conditional value at risk (CVaR) in quantifying financial and insurance risks, the study of risk measure based optimal reinsurance problems has attracted great attention, such as (Cai et al. 2008).Not only stochastic models based on current situation was discussed (Zeng and Luo 2013a), dynamic programming has also been utilized to model multi-period optimization in reinsurance, such as (Chunxiang and Li 2015) and (Guan and Liang 2014).

B.2.2 Utility theory and decision modelling.
Friefelder, as cited in Samson and Thomas (1983), suggests that mean-variance approach which most operations research models were based have obvious weakness and that utility theory is the best method in determining property and liability insurance rates.Samson and Thomas (1983) (Samson and Thomas 1983) first approaches the topic of decision making in reinsurance using utility theory, formulation of which suggests that the choice of an appropriate utility function is important to decision making process.They later on suggest using decision analysis tools such as decision tree structure in modelling ceding company's buying quantity of excess-of-loss treaty reinsurance (Samson & Thomas, 1985).A rough reproduction of the simulation model using DPL software is shown in Figure B1.This model could be further improved using continuous distribution in modelling loss profile.This model illustrates the feasibility of using decision analysis in choosing whether to buy Excess-of-loss reinsurance and the quantity of reinstatement to buy.Smith and Nau (1995) discussed thoroughly how option pricing techniques could be used to simplify investment decision making when risk-hedging options are available.Given that reinsurance itself is a risky project, Brotman (1987) suggests that reinsurance treaty arrangement and futures market share similar characteristics and that hedging strategies using options to buy or sell reinsurance could meet special need of insurance companies.

B.2.4 Multi-attribute decision making.
All the reinsurance optimization models in previous sections mostly aid in decision making using single criteria.However, Basak and David (Bulut Karageyik and Dickson 2016) provides new insights to approach optimal retention level in proportional quota share reinsurance using multi attribute decision making (MADM).In their discussion, "released capital, expected profit and expected utility of resulting wealth" are the competing criteria of choice, and compare the results with that of single attribute decision making.Following this study, using MADM in determining optimal retention level for proportional reinsurance has been extensively evaluated by Basak and Sule (Bulut Karageyik and Şahin 2017).The authors use measures of Expected Shortfall, expected profit and ruin probability taking the perspective of reinsurance buyer, and they conclude that as relationship between criteria is too small, there is no significant difference in results using different MADM techniques such as between TOPSIS-Euclidean and TOPSIS-Mahalanobis.

B.3 Optimal reinsurance considering both parties
Recently, intriguing models started to address the promising and practical issue of reaching a "win-win" policy, such as (Hürlimann 2011) and (Bazaz and Najafabadi 2015) taking views from both cedent and reinsurer.and applying Conditional tail expectation (cVaR, CTE) in deciding optimal stop-loss limits that maximizes joint party profit while controlling joint party risks.Dimitrova & Kaishev (Dimitrova and Kaishev 2010) based their analysis on maximizing joint survival probability to get the efficient frontier reinsurance design.Other mathematical models such as game theory could also be applied to reinsurance negotiation as demonstrated by Zeng and Luo (2013) in seeking for a Pareto Optimal proportional reinsurance policy.A recent research (Payandeh-Najafabadi and Panahi-Bazaz 2017) looks into the joint party optimal reinsurance design with combination of proportional and stop-loss reinsurance contract.By using Bayesian estimator, the research seeks to achieve maximum ending wealth for both party.We could see that under this category, almost all existing studies are using at most two decision criteria.

Figure 5 .
Figure 5. Case study flowchart in modeling real-world reinsurance deals 3.1 Loss Distribution Modeling Under treaty reinsurance which covers the entire line of insurance business handled by the insurance company, the aggregated claim is a compounding distribution of single risks or claims incurred during time period .Following the majority of research study on modeling loss or claim, such as (Samson and Thomas 1985), (Bazaz and Najafabadi 2015), (Payandeh-Najafabadi and Panahi-Bazaz 2017) and (Bulut Karageyik and Şahin 2017), our case study chooses to model individual claim as exponential loss model with parameter = 100, i.e.

Figure 6 .
Figure 6.Optimal Pairs of reinsurance design given constant ceding portion

Figure 7 .
Figure 7. Screenshot of criteria calculation using Monte Carlo Simulation with @Risk

Figure 8 .
Figure 8. Build decision matrix for preparation of TOPSIS

PreprintsFigure 9 .
Figure 9. Ranking of the alternatives based on TOPSIS From the TOPSIS output, we could identify that Alternative 18 (0.27,100) is the best choice followed closely by Alternative 17 (0.26,96).Alternative 19 (0.28,103) is not far away as the third best alternative identified.Alternative 35 (0.44, 149) is the furthest from ideal solutions.Multiple trials were tested fixing a at a=0.75 (Trial 2), a=0.9 (Trial 3) and a=1 (Trial 4).Trial 4 resembles pure Excessof-loss Reinsurance to compare and contrast decision differences under different ceding portion in Proportional-Stop-loss Reinsurance design.Results of these trials are included in Appendix C. From the result, we could draw insightful managerial implications.

PreprintsFigure 10 .
Figure 10.Trial 5 Pure proportional reinsurance selection4.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 bankrupt if large losses are incurred.As a 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, note the high ruin probability which suggests a much higher risk of bankrupting, the insurance company will often seek for reinsurance to keep ruin probability low. 5. Furthermore, through the simulation process, the variance and profitability of the reinsurer are also being observed and calculated (as could be seen from Figure10.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 design that only differs in parameter a.

Figure B1 .
Figure B1.Screenshot of excess of loss decision model using DPL B.2.3 Options theory.

Preprints 21 Figure
Figure C3.Trial 3 Screen-shot of crucial TOPSIS output parameters

Table 1 .
Key variable definitions