#### 3.1. Decision Flow

As reinsurance decision-making involves the selling party (the reinsurance company, or the reinsurer) and the buying party (the insurance company, or the reinsured party), it could be safely viewed as a two-sided trade process, which involves negotiation between the selling and buying parties. Furthermore, reinsurance deals could be viewed as the 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 the reinsurer and the reinsured party exchange information through the broker. Previous research on two-sided optimality could largely be viewed as P1 when both firms make decisions simultaneously. In this case, an equilibrium strategy either optimizing one side or both side’s benefits could be reached (see

Borch 1960;

Wang 2003;

Cai et al. 2017), while few works in literature have discussed reinsurance deals settled under procedure P2. Under P2, the seller (the reinsurer) will provide several plans for the buyer to choose from. Noting the prevalence of procedure P2 in reinsurance industry practice, this study attempts to model reinsurance scenarios under P2 by using classic MODM in providing reinsurance alternatives and by using MADM in selecting an appropriate reinsurance design for the reinsured party. The rest of the section follows this decision flow.

Section 3.2 defines the proportional-stop-loss reinsurance form;

Section 3.4 constructs one simplified set of reinsurer’s offerings using MODM.

Section 3.5 defines each of the key criteria the reinsured may consider. Finally,

Section 3.6 presents the application of MADM in aiding the decision of ranking the reinsurance alternatives.

#### 3.2. 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

$\left(a,M\right)$, 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 party will pay the rest of the claim

${X}_{i}=X-{X}_{r}$, where

${X}_{r}$ represents the amount of claim paid by the reinsurer. 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.

#### 3.4. Simulating Alternatives Using MODM

Considering the reinsurance practices and following previous research on the joint-party reinsurance problem, we first attempt to model the reinsurer pricing objectives. As such, we provide a list of alternatives for the reinsured party to choose from. We attempt to formulate a model maximizing the reinsurer’s expected profit while minimizing the variance of profit. In deciding a reinsurance design, the reinsurer needs to specify the premium and the arrangement of the reinsurance claim amount, in other words, the reinsurance premium loading factor

$\zeta $ and the reinsurance design parameter

$\left(a,M\right)$. Under the expected value premium principle, the insurance premium must be at least greater than the expected individual loss (

Karageyik and Şahin 2017). Thus, the bi-objective model is formulated as:

where

${S}_{r}$ is defined as the aggregated claim of loss (compounded from individual loss

${X}_{r}$), and

${c}_{r}$ is the premium paid to the reinsurance company per unit of time, defined according to the expected value premium principle (formulas are in

Section 3.5.1).

Clearly, there is conflict between two objectives and there is no single design of

$\left(\zeta ,a,M\right)$ that can achieve all objectives. The closed-form derivations (

Hürlimann 2011) are omitted and the optimal solution set 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

$a$ while fixing the pair of

$\left(\zeta ,M\right)$. This is in line with the work of

Payandeh-Najafabadi and Panahi-Bazaz (

2017) where it is suggested that optimal design

$\left(a,M,\zeta \right)$ depends on the loss distribution (in our case,

$\lambda $) but not on the market premium (

$\theta $) , and does not depend on the portion retained (

$a$). Thus, it would be flexible for the reinsurance company to select an appropriate ceding portion

$a$ given their risk appetite and their financial capability (which is often not necessarily known by the broker). In

Section 5, we will briefly discuss the resulting effects of choosing different ceding portions of

$a$, based on a numerical case study.

Thus, the alternatives provided by the reinsurance firm will be in the form of $\left(a,M,\zeta \right)$. These are inputing alternatives we will use to apply MADM.

#### 3.5. 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 these factors are calculated taking the viewpoints of reinsurance buyers (the reinsured parties).

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

In general, the expected profit of the reinsured party 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:

The net insurance profit after considering the reinsurance arrangement is:

Our objective is to maximize the expected profit of the insurance company.

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

Expected shortfall is calculated under value at risk (VaR) measurement. VaR, given a confidence level of

$\alpha \in \left(0,1\right)$, is defined as the smallest

$l$, such that the probability of loss

$L<{l}^{\ast}$ is at least

$\alpha $ (

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

$Va{R}_{\alpha}$ 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 will release the insurer from burden and thus will decrease the amount of liability held by the reinsured party. Our objective is to find the optimal (a,M) pair that could minimize the expected shortfall of the insurance company.

#### 3.5.3. Ruin Probability

The ruin probability criterion is based on definitions of finite time ruin probability measurement. The insurer’s asset is represented as

$W\left(t\right)$ and is defined by:

In Equation (8)

${c}^{\ast}$ 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,

$\psi \left({w}_{0},t\right)$, is given as:

In our study, the ruin probability is approximated through a 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 $\psi \left(w,t\right)$ such that the insurance company would be less likely to go bankrupt if there is a large loss incurred.

#### 3.5.4. 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, decisions could be made through MADM. In deciding the ranking of reinsurance alternatives, one of our objectives is to maximize the expected utility of the insurance company.

#### 3.6. 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 in prior works (

Bazaz and Najafabadi 2015) is the most popular MADM technique and is the most suitable for pure numerical criteria. TOPSIS was originally developed by

Hwang and Yoon (

1981), and was later developed by

Yoon (

1987) and

Hwang et al. (

1993).

TOPSIS is thus applied to the reinsurance selection problem. Furthermore, as suggested by

Karageyik and Şahin (

2017), the correlation amongst 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 a previous study (

Bazaz and Najafabadi 2015;

Ameri Sianaki 2015;

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 a 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.

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:

Normalize the decision matrix using the following formula:

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:

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:

The distance between alternative and the negative ideal solutions are:

Calculate the relative closeness coefficient of each alternative represented as:

Rank the alternatives according to ${C}_{i}$. The alternative with higher ${C}_{i}$ value is preferred over lower ${C}_{i}$ alternatives.

A graphical representation of TOPSIS is shown in

Figure 3. 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 four selection criteria, it would be hard to visualize the alternatives in 3-dimensional space, thus the calculation of distance is coded using MATLAB.