# Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms

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## Abstract

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## 1. Introduction

## 2. Black-Box Optimization Problems in Reinsurance

- $\mathcal{S}$ is a search space, formed by feasible elements $\mathbf{x}\in \mathcal{S}$.
- $f\left(\mathbf{x}\right)$ is an objective function $\mathcal{S}\to \mathbb{R}$, to be optimized (maximized or minimized).

#### 2.1. Problem 1: Excess of Loss Reinsurance

#### 2.2. Problem 2: Stop-Loss Reinsurance

#### 2.3. Problem 3: Threshold Proportional Reinsurance

## 3. Evolutionary-Based Algorithms

#### 3.1. Evolutionary Algorithms: Evolutionary Programming

- Generate an initial population of μ individuals (solutions). Let t be a counter for the number of generations; set it to $t=1$. Each individual is taken as a pair of real-valued vectors, $\left(\right)open="("\; close=")">{\mathbf{x}}_{i},{\sigma}_{i}$, $\forall i\in \left(\right)open="\{"\; close="\}">1,\cdots ,\mu $, where ${\mathbf{x}}_{i}$’s are objective variables and ${\sigma}_{i}$’s are standard deviations forGaussian mutations.
- Evaluate the fitness value for each individual (${\mathbf{x}}_{i}$, ${\sigma}_{i}$) (using the problem’s objective function).
- Each parent (${\mathbf{x}}_{i}$, ${\sigma}_{i}$), $\{i=1,\cdots ,\mu \}$, then creates a single offspring (${\mathbf{x}}_{i}^{\prime}$, ${\sigma}_{i}^{\prime}$) as follows (j denotes components of the i-th vector):$${{\mathbf{x}}^{\prime}}_{i}\left(j\right)={\mathbf{x}}_{i}\left(j\right)+{\sigma}_{i}\left(j\right)\xb7{N}_{j}(0,1)$$$${\sigma}_{i}^{\prime}\left(j\right)={\sigma}_{i}\left(j\right)\xb7exp({\tau}^{\prime}\xb7N(0,1)+\tau \xb7{N}_{j}(0,1))$$
- If ${x}_{i}\left(j\right)>lim\_sup$, then ${x}_{i}\left(j\right)=lim\_sup$, and if ${x}_{i}\left(j\right)<lim\_inf$, then ${x}_{i}\left(j\right)=lim\_inf$.
- Calculate the fitness values associated with each offspring (${\mathbf{x}}_{i}^{\prime}$,${{\sigma}^{\prime}}_{i}),\phantom{\rule{3.33333pt}{0ex}}\forall i\in \{1,\cdots ,\mu \}$.
- Conduct pairwise comparison over the union of parents and offspring: for each individual, p opponents are chosen uniformly at random from all the parents and offspring. For each comparison, if the individual’s fitness is better than the opponent’s, it receives a “win”.
- Select the μ individuals out of the union of parents and offspring that have the most “wins” to be parents of the next generation.
- Stop if the halting criterion is satisfied, and if not, set $t=t+1$ and go to Step 3.

#### 3.2. Particle Swarm Optimization

## 4. Numerical Results

**Figure 1.**The objective function and zoom of the excess of the loss reinsurance optimization problem.

#### 4.1. Results in Problem 1

**Table 1.**Results in Problem 1 (an excess of the loss reinsurance model) using the evolutionary programming (EP) and particle swarm optimization (PSO) algorithms.

Algorithm | ${\psi}_{I,R}\left({c}_{R}\right)$ | ${c}_{R}$ | Computation time (s) |
---|---|---|---|

EP | 0.57372111552 | 0.1200013 | 3.5 |

PSO | 0.5737211153 | 0.1200014 | 2.2 |

#### 4.2. Results in Problem 2

Algorithm | $f\left({c}_{R}\right)$ | ${c}_{R}$ | Computation time (s) |
---|---|---|---|

EP | 0.0014486748 | 0.14 | 2.9 |

PSO | 0.0014486748 | 0.14 | 2.4 |

#### 4.3. Results in Problem 3

**Table 3.**Results in Problem 3 (the threshold proportional reinsurance model) using the EP and PSO algorithms.

Algorithm | ${\psi}_{I}({k}_{1},{k}_{2},b)$ | ${k}_{1}$ | ${k}_{2}$ | b | Computation time (s) |
---|---|---|---|---|---|

Exponential | |||||

EP | 0.4980669653 | 1.0 | 0.7596477801 | 3.2688654179 | 9.6 |

PSO | 0.4980669664 | 1.0 | 0.7596477914 | 3.2688441178 | 8.5 |

Erlang(2,2) | |||||

EP | 0.415635 | 1.0 | 0.761572 | 1.9871 | 9.5 |

PSO | 0.415641 | 1.0 | 0.761564 | 1.9866 | 8.5 |

#### 4.4. Discussion

Problem # | Optimal solution | Computation time (s) |
---|---|---|

Problem 1 | ${c}_{R}=0.1200000$; ${\psi}_{I,R}\left({c}_{R}\right)=0.573721115524965$ | 2100 |

Problem 2 | ${c}_{R}=0.1400000$; $f\left({c}_{R}\right)=0.001448674863134$ | 2840 |

Problem 3 (Exponential) | ${k}_{1}=1.0000000000$; ${k}_{2}=0.759647780145614$; | 3700 |

Problem 3 (Erlang) | ${k}_{1}=1.000000$; ${k}_{2}=0.761572$; $b=1.9871$; ${\psi}_{I}({k}_{1},{k}_{2},b)=0.415635$ | 3850 |

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Salcedo-Sanz, S.; Carro-Calvo, L.; Claramunt, M.; Castañer, A.; Mármol, M.
Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms. *Risks* **2014**, *2*, 132-145.
https://doi.org/10.3390/risks2020132

**AMA Style**

Salcedo-Sanz S, Carro-Calvo L, Claramunt M, Castañer A, Mármol M.
Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms. *Risks*. 2014; 2(2):132-145.
https://doi.org/10.3390/risks2020132

**Chicago/Turabian Style**

Salcedo-Sanz, Sancho, Leo Carro-Calvo, Mercè Claramunt, Ana Castañer, and Maite Mármol.
2014. "Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms" *Risks* 2, no. 2: 132-145.
https://doi.org/10.3390/risks2020132