Combining chain-ladder claims reserving with fuzzy numbers

In this paper we extend the classical chain-ladder claims reserving method using fuzzy methods. Therefore, we derive new estimators for the claims development factors as well as new predictors for the ultimate claims. The advantage in using fuzzy numbers lies in the fact that the model uncertainty is directly included in and can be controlled by the “new” fuzzy claims development factors. We also provide an estimator for the uncertainty of the ultimate claims for single accident years and for aggregated accident years.     

基于TOPSIS的模糊群体多属性决策方法

针对模糊群体多属性决策问题,给出一种基于理想点法(TOPSIS)的多属性决策方法.方法先用三角模糊数的形式表示专家评价值的模糊性和不确定性,而后考虑了专家在不同评价属性中的重要程度和意见的相似度,并将专家意见进行集结得到专家群体关于方案集的模糊决策矩阵,最后定义了三角模糊数形式的正负理想方案,通过计算各方案与正负理想方案的距离以及各方案与理想点的相对接近度,最终确定最优方案.通过实例分析说明了该方法的可行性和有效性.     

A micro-level claim count model with overdispersion and reporting delays

The accurate estimation of outstanding liabilities of an insurance company is an essential task. This is to meet regulatory requirements, but also to achieve efficient internal capital management. Over the recent years, there has been increasing interest in the utilisation of insurance data at a more granular level, and to model claims using stochastic processes. So far, this so-called ‘micro-level reserving’ approach has mainly focused on the Poisson process.In this paper, we propose and apply a Cox process approach to model the arrival process and reporting pattern of insurance claims. This allows for over-dispersion and serial dependency in claim counts, which are typical features in real data. We explicitly consider risk exposure and reporting delays, and show how to use our model to predict the numbers of Incurred-But-Not-Reported (IBNR) claims. The model is calibrated and illustrated using real data from the AUSI data set.     

复模糊数及其收敛性

在新的模糊数序关系意义下,将定义在所有实模糊数上的模糊距离推广到所有复模糊数集上的模糊距离.并以此为基础,定义了复模糊数列的极限,讨论了复模糊数列的保号性、有界性等收敛性质及C auchy收敛判别法等.     

具有模糊支付的Moran过程演化博弈动态

以往对演化博弈的研究都假设个体从博弈中获得的支付是确定的并以精确的数来表示。然而由于受环境中各种不确定因素的影响,个体博弈时所获得的支付并不是一个精确的数值,而需要用一个模糊数来表示。本文研究模糊支付下2×2的对称博弈, 利用模糊数的运算, 分析具有模糊支付的有限种群Moran过程演化动态。在弱选择下以梯形模糊数和三角模糊数表示博弈支付,计算策略的模糊扎根概率,分析自然选择有利于策略扎根及策略成为模糊演化稳定策略的条件。将经典博弈推广到模糊环境中丰富了演化博弈理论,更具有现实意义。     

Parameter Estimation of Multivariate Regression Model with Fuzzy Random Errors

In many real-world problems, observations are usually described by approximate values due to fuzzy uncertainty, unlikeprobabilistic uncertainty that has nothing to do with experimentation. The combination of statistical model and fuzzy set theory is helpful to improve the identification and analysis of complex systems. As an extension ofstatistical techniques, this study is an investigation of the relationship between fuzzy multiple explanatory variables and fuzzy response with numeric coefficients and the fuzzy random error term. In this work we describe a parameter estimation procedure carrying out the least-squares method in a complete metric space of fuzzy numbers to determine the coefficients based on the extension principle. We demonstrate how the fuzzy least squares estimators present large sample statistical properties, including asymptotic normality, strong consistency and confidence region. The estimators are also examined via asymptotic relative efficiency concerning traditional least squares estimators. Different from the construction of error term in Kim et al.\cite{21}, it is more reasonable in the proposed model since the problems of inconsistency in referring to fuzzy variable and producing the negative spreads may be avoided. The experimental study verifies that the proposed fuzzy least squares estimators achieve the meaning consistent with the theory identification for large sample data set and better generalization regarding one single variable model.     

A machine learning approach for individual claims reserving in insurance

Accurate loss reserves are an important item in the financial statement of an insurance company and are mostly evaluated by macrolevel models with aggregate data in run‐off triangles. In recent years, a new set of literature has considered individual claims data and proposed parametric reserving models based on claim history profiles. In this paper, we present a nonparametric and flexible approach for estimating outstanding liabilities using all the covariates associated to the policy, its policyholder, and all the information received by the insurance company on the individual claims since its reporting date. We develop a machine learning–based method and explain how to build specific subsets of data for the machine learning algorithms to be trained and assessed on. The choice for a nonparametric model leads to new issues since the target variables (claim occurrence and claim severity) are right‐censored most of the time. The performance of our approach is evaluated by comparing the predictive values of the reserve estimates with their true values on simulated data. We compare our individual approach with the most used aggregate data method, namely, chain ladder, with respect to the bias and the variance of the estimates. We also provide a short real case study based on a Dutch loan insurance portfolio.     

A marked Cox model for the number of IBNR claims: Theory

Incurred but not reported (IBNR) loss reserving is an important issue for Property & Casualty (P&C) insurers. To calculate IBNR reserve, one needs to model claim arrivals and then predict IBNR claims. However, factors such as temporal dependence among claim arrivals and environmental variation are often not incorporated in many of the current loss reserving models, which may greatly affect the accuracy of IBNR predictions.In this paper, we propose to model the claim arrival process together with its reporting delays as a marked Cox process. Our model is versatile in modeling temporal dependence, allowing also for natural interpretations. This paper focuses mainly on the theoretical aspects of the proposed model. We show that the associated reported claim process and IBNR claim process are both marked Cox processes with easily convertible intensity functions and marking distributions. The proposed model can also account for fluctuations in the exposure. By an order statistics property, we show that the corresponding discretely observed process preserves all the information about the claim arrivals. Finally, we derive closed-form expressions for both the autocorrelation function (ACF) and the distributions of the numbers of reported claims and IBNR claims. Model estimation and its applications are considered in a subsequent paper, Badescu et al. (2015b).     

Application of possibility theory to investment decisions

Carlson and Fuller (2001, Fuzzy Sets and Systems, 122, 315–326) introduced the concept of possibilistic mean, variance and covariance of fuzzy numbers. In this paper, we extend some of these results to a nonlinear type of fuzzy numbers called adaptive fuzzy numbers (see Bodjanova (2005, Information Science, 172, 73–89) for detail). We then discuss the application of these results to decision making problems in which the parameters may involve uncertainty and vagueness. As an application, we develop expression for fuzzy net present value (FNPV) of future cash flows involving adaptive fuzzy numbers by using their possibilistic moments. An illustrative numerical example is given to illustrate the results.     

On theoretical pricing of options with fuzzy estimators

In this paper we present an application of a new method of constructing fuzzy estimators for the parameters of a given probability distribution function, using statistical data. This application belongs to the financial field and especially to the section of financial engineering. In financial markets there are great fluctuations, thus the element of vagueness and uncertainty is frequent. This application concerns Theoretical Pricing of Options and in particular the Black and Scholes Options Pricing formula. We make use of fuzzy estimators for the volatility of stock returns and we consider the stock price as a symmetric triangular fuzzy number. Furthermore we apply the Black and Scholes formula by using adaptive fuzzy numbers introduced by Thiagarajah et al. [K. Thiagarajah, S.S. Appadoo, A. Thavaneswaran, Option valuation model with adaptive fuzzy numbers, Computers and Mathematics with Applications 53 (2007) 831–841] for the stock price and the volatility and we replace the fuzzy volatility and the fuzzy stock price by possibilistic mean value. We refer to both cases of call and put option prices according to the Black & Scholes model and also analyze the results to Greek parameters. Finally, a numerical example is presented for both methods and a comparison is realized based on the results.     

Ranking fuzzy numbers with integral value

Ranking fuzzy numbers is important in decision making. Since very often the alternatives are evaluated by fuzzy numbers in a vague environment, a comparison between these fuzzy numbers is indeed a comparison between alternatives. This paper proposes a method of ranking fuzzy numbers with integral value. The method, which is independent of the type of membership functions used and the normality of the functions, can rank more than two fuzzy numbers simultaneously. It is relatively simple in computation, especially in ranking triangular and trapezoidal fuzzy numbers. Further, an index of optimism is used to reflect the decision maker's optimistic attitude. Discussion on comparative advantages is included.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

Ranking generalized fuzzy numbers in fuzzy decision making based on the left and right transfer coefficients and areas

Although a number of recent studies have proposed ranking fuzzy numbers based on the deviation degree, most of them have exhibited several shortcomings associated with non-discriminative and counter-intuitive problems. In fact, none of the existing deviation degree methods has guaranteed consistencies between the ranking of fuzzy numbers and that of their images under all situations. They have also ignored decision maker’s attitude toward risk, which significantly influences final ranking result. To overcome the above-mentioned drawbacks, this study proposes a new approach for ranking fuzzy numbers that ensures full consideration for all information of fuzzy numbers. Accordingly, an overall ranking index is obtained by the integration of the information from the left and the right (LR) areas between fuzzy numbers, the centroid points of fuzzy numbers and the decision maker’s attitude toward risk. This new method is efficient for evaluating generalized fuzzy numbers and distinguishing symmetric fuzzy numbers. It also overcomes the shortcomings of the existing approaches based on deviation degree. Several numerical examples are provided to illustrate the superiority of the proposed approach. Lastly, a new fuzzy MCDM approach for generalized fuzzy numbers is proposed based on the proposed ranking approach and the concept of generalized fuzzy numbers. The proposed fuzzy MCDM approach does not require the normalization process and thus avoids the loss of information results from transforming generalized fuzzy numbers to normal form.     

基于集对分析联系数的信息不完全直觉模糊多属性决策

信息不完全直觉模糊多属性决策是一类不确定性决策问题,其不确定性来自属性权重信息不完全和属性值的直觉模糊数表示.为了系统地刻画直觉模糊多属性决策中的不确定性,避免直觉模糊多属性决策中利用得分函数做决策的片面性和不准确性,可以将信息不完全的权重和直觉模糊数表示的属性值转化成集对分析理论中的联系数,并建立信息不完全直觉模糊多属性决策模型,通过对不确定性进行分析后作出决策.实例应用表明该决策方法具有合理性和可行性.     

Assessing inflation risk in non-life insurance

Inflation risk is of high relevance in non-life insurers’ long-tail business and can have a major impact on claims reserving. In this paper, we empirically study claims inflation with focus on automobile liability insurance based on a data set provided by a large German non-life insurance company. The aim is to obtain empirical insight regarding the drivers of claims inflation risk and its impact on reserving. Toward this end, we use stepwise multiple regression analysis to identify relevant drivers based on economic indices related to health costs and consumer prices, amongst others. We further study the impact of (implicitly and explicitly) predicting calendar year inflation effects on claims reserves using stochastic inflation models. Our results show that drivers for claims inflation can considerably vary for different lines of business and emphasize the importance of explicitly dealing with (stochastic) claims inflation when calculating reserves.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

An individual loss reserving model with independent reporting and settlement

The main purpose of this paper is to assess and demonstrate the advantage of claims reserving models based on individual data in forecasting future liabilities over traditional models on aggregate data both theoretically and numerically. The available information consists of the reporting delays, settlement delays and claim payments. The model settings include Poisson distributed frequency of claims produced by each policy, claims payable at the settlement time, and the amount of payment depending only on its settlement delay. While such settings are applicable to certain but not all practical cases, the principal purpose of the paper is to examine the efficiency of individual data against aggregate data. We refer to loss reserving as to estimate the projections of the outstanding liabilities on observed information. The efficiency of the individual loss reserving against classical aggregate loss reservings, namely Chain-Ladder (C-L) and Bornhuetter–Ferguson (B–F), is assessed by comparing the asymptotic variances of the errors in estimating the conditional expectation (projection) of the outstanding liability between individual, C-L and B–F reservings. The research shows a significant increase in the accuracy of loss reserving by using individual data compared with aggregate data.     

一种新的梯形模糊数间距离及在投资组合模型选择中的应用

给出一种新的模糊数间的距离公式.用新定义的距离公式来度量给定优先资产的条件下投资组合的分散度.用可能性均值度量投资组合的收益,可能性半方差度量投资组合的风险,在收益和风险满足一定的条件下构造分散度模型.通过实例分析,给出的方法不仅分散度更好,而且资产分配多元化程度更高,计算也更简单.     

基于Hausdauff度量的模糊TOPSIS方法研究

针对模糊多属性决策中的模糊 TOPSIS方法 ,提出了一种基于 Hausdauff度量的模糊 TOPSIS方法 .首先由模糊极大集与模糊极小集确定模糊多属性决策问题的理想解与负理想解 ,进而由 Hausdauff度量获得不同备选方案到理想解与负理想解的距离及其贴近度 ,根据贴近度指标对方案进行排序 ,为决策者提供决策支持 .最后以 L-R梯形模糊数为例进行了实例研究 .     

Duality Theory in Fuzzy Linear Programming Problems with Fuzzy Coefficients

The concept of fuzzy scalar (inner) product that will be used in the fuzzy objective and inequality constraints of the fuzzy primal and dual linear programming problems with fuzzy coefficients is proposed in this paper. We also introduce a solution concept that is essentially similar to the notion of Pareto optimal solution in the multiobjective programming problems by imposing a partial ordering on the set of all fuzzy numbers. We then prove the weak and strong duality theorems for fuzzy linear programming problems with fuzzy coefficients.