Addendum to ‘The multi-year non-life insurance risk in the additive reserving model’ [Insurance Math. Econom. 52(3) (2013) 590–598]: Quantification of multi-year non-life insurance risk in chain ladder reserving models

This is the first study to derive closed-form analytical expressions for multi-year non-life insurance risk in the chain ladder model. Extending on previous research on the additive reserving model, we define multi-year risk via prediction errors of multi-year claims development results including both observed and future accident years. A resampling argument and a first-order Taylor approximation address the quantification of estimation errors and multiplicative dependencies in the chain ladder framework, respectively. From our generalized multi-year approach, we deduce estimators for reserve and premium risks in multi-year view and their implicit correlation. We reproduce well-known results from literature for the special cases of one-year and ultimo view. Further, we comment on how to obtain estimators for generalized versions of the chain ladder method. A case study demonstrates the applicability of our analytical formulae.     

A stochastic Nash equilibrium portfolio game between two DC pension funds

In this paper, we study the stochastic Nash equilibrium portfolio game between two pension funds under inflation risks. The financial market consists of cash, bond and two stocks. It is assumed that the price index is derived through a generalized Fisher equation while the bond is related to the price index to hedge the risk of inflation. Besides, these two pension managers can invest in their familiar stocks. The goal of the pension managers is to maximize the utility of the weighted terminal wealth and relative wealth. Dynamic programming method is employed to derive the Nash equilibrium strategies. In the end, a numerical analysis is presented to reveal the economic behaviors of the two DC pension funds.     

Confidence band for expectation dependence with applications

Motivated by the applications of the concept of expectation dependence in economics and finance, we propose a method to construct uniform confidence band for expectation dependence. It is derived based on Hoeffding’s inequality. Our proposed confidence band can be explicitly expressed and thus it is very easy to implement. Our method has applications to demand for a risky asset and first-order risk aversion problems. Simulations suggest our proposed confidence interval can control the coverage probabilities very well, and the average lengths are very short. Two empirical applications are presented to illustrate the usefulness of the constructed confidence band of expectation dependence.     

Hedging pure endowments with mortality derivatives

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. (2009). The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.     

Marginal Indemnification Function formulation for optimal reinsurance

In this paper, we propose to combine the Marginal Indemnification Function (MIF) formulation and the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure and distortion reinsurance premium principle. The MIF method exploits the absolute continuity of admissible indemnification functions and formulates optimal reinsurance model into a functional linear programming of determining an optimal measurable function valued over a bounded interval. The MIF method was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is applied to combine with MIF to solve for optimal reinsurance solutions under premium budget constraint. Compared with the existing literature, the proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the practicality of the proposed method, analytical solution is derived on a particular reinsurance model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion principle.     

基于模糊数对高新技术企业决策风险的影响因素排序

针对高新技术企业决策风险的特点,结合高新技术企业决策风险体系的构成要素,基于模糊数对决策风险的影响因素两茂比较,建立三角模糊数互补判断矩阵,最后转化为模糊互补一致矩阵,得到决策风险影响因素指标的权重值,并对决策风险的影响因素进行排序,为高新技术企业决策者降低决策风险提供理论依据.     

产品质量风险监控绩效评估的结构方程模型

产品质量安全风险监控体系是一个涉及诸多要素的复杂系统,其绩效评估指标众多且关联度较大,使用常规方法不能得到最佳评估效果。通过结构方程模型对产品质量安全风险监控体系的绩效评估指标进行模型构建,结合AMOS 7.0对其求解,并做拟合验证,最后得出绩效评估结果。结果显示,结构方程模型可以有效地处理复杂的绩效数据,并能通过模型拟合及修正以使评估结果更接近客观反映。     

基于极值统计和高维动态C藤Copula的股市行业集成风险计算

股市诸多行业风险之间存在着波动相依性,集成计量多维风险对投资决策意义重大。藤Copula是Copula函数高维化拓展的一个方向,其动态化是新的研究前沿。将极值理论的GPD模型和高维动态C藤Copula方法结合起来研究沪深300指数中地产、基建、银行和运输四个行业风险,能够有效描述尾部极值形态,突出关键变量的作用。再运用动态Pair-Copula分解,刻画高维行业风险变量间的动态关系,以仿真出动态集成风险变量VaR序列。VaR计算结果通过了回溯检验和稳定性测试,表明高维动态C藤Copula模型可以作为风险集成计量的一种新的有效方法。