Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

Fitting asset returns to skewed distributions: Are the skew-normal and skew-student good models?

Vernic (2006), Bolancé et al. (2008), and Eling (2012) identify the skew-normal and skew-student as promising models for describing actuarial loss data. In this paper, we change the focus from the liability to the asset side and ask whether these distributions are also useful for analyzing the investment returns of insurance companies. To answer this question, we fit various parametric distributions to capital market data which has been used to describe the investment set of insurance companies. Our results show that the skew-student is an especially promising distribution for modeling asset returns such as those of stocks, bonds, money market instruments, and hedge funds. Combining the results of Vernic (2006), Bolancé et al. (2008), Eling (2012), and this paper, it appears that the skew-student is a promising actuarial tool since it describes both sides of the insurer’s balance sheet reasonably well.     

On distributional robust probability functions and their computations

Consider a random vector, and assume that a set of its moments information is known. Among all possible distributions obeying the given moments constraints, the envelope of the probability distribution functions is introduced in this paper as distributional robust probability function. We show that such a function is computable in the bi-variate case under some conditions. Connections to the existing results in the literature and its applications in risk management are discussed as well.     

Medium range optimization of copper extraction planning under uncertainty in future copper prices

Deterministic mine planning models along a time horizon have proved to be very effective in supporting decisions on sequencing the extraction of material in copper mines. Some of these models have been developed for, and used successfully by CODELCO, the Chilean state copper company. In this paper, we wish to consider the uncertainty in a very volatile parameter of the problem, namely, the copper price along a given time horizon. We represent the uncertainty by a multistage scenario tree. The resulting stochastic model is then converted into a mixed 0–1 Deterministic Equivalent Model using a compact representation. We first introduce the stochastic model that maximizes the expected profit along the time horizon over all scenarios (i.e., as in a risk neutral environment). We then present several approaches for risk management in a risk averse environment. Specifically, we consider the maximization of the Value-at-Risk and several variants of the Conditional Value-at-Risk (one of them is new), the maximization of the expected profit minus the weighted probability of having an undesirable scenario in the solution provided by the model, and the maximization of the expected profit subject to stochastic dominance constraints recourse-integer for a set of profiles given by the pairs of target profits and bounds on either the probability of failure or the expected profit shortfall. We present an extensive computational experience on the actual problem, by comparing the risk neutral approach, the tested risk averse strategies and the performance of the traditional deterministic approach that uses the expected value of the uncertain parameters. The results clearly show the advantage of using the risk neutral strategy over the traditional deterministic approach, as well as the advantage of using any risk averse strategy over the risk neutral one.     

Stochastically weighted stochastic dominance concepts with an application in capital budgeting

The problem of comparing random vectors arises in many applications. We propose three new concepts of stochastically weighted dominance for comparing random vectors X and Y. The main idea is to use a random vector V to scalarize X and Y   as V^TX$V^{T}X$ and V^TY$V^{T}Y$, and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison. For the case where the distributions of X, Y and V have finite support, we give (mixed-integer) linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized. Some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example.     

Portfolio insurance: Gap risk under conditional multiples

The research on financial portfolio optimization has been originally developed by Markowitz (1952). It has been further extended in many directions, among them the portfolio insurance theory introduced by Leland and Rubinstein (1976) for the “Option Based Portfolio Insurance” (OBPI) and Perold (1986) for the “Constant Proportion Portfolio Insurance” method (CPPI). The recent financial crisis has dramatically emphasized the interest of such portfolio strategies. This paper examines the CPPI method when the multiple is allowed to vary over time. To control the risk of such portfolio management, a quantile approach is introduced together with expected shortfall criteria. In this framework, we provide explicit upper bounds on the multiple as function of past asset returns and volatilities. These values can be statistically estimated from financial data, using for example ARCH type models. We show how the multiple can be chosen in order to satisfy the guarantee condition, at a given level of probability and for various financial market conditions.     

Multistage decision-making fuzzy methodology for optimal investments based on experts’ evaluations

A new methodology of making a decision on an optimal investment in several projects is proposed. The methodology is based on experts’ evaluations and consists of three stages. In the first stage, Kaufmann’s expertons method is used to reduce a possibly large number of applicants for credit. Using the combined expert data, the credit risk level is determined for each project. Only the projects with low risks are selected.     

Inverse portfolio problem with mean-deviation model

A Markowitz-type portfolio selection problem is to minimize a deviation measure of portfolio rate of return subject to constraints on portfolio budget and on desired expected return. In this context, the inverse portfolio problem is finding a deviation measure by observing the optimal mean-deviation portfolio that an investor holds. Necessary and sufficient conditions for the existence of such a deviation measure are established. It is shown that if the deviation measure exists, it can be chosen in the form of a mixed CVaR-deviation, and in the case of n risky assets available for investment (to form a portfolio), it is determined by a combination of (n + 1) CVaR-deviations. In the later case, an algorithm for constructing the deviation measure is presented, and if the number of CVaR-deviations is constrained, an approximate mixed CVaR-deviation is offered as well. The solution of the inverse portfolio problem may not be unique, and the investor can opt for the most conservative one, which has a simple closed-form representation.     

基于GIS与数值模拟的三峡水库库首区诱发地震危险性评价

采用GIS与数值模拟相结合的方法对三峡水库库首区诱发地震危险性进行评价。通过研究水库诱发地震的影响因素,建立模型进行数值模拟。在GIS平台上,综合数值模拟结果与诱发地震监测结果,建立危险性评价数据库。在GIS支持下,采用最大拉应变准则判别拉伸破坏,采用摩尔-库仑准则判别剪切破坏,对单元进行评价。利用诱发地震监测资料,以发生诱发地震单元的第一主应力、第三主应力、位移作为危险单元的标准,对单元进行二次评价。研究结果表明,建立同一GIS平台下的水库区地质环境、监测资料数据库是建立多因素综合评价模型的基础。采用地震评估模型对模拟结果进行评估,结合监测资料对评估成果进行修正,在三峡库首区水库诱发地震危险性评价中具有一定的可信度。