Joël Wagner

Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented.

The Impact of Introducing Insurance Guaranty Schemes on Pricing and Capital Structure

The introduction of an insurance guaranty scheme can have significant influence on the pricing and capital structures in a competitive market. The aim of this article is to study this effect on competitive equity–premium combinations while considering a framework with policyholders and equity holders where guaranty fund charges are volume‐based, as levied in existing schemes. Several settings with regard to the origin of the fund contributions are assessed and the immediate effects on the incentives of the policyholders and equity holders are analyzed through a one‐period contingent claim approach. One result is that introducing a guaranty scheme in a market with competitive conditions entails a shift of equity capital towards minimum solvency requirements. Hence, adverse incentives may arise with regard to the overall security level of the industry.

Under What Conditions is an Insurance Guaranty Fund Beneficial for Policyholders

In this article, we derive conditions in an imperfect market setting, under which the introduction of a self‐supporting insurance guaranty fund improves the position of the policyholders. When a guaranty fund is advantageous given homogeneous firms in the market, all policyholders benefit from it to the same extent, if they have the same underlying risk preferences and are charged identical premiums. In a more realistic heterogeneous setting, the introduction of an insurance guaranty fund is in general no longer beneficial for all policyholders in the same manner. Hence, systematic wealth transfers take place between the policyholders of different insurance companies. As a possible solution, and in order to counteract this effect, we introduce a framework for utility‐based fund charges.

The risk of model misspecification and its impact on solvency measurement in the insurance sector

Purpose - The purpose of this paper is to study the risk of misspecifying solvency models for insurance companies. Design/methodology/approach - Based on a basic solvency model, the authors examine the sensitivity of different risk measures with respect to model misspecification. An analysis considers the effects of introducing stochastic jumps and linear, as well as non-linear dependencies into the basic setting on the solvency capital requirements, shortfall probability and expected policyholder deficit. Additionally, the authors take a regulatory view and consider the degree to which the deviations in risk measures, due to the different model specifications, can be diminished by means of requiring interim financial reports. Findings - The simulation results suggest that the sensitivity of solvency capital as a risk measure – as it is in regulatory practice – underestimates the actual misspecification risk that policyholders are exposed to. It is also found that semi-annual mandatory interim reports can already reduce the model uncertainty faced by a regulator, significantly. This has important implications for the design of risk-based capital standards and the implementation of internal solvency models. Originality/value - The results from the Monte Carlo simulation show that changes in the specification of a solvency model have a much greater impact on shortfall probabilities and expected policyholder deficits than they have on capital requirements. The shortfall risk measures react much more sensitively to small changes in the model assumptions, than the capital requirements. This leads us to the conclusion that regulators should not solely rely on capital requirements to monitor the solvency situation of an insurer, but should additionally consider shortfall risk measures. More precisely, an analysis of model risk focusing on the sensitivity of capital requirements will typically underestimate the relevant risk of model misspecification from a policyholders perspective. Finally, the simulation results suggest that mandatory interim reports on the solvency and financial situation of an insurance company are a powerful tool in order to reduce the model uncertainty faced by regulators.

Comparison of Stakeholder Perspectives on Current Regulatory and Reporting Reforms

In the European insurance industry, regulatory and reporting frameworks are currently subject to far‐reaching reforms. We focus on four of these frameworks, namely the Solvency II framework, insurance guaranty systems, the proposed IFRS 4 Phase II international accounting standards, and Market Consistent Embedded Value reporting. We present these frameworks, analyze them from the insurance companys management, investors, and policyholder perspectives, and compare them. Our analysis implies that the four frameworks need to be considered jointly, due to various interrelations and interactions. We argue that a coordinated introduction will be necessary to ensure that the regulatory burden is reduced and synergies can be utilized in the event of all four frameworks being implemented as planned. Furthermore, we analyze the challenges of a holistic, comprehensive approach to insurance reporting and regulation and its implementation in order to achieve the goals set by the frameworks.

Finite element methods with patches and applications

Theoretical and numerical aspects of multi-scale problems are investigated. On one hand, mathematical analysis is done on a new method for numerically solving problems with multi-scale behavior using multiple levels of not necessarily nested grids. A particularly flexible multiplicative Schwarz method is presented, requiring no conformity between the meshes at the different scales. The relaxed iterative method consists in calculating successive corrections to the solution in regions where the variations of a problem are too strong to be captured by a coarse initial mesh. In these sub-domains patches of finite elements are applied. A priori and a posteriori error estimates are given and an exact spectral analysis of the iteration operator describing the algorithm is presented. Computational issues are addressed and numerical methods to obtain optimal convergence are given. Crucial implementation matters are discussed with special regard to usage of memory and CPU-time. On the other hand, the efficiency of the introduced correction method is demonstrated on Laplace model problems, either with changing Dirichlet-Neumann boundary conditions or in a polygonal domain with entrant corner. The regularity of the solutions is studied as well as the improvement of the convergence order in the mesh size using various sizes of patches. The correction algorithm is also applied to improve the accuracy in the simulation of the stress field in glacier modeling. A simple model to obtain the effective stress field in the ice mass of a glacier is presented and concluding results are obtained using patches in the regions where changes in the basal boundary conditions are involved.

Multiscale algorithm with patches of finite elements

We develop a discretization and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localized regions. We start with a coarse finite element discretization with a mesh size H, and we superpose to it local patches of finite elements with finer mesh size h@?H to capture local behaviour of the solution. We discuss the implementation and illustrate the method on an industrial example.

A note on the appropriate choice of risk measures in the solvency assessment of insurance companies

Purpose - – The concept of value at risk is used in the risk-based calculation of solvency capital requirements in the Basel II/III banking regulations and in the planned Solvency II insurance regulation framework planned in the European Union. While this measure controls the ruin probability of a financial institution, the expected policyholder deficit (EPD) and expected shortfall (ES) measures, which are relevant from the customers perspective as they value the amount of the shortfall, are not controlled at the same time. Hence, if there are variations in or changes to the asset-liability situation, financial companies may still comply with the capital requirement, while the EPD or ES reach unsatisfactory levels. This is a significant drawback to the solvency frameworks. The paper aims to discuss these issues. Design/methodology/approach - – The author has developed a model framework wherein the author evaluates the relevant risk measures using the distribution-free approach of the normal power approximation. This allows the author to derive analytical approximations of the risk measures solely through the use of the first three central moments of the underlying distributions. For the case of a reference insurance company, the author calculates the required capital using the ruin probability and EPD approaches. For this, the author performs sensitivity analyses considering different asset allocations and different liability characteristics. Findings - – The author concludes that only a simultaneous monitoring of the ruin probability and EPD can lead to satisfactory results guaranteeing a constant level of customer protection. For the reference firm, the author evaluates the relative changes in the capital requirement when applying the EPD approach next to the ruin probability approach. Depending on the development of the assets and liabilities, and in the cases the author illustrates, the reference company would need to provide substantial amounts of additional equity capital. Originality/value - – A comparative assessment of alternative risk measures is relevant given the debate among regulators, industry representatives and academics about how adequately they are used. The author borrows the approach in parts from the work of Barth. Barth compares the ruin probability and EPD approach when discussing the RBC formulas of the US National Association of Insurance Commissioners introduced in the 1990s. The author reconsiders several of these findings and discusses them in the light of the new regulatory frameworks. More precisely, the author first performs sensitivity analyses for the risk measures using different parameter configurations. Such analyses are relevant since in practice parameter values may differ from estimates used in the model and have a significant impact on the values of the risk measures. Second, the author goes beyond a simple discussion of the outcomes for each risk measure, by deriving the firm conclusion that both the frequency and magnitude of shortfalls need to be controlled.

What transaction costs are acceptable in life insurance products from the policyholders’ viewpoint?

Purpose The purpose of this paper is to analyze what transaction costs are acceptable for customers in different investments. In this study, two life insurance contracts, a mutual fund and a risk-free investment, as alternative investment forms are considered. The first two products under scrutiny are a life insurance investment with a point-to-point capital guarantee and a participating contract with an annual interest rate guarantee and participation in the insurer’s surplus. The policyholder assesses the various investment opportunities using different utility measures. For selected types of risk profiles, the utility position and the investor’s preference for the various investments are assessed. Based on this analysis, the authors study which cost levels can make all of the products equally rewarding for the investor. Design/methodology/approach The paper notes the risk-neutral valuation calibration using empirical data utility and performance measurement dynamics underlying: geometric Brownian motion numerical examples via Monte Carlo simulation. Findings In the first step, the financial performance of the various saving opportunities under different assumptions of the investor’s utility measurement is studied. In the second step, the authors calculate the level of transaction costs that are allowed in the various products to make all of the investment opportunities equally rewarding from the investor’s point of view. A comparison of these results with transaction costs that are common in the market shows that insurance companies must be careful with respect to the level of transaction costs that they pass on to their customers to provide attractive payoff distributions. Originality/value To the best of the authors’ knowledge, their research question – i.e. which transaction costs for life insurance products would be acceptable from the customer’s point of view – has not been studied in the above described context so far.

The pricing of hedging longevity risk with the help of annuity securitizations: An application to the German market

Purpose - – The purpose of this paper is to develop a model to hedge annuity portfolios against increases in life expectancy. Across the globe, and in the industrial nations in particular, people have seen an unprecedented increase in their life expectancy over the past decades. The benefits of this apply to the individual, but the dangers apply to annuity providers. Insurance companies often possess no effective tools to address the longevity risk inherent in their annuity portfolio. Securitization can serve as a substitute for classic reinsurance, as it also transfers risk to third parties. Design/methodology/approach - – This paper extends on methods insurers can use to hedge their annuity portfolio against longevity risk with the help of annuity securitization. Future mortality rates with the Lee-Carter-model and use the Wang-transformation to incorporate insurance risk are forecasted. Based on the percentile tranching method, where individual tranches are aligned to Standard & Poors ratings, we price an inverse survivor bond. This bond offers fix coupon payments to investors, while the principal payments are at risk and depend on the survival rate within the underlying portfolio. Findings - – The contribution to the academic literature is threefold. On the theoretical side, building on the work of Kim and Choi (2011), we adapt their pricing model to the current market situation. Putting the principal at risk instead of the coupon payments, the insurer is supplied with sufficient capital to cover additional costs due to longevity. On the empirical side, the method for the German market is specified. Inserting specific country data into the model, price sensitivities of the presented securitization model are analyzed. Finally, in a case study, the procedure to the annuity portfolio of a large German life insurer is applied and the price of hedging longevity risk is calculated. Practical implications - – To illustrate the implication of this bond structure, several sensitivity tests were conducted before applying the pricing model to the retail sample annuity portfolio from a leading German life insurer. The securitization structure was applied to calculate the securitization prices for a sample portfolio from a large life insurance company. Social implications - – The findings contribute to the current discussion about how insurers can face longevity risk within their annuity portfolios. The fact that the rating structure has such a severe impact on the overall hedging costs for the insurer implies that companies that are willing to undergo an annuity securitization should consider their deal structure very carefully. In addition, we have pointed out that in imperfect markets, the retention of the equity tranche by the originator might be advantageous. Nevertheless, one has to bear in mind that by this behavior, the insurer is able to reduce the overall default risk in his balance sheet by securitizing a life insurance portfolio; however, the fraction of first loss pieces from defaults increases more than proportionally. The insurer has to take care to not be left with large, unwanted remaining risk positions in his books. Originality/value - – In this paper, we extend on methods insurers can use to hedge their annuity portfolio against longevity risk with the help of annuity securitization. To do so, we take the perspective of the issuing insurance company and calculate the costs of hedging in a four-step process. On the theoretical side, building on the work of Kim and Choi (2011), we adapt their pricing model to the current market situation. On the empirical side, we specify the method for the German market. Inserting specific country data into the model, price sensitivities of the presented securitization model are analyzed.