Majorisation with applications to the calculus of variations

This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of possible non-polyconvex but multiplicative quasiconvex isotropic functions. This class is not contained in a well known theorem of Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and objective function to be polyconvex. We show here that there is a new way to prove directly the quasiconvexity (in the multiplicative form). Relevance of Schur convexity for the description of rank one convex hulls is explained.Comment: 13 page

Heterogeneous volatility cascade in financial markets

Using high frequency data, we have studied empirically the change of volatility, also called volatility derivative, for various time horizons. In particular, the correlation between the volatility derivative and the volatility realized in the next time period is a measure of the response function of the market participants. This correlation shows explicitly the heterogeneous structure of the market according to the characteristic time horizons of the differents agents. It reveals a volatility cascade from long to short time horizons, with a structure different from the one observed in turbulence. Moreover, we have developed a new ARCH-type model which incorporates the different groups of agents, with their characteristic memory. This model reproduces well the empirical response function, and allows us to quantify the importance of each group.Comment: 10 pages, 2 figures, To be published in Physica

Extreme Moves in Foreign Exchange Rates and Risk Limit Setting

Foreign exchange rates can be subject to considerable daily fluctuations (up to 5 percent within one day). This can, in certain cases, cause serious losses on open overnight positions. Given a maximum tolerable loss for a company, limits have to be set on open overnight positions in foreign currencies. Usually, these limits are determined by using a normal ("Gaussian") model for the daily fluctuations. In our study we illustrate how this common model sometimes quite strongly underestimates the actual extreme risks and, based on methods from the Extreme Value Theory (EVT), we propose and justify a more accurate model.extreme value theory, risk management, foreign exchange, time series analysis

The impact of systemic risk on the diversification benefits of a risk portfolio

Risk diversification is the basis of insurance and investment. It is thus crucial to study the effects that could limit it. One of them is the existence of systemic risk that affects all the policies at the same time. We introduce here a probabilistic approach to examine the consequences of its presence on the risk loading of the premium of a portfolio of insurance policies. This approach could be easily generalized for investment risk. We see that, even with a small probability of occurrence, systemic risk can reduce dramatically the diversification benefits. It is clearly revealed via a non-diversifiable term that appears in the analytical expression of the variance of our models. We propose two ways of introducing it and discuss their advantages and limitations. By using both VaR and TVaR to compute the loading, we see that only the latter captures the full effect of systemic risk when its probability to occur is lowComment: 17 pages, 5 tableau

On the best constant in {G}affney inequality

We discuss the value of the best constant in Gaffney inequality namely \lVert \nabla \omega \rVert_{L^{2}}^{2}\leq C\left( \lVert d\omega\rVert_{L^{2}}^{2}+\lVert \delta\omega\rVert_{L^{2}% }^{2}+\lVert \omega\rVert_{L^{2}}^{2}\right) when either $\nu\wedge\omega=0$ or $\nu\,\lrcorner\,\omega=0$ on $\partial\Omega.

Estimating the risk-adjusted capital is an affair in the tails

(Re)insurance companies need to model their liabilities' portfolio to compute the risk-adjusted capital (RAC) needed to support their business. The RAC depends on both the distribution and the dependence functions that are applied among the risks in a portfolio. We investigate the impact of those assumptions on an important concept for (re)insurance industries: the diversification gain. Several copulas are considered in order to focus on the role of dependencies. To be consistent with the frameworks of both Solvency II and the Swiss Solvency Test, we deal with two risk measures: the Value-at-Risk and the expected shortfall. We highlight the behavior of different capital allocation principles according to the dependence assumptions and the choice of the risk measure.Capital Allocation, Copula, Dependence, Diversification Gain, Model Uncertainty, Monte Carlo Methods, Risk-Adjusted Capital, Risk Measure

Dynamic Financial Analysis - Understanding Risk and Value Creation in Insurance

The changing business environment in non-life insurance and reinsurance has raised the need for new quantitative methods to analyze the impact of various types of strategic decisions on a company’s bottom line. Dynamic Financial Analysis («DFA») has become popular among practitioners as a means of addressing these new requirements. It is a systematic approach based on large-scale computer simulations for the integrated financial modeling of non-life insurance and reinsurance companies aimed at assessing the risks and the benefits associated with strategic decisions. DFA allows decision makers to understand and quantify the impact and interplay of the various risks that their company is exposed to, and – ultimately – to make better informed strategic decisions. In this brochure, we provide an overview and assessment of the state of the industry related to DFA. We investigate the DFA value proposition, we explain its elements and we explore its potential and limitations.reinsurance, dynamic financial analysis, insurance

Credit Risk Models - Do They Deliver Their Promises? A Quantitative Assessment

We develop a framework to assess the statistical significance of expected default frequency as calculated by credit risk models. This framework is then used to analyze the quality of two commercially available models that have become popular among practitioners: KMV Credit Monitor and RiskCalc from Moody's. Using a unique database of expected default probability from both vendors, we study both the consistency of predictions and their timeliness. We introduce the concept of cumulative accuracy profile (CAP), which allows to see in one curve the percentage of companies whose defualts were captured by the models one year in advance. We also use the Miller's information test to see if the models add information to the S&P rating. The result of the analysis indicates that these models indeed add relevant information not accounted for by rating alone. Moreover, with respect to rating agencies, the models predict defaults more than ten months in advance on average.credit risk models, cumulative accuracy profile, risk modeling