Gian Paolo Clemente

Hierarchical structures in the aggregation of premium risk for insurance underwriting

In the valuation of the Solvency II capital requirement, the correct appraisal of risk dependencies acquires particular relevance. These dependencies refer to the recognition of risk diversification in the aggregation process and there are different levels of aggregation and hence different types of diversification. For instance, for a non-life company at the first level the risk components of each single line of business (e.g. premium, reserve, and CAT risks) need to be combined in the overall portfolio, the second level regards the aggregation of different kind of risks as, for example, market and underwriting risk, and finally various solo legal entities could be joined together in a group. Solvency II allows companies to capture these diversification effects in capital requirement assessment, but the identification of a proper methodology can represent a delicate issue. Indeed, while internal models by simulation approaches permit usually to obtain the portfolio multivariate distribution only in the independence case, generally the use of copula functions can consent to have the multivariate distribution under dependence assumptions too. However, the choice of the copula and the parameter estimation could be very problematic when only few data are available. So it could be useful to find a closed formula based on Internal Models independence results with the aim to obtain the capital requirement under dependence assumption. A simple technique, to measure the diversification effect in capital requirement assessment, is the formula, proposed by Solvency II quantitative impact studies, focused on the aggregation of capital charges, the latter equal to percentile minus average of total claims amount distribution of single line of business (LoB), using a linear correlation matrix. On the other hand, this formula produces the correct result only for a restricted class of distributions, while it may underestimate the diversification effect. In this paper we present an alternative method, based on the idea to adjust that formula with proper calibration factors (proposed by Sandström (2007)) and appropriately extended with the aim to consider very skewed distribution too. In the last part considering different non-life multi-line insurers, we compare the capital requirements obtained, for only premium risk, applying the aggregation formula to the results derived by elliptical copulas and hierarchical Archimedean copulas.

A New Lower Bound for the Kirchhoff Index using a numerical procedure based on Majorization Techniques

Abstract In this note, we use a procedure, proposed in [Bianchi, M., and A. Torriero, Some localization theorems using a majorization technique, Journal of Inequalities and Applications 5 (2000), 433–446], based on a majorization technique, which localizes real eigenvalues of a matrix of order n. Through this information, we compute a lower bound for the Kirchhoff index (see [Bianchi M., A. Cornaro, J.L. Palacios and A. Torriero, Bounds for the Kirkhhoff index via majorization techniques, Journal of Mathematical Chemistry, (2012) online first]) that takes advantage of additional eigenvalues bounds. An algorithm has been developed with MATLAB software to evaluate the above mentioned bound. Finally, numerical examples are provided showing how tighter results can be obtained.

Directed clustering in weighted networks: A new perspective

Abstract Several definitions of clustering coefficient for weighted networks have been proposed in literature, but less attention has been paid to both weighted and directed networks. We provide a new local clustering coefficient for this kind of networks, starting from those already existing in the literature for the weighted and undirected case. Furthermore, we extract from our coefficient four specific components, in order to separately consider different link patterns of triangles. Empirical applications on several real networks from different frameworks and with different order are provided. The performance of our coefficient is also compared with that of existing coefficients.

New bounds for the sum of powers of normalized Laplacian eigenvalues of graphs

For a simple and connected graph, a new graph invariant s α  *  ( G ) , defined as the sum of α powers of the eigenvalues of the normalized Laplacian matrix, has been introduced by Bozkurt and Bozkurt (2012). Lower and upper bounds for this index have been proposed by the authors. In this paper, we localize the eigenvalues of the normalized Laplacian matrix by adapting a theoretical method, proposed in Bianchi and Torriero (2000), based on majorization techniques. Through this approach we derive upper and lower bounds of s α  *  ( G ) . Some numerical examples show how sharper results can be obtained with respect to those existing in literature.

A Risk-Theory Model to Assess the Capital Requirement for Mortality and Longevity Risk

Abstract There is considerable uncertainty regarding the future development of life expectancy that leads to significant change in many fields of the insurance market. Pricing annuity products and mortality-linked securities seem primary goals of actuarial literature. At the same time, the valuation of non-hedgeable liabilities (as technical provisions for contracts where risk is not entirely borne by the policyholders) and the estimation of capital requirement appear very important issues in Solvency II framework. In this context, we propose a model based on Risk Theory in order to evaluate the capital requirement for mortality and longevity risk. We assume a life portfolio characterized by traditional and with-profit products divided in several homogeneous generations of contracts. Each cohort includes equal contracts that differ only by the insured sum with the aim to consider the effect of variability coefficient. Some assumptions allow to obtain closed formulae for the exact characteristics of demographic profit distribution regardless of contract types (i.e either with survival or death benefits). Furthermore Monte-Carlo methods provide the simulated distribution of mortality and longevity profit for each generation. Some case studies show the moments and the capital requirements for different life portfolios. Finally, further research will regard both the aggregation effect between several generations and a valuation of liabilities consistent to Solvency II context.

Actuarial Improvements of Standard Formula for Non-life Underwriting Risk

Solvency II Directive introduced a new framework in order to develop new risk management practices to manage risk and to define a minimum capital requirement. To this aim, Commission Delegated Regulation provided the final version of the standard formula. Capital requirement is obtained via a modular structure where each source of risk must be first measured and then aggregated under a linear correlation assumption. As the results of main Quantitative Impact Studies have shown, premium and reserve risks represent a key driver for non-life insurers. In this regard, we focus here on the valuation of the capital requirement for this specific sub-module. Some inconsistencies of the approach provided by Solvency II will be highlighted. We show that some assumptions of the standard formula may lead to an underestimation of the capital requirement for small insurers.

Model selection for forecasting mortality rates

Abstract The increasing life expectancy, driven mainly by improvements in sanitation, housing and education, is a positive change for individuals and a substantial social achievement but, at the same time, the implications for public spending need to be properly evaluated. Forecasting mortality appears indeed a key issue in different fields of insurance and financial markets as pricing annuity, evaluating mortality-linked securities and quantifying longevity risk. Lee-Carter proposed in 1992 a model widely used in order to forecast mortality rates. Actuarial literature has subsequently provided several extensions with the aim to correct some weaknesses of the original model. The purpose of this paper is to compare a range of both new and existing methodologies proposed over years. The study pays considerable attention to both consistency with historical data and robustness relative to the range of data employed in order to find a good balance between the goodness of fit, the simplicity of the model and the robustness of projections. At this regard, several nested methods are here compared by an application to five different countries and by analyzing several data-sets. Results confirm that some models perform better than others, but no single model can be defined as the best method. Furthermore the original version of Lee-Carter provides a good fit when a cohort effect is not significant leading to the strictest confidence interval.

Bounding the HL-index of a graph: a majorization approach

In mathematical chemistry, the median eigenvalues of the adjacency matrix of a molecular graph are strictly related to orbital energies and molecular orbitals. In this regard, the difference between the occupied orbital of highest energy (HOMO) and the unoccupied orbital of lowest energy (LUMO) has been investigated (see Fowler and Pisansky in Acta Chim. Slov. 57:513-517, 2010). Motivated by the HOMO-LUMO separation problem, Jaklič et al. in (Ars Math. Contemp. 5:99-115, 2012) proposed the notion of HL-index that measures how large in absolute value are the median eigenvalues of the adjacency matrix. Several bounds for this index have been provided in the literature. The aim of the paper is to derive alternative inequalities to bound the HL-index. By applying majorization techniques and making use of some known relations, we derive new and sharper upper bounds for this index. Analytical and numerical results show the performance of these bounds on different classes of graphs.