Nino Savelli

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 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.