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Gradient capital allocation, also known as Euler allocation, is a technique used to redistribute diversified capital requirements among different segments of a portfolio. The method is commonly employed to identify dominant risks, assessing the risk-adjusted profitability of segments, and installing limit systems. However, capital allocation can be misleading in all these applications because it only accounts for the current portfolio composition and ignores how diversification effects may change with a portfolio restructuring. This paper proposes enhancing the gradient capital allocation by adding “orthogonal convexity scenarios” (OCS). OCS identify risk concentrations that potentially drive portfolio risk and become relevant after restructuring. OCS have strong ties with principal component analysis (PCA), but they are a more general concept and compatible with common empirical patterns of risk drivers being fat-tailed and increasingly dependent in market downturns. We illustrate possible applications of OCS in terms of risk communication and risk limits.
Most insurers in the European Union determine their regulatory capital requirements based on the standard formula of Solvency II. However, there is evidence that the standard formula inaccurately reflects insurers’ risk situation and may provide misleading steering incentives. In the second pillar, Solvency II requires insurers to perform a so-called “Own Risk and Solvency Assessment” (ORSA). In their ORSA, insurers must establish their own risk measurement approaches, including those based on scenarios, in order to derive suitable risk assessments and address shortcomings of the standard formula. The idea of this paper is to identify scenarios in such a way that the standard formula in connection with the ORSA provides a reliable basis for risk management decisions. Using an innovative method for scenario identification, our approach allows for a simple but relatively precise assessment of marginal and even non-marginal portfolio changes. We numerically evaluate the proposed approach in the context of market risk employing an internal model from the academic literature and the Solvency Capital Requirement (SCR) calculation under Solvency II.