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J. Risk Financial Manag. 2017, 10(2), 11; doi:10.3390/jrfm10020011

# The Solvency II Standard Formula, Linear Geometry, and Diversification

R+V Lebensversicherung AG, Raiffeisenplatz 2, 65189 Wiesbaden, Germany
Received: 28 February 2017 / Revised: 8 May 2017 / Accepted: 10 May 2017 / Published: 18 May 2017

# Abstract

The core of risk aggregation in the Solvency II Standard Formula is the so-called square root formula. We argue that it should be seen as a means for the aggregation of different risks to an overall risk rather than being associated with variance-covariance based risk analysis. Considering the Solvency II Standard Formula from the viewpoint of linear geometry, we immediately find that it defines a norm and therefore provides a homogeneous and sub-additive tool for risk aggregation. Hence, Euler’s Principle for the reallocation of risk capital applies and yields explicit formulas for capital allocation in the framework given by the Solvency II Standard Formula. This gives rise to the definition of diversification functions, which we define as monotone, subadditive, and homogeneous functions on a convex cone. Diversification functions constitute a class of models for the study of the aggregation of risk and diversification. The aggregation of risk measures using a diversification function preserves the respective properties of these risk measures. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Each $L p$ norm has this property, and any scalar product given by a non-negative positive semidefinite matrix does as well. In particular, the Standard Formula is a diversification function and hence a risk measure that preserves homogeneity, subadditivity and convexity. View Full-Text
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Paulusch, J. The Solvency II Standard Formula, Linear Geometry, and Diversification. J. Risk Financial Manag. 2017, 10, 11.

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