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

Version 1
: Received: 10 October 2016 / Approved: 10 October 2016 / Online: 10 October 2016 (11:53:00 CEST)

Version 2 : Received: 28 February 2017 / Approved: 1 March 2017 / Online: 1 March 2017 (08:53:59 CET)

Version 2 : Received: 28 February 2017 / Approved: 1 March 2017 / Online: 1 March 2017 (08:53:59 CET)

A peer-reviewed article of this Preprint also exists.

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

Journal reference: J. Risk Financial Manag. 2017, 10, 11

DOI: 10.3390/jrfm10020011

## 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*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.^{p}## Keywords

Solvency II; Standard Formula; Risk Measure; Diversification; Aggregation; Monotony; Homogeneity; Subadditivity; Euler's Principle; Capital Allocation

## Subject

MATHEMATICS & COMPUTER SCIENCE, Applied Mathematics

Copyright: 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.

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