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

*Joachim Paulusch* ()

Additional contact information

Joachim Paulusch: R+V Lebensversicherung AG, Raiffeisenplatz 2, 65189 Wiesbaden, Germany

*Journal of Risk and Financial Management*, 2017, vol. 10, issue 2, 1-12

**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.

**Keywords:** Solvency II; standard formula; risk measure; diversification; aggregation; monotony; homogeneity; subadditivity; Euler’s Principle; capital allocation (search for similar items in EconPapers)

**JEL-codes:** C E F2 F3 G (search for similar items in EconPapers)

**Date:** 2017

**References:** View references in EconPapers View complete reference list from CitEc

**Citations:** View citations in EconPapers (1) Track citations by RSS feed

**Downloads:** (external link)

https://www.mdpi.com/1911-8074/10/2/11/pdf (application/pdf)

https://www.mdpi.com/1911-8074/10/2/11/ (text/html)

**Related works:**

This item may be available elsewhere in EconPapers: Search for items with the same title.

**Export reference:** BibTeX
RIS (EndNote, ProCite, RefMan)
HTML/Text

**Persistent link:** https://EconPapers.repec.org/RePEc:gam:jjrfmx:v:10:y:2017:i:2:p:11-:d:98991

Access Statistics for this article

Journal of Risk and Financial Management is currently edited by *Prof. Dr. Michael McAleer*

More articles in Journal of Risk and Financial Management from MDPI, Open Access Journal

Bibliographic data for series maintained by XML Conversion Team ().