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European Actuarial Journal
Published in: European Actuarial Journal 1/2019

22-02-2019 | Original Research Paper

Practical aspects of the aggregation of two risks in the Solvency II standard formula

Author: Magnus Carlehed

Published in: European Actuarial Journal | Issue 1/2019

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Abstract

The Solvency II standard formula rests on aggregation of capital requirements using prescribed so-called correlation parameters. The theme of this note is a comparison between the dependence structure of the underlying risk factors on one hand and parameters used for aggregation of the capital requirements on the other hand. As soon as the portfolio value depends non-linearly on the risk factors these clearly differ, and we investigate their interdependence in a couple of examples, related to savings products in life insurance business. In particular, we treat the case of a unit-linked life insurance company whose profits to large extent are generated from fees related to the value of Assets under Management, a type of business that is becoming more and more common in several markets across Europe. We show that the standard formula largely overestimates the capital need in this case.
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Appendix
Available only for authorised users
Footnotes
1
As an example, define \(f(x,y)=(x+y)^3\) with independent X and Y. Then a calculation shows that \(\alpha =3\). It is easy to see that a necessary and sufficient condition for the existence of a solution in the desired interval is \(|C_X-C_Y|\le C\le C_X+C_Y\).
 
2
This assumes that a and b have the same sign; however, this is not essential as we can always change sign on X and/or Y. See also the interesting paper [3].
 
3
According to Solvency II regulation, the lapse should only be performed for policies where the lapse event leads to a loss, a restriction that is neglected in our calculations. Also, note that after the instant lapse scenario, we make no update of the best estimate assumption of future lapse rates used for calculating the OFFP.
 
4
The distribution is not adapted to the portfolios UL-G2; for instance, it has lower average lapse rate than UL. We still deem that the distribution has a shape that could be used for a simulation of our three portfolios.
 
5
The quadratic term \(E^2\) in the approximating polynomials obviously becomes positive for negative E, corresponding to a market upturn, and if this term dominates, an upturn scenario might wrongfully give raise to a capital need. We have solved this technical issue by only looking at the negative equity scenarios, i.e. where \(E>0\).
 
6
A more accurate but also more complicated method is to use nonlinear regression to minimize the full residual vector, given the risk matrix.
 
7
Note that the distributional assumption in Sect. 4.1 is different from Sect. 5. We have rerun the simulation for UL using lognormal distributions, but the result does not differ much.
 
Literature
1.
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3.
Devineau L, Loisel S (2009) Risk aggregation in Solvency II: how to converge the approaches of the internal models and those of the standard formula? Bulletin Français d’Actuariat 9(18):107–145
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Metadata
Title
Practical aspects of the aggregation of two risks in the Solvency II standard formula
Author
Magnus Carlehed
Publication date
22-02-2019
Publisher
Springer Berlin Heidelberg
Published in
European Actuarial Journal / Issue 1/2019
Print ISSN: 2190-9733
Electronic ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-019-00196-z

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