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Supervisory Insurance Accounting: Mathematics for Provision – and Solvency Capital – Requirements*

  • Philippe Artzner (a1) and Karl-Theodor Eisele (a2)


This paper aims at providing a mathematical foundation for the terms of the well spread supervisory rule “initial market value of assets must be at least equal to provision plus solvency capital”.

It starts with a risk-adjusted assessment — given by a set of test probabilities — of the future cash-flows coming from a company business plan and attempts to define terms of a supervisory accounting mode.

First, inspired by the idea of “representation” of obligations by “equivalent” assets, we define the supervisory provision (or “liability”) attached to existing obligations. This provision is market consistent according to the mathematical definition by Cheridito, Filipovic and Kupper and satisfies a property of equilibrium between supervision's wish for stress testing and management's possibility for appropriate choice of assets.

The comparison between the initial market price of assets and the supervisory provision defines “solvability” of existing obligations.

In a second step the paper defines a required solvency capital as related to the level of discrepancy between assets and obligations of a company. Solvency of a business plan is defined by requiring as initial market value an additional amount over the one needed for solvability: this is the required solvency capital. A business plan with zero required solvency capital is said to have an optimal replicating asset portfolio.

It is shown that — under a natural additional condition, that of a market prudent set of test probabilities — solvability of an obligation allows for solvency of a related business plan, by choice of the asset portfolio.

The paper emphasizes the distinction between supervisory and market oriented accounting hinted to in the CEIOPS CP 20 consultative paper.



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Partial support from AERF/CKER, The Actuarial Foundation is gratefully acknowledged.


An earlier version of this paper has been presented at the 2b) or not 2b) Conference in honour of Professor Hans Gerber, University of Lausanne, June 2-3, 2009. Thanks are due to F. Delbaen and J.-L. Netzer for useful discussions and to a Referee for his/her encouraging remarks which led to further investigations on provision and to some institutional precisions.



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Artzner, Ph., Delbaen, F., Eber, J.-M. and Heath, D. (1999) Coherent Risk Measures, Mathematical Finance, 9, 203228.
Artzner, Ph., Delbaen, F. and Koch-Medina, P. (2009) Risk Measures and Efficient Use of Capital, Astin Bulletin, 39, 101116.
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Cheridito, P., Filipovic, D. and Kupper, M. (2008) Dynamic Risk Measures, Valuations and Optimal Dividends for Insurance, Paper presented at the Oberwolfach Mini Workshop on the Mathematics of Solvency, February 15.
Code des Assurances, Réglementation des placements et autres éléments d'actif, available at ‘’.
Committee of European Insurance and Occupational Pensions Supervisors (2007) Advice to the European Commission in the Framework of the Sovency II Project on Pillar I issues – further advice, CEIOPS-DOC-08/07, March, formerly Consultative Paper CP 20.
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Eisele, K.-Th. and Artzner, Ph. (2010) Time-Consistent Supervisory Accounting, Paper presented at Risk Day 2010, ETH Zurich, available at ‘http//’.
Federal Office of Private Insurance (2006) The Swiss Experience with Market Consistent Technical Provisions – the Cost of Capital Approach.
Föllmer, H. and Schied, A. (2004) Stochastic Finance, 2nd ed., de Gruyter, Berlin.
Klöppel, S. and Schweizer, M. (2007) Dynamic indifference valuation via convex risk measures, Mathematical Finance, 17, 599627.
Wüthrich, M.V., Bühlmann, H. and Furrer, H. (2008) Market-Consistent Actuarial Valuation, Springer EAA Lecture Notes.


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Supervisory Insurance Accounting: Mathematics for Provision – and Solvency Capital – Requirements*

  • Philippe Artzner (a1) and Karl-Theodor Eisele (a2)


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