Hostname: page-component-cb9f654ff-r5d9c Total loading time: 0 Render date: 2025-08-27T23:26:11.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic risk measures for stochastic asset processes from ruin theory

Published online by Cambridge University Press:  19 February 2018

Yasutaka Shimizu  and
Shuji Tanaka
Yasutaka Shimizu*
Affiliation:
Department of Applied Mathematics, Waseda University
Shuji Tanaka
Affiliation:
Department of Mathematics, College of Humanities and Sciences, Nihon University
*
*Correspondence to: Yasutaka Shimizu, Department of Applied Mathematics, Waseda University, Shinjuku-ku, Tokyo 169-8555, Japan. E-mail: shimizu@waseda.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

This article considers a dynamic version of risk measures for stochastic asset processes and gives a mathematical benchmark for required capital in a solvency regulation framework. Some dynamic risk measures, based on the expected discounted penalty function launched by Gerber and Shiu, are proposed to measure solvency risk from the company’s going-concern point of view. This study proposes a novel mathematical justification of a risk measure for stochastic processes as a map on a functional path space of future loss processes.

Keywords

Insolvency riskGerber–Shiu functionDynamic risk measureSolvency capital requirementAsset/liability management

JEL classification

C02: Mathematical Methods G22: Insurance • Insurance Companies • Actuarial Studies G32: Financing Policy • Financial Risk and Risk Management • Capital and Ownership Structure • Value of Firms • Goodwill

Information

Type
Paper
Information
Annals of Actuarial Science , Volume 12 , Issue 2 , September 2018 , pp. 249 - 268
Copyright
© Institute and Faculty of Actuaries 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Artzner, P., Delbaen, F., Eber, J. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–228.Google Scholar
Artzner, P. & Eisele, K. (2010). Supervisory insurance accounting: mathematics for provision – and solvency capital – requirements. ASTIN Bulletin, 40(2), 569–585.Google Scholar
Cheridito, P. & Kupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. International Journal of Theoretical and Applied Finance, 14(1), 137–162.Google Scholar
Cojocaru, I., Garrido, J. & Zhou, X. (2014). On the finite-time Gerber-Shiu function (Not yet published).Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. & Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons Ltd, Pondicherry, India.Google Scholar
Eisenberg, J. & Schmidli, H. (2011). Minimising expected discounted capital injections by reinsurance in a classical risk model. Scandinavian Actuarial Journal, 2011(3), 155–176.Google Scholar
Feng, R. & Shimizu, Y. (2013). On a generalization from ruin to default in a Lévy insurance risk model. Methodology and Computing in Applied Probability, 15(4), 773–802.Google Scholar
Garrido, J. (2010). Five easy pieces on Gerber-Shiu analysis. 3rd International Gerber-Shiu Workshop, University of Waterloo, Ontario, Canada, 14–16 June.Google Scholar
Garrido, J. (2013). Is the finite-time Gerber-Shiu function a risk measure? The 17th International Congress on Insurance: Mathematics and Economics, Copenhagen, Denmark, 1–3 July.Google Scholar
Gerber, H.U. & Loisel, S. (2012). Why ruin theory should be of interest for insurance practitioners and risk managers nowadays? Actuarial and Financial Mathematics, Feb 2012, Bruxelles, Belgium. 17–21.Google Scholar
Gerber, H.U. & Shiu, E.S.W. (1998). On the time value of ruin; with discussion and a reply by the authors. North American Actuarial Journal, 2(1), 48–78.Google Scholar
Hardy, M.R. & Wirch, J.L. (2004). The iterated CTE: a dynamic risk measure. North American Actuarial Journal, 8(4), 62–75.Google Scholar
Kriele, M. & Wolf, J. (2014). Value-Oriented Risk Management of Insurance Companies. European Actuarial Academy (EAA) Series. Springer, London.Google Scholar
Kuznetsov, A. & Morales, M. (2014). Computing the finite-time expected discounted penalty function for a family of Lévy risk processes. Scandinavian Actuarial Journal, 2014(1), 1–31.Google Scholar
Loisel, S. & Trufin, J. (2014). Properties of a risk measure derived from the expected area in red. Insurance: Mathematics and Economics, 55, 191–199.Google Scholar
Mitric, I.-R. & Trufin, J. (2015). On a risk measure inspired from the ruin probability and the expected deficit at ruin. Scandinavian Actuarial Journal, 2016(10), 932–951.Google Scholar
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Annals of Applied Probability, 12(3), 890–907.Google Scholar
Schmidli, H. (2014). A note on Gerber-Shiu functions with an application. In D. Silvestrov & A. Martin-Löf, (Eds.), Modern Problems in Insurance Mathematics (pp. 21–36). Springer, Cham.Google Scholar
Trufin, J., Albrecher, H. & Denuit, M.M. (2011). Properties of a risk measure derived from ruin theory. The Geneva Risk and Insurance Review, 36(2), 174–188.Google Scholar
Tsai, C.C. & Willmot, G.E. (2002). A generalized defective renewal equation for the surplus process perturbed by diffusion. Insurance: Mathematics and Economics, 30, 51–66.Google Scholar
Wüthrich, M.V. & Merz, M. (2013). Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer-Verlag, Berlin Heidelberg.Google Scholar