On the diversification benefit of reinsurance portfolios

Limani, Jeta and Bettinger, Régis and Dacorogna, Michel M (2017): On the diversification benefit of reinsurance portfolios.

Abstract

In this paper we compare the diversification benefit of portfolios containing excess-of-loss treaties and portfolios containing quota-share treaties, when the risk measure is the (excess) Value-at-Risk or the (excess) Expected Shortfall. In a first section we introduce the set-up under which we perform our investigations. Then we show that when the losses are continuous, independent, bounded, the cover unlimited and when the risk measure is the Expected Shortfall at a level alpha close to 1, a portfolio of n excess-of-loss treaties diversifies better than a comparable portfolio of n quota-share treaties. This result extends to the other risk measures under additional assumptions. We further provide evidence that the boundedness assumption is not crucial by deriving analytical formulas in the case of treaties with i.i.d. exponentially distributed original losses. Finally we perform the comparison in the more general setting of arbitrary continuous joint loss distributions and observe in that case that a finite cover leads to opposite results, i.e. a portfolio of n quota-share treaties diversifies better than a comparable portfolio of n excess-of-loss treaties at high quantile levels.

• Item Type: MPRA Paper
• Language: English
• Keywords: Diversification benefit, risk measures, portfolio, excess-of-loss treaties
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C00 - General; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General

References

M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Tenth Printing, with corrections, 1972. URL http://people.math.sfu.ca/~cbm/aands/abramowitz_and_ stegun.pdf.

P. Artzner, F. Delbaen, J. M. Eber, and D. Heath. Coherent measures of risk. Mathematical Finance, 9(3):203–228, 1999.

R. Bettinger, M. Dacorogna, T. Domergue, and M. Topuzu. Group Internal Model or Solvency II Standard Formula. What is best to represent the risk of a reinsurer? SCOR Working Paper, 2015.

D. M. Bradley and C. R. Gupta. On the distribution of the sum of n non-identically distributed uniform random variables. Ann Inst Stat Math, 54(3):689–700, 2002.

R. Bürgi, M. Dacorogna, and R. Iles. Risk aggregation, dependence structure and diversification benefit. In D. Rösch and H. Scheule, editors, Stress Testing for Financial Institutions, pages 265–306. Riskbooks, Incisive Media, London, 2008.

M. Busse, M. Dacorogna, and M. Kratz. Does risk diversification always work? The answer through simple modelling. SCOR Paper No. 24. 2013. URL http://www.scor.com/en/sgrc/scor-publications/scor-papers.html.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. On the Lambert W function. Advances in Computational Mathematics, 5:329–359, 1996.

M. G. Cruz, G. W. Peters, and P. V. Shevchenko. Fundamental Aspects of Operational Risk and Insurance Analytics and Advances in Heavy Tailed Risk Modeling: Handbooks of Operational Risk Set. Wiley Handbooks in Financial Engineering and Econometrics. Wiley, 2015.

M. Dacorogna and C. Hummel. Capital at Risk. Global Reinsurance, pages 19–20, 2005.

P. Embrechts and R. Wang. Seven Proofs for the Subadditivity of Expected Shortfall. Dependence Modeling, 3:126–140, 2015.

P. Embrechts, A. J. McNeil, and D. Straumann. Correlation and dependency in risk management: properties and pitfalls. In M. A. H. Dempster, editor, Risk Management: Value at Risk and Beyond, pages 176–223. Cambridge University Press, 2002.

W. Feller. An Introduction to Probability Theory and its Applications, volume II. Wiley, 1971.

I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press, 7th edition, 2007.

L. Lakatos, L. Szeidl, and M. Telek. Introduction to Queueing Systems with Telecommunication Applications. Springer, 2013.

Background J. Limani. On dependence modelling and risk diversification. Master’s thesis, ETH Zurich, 2015.

A. J. McNeil, R. Frey, and P. Embrechts. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, 2015.

P. E. Pfeiffer. Probability for Applications. Springer, 1990.

R. Rees and A. Wambach. The Microeconomics of Insurance. Foundations and Trends in Microeconomics, 4(1-2):1–163, 2008.