Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T16:31:25.221Z Has data issue: false hasContentIssue false
  • English
  • Français

PROPERTY GRAPHS – A STATISTICAL MODEL FOR FIRE AND EXPLOSION LOSSES BASED ON GRAPH THEORY

Published online by Cambridge University Press:  27 March 2019

Pietro Parodi  and
Peter Watson
Pietro Parodi*
Affiliation:
SCOR Global P&C London, UK E-Mail: [email protected]
Peter Watson
Affiliation:
SCOR Global P&C London, UK E-Mail: [email protected]
*
E-Mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is rare that the severity loss distribution for a specific line of business can be derived from first principles. One such example is the use of generalised Pareto distribution for losses above a large threshold (or more accurately: asymptotically), which is dictated by extreme value theory. Most popular distributions, such as the lognormal distribution or the Maxwell-Boltzmann-Bose-Einstein-Fermi-Dirac (MBBEFD), are convenient heuristics with no underlying theory to back them. This paper presents a way to derive a severity distribution for property losses based on modelling a property as a weighted graph, that is, a collection of nodes and weighted arcs connecting these nodes. Each node v (to which a value can also be assigned) corresponds to a room or a unit of the property where a fire can occur, while an arc (v, v′; p) between vertices v and v′ signals that the probability of the fire propagating from v to v′ is p. The paper presents two simple models for fire propagation (the random graph approach and the random time approach) and a model for explosion risk that allow one to calculate the loss distribution for a given property from first principles. The MBBEFD model is shown to be a good approximation for the simulated distribution of losses based on property graphs for both the random graph and the random time approach.

Keywords

Random graphexposure curveMBBEFDcritical threshold
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 2 , May 2019 , pp. 263 - 297
Copyright
Copyright © Astin Bulletin 2019 

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.)

References

Bernegger, S. (1997) The Swiss Re exposure curves and the MBBEFD distribution class. ASTIN Bulletin, 27(1), 99–111.CrossRefGoogle Scholar
Bollobas, B. (2011) Random Graphs. Cambridge: Cambridge University Press.Google Scholar
Brehm, J.J. and Mullin, W.J. (1989) Introduction to the Structure of Matter. New York: Wiley.Google Scholar
Center for Chemical Process Safety. (2010) Layer of Protection Analysis: Simplified Process Risk Assessment. New York: Wiley.Google Scholar
Chhabra, A. and Parodi, P. (2010) Dealing with sparse data. Proceedings of GIRO 2010, Newport.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Freund, J. E. (1999) Mathematical Statistics. 6th Edition. New Jersey: PrenticeHall International, Inc.Google Scholar
Frieze, A. (2015) Introduction to Random Graphs. Cambridge: Cambridge University Press.Google Scholar
Guggisberg, D. (2004) Exposure Rating. Zurich: Swiss Re. http://www.swissre.com.Google Scholar
Hagberg, A., Schult, D., Swart, P. (2018) NetworkX reference, release 2.1, https://networkx.github.io/documentation/stable/_downloads/networkx_reference.pdf.Google Scholar
Kraft, D. (1988) A software package for sequential quadratic programming. Tech. Rep. DFVLRFB 88-28, Köln, Germany: DLR German Aerospace Center – Institute for Flight Mechanics.Google Scholar
Mandelbrot, B. B. (1982) The Fractal Geometry of Nature. San Francisco: W.H. Freeman and Co.Google Scholar
National Archives. (2011) Office For National Statistics. Part of 2011 Census. Detailed Characteristics on Housing for Local Authorities in England and Wales Release. Adapted from data from the Office for National Statistics licensed under the Open Government Licence v.3.0. Crown Copyright 2015. http://webarchive.nationalarchives.gov.uk/20160105160709/http://www.ons.gov.uk/ons/rel/census/2011-census/detailed-characteristics-on-housing-for-local-authorities-in-england-and-wales/sty-households-in-england-and-wales.html.Google Scholar
Pareto, V. (1964) Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, pp. 299–345. Geneva: Librairie Droz.CrossRefGoogle Scholar
Parodi, P. (2014) Pricing in General Insurance. New York: CRC Press.CrossRefGoogle Scholar
Rath, B. and Toth, B. (2009) Erdos–Renyi random graphs + forest fires = self-organized criticality. Electronic Journal of Probability, 14, Paper no. 45, 1290–1327. doi:10.1214/EJP.v14-653. https://projecteuclid.org/euclid.ejp/1464819506.CrossRefGoogle Scholar
Reason, J. (1990) The contribution of latent human failures to the breakdown of complex systems. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 327(1241), 475–484.CrossRefGoogle ScholarPubMed
Riegel, U. (2010) On fire exposure rating and the impact of the risk profile type. ASTIN Bulletin, 40(2), 727–777.Google Scholar
Slade, G. (2008) Probabilistic models of critical phenomena. In The Princeton Companion to Mathematics (ed. Gowers, et al.), pp. 657–670. Princeton: Princeton University Press.Google Scholar
Trudeau, R.J. (2003) Introduction to Graph Theory. New York: Dover Publications Inc.Google Scholar
UK Government. (2010) The building regulations 2010. Fire safety. Approved Document B. Vol. 1.—Dwelling Houses: Crown Copyright (2011). https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/485420/BR_PDF_AD_B1_2013.pdf.Google Scholar
Willey, R.J. (2014) Layer of protection analysis. Procedia Engineering, 84, 12–22. https://www.sciencedirect.comCrossRefGoogle Scholar