# Tail Risk in Commercial Property Insurance

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## Abstract

**:**

## 1. Introduction

## 2. Data

Region | Country | Risk Code | Occupancy 1 | Occupancy 2 | Occupancy 3 |
---|---|---|---|---|---|

NoA | US | P2 | RE | R | 51 |

(Physical damage; primary layer property; USA; excluding binders) | (residential) | (residential) | (Large Hotels) |

Risk Code | Definition |
---|---|

B2 | Physical damage; private property; USA; binder |

B3 | Physical damage; commercial property; USA; binder |

B4 | Physical damage; private property; excluding USA; binder |

B5 | Physical damage; commercial property; excluding USA; binder |

P2 | Physical damage; primary layer property; USA; excluding binders |

P3 | Physical damage; primary layer property; excluding USA; excluding binders |

P4 | Physical damage; full value of property; USA; excluding binders |

P5 | Physical damage; full value of property; excluding USA; excluding binders |

P6 | Physical damage; excess layer property; USA; excluding binders |

P7 | Physical damage; excess layer property; excluding USA; excluding binders |

PG | Operational power generation, transmission, and utilities; excluding construction |

Code | Definition | Code | Definition |
---|---|---|---|

A | Miscellaneous | Q | Offices/Banks |

B | Manufacturers/Processors | R | Residential |

C | Chemicals/Pharmaceuticals | T | Transport |

D | Bridges/Dams/Tunnels/Piers | U | Utilities |

E | Conglomerates | V | Telecoms and Data Processing |

F | Food | W | Woodworkers (Sawmills, Papermills) |

G | Grain | X | Onshore Crude |

H | General Mercantile/Shops | Y | Onshore GasPlants |

J | Mines | Z | Onshore Construction |

K | Crops | 2 | Hospital/Health care centres |

L | Auto | 4 | Semiconductor/Fabs |

M | Metals | 5 | Motor Manufaturers |

O | Municipal Property | 6 | Warehouses |

P | Energy (Oil Refineries/Petrochemicals) |

Lloyd’s Code | Description | |
---|---|---|

AF | Africa | 22 |

AS | Asia - Pacific | 54 |

CA | Central Asia | 21 |

EU | Europe | 78 |

LA | Latin America - Carribean | 78 |

ME | Middle East | 13 |

NA | North America | 1576 |

OC | Oceania | 71 |

WW | World Wide | 1258 |

Total | 3171 |

Years | RE | CO | MA |
---|---|---|---|

2000 | 5 | 116 | 69 |

2001 | 20 | 110 | 58 |

2002 | 17 | 97 | 92 |

2003 | 69 | 105 | 138 |

2004 | 41 | 96 | 112 |

2005 | 12 | 129 | 89 |

2006 | 22 | 74 | 55 |

2007 | 143 | 105 | 153 |

2008 | 51 | 76 | 100 |

2009 | 154 | 55 | 83 |

2010 | 110 | 97 | 108 |

2011 | 23 | 60 | 51 |

2012 | 2 | 4 | 0 |

Total | 669 | 1124 | 1108 |

## 3. Methodology

#### 3.1. Hill Estimator.

#### 3.2. LLRS Regression

#### 3.3. Alternative Methodologies

## 4. Empirical Evidence

**Figure 2.**Tail index estimates and 90% confidence bands for Occupancy Level 1 RE (residential): comparison of Hill and LLRS-1/2 regression methods for different estimation thresholds (from 10-th to 40-th percentile of the data).

**Figure 3.**Tail index estimates and 90% confidence bands (LLRS-1/2 regression method) for all Occupancy Level 1 types (CO, MA, RE).

**Table 6.**Comparison of estimates and 90% confidence bands for the Hill estimator given in Equation (1), for the LLRS-1/2 estimator obtained from the OLS estimate $\widehat{d}$ of the regression $lnl\left(i\right)=c-dln(i-1/2)$, and for the weighted Hill estimator ${\widehat{\gamma}}_{w}$ given in Equation (4). Occupancy Level 1 data: RE (residential), CO (commercial), and MA (manufacturing).

RE | CO | MA | |||||||
---|---|---|---|---|---|---|---|---|---|

Obs. | 5% | ${\widehat{\gamma}}_{Hill}$ | 95% | 5% | ${\widehat{\gamma}}_{Hill}$ | 95% | 5% | ${\widehat{\gamma}}_{Hill}$ | 95% |

10% | 0.905 | 1.225 | 1.546 | 0.593 | 0.731 | 0.868 | 0.829 | 1.055 | 1.281 |

15% | 1.014 | 1.292 | 1.570 | 0.676 | 0.798 | 0.920 | 1.006 | 1.219 | 1.432 |

20% | 1.129 | 1.388 | 1.646 | 0.703 | 0.811 | 0.918 | 0.877 | 1.032 | 1.188 |

25% | 1.219 | 1.463 | 1.706 | 0.759 | 0.861 | 0.963 | 0.899 | 1.039 | 1.180 |

30% | 1.271 | 1.498 | 1.725 | 0.820 | 0.919 | 1.018 | 1.084 | 1.237 | 1.389 |

35% | 1.278 | 1.486 | 1.695 | 0.840 | 0.933 | 1.027 | 1.155 | 1.304 | 1.453 |

40% | 1.417 | 1.632 | 1.846 | 0.881 | 0.972 | 1.063 | 1.322 | 1.480 | 1.638 |

Obs. | 5% | ${\widehat{\gamma}}_{LLRS-1/2}$ | 95% | 5% | ${\widehat{\gamma}}_{LLRS-1/2}$ | 95% | 5% | ${\widehat{\gamma}}_{LLRS-1/2}$ | 95% |

10% | 0.572 | 0.908 | 1.244 | 0.486 | 0.662 | 0.838 | 0.493 | 0.707 | 0.920 |

15% | 0.745 | 1.071 | 1.396 | 0.556 | 0.709 | 0.863 | 0.639 | 0.848 | 1.057 |

20% | 0.840 | 1.140 | 1.440 | 0.598 | 0.736 | 0.873 | 0.723 | 0.919 | 1.115 |

25% | 0.912 | 1.193 | 1.473 | 0.629 | 0.755 | 0.881 | 0.760 | 0.939 | 1.119 |

30% | 0.976 | 1.242 | 1.509 | 0.658 | 0.777 | 0.895 | 0.809 | 0.980 | 1.150 |

35% | 1.024 | 1.278 | 1.531 | 0.685 | 0.798 | 0.911 | 0.855 | 1.019 | 1.184 |

40% | 1.070 | 1.314 | 1.559 | 0.708 | 0.816 | 0.924 | 0.906 | 1.067 | 1.228 |

Obs. | 5% | ${\widehat{\gamma}}_{w}$ | 95% | 5% | ${\widehat{\gamma}}_{w}$ | 95% | 5% | ${\widehat{\gamma}}_{w}$ | 95% |

10% | 0.328 | 0.436 | 0.544 | 0.61 | 0.657 | 0.704 | 0.493 | 0.707 | 0.92 |

15% | 0.5 | 0.656 | 0.813 | 0.544 | 0.586 | 0.628 | 0.639 | 0.848 | 1.057 |

20% | 0.691 | 0.883 | 1.075 | 0.596 | 0.639 | 0.683 | 0.723 | 0.919 | 1.115 |

25% | 0.793 | 0.986 | 1.18 | 0.623 | 0.668 | 0.713 | 0.76 | 0.939 | 1.119 |

30% | 0.836 | 1.024 | 1.212 | 0.617 | 0.664 | 0.71 | 0.809 | 0.98 | 1.15 |

35% | 0.906 | 1.088 | 1.269 | 0.618 | 0.665 | 0.712 | 0.855 | 1.019 | 1.184 |

40% | 0.91 | 1.092 | 1.275 | 0.635 | 0.681 | 0.727 | 0.906 | 1.067 | 1.228 |

**Figure 4.**Occupancy level 3 type “Large Hotel”: tail index estimation (both Hill and LLRS-1/2 regression methods), with 90% confidence bands.

**Figure 5.**Occupancy level 3 types corresponding to dwellings (single family and multi family), institutional housing, condos, housing associations, : tail index estimation (both Hill and LLRS-1/2 regression methods), with 90% confidence band.

**Figure 6.**Occupancy level 2 types C (chemicals), J (metals), and M (mines) aggregated: tail index estimation (both Hill and LLRS-1/2 regression methods), with 90% confidence band.

**Table 7.**Tail regression results for the exponential model $\tilde{\alpha}=exp\left({\theta}^{\prime}X\right)$. The vector of covariates, $X={({X}_{1},{X}_{2},{X}_{3})}^{\prime}$, includes Occupancy Level 1 indicators for classes RE (residential), CO (commercial), and MA (manfacturing). The corresponding estimated loadings are indicated by $\widehat{\theta}={({\widehat{\theta}}_{1},{\widehat{\theta}}_{2},{\widehat{\theta}}_{3})}^{\prime}$. Following Wang and Tsai [20], the optimal sample fraction corresponds to 10% of the largest losses in the entire dataset. We set ${\widehat{\alpha}}_{i}:=exp\left({\widehat{\theta}}_{i}\right)$ for the tail index estimate resulting from considering only the loading estimate ${\widehat{\theta}}_{i}$ pertaining to each individual occupancy type.

5% | ${\widehat{\theta}}_{i}$ | 95% | ${\widehat{\alpha}}_{i}$ | |
---|---|---|---|---|

CO | 0.177 | 0.196 | 0.215 | 1.217 |

RE | 0.070 | 0.082 | 0.093 | 1.085 |

MA | −0.032 | −0.019 | −0.006 | 0.981 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Swiss Re. “Insuring Ever-evolving Commercial Risks.” In SIGMA. 2012, Volume 5, Zurich, Switzerland: Swiss Reinsurance Company. [Google Scholar]
- U. Riegel. “On Fire Exposure Rating and the Impact of the Risk Profile Type.” ASTIN Bull. 40 (2010): 727–777. [Google Scholar]
- S. Desmedt, M. Snoussi, X. Chenut, and J. Walhin. “Experience and Exposure Rating for Property per Risk Excess of Loss Reinsurance Revisited.” ASTIN Bull. 42 (2012): 233–270. [Google Scholar]
- J. Buchanan, and M. Angelina. The Hybrid Reinsurance Pricing Method: A Practitioner’s Guide. Technical Report; New York City, NY, USA: ISO Verisk, 2014. [Google Scholar]
- N. Michaelides, P. Brown, F. Chacko, M. Graham, J. Haynes, D. Hindley, S. Howard, H. Johnson, K. Morgan, C. Pettengell, and et al. “The Premium Rating of Commercial Risks. Technical Report, Working Party on Premium Rating of Commercial Risks.” In Proceedings of the General Insurance Convention, Blackpool, UK, 15–18 October 1997.
- X. Gabaix, and R. Ibragimov. “Rank- 1/2: A simple way to improve the OLS estimation of tail exponents.” J. Bus. Econ. Stat. 29 (2011): 24–39. [Google Scholar] [CrossRef]
- R. Huisman, K.G. Koedijk, C.J.M. Kool, and F. Palm. “Tail-index estimates in small samples.” J. Bus. Econ. Stat. 19 (2001): 208–216. [Google Scholar] [CrossRef]
- P. Embrechts, C. Klueppelberg, and T. Mikosch. Modelling Extremal Events: For Insurance and Finance. Berlin, Germany: Springer Verlag, 1997. [Google Scholar]
- J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels. Statistics of Extremes: Theory and Applications. Hoboken, NJ, USA: John Wiley & Sons, 2006. [Google Scholar]
- J. Nešlehová, P. Embrechts, and V. Chavez-Demoulin. “Infinite mean models and the LDA for operational risk.” J. Oper. Risk 1 (2006): 3–25. [Google Scholar]
- M. Degen, P. Embrechts, and D.D. Lambrigger. “The quantitative modeling of operational risk: Between g-and-h and EVT.” ASTIN Bull. 37 (2007): 265. [Google Scholar] [CrossRef]
- G. Silverberg, and B. Verspagen. “The size distribution of innovations revisited: An application of extreme value statistics to citation and value measures of patent significance.” J. Econ. 139 (2007): 318–339. [Google Scholar] [CrossRef]
- P. Artzner, F. Delbaen, J.M. Eber, and D. Heath. “Coherent measures of risk.” Math. Financ. 9 (1999): 203–228. [Google Scholar] [CrossRef]
- R. Ibragimov. “Portfolio diversification and value at risk under thick-tailedness.” Quant. Financ. 9 (2009): 565–580. [Google Scholar] [CrossRef]
- R. Ibragimov. “Heavy-tailed densities.” In The New Palgrave Dictionary of Economics, Online Edition. Edited by Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 2009, Available online: http://www.dictionaryofeconomics.com/article?id=pde2009_H000191 (accessed on August 3, 2014).
- A.J. McNeil. “Estimating the tails of loss severity distributions using extreme value theory.” ASTIN Bull. 27 (1997): 117–137. [Google Scholar] [CrossRef]
- B. Hill. “A simple general approach to inference about the tail of a distribution.” Ann. Stat. 3 (1975): 1163–1174. [Google Scholar] [CrossRef]
- J. Beirlant, G. Dierckx, Y. Goegebeur, and G. Matthys. “Tail index estimation and an exponential regression model.” Extremes 2 (1999): 177–200. [Google Scholar] [CrossRef]
- J. Beirlant, and Y. Goegebeur. “Regression with response distributions of Pareto-type.” Comput. Stat. Data Anal. 42 (2003): 595–619. [Google Scholar] [CrossRef]
- H. Wang, and C.L. Tsai. “Tail index regression.” J. Am. Stat. Assoc. 104 (2009): 1233–1240. [Google Scholar] [CrossRef]
- P. Hall. “Using the Bootstrap to Estimate Mean Square Error and Select Smoothing Parameters in Non-parametric Problems.” J. Multivar. Anal. 32 (1990): 177–203. [Google Scholar] [CrossRef]
- M.M. Dacorogna, U.A. Müller, O.V. Pictet, and C.G. de Vries. The Distribution of Extremal Foreign Exchange Rate Returns in Extremely Large Data Sets. Technical Report; Rotterdam, Netherlands: Tinbergen Institute, 1995. [Google Scholar]
- S. Csorgo, P. Deheuvels, and D. Mason. “Kernel estimates of the tail index of a distribution.” Ann. Stat. 13 (1985): 1050–1077. [Google Scholar] [CrossRef]
- Y. Goegebeur, J. Beirlant, and T. de Wet. “Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation.” Stat. J. 6 (2008): 51–69. [Google Scholar]
- A.L. Dekkers, J.H.J. Einmahl, and L.D. Haan. “A moment estimator for the index of an extreme-value distribution.” Ann. Stat. 17 (1989): 1833–1855. [Google Scholar] [CrossRef]
- J. Diebolt, A. Guillou, and I. Rached. “Approximation of the distribution of excesses through a generalized probability-weighted moments method.” J. Stat. Plan. Inference 137 (2007): 841–857. [Google Scholar] [CrossRef]
- J. Beirlant, P. Vynckier, and J.L. Teugels. “Tail Index Estimation, Pareto Quantile Plots Regression Diagnostics.” J. Am. Stat. Assoc. 91 (1996): 1659–1667. [Google Scholar]
- I. Gomes, L. de Haan, and L.H. Rodrigues. “Tail index estimation for heavy-tailed models: Accommodation of bias in weighted log-excesses.” J. Roy. Stat. Soc. B 70 (2008): 31–52. [Google Scholar]
- T. Nguyen, and G. Samorodnitsky. “Multivariate tail estimation with application to analysis of COVAR.” ASTIN Bull. 43 (2013): 245–270. [Google Scholar] [CrossRef]

^{1}According to the estimates reported in Swiss Re [1], 29% of direct insurance premiums in 2010 were written for property and business interruption risks.^{2}In terms of global direct premiums written in 2010, the shares of these lines of business were as follows: 25% (liability insurance), 19% (commercial auto insurance), 17% (specialty lines such as offshore energy), and 11% (workers’ compensation); see Swiss Re [1].^{3}With USD 1.1 billion of direct premiums written in 2010; see Swiss Re [1].^{4}Although the domestic UK market is not as large as, for example, the German and French markets, London is the main marketplace for international commercial (re)insurance risks. When counting foreign business, the UK jumps to second place; see Swiss Re [1].^{5}These long standing issues were already pointed out, for example, in Michaelides et al. [5].^{6}The claim overlap rate between the two syndicates is roughly 50%. We estimated that an additional large syndicate would have led to an overlap rate higher than 70%.^{8}Commercial property insurance providers in the London market, for example, often quantify reserves by inflating premiums by a factor (say 30%) of the standard deviation of the losses.^{9}As an example, consider the well-known dataset of Danish fire losses presented in McNeil [16]; it provides information on 2157 fire losses occurred between 1980 and 1990, with a 1985-equivalent value above 1 million Danish Krone (GBP 197k in current, RPI-adjusted money terms). Data were made available by Copenhagen Re with a breakdown of total loss data into building, content, and profit losses.^{10}We are grateful to John Buchanan and Chris Kent at ISO Verisk for making this validation exercise possible. The exercise demonstrates consistent results in terms of average excess severity of losses above USD 1m and 5m across occupancy types (see Figure 1). The Imperial-IICI dataset provides larger coverage of manufacturing exposures, which give rise to the largest losses also in the data compiled by ISO Verisk.^{11}Claims are available in original currency. For comparability, we converted claims in USD by using Lloyd’s year-end currency conversion rates.^{12}These include kernel estimators (e.g., [23,24]), moment estimators (e.g., [25]), probability weighted moment estimators (e.g., [26]), and weighted least squares estimators (e.g., [27]). See also Gomes et al. [28] for weighted log-excesses, Nguyen and Samorodnitsky [29] for the multivariate case, and the monographies by Embrechts et al. [8], Beirlant et al. [9] for a general overview.^{13}We exclude on-shore energy exposures, and do not study the split between physical damage and business interruption in our claims.

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Biffis, E.; Chavez, E.
Tail Risk in Commercial Property Insurance. *Risks* **2014**, *2*, 393-410.
https://doi.org/10.3390/risks2040393

**AMA Style**

Biffis E, Chavez E.
Tail Risk in Commercial Property Insurance. *Risks*. 2014; 2(4):393-410.
https://doi.org/10.3390/risks2040393

**Chicago/Turabian Style**

Biffis, Enrico, and Erik Chavez.
2014. "Tail Risk in Commercial Property Insurance" *Risks* 2, no. 4: 393-410.
https://doi.org/10.3390/risks2040393