Understanding Damage from Landfalling Tropical Cyclones

Power-Law distributions have shown to be useful tools for describing rare, high impact “black swan” events (Hernandez 2014). The Pareto distribution (e.g., Hardy 2010) describes the tail of the distribution of historical hurricane losses. A separate Power-Law distribution, the Zipf distribution (Newman 2005), fits the tail of the size distribution of populated places. The sizes of the largest populated places are inversely proportional to their rank so that the Zipf distribution is effectively a Pareto distribution with unit exponent. Hernandez (2014) uses an idealized hurricane catastrophe model (e.g., Grossi and Kunreuther 2005), Z-CAT, to simulate damage on a Zipf distributed coastal population. Z-CAT demonstrates that the tail of the distribution (the large impacts) inherits its shape from the distribution of assets along the coast. These losses account for approximately 2/3 of historical losses but only ~10% of damaging events.

Statistical analysis of damage shows that nationally aggregated US hurricane losses normalized for inflation, population and individual wealth has been constant since 1900. Detrended nominal damage increases at approximately the same rate as the US Gross Domestic Product; ~ 6% increase. Yet, normalized and detrended US damage are approximately log-normal, but somewhat leptokurtic and negatively skewed with standard deviations of 10 to 12 about their geometric means. Z-CAT cannot reproduce this large variability. Current population size distributions show log-normality much like the distribution of losses but with a smaller variance. Here we reassess US aggregated loss under the hypothesis that the log normal distribution of losses is inherited from a log-normal distribution of assets rather than from a Zipf distribution.

A modified version of Z-CAT, based upon a log normal distribution of populated places will be more realistic and provide insight to what controls the shape of the nationally aggregated damage exceedance probability curve; a negatively skewed, leptokurtic distribution. The application of a hurricane catastrophe model that uses the actual distribution of coastal cities as well as geographic, inter-seasonal, multi-decadal or secular variations of landfall intensity and frequency will also aid in understanding the peril. A salient result from the previous study was that damage would need to double on a century timescale to attain significance using standard nonparametric tests. Yet, increasing hazards may become financially significant before they become statistically significant. An example is the impact of Hurricane Maria on Puerto Rico. How would using the log normally distributed assets change this? How rapidly would hurricanes characteristics need to change to produce significant trends in losses? How does the intensity and frequency of the peril affect the damage trend? If more threatening peril becomes more frequent, how would building standards have to improve to compensate? How would insurers, regulators, and policy holders address the conflict between the actual changes in peril that may or may not be masked by long term natural variability such as ENSO and AMO?

References

Grossi, P., and Kunreuther, H., 2005: Catastrophe Modeling: A New Approach to Managing Risk, Springer, New York, 245 pp.

Hardy,M., 2010: Pareto's Law, Mathematical Intelligencer, 32(3),38–43. doi: 10.1007/s00283-010-9159-2

Hernandez, Javiera I., 2014: Does the Pareto distribution of hurricane damage inherit its fat tail from a Zipf distribution of assets at hazard?; FIU Electronic Theses and Dissertations. Paper 1488. http://digitalcommons.fiu.edu/etd/1488

M.E. J. Newman, 2005: Power laws, Pareto distributions and Zipf's law, Contemporary Physics 46, pages 323–351. doi:10.1080/00107510500052444 http://arxiv.org/abs/cond-mat/0412004v3