An Imperfect Storm: Fat-Tailed Hurricane

Damages, Insurance, and Climate Policy∗

Marc N. ConteDepartment of Economics

Fordham University

David L. KellyDepartment of Economics

University of Miami

February 2, 2016

Abstract

We perform two tests that estimate the thickness of the tails of the distri-bution of aggregate US hurricane damages. Both tests reject the hypothesisthat the distribution of damages is thin tailed at the 95% confidence level, evenafter correcting for inflation, population, and per capita income growth. Ourpoint estimates of the shape parameter of the damage distribution indicatethat the distribution has finite mean, but infinite variance.

In the second part of the paper, we develop a microfoundations modelof insurance and storm size that generates fat tails in aggregate hurricanedamages. In the model, the distribution of the number properties within arandom geographical area that lies in the path of a hurricane drives fat tailsin hurricane damages, and we confirm that the distribution of coastal citypopulation is fat tailed in the US. We show empirically and theoretically thatother random variation, such as the distribution of hurricane strength and thedistribution of damages across individual properties do not generate fat tails.We consider policy options such as climate change mitigation, policies whichencourage adaptation, reducing subsidies for coastal development, and disasterrelief policies, which distort insurance markets. Such policies can reduce thethickness of the tail, but do not affect the shape parameter or the existence ofthe fat tail.

Keywords: natural disasters, fat tails, hurricanes, adaptation, disaster aid, propertyinsurance.JEL Codes: Q54, H84, Q58, R11

∗We would like to thank seminar participants at the 15th OccasionalWorkshop on Environmentaland Resource Economics, the 2015 American Economics Association Meetings, the 2015 SummerMeeting of the Association of Environmental and Resource Economists, Georgia State University,and the University of Miami for useful comments and suggestions.

1 Introduction

Hurricanes are the most costly natural disasters in the US (Smith and Katz2013). Indeed, several recent storms (e.g. Katrina and Sandy) were catas-trophic, causing damages on the order of tens of billions of dollars, as wellas loss of life, dislocations, and suppression of economic activity. Despite thelarge losses, little is known about the ultimate cause of catastrophic hurricanedamages and to what extent government policies can reduce the welfare lossescaused by these natural disasters.

Although some hurricanes are catastrophic, the majority of storms causelittle or no damage. Hence, one might suspect that the distribution of hur-ricane damages is fat tailed. A growing body of literature shows that thedistribution of total damages from natural disasters is often fat tailed. Forexample, Kousky and Cooke (2009) find evidence of fat-tailed aggregate dam-ages for several varieties of natural disasters, including crop indemnities andNational Flood Insurance claims.1

In the first part of this paper, we provide empirical evidence that the dis-tribution of total hurricane damages is indeed fat tailed, even after controllingfor inflation and growth in population and per capita income. In the secondpart of the paper, we develop a microfoundations model that generates a fat-tailed aggregate damage distribution, and show how public policies affect theupper tail of the distribution.

We perform two tests to estimate the thickness of the tails of the damagedistribution. The first test estimates the parameters of the generalized Paretodistribution (GPD) using maximum likelihood. The GPD nests both fat andthin tails and is a commonly used test for fat tails (Brazouskas and Kleefeld2009). The second test estimates the slope of the mean excess function: theexpectation conditional on a realization being larger than a threshold. Theslope of the mean excess as a function of the threshold is positive if and onlyif the distribution is fat tailed (Ghosh 2010).

The two tests produce similar estimates of the shape parameter of thedamage distribution in the range of 0.66 to 0.8, rejecting the hypothesis thatthe distribution of damages is thin tailed at the 95% confidence level. Thepoint estimates of the shape parameter suggest that the damage distributionis fat tailed with finite mean but infinite variance.2

1In addition, fat-tailed uncertainty has been shown to pose a particular challenge topolicy makers in the context of climate change (Weitzman 2014; Kelly and Tan 2015).

2The shape parameter measures the thickness of the tail of the distribution. The meanand variance of the distribution are finite if the shape parameter is less than or equal toone and one half, respectively. See Cooke and Nieboer (2011) for a primer on fat-tailed

1

A fat-tailed hurricane damage distribution poses a number of challenges.The estimated mean damage depends closely on the damage from catastrophicstorms, for which relatively few observations exist. The mean damage, uponwhich insurance is priced, is more difficult to estimate when damages aredrawn from a fat-tailed distribution. Similarly, the expected damage from acatastrophic storm (e.g., the damage from a once-in-a-hundred-years storm),is also more difficult to estimate, meaning that regulators will have difficultyestablishing minimum levels of reserve capital, increasing the likelihood of acatastrophic storm for which reserves are inadequate and adding risk to thehouseholds. Insurance firms must also hold more costly reserves. Policiesthat reduce the mass of the upper tail of the damage distribution are poten-tially welfare increasing. Identifying such policies requires an empirical andtheoretical understanding of the determinants of the fat tail in the damagedistribution.

Hurricanes that cause catastrophic damages feature extraordinary stormintensity (e.g., Andrew, which had a maximum sustained wind speed of 170mph), make landfall in areas with an extraordinarily large population andproperty base (e.g., Sandy, which struck the New York metropolitan area), andcause extraordinary damage to individual properties. Aggregate damages are afunction of each of these random variables. A fat tail in the distribution of oneor more of these random variables can imply a fat-tailed damage distribution.

One possible cause of fat-tailed hurricane damages is the distribution ofstorm intensity. However, we show empirically that storm wind speed is thintailed.3 This result is intuitive: the expected wind speed conditional on ex-ceeding a threshold should decline with the threshold, because the extra energyrequired to generate and sustain larger storms is not readily available. Ourresults show that the wind speed distribution cannot cause a fat tail in thedamage distribution.

Aggregate damages are the sum of individual property damages. Becauseproperty damages are bounded and therefore have finite variance, the centrallimit theorems apply. The sum of property damages becomes approximatelynormally distributed (thin tailed) as the number of properties becomes large;random variation in the individual damage across properties cannot be thesource of fat tails in the damage distribution.

Considerable evidence exists that the distribution of city population in theUS is fat tailed (see for example, Gabaix 1999; Gabaix and Ibragimov 2011;

distributions.3Maximum wind speed is a commonly used measure of storm intensity (Nordhaus 2010).

Other measures of storm intensity, such as size, are correlated with wind speed (Holland1980).

2

Gonzalez-Val et al. 2015). Gabaix and Ibragimov (2011) and others find theexponent of the Pareto distribution is close to one (known as Zipf’s law) forUS metropolitan areas above a minimum population.4 However, Gonzalez-Valet al. (2015) considers the GPD and finds a range of potential exponents.

We test the coastal population distribution for fat tails and find that thecoastal population distribution is also fat tailed, with a shape parameter inthe range of 0.6 to 0.8, very close to the shape parameter of the aggregatedamage distribution.5 Our empirical results indicate that a determinant of fattails in the damage distribution is fat tails in the geographic distribution ofproperty. Most hurricanes intersect geographic areas with little or no property.However, hurricanes intersect areas with large amounts of property more oftenthan expected given a normal distribution, causing the fat tail in the damagedistribution.

Having demonstrated empirically that hurricane damages are fat tailed, wepresent a model of homeowner behavior that links policies to the mass of theupper tail of the hurricane damage distribution.6 In the model, householdsdevelop both coastal and inland properties using constant marginal cost tech-nologies. The cost of property insurance raises the total cost of ownership forcoastal properties, thus discouraging development.7 A disaster relief agencyexists which reimburses a fraction of household losses.

The total coastal property at risk as determined by household choices isdistributed geographically into population centers. Following the empiricalresults, the size distribution of the population centers has a fat tail. Finally,a model of hurricane size, wind speed, and adaptations (e.g. building codes)determines the area of damage. In the model, a hurricane of random (thintailed) size and wind speed may intersect a population center of random (fattailed) size, in which case individual properties experience random, bounded

4Gabaix (1999) proposes a theory for city sizes that generates a Pareto distribution. Intheir model, wages adjust to offset city-specific amenity shocks, which keeps the growth rateof cities independent of the size, which implies a power law steady state distribution.

5Our interest is in coastal property, not the US population. Nonetheless, the coastalproperty distribution is correlated with the coastal population distribution.

6While we are not aware of any other attempt to create a microfoundations model of fattailed aggregate damages, in a recent paper Jones (2015) creates a model that generates fattails in the distribution of income. In a theory in finance, fat tails arise in financial returndata through random news events (for example, Clark 1973; Haas and Pigorsch 2011).

7In practice, households may not have correct risk perceptions, and evidence of home-owners updating their risk perceptions in response to natural disasters is mixed. See Binand Polasky (2004); Hallstrom and Smith (2005); Kelly et al. (2012) for analyses of changesin risk perceptions following hurricanes. Beron et al. (1997) studies risk perceptions follow-ing an earthquake and Gallagher (2014) finds that enrollment in flood insurance programsincreases following major flooding events and then subsides until the next such event.

3

damages.We show that, as the number of potentially affected properties becomes

large, the damage distribution converges to a distribution that is approxi-mately fat tailed, with tail index equal to the tail index of the property dis-tribution, which is consistent with our close empirical estimates of the shapeparameters for damage and coastal population.8 Although aggregate damagesare the sum of individual damages with finite variance, the aggregate dam-age distribution is not normal because the number of terms in the sum (thenumber of properties in the population center) is random and fat tailed.

The costs of fat-tailed damages motivate public policies to thin the tail ofthe damage distribution. In the model, the distribution of hurricane damagesis a function of several policies, including climate policy (which affects thefrequency of large storms), disaster relief and development subsidies (whichaffect the total property at risk), and adaptations (which affect the minimumwind speed that causes damage). Each of these policies affects the damagefrom a catastrophic storm in different ways.

One important policy is disaster relief and other subsides to coastal de-velopment. Indeed, the rise in coastal development has led to concern thatdisaster relief increases hurricane damage (Bunby 2006).9 After a hurricane,the Federal Emergency Management Agency (FEMA) may declare the af-fected area a disaster zone and provide relief from uninsured damages. Thisintervention is meant to reduce the welfare losses from hurricanes.

We show that moral hazard created by disaster relief adds to the totalproperty at risk, both on the extensive (more coastal land is developed versusinland, ceteris parabis) and intensive (developed coastal land has more prop-erty than developed inland ceteris parabis) margins. An increase in the totalproperty at risk implies population centers have more property, increasing thedamage from a once-in-a-hundred-years storm.

Adaptations such as stronger building codes affect the damage distribution.Hurricane strength declines with the distance from the eye of the storm (Hol-land 1980). If a random hurricane path intersects a population center awayfrom the eye of the storm, then little or no damage will result if building codesare sufficiently strong. Much of the literature on adaptations (for example,Kelly et al. 2005; Mendelsohn et al. 1994) shows empirically how adaptationscan reduce the point estimate of damages. Our complementary analysis pro-vides theoretical insight into the process whereby adaptation policies affect

8For the GPD distribution, the tail index is the inverse of the shape parameter.9Populations in coastal communities at risk of hurricane damages are growing more

rapidly than populations in the rest of the United States (Blake et al. 2011).

4

the tail of the damage distribution.We show that an increase in the strength of adaptations reduces the ge-

ographic area of damage in two ways. First, adaptations, such as strongerbuilding codes, reduce the radius of winds that are strong enough to causedamage, as the wind speed beyond a certain distance from the eye is no longerstrong enough to cause damage. Second, adaptations decrease the time fromlandfall during which the hurricane is sufficiently strong to cause damages. Themaximum wind speed declines over land, reaching the point where adaptationsare strong enough to prevent damage more quickly. In turn, the reduction inthe geographic area of damage implies that more hurricanes miss populationcenters and cause no damage. Because a sample of hurricanes includes morestorms with little or no damage, adaptations reduce the damages of a once-in-a-hundred-years storm.

Climate change mitigation is another policy lever that affects the aggre-gate damage distribution. Some research suggests that a positive link existsbetween greenhouse gas emissions and hurricane frequency (Holland and Web-ster 2007; Mann and Emanuel 2006; Emanuel 2005). However, there is not aconsensus on this point, as other studies fail to support that conclusion (Vec-chi et al. 2008; Vecchi and Knutson 2008; Knutson et al. 2008). One recentstudy finds that the number of intense storms (category 4 or 5 hurricanes) maydouble by the end of the 21st century, even as the overall frequency of tropicalcyclones decreases (Bender et al. 2010), suggesting that changing climate mayprimarily affect the tails of the distribution of storm size.10

We show that the effect of climate change mitigation is similar to that ofadaptation. Limiting climate change reduces the increase in energy availablefor storm formation. By reducing the likelihood of intense hurricanes, thegeographic area of damage decreases, reducing the damage from a once-in-a-hundred-years storm.

Finally, we show that none of the policies we consider affect the tail in-dex or the existence of the fat tail. While the policies can thin the tail ofthe distribution, the policies cannot make a fat tailed distribution thin tailed.Specifically, the model predicts that the fat tail is present regardless of thestrength of adaptations (e.g. both less and more developed countries shouldsee catastrophic storms more frequently than expected from a normal distri-bution).

The next section introduces the data used in the empirical analyses. Section

10Pielke et al. (2008), fail to identify an observable linear trend in the mean damage asso-ciated with each storm making landfall in the United States from 1900-2008 after controllingfor population growth and inflation. In contrast, our focus is on changes in other moments,such as the risk of catastrophic storms.

5

3 describes the empirical methods. Section 4 presents the empirical results.Section 5 presents a model of property development, insurance, and hurricanestrength. Section 6 derives the aggregate damage distribution and the policyimplications. Section 7 concludes.

2 Data

The distributions of hurricane damages, hurricane strength, and coastal pop-ulation are explored using three data sets. The first data set was downloadedfrom the ICAT insurance firm (http://www.icatdamageestimator.com) web-site, which provides damages from all storms making landfall in the UnitedStates between 1900 and 2012. The second data set is provided by a NASAwebsite, which provides detailed storm-location and characteristic information(e.g., minimum pressure and maximum sustained wind speed) for all stormsthat originated in the Northern Atlantic Basin between 1850 and 2012. Givenchanges in satellite technology, only data since 1950 is used in the below anal-ysis of storm strength. The third data set of coastal city populations is fromthe 2010 Census conducted by the US Census Bureau.

The ICAT data set aggregates data contained in the Monthly Weather Re-views, published since 1872, which provide summaries of the storms that haveoccurred in the North Atlantic Basin each year. These reports are currentlypublished by employees of the National Hurricane Center, which is an officewithin the National Weather Service and the National Oceanic and Atmo-spheric Administration. The raw data includes storm name and the economicdamages associated with the storm.11 Several adjustments are made to thereported storm-year damages to increase the degree of comparability for dam-ages across different years.

A number of features of coastal communities in the United States havechanged through time in ways that might lead the economic damages causedby a storm to depend on the year in which the storm made landfall. Of prin-ciple importance is the amount of physical capital located in at-risk coastalcommunities. The total property in coastal communities has varied throughtime, though the value of this property is not historically recorded, confoundingefforts to directly control for temporal variation in at-risk capital. Other re-lated socioeconomic characteristics, such as per capita income and populationare historically available. Using 2005 country-level data from the World Bank,Bakkensen (2014) find that capital scales proportionately with per capita in-

11According to the ICAT website, since 1987, the National Hurricane Center has calcu-lated the total economic damages caused by a storm event as twice the insured damages.

6

come, estimating that capital stock is 2.65 times the per-capita income in anation. In the absence of a time series of the capital stock of coastal com-munities in the United States between 1900 and 2012, we will rely on thepopulation and income level of coastal communities to control for variation incapital stocks across time.

We adopt the methodology of Pielke et al. (2008), who controls for temporalchanges in prices due to inflation and the population and wealth of coastalcommunities, by normalizing damages relative to 2012 prices, income, andpopulation for each storm from 1900-2012 according to:

Di,2012 = Di,y × Iy/2012 × RWPCy/2012 × P2012/y , (1)

where Di,2012 is the normalized damage of hurricane i in 2012 dollars; Di,y

is the reported damage in year y dollars; I2012/y is the inflation adjustment;RWPC2012/y is a real per capita income adjustment; and P2012/y is the coastalcounty population adjustment. Note that controlling for per capita income as-sumes that the income elasticity of hurricane damages is one. This assumptionimplies no changes in adaptation to the threat of hurricane damages as wealthin coastal communities within the United States increases. This assumptionis consistent with the finding from Bakkensen (2014), who find no evidenceof changes in adaptation with respect to income in the United States overthe period 1960 to 2010. If a policy-driven or market response to hurricanedamage existed that was influenced only by per capita income, and not stormfrequency or intensity, then our approach might overestimate the damagescaused by recent storms, when income levels are highest. While a number ofcatastrophic storms have occurred in recent decades, six of the top 10 stormsas measured by damages in 2012 dollars using the above methodology occurredprior to the end of World War II, when per capita income in coastal areas wasmuch lower than modern levels.

Other time-varying characteristics of coastal communities are building codesand materials. Any improvements in building codes and materials over timetend to reduce the damages associated with more recent storms. Changes inbuilding materials include reductions in the use of traditional building ma-terial such as wood in favor of reinforced concrete. Building codes in manycoastal communities now specify preferred building footprints, materials, andtechniques related to both energy efficiency and building stability.12 Many of

12Since 2000, the International Residential Code has been updated each three years withthe latest recommendations regarding construction materials and techniques (InternationalCode Council 2003).

7

these techniques, such as tighter nailing patterns for interior panels in highwind regions, increase the cost of construction relative to regions not at riskof storms. Higher construction costs are an additional cost of the risk of hur-ricanes that is not captured by the ICAT data set.

The distribution of storm strength will play an important role in the theo-retical model and in the effects of adaptation, climate change mitigation, andcoastal development policies on aggregate damages. Therefore, we also usestorm characteristic information on all cyclones formed in the Atlantic basinbetween 1950 and 2012, gathered from the Tropical Storm Tracks databasemaintained by the NASA Earth Science Data and Information System GlobalHydrology Resource Center using data published by the National HurricaneCenter.13 The database provides location, category, minimum pressure, andmaximum wind speed data for Atlantic basin storms from 1851.

The distribution of coastal property is also important for the theoreticalmodel. We use coastal population data from the 2010 Census conducted by theUS Census Bureau as a proxy for coastal property. The data gives populationsize within cities located in coastal counties of states in which a storm has ledto damages at least once between 1900 and 2012.

3 Empirical Methods

The goal of the empirical analysis is to detect whether or not the distributionof hurricane damages has fat tails, which has implications for the efficacy ofclimate change mitigation, adaptation, and development policies in coastalcommunities dealing with the threat of damage-inducing storms. Additionalanalyses explore the potential sources of the fat tails in the aggregate damagesdistribution.

The first step in the empirical methodology is the definition of fat tails. Nosingle definition of fat tails exists in the literature. Here we say a distributionis fat tailed if it is asymptotically equivalent to a Pareto distribution (Cookeand Nieboer 2011). That is:

limD→∞

f (D)

αD−α−1→ 1 , α > 0. (2)

The above definition particularly convenient because the tail index of a fat-tailed distribution is α. The kth moment is infinite if and only if α < k.

13Tropical Storm Tracks. NASA EOSDIS GHRC DAAC, Huntsville, AL. http://ghrc.nsstc.nasa.gov/storms/. Accessed on December 27, 2013.

8

3.1 Generalized Pareto Distribution

The GPD is a popular choice for extreme value analysis, with applicationsin a variety of disciplines germane to the current effort, including; actuarialscience (e.g., Brazouskas and Kleefeld 2009; Cebrian et al. 2003), climatology(e.g., Nadarajah 2008), and meteorology (e.g., Holmes and Moriarty 1999).The distribution and density functions for the GPD, with shape parameter, ξ,and scale parameter, σ, are:

FGPD(D; σ, ξ) =

1− (1 + ξDσ)−1/ξ if D > 0 , ξ 6= 0

1− e−Dσ if ξ = 0

(3)

fGPD(D; σ, ξ) =

( 1σ)(1 + ξD

σ)−1

ξ−1 if D > 0 , ξ 6= 0

( 1σ)e−

Dσ if ξ = 0

, (4)

respectively. Note that 0 ≤ D < ∞ if ξ ≥ 0, and 0 ≤ D < −σ/ξ if ξ < 0.14

The shape parameter affects the tail of the distribution as well as theexistence of the moments for the distribution; the tail index is α = 1/ξ, so thekth moment exists if and only if ξ < 1

k(Hosking and Wallis 1987). The mean

of the distribution, σ/(1 − ξ), is finite if and only if ξ < 1. When the meanis infinite, efforts to estimate distribution parameters based on maximizedlikelihood yield unreliable results. The variance, σ2/(1 − ξ)2(1− 2ξ), is finiteif and only if ξ < 1

2. Maximum likelihood estimation of GPD parameters is

feasible when the distribution has infinite variance. For 0 < ξ ≤ 0.5, thedistribution decays at a rate slower than the normal distribution, but thevariance is finite, and the central limit theorems still apply. When ξ < 0, thedistribution is bounded above by −1

ξ. When ξ = 0, the distribution is the

exponential distribution. The area under the tail is monotonically increasingin ξ. The probability of observing extreme values is greater than with a normaldistribution if and only if ξ > 0. Therefore, the GPD nests thin and fat tailsdepending on ξ. Figure 1 illustrates how the area under the tail changes as ξtakes on values between -1 and 1. Figure 1 also gives an example with infinitevariance (ξ = 2/3) and infinite mean (ξ = 1).

14Some versions of the GPD specify an additional location parameter, which correspondsto the lower bound of the support of the distribution. Here we follow the literature citedabove and naturally set the location parameter equal to zero. The estimate of the shapeparameter is not affected by the value of the location parameter (Pickands 1975).

9

3.2 Tests for Fat Tails

Our interest is in estimation of the shape of the tail of the distribution. Mostof the literature uses subsets of the data that lie beyond threshold values (seede Zea Bermudez and Kotz 2010, for a review of available techniques). Re-stricting the data set to values above a threshold is appropriate for two reasons.First, because extreme values are by definition rare, using the entire data setmay result in an estimate of ξ that fits most of the empirical distributionwell but is a poor fit for the tail (DuMouchel 1983). Second, a fairly generaltheorem (Pickands 1975) shows that the tail of the GPD approximates thetail of any fat-tailed distribution (in particular, a sufficiently large thresholdexists such that the GPD approximates any distribution satisfying a technicalcondition above the threshold to an arbitrarily small approximation error).Therefore, the GPD is always appropriate for estimation of the tail, but maynot be appropriate for estimation of the entire distribution.

A tradeoff exists regarding the choice of threshold value: thresholds thatare too high result in small sample sizes and high variance for the estimatedparameters,15 while thresholds that are too low include ranges of the distribu-tion for which the GPD may not be a good fit, leading to biased parameterestimates (Smith 1987). Although a number of techniques based on peaks-over-threshold approaches exist, no clear rule exists for the selection of thethreshold (Davison and Smith 1990).16

The analysis explores the presence of fat tails in the hurricane damagedistribution using two methods. The first method directly estimates of theshape parameter of the GPD via maximum likelihood. The second methodestimates the slope of the mean-excess function via weighted least squares(WLS), which is a function of the shape parameter (Ghosh 2010). The testsutilize damage data from storms that made landfall between 1900 and 2012.

3.2.1 Direct MLE estimation

Consistent and asymptotically normal maximum likelihood estimators (MLEs)of the shape and scale parameters exist for −1/2 < ξ < 1. Of course, the mag-

15Giles et al. (2011) derives analytic expressions for the small-sample bias of MLE of theparameters of the GPD. Giles et al. (2011) also finds that, for positive shape parameters, thesmall-sample bias decreases as the true value of the shape parameter increases, for samplesof 50 and 100 observations.

16In a sense, the empirical dilemma of choosing a threshold mirrors the difficulty of pricinginsurance when the damage distribution may have fat tails: a dearth of extreme values andsensitivity of the shape parameter to the choice of threshold makes estimating the probabilityof extreme events difficult.

10

nitude and sign of the shape parameter are not known a priori, making estima-tion of the GPD parameters via maximum likelihood estimation challengingin practice. The estimation uses storm level damage data.17

3.2.2 Mean-Excess Estimation

The distinguishing feature of a fat-tailed distribution, relative to a thin-taileddistribution, is that the expected size of a draw larger than any draw yetobserved is much larger than the largest draw to date. Given a threshold u,the mean excess function is:

M(u) ≡ E [X − u|X > u] , (5)

as long as E[X+] < ∞. A distribution is fat tailed if M(u) is increasing inu. The implication is that previous realizations from a fat-tailed distributionare poor predictors of the magnitude of observations from the tails of thedistribution.18 Distributions can be identified as fat tailed through the mean-excess function.

The empirical estimate of the mean-excess is:

M(u) =

∑ni=1(Xi − u)IXi>u∑n

i=1 I[Xi−u]

. (6)

The slope of the mean-excess function can be used to estimate ξ, as the mean-excess function of a GPD is linear in u (Ghosh 2010):

M(u) =σ

1− ξ+

ξ

1− ξu. (7)

The slope of the mean excess function for a GPD random variable is positiveif the GPD has fat tails and finite expected value (0 < ξ < 1). Figure 2illustrates the mean excess function for the GPD when ξ = 2/3.

We use WLS to estimate the mean-excess function (7). The data set con-sists of a set of thresholds and empirical mean excess values computed via

17Given that some insurance contracts are annual, MLE of annual damages may be the-oretically more relevant than data at the individual storm level. However, evidence existsthat prospective homeowners may make choices about property purchases and insurancecoverage in response to the damages caused by a single storm (Gallagher 2014; Bin andPolasky 2004; Beron et al. 1997).

18In contrast, new records in, for example, athletic events tend to be close to old records,indicating that results from athletic events are thin-tailed.

11

equation (6). The maximum threshold must not be too large, as data pointswith too few terms result in the average (6) being a noisy estimate of the truemean excess (Ghosh 2010).

The number of data points greater than the threshold is monotonicallydecreasing in the threshold, so the mean excess of smaller thresholds are moreprecisely estimated. The standard approach using WLS is to assign weightsbased on the precision of the estimated mean-excess. Then the weights mono-tonically decrease as the threshold increases, which means that WLS puts littleemphasis on fitting the tail. Therefore, we use two weighting schemes to dealwith the heteroskedasticity in the mean excess data. The first weights by thenumber of observations used to calculate the mean excess, while the secondweights by the inverse of the number of observations.

4 Results

4.1 Fat-tailed Damages

Tables 1 and 2 report estimates of the shape parameter for storm level damagedata, during the period 1900 to 2012, using MLE and WLS, respectively.

Given the lack of guidance about the appropriate threshold for use in esti-mation of the shape parameter, Table 1 reports MLEs of ξ for several thresh-olds. Table 1 indicates the MLE of the shape parameter is sensitive to thethreshold. Nonetheless, all point estimates are in the region indicating a fattailed distribution with finite mean but infinite variance, 0.5 < ξ < 1. Further,a thin-tailed distribution (ξ ≤ 0) is rejected at the 95% confidence level for allthreshold choices.

For lower thresholds, more data is available, increasing the precision ofthe estimates. However, the additional data implies the shape parameter ischosen to fit a wider range of the tail of the damage distribution. In fact, alarger shape parameter fits the middle part of the damage distribution best.In contrast, a smaller shape parameter fits the rightmost part of the tail of thedamage distribution best. Restricting the data set to larger thresholds comesat a price of reducing the number of observations. Therefore, the standarderrors widen and finite variance cannot be rejected at the 95% confidencelevel for thresholds equal to the 60th-80th percentiles. Nonetheless, the pointestimates continue to indicate fat tails with finite mean but infinite variance,even at the 80th percentile.

The results in Table 2 provide even stronger support for the conclusion thatthe distribution of hurricane damages is fat tailed. When the upper thresholdis chosen at the 70th or 75th percentile, so that each mean excess is calculated

12

using at least 49 observations, the point estimates of ξ lie in a tight range, from0.69 to 0.74. These point estimates indicate the distribution is fat tailed, witha finite mean and infinite variance. Further, the bootstrapped 95% confidenceintervals are all between 0.5 and 1, indicating thin tails, infinite mean, andfinite variance are all rejected at the 95% confidence level.

Table 2 also reports results generated assuming the upper threshold equalsthe second largest data point. For the second largest data point, the largestmean excess is estimated using a single observation and the larger mean excessvalues have very few data points. The resulting additional noise in the meanexcess estimates increases the standard errors by almost a factor of 10 andcauses the point estimates to vary across specifications. Nonetheless, a thin tailis rejected at the 95% confidence level for both specifications. However, bothpoint estimates indicate that the variance is finite when the upper thresholdequals the second largest data point.

4.2 Distribution of Windspeed

We next perform empirical tests for fat tails identical to section 4.1, but usingdata on hurricane strength, measured by maximum wind speed,19 rather thandamages. Testing the wind speed data provides a convenient falsification test.Given the underlying physical system changes at most slowly over time, thewind speed distribution is likely to be thin-tailed. The mean wind speed condi-tional on the speed exceeding a threshold is likely to be constant or decreasingin the threshold. More importantly, showing that the wind speed distributionis thin tailed eliminates one possible cause of fat tails in the theoretical modelof Section 5.

Table 3 reports the results from WLS estimation of the mean excess (theMLE results are similar). Table 3 shows that, regardless of the lower and upperthresholds, the point estimate of the tail index is always negative indicatingthe wind speed distribution has thin tails. In all specifications, a fat tail isrejected at the 95% confidence level.

19Maximum wind speed is a commonly used measure of the strength of storms (see, forexample Nordhaus 2010). Other measures of hurricane strength, such as size, are correlatedwith wind speed. Further, we find that the distributions of storm surge, storm tides, and thecombination of storm surge and tide are all thin-tailed using the mean-excess methodologyon a Gulf of Mexico storm surge data set (Needham and Keim 2012). Details are availableon request.

13

4.3 Coastal Population Distribution

Another possible cause of fat tailed damages in the theoretical model of section5 is a fat tail in the property distribution. Table 4 reports the results of WLSestimation of the coastal property data. Table 4 reports shape parameter esti-mates between 0.66 and 0.73, again rejecting thin tails at the 95% confidencelevel. Further, the shape parameter estimates are very close to the shape pa-rameter estimates for the aggregate damages. In section 6, we show that thetheoretical model predicts the fat tail in the damage distribution arises from afat tail in the property distribution, and that the shape parameter estimatesof damages and the property distributions are equal.

5 Theoretical Model

We consider a theoretical model of aggregate damages which consists of threemajor parts. The first part consists of a model of household behavior. House-holds decide whether or not to live on the coast, and how much insuranceand property to purchase. In the second part of the model, households livingon the coast organize into population centers. Finally, a model of hurricanestrength relative to the strength of adaptations determines the probability thata population center is in the path of a hurricane. Figure 3 illustrates the threeparts of the model.

5.1 Household Behavior

A continuum of households with measure one have preferences for consump-tion, c, and property. “Property” here refers to the structures on the land,not the land itself. We denote the quantity of property for household i ∈ [0, 1]as Aj,i, where j ∈ {K, I} denotes the property type (coastal, K, or inland, I).Households have heterogeneous preferences for property. The period utilityfunction for household i is:

Ui = u [ci] + v [φ (i)AK,i + AI,i] . (8)

Equation (8) assumes property types are perfect substitutes, which implieshouseholds will either buy coastal or inland properties, but not both. Propertyis homogeneous except for type. Without loss of generality, we order house-holds from least to strongest preference for the coastal property, so that φ (i)is increasing. We assume φ (0) = 0 and 2qr ≤ φ (1) < ∞, where qr = qK/qI isthe price ratio (coastal over inland property). The assumptions on φ ensure

14

that the household with the smallest preference for coastal property lives in-land and the household with the strongest preference lives on the coast. Wefurther assume the Inada conditions hold for u and v, so that total property(coastal plus inland) is positive for each household.

5.2 Hurricane Damage and Insurance

Only the coastal property is at risk of damage from hurricanes. Let h be arandom variable which equals one if a unit of coastal property suffers a lossfrom a hurricane and zero otherwise. Conditional on h = 1, the fractional lossof property the household suffers is d ∼ fd (d). We assume that d and h areindependent.20

Let qK and qI denote the price per unit of coastal and inland properties,respectively. We assume both types of property can be produced at constantmarginal cost. The cost of coastal property includes any adaptation costs suchas the cost of compliance with stricter building codes. Perfect competitionamong suppliers of property implies qK = MCK and qI = MCI .

The total value of assets at risk is therefore qK∫ 1

0Ak,idi = qKAK ≡ qKn.

It is convenient to treat each unit of property (as opposed to each individual)as potentially suffering a loss. That is, some units of an individual’s propertymay suffer a loss, while other units do not. Let N ∈ [0, 1, . . . , n] be the(random) total units of property which suffer any loss. If N has a continuousdistribution, then for any realization, N , we take the highest integer less thanor equal to N . Total losses are then:

D = qK

N∑

i=1

di. (9)

According to equation (9), each of N units of coastal property experience aloss, and the losses are realizations di. One could equivalently (given indepen-dence) sum over households and let the distribution of d grow as householdsacquire more property. However, the above formulation simplifies the limitingarguments presented later.

Households may purchase an insurance contract which reimburses the house-hold in the event of a loss from a hurricane. Let p (x) be the insurance premiumper unit of insured property, where x is the co-insurance rate. A large number

20Presumably both the number of properties damaged and the amount of damage increasewith the strength of the storm. Positive correlation between d and h would strengthen theresults, but at a cost of considerable complexity.

15

of risk neutral firms provide insurance in a competitive market. Total expectedprofits equal premiums less expected losses:

E [π] = p (x)n− xE [D] . (10)

Applying Wald’s lemma to (9) gives:

E [π] = p (x)n− xqKE[N ]E [d] = p (x)n− xqKnhd. (11)

Here d is the expected fraction of loss a damaged property receives. Equation(11) assumes that the expected fraction of total properties which suffer damage(h) is independent of n.21

Perfect competition implies profits are zero, in which case the premiumequals the expected reimbursement per unit of property:

p (x) = xqK hd. (12)

5.3 Government

A government agency exists which reimburses all coastal property owners afraction τ of their losses. The government agency is funded by a lump sum taxon all households (irrespective of whether the households own coastal prop-erty). In addition, the government subsidizes the cost of coastal property atrate λ per dollar.

The government budget constraint then sets tax revenue, T , equal to totalreimbursement expenditures, τD plus total coastal property subsidies sqKAK :

T = τD = τqK

N∑

j=1

dj + λqKAK . (13)

The reimbursement of losses can be thought of as a disaster relief agency,such as FEMA. The loss subsidy is equivalent to direct government provisionof insurance with co-insurance rate τ at no cost. Examples of government

21That is, adding more property does not change the probability that another existingproperty is damaged by a hurricane. If h was a decreasing function of n, then property in-surance becomes less expensive as additional properties are added. This creates a feedbackmechanism, where disaster relief encourages the purchase of additional coastal property,which decreases the cost of insurance, which encourages still further coastal property build-ing. Such a feedback mechanism is not likely to be large, but adds considerable complexityto the problem.

16

provision of property insurance include the National Flood Insurance Program,NFIP and Florida’s Citizen’s Property Insurance. Jenkins (2005) estimatessome NFIP premiums are 35-40% below actuarially fair levels. The propertysubsidy captures development subsidies.

Total reimbursement is a random variable, so equation (13) implies taxpayments are random. In practice, FEMA keeps a reserve fund to smooth taxesover time. Further, the government can borrow (and has in fact borrowed) tosmooth tax payments in the event the fund is exhausted. Here we assumeFEMA can perfectly smooth tax payments over time. Therefore, taxes equalthe mean of total disaster reimbursements plus total coastal property subsidies:

T = τE [D] = τnqK hd+ λqKAK . (14)

5.4 Household Problem

The budget constraint sets household endowment income ω plus disaster reliefequal to expenses, which are consumption, the tax, the uninsured hurricanelosses, the insurance premium, and the purchase of property.

ω+τqK

AK,i∑

j=1

hjdj = ci+T+(1− x) qK

AK,i∑

j=1

hjdj+p (x)AK,i+qK (1− λ)AK,i+qIAI,i.

(15)Note that the property subsidy applies only to the initial purchase, not therepair cost paid by either the insurance company or household after the dam-age.22 The timing of the problem is such that at the beginning of the periodthe household purchases insurance and enjoys property. Then losses are real-ized. The household must spend endowment income replacing any damagedproperty not already reimbursed by insurance or disaster relief.23 At the endof the period, the household consumes the remaining endowment.

Substituting the actuarially fair insurance premium (12) into the budgetconstraint and the budget constraint into the utility function (8) results in the

22The model allows for subsidizing repair costs by appropriately modifying τ .23In practice, a household might elect not to replace all damaged property.

17

household problem:

maxE [U ] = maxx,AK,i,AI,i

Ehj ,dj

u

ω − T + (τ − (1− x)) qK

AK,i∑

j=1

hjdj − xqK hd

−qK (1− λ)AK,i − qIAI,i

)]

+ v [φ (i)AK,i + AI,i] . (16)

In problem (16) each unit of coastal property is subject to an iid distributedloss. Hence the households take expectations with respect to each unit ofproperty at risk. Problem (16) implies the disaster relief agency effectivelysubsidizes the acquisition of coastal property.

An equilibrium given a government policy τ is a set of household decisions[x,AK,i, AI,i], prices [p, qK , qI ], and a government tax rate T such that house-holds and firms optimize, the government budget constraint holds, and supplyequals demand in the property markets: qK = MCK and qI = MCI .

5.5 Results: Household Problem

We first derive a set of preliminary results for the household problem. Thegovernment insures part of the risk through the disaster relief agency. We firstshow that the household buys enough insurance to remove any remaining risk.The first order condition for the optimal co-insurance rate is:

Ehj ,dj

u′

ω − T + (τ − (1− x)) qK

AK,i∑

j=1

hjdj − xqK hd− qK (1− λ)AK,i

−qIAI,i

)

qK

AK,i∑

j=1

hjdj − hdqKAK,i

= 0. (17)

We hypothesize that the household insures all risk not covered by the gov-ernment agency. That is, we hypothesize that x = 1−τ satisfies the first ordercondition (17) and is therefore the optimal co-insurance rate. Substituting the

18

proposed solution into the first order condition results in:

u′

[

ω − T − qK (1− τ)AK,ihd− qK (1− λ)AK,i − qIAI,i

]

· Ehj ,dj

AK,i∑

j=1

hjdj − hdAK,i

= 0,

(18)

because marginal utility is not stochastic under the proposed solution. Further,hj and dj are iid, so equation (18) simplifies to:

AK,i∑

j=1

E [hj] E [dj] = hdAK,i. (19)

Equation (19) holds, which verifies the assumed behavior is optimal. There-fore, the household insures all risk not covered by the disaster relief agency.

We next derive the purchase of coastal versus inland properties. Substi-tuting the optimal insurance rate x = 1 − τ into the problem (16) resultsin:

maxAK,i,AI,i

u

[

ω − T − (1− τ) hdqKAK,i − qK (1− λ)AK,i − qIAI,i

]

+v [φ (i)AK,i + AI,i] .

(20)The next proposition shows that a cutoff household exists such that all

households with stronger preferences for coastal property than the cutoff house-hold will buy only coastal property. All other households will buy only inlandproperty.

Proposition 1. Assume the Inada conditions hold for u and v, φ (0) = 0,2qr < φ (1) < ∞, and φ is continuous and increasing. Then there exists acutoff household i∗ ∈ (0, 1) such that:

φ (i∗) = qr(

1− λ+ (1− τ) hd)

. (21)

Further,

1. AI,i = 0 for all i ∈ (i∗, 1].

2. AK,i = 0 for all i ∈ [0, i∗).

Given that each household holds only one type of property, the first-order

19

conditions become:

u′[

ω − T − (1− τ) hdqKAK,i − qK (1− λ)AK,i

]

qK(

1− λ+ (1− τ) hd)

=

φ (i) v′ [φ (i)AK,i] , i > i∗, (22)

u′ [ω − T − qIAI,i] qI = v′ [AI,i] , i < i∗. (23)

For i < i∗, the first-order condition is independent of i. Therefore, house-holds purchase identical amounts of inland property: AI = i∗AI,i. The demandfor coastal property may vary by household. General comparative statics andanalytical solutions for special cases are available for equation (22):

Proposition 2. Let the assumptions of Proposition 1 hold. Then:

2.1. The fraction of households living on the coast, 1− i∗, is increasing in τand λ.

2.2. The quantity of coastal property, AK,i, is increasing in τ and λ for alli ∈ [i∗, 1].

2.3. The total replacement cost of coastal property at risk, qKn = qK∫

AK,idi,is increasing in τ and λ.

2.4. For the special case of Ui = logCi + log (φ (i)AK,i + AI,i), total propertyat risk is:

n =(w − T ) (1− i∗)

2(

1− λ+ (1− τ) hd) . (24)

2.5. Per household purchase of inland property, AI,i, is independent of τ andλ.

Proposition 2 shows that disaster relief causes households living along thecoast to buy more property, which directly increases total property at risk.Further, disaster relief causes more households to locate on the coast, whichalso increases total property at risk. Nonetheless, disaster relief, and indeedall parameters governing the insurance market, do not affect the damage dis-tribution of an individual property. Adding property at risk does not changethe damage distribution for another property already at risk.

Most of the assumptions are relatively innocuous. For example, we assumeindependence both across h and d and across properties. Adding positive cor-relation would not, for sufficiently large n, affect the distribution to which Dconverges, in the same way that the central limit theorems hold when obser-vations are correlated.

20

5.6 Population Centers

We let all property be divided into S population centers. Following the re-sults of section 4.3, we assume each population center s contains a quan-tity of property, ns, drawn from a truncated GPD.24 In particular, we letns ∼ fGPD (ns; σn, ξ) /FGPD (n; σn, ξ). Here equations (3) and (4) define FGPD

and fGPD, respectively. If σ satisfies:

n

(

σ

1− ξ− Γ (σ)

)

=n

S, (25)

the mean total property in a population center is n/S and the mean totalamount of property is n.25 The truncation of the distribution ensures that themaximum amount of property in a population center is n.

Population centers have equal geographic area, and are distributed ran-domly along the coast, which has total area Jc. If a hurricane path over landcovers an area equal to JD (determined below), then the probability that arandomly placed population center is in the hurricane path equals ρ = JD/Jc.For simplicity, we assume at most one population center is in the hurricanepath.26 If a population center is in the hurricane path, then each property inthe population center suffers the random loss di. Therefore:

N =

{

ni ∼ fGPD (ni; σn, ξ) /FGPD (n; σn, ξ) w.p. ρ

0 w.p. 1− ρ, (26)

24Gabaix (1999) proposes a theory for city sizes that generates a Pareto distribution. Intheir model, wages adjust to offset city-specific amenity shocks, which keeps the growthrate of cities independent of the size, which implies a power law steady state distribution.Here the property distribution is exogenous, since most of the policies of interest affecttotal property or the relationship between hurricane strength and damage, not the propertydistribution.

25However, the total amount of property is not equal to n with probability one. Smalldifferences between

∑S

s=1ns and n do not affect the results for large n and scaling the

realizations of the population center sizes so that the total property equals n would vastlycomplicate the distribution of ns.

26If instead the assumption was that a hurricane could intersect multiple populationcenters, then N would equal the sum of a random number of GPD random variables. Sincethe sum of Pareto random variables converges to a stable distribution with the same tailindex, the restriction that the hurricane may intersect only a single population center isunlikely to alter the tail index of N , or the conclusion that N is fat tailed.

21

N ∼

{

ρfGPD

(

N ; σn, ξ)

/FGPD (n; σn, ξ) if N > 0

1− ρ if N = 0. (27)

Therefore N is a GPD random variable, conditional on a population centerbeing in the hurricane path.

5.7 Hurricane Properties

A variety of hurricane-specific factors influence total damages, including windspeed, the size of the storm, and the path of the storm over land. We modelthese influences in a simple way. Let r denote the distance from the center ofa storm, with t the time since landfall. Holland (1980) studies a model of thewind speed, wr (t), of a hurricane:

wr (t) = rθmwrm (t)

rθr ≥ rm. (28)

Here rm is the distance between the center of the storm and the edge of theeye of the hurricane, where the maximum wind speed occurs. Holland (1980)cites estimates of θ ∈ [0.4, 0.6]. The wind speed decreases as the distance fromthe eye of the storm increases.

We assume that building codes and adaptations are such that no damageoccurs if the wind speed is less than w.27 Therefore, the radius of the stormbeyond which no damage occurs, r, is:

r (t) = rm

(

wrm (t)

w

)1

θ

. (29)

Hurricanes lose energy over land, causing wind speed to decline. FollowingKaplan et al. (2007), we assume maximum winds decay exponentially withtime since landfall:

wrm (t) = wrm (0) exp (−δt) . (30)

The hurricane causes no further damage when wrm (t) = w, that is, when themaximum winds are no longer strong enough to cause damage. This occurs at

27Nordhaus (2010) finds support for a similar assumption, based on the idea that somebuilding materials may withstand a maximum amount of wind with little or no damage,and then break all at once when the wind speed crosses a threshold. An alternative is to letthe distribution of damage d be a function of the wind speed.

22

time:

t =1

δlog

(

wrm (0)

w

)

. (31)

Further, combining equations (29) and (30) yields:

r (t) = rm

(

wrm (0)

w

)1

θ

exp

(

−δ

θt

)

. (32)

Equation (32) gives the radius of damage as a function of the wind speed atlandfall and the time since landfall.

Assume that the hurricane travels at a constant velocity ν, which has unitsof distance, m, per unit of time. Then the area of damage is:

JD =

∫ m(t)

0

2r (t (m)) dm =

∫ νt

0

2r(m

ν

)

dm, (33)

= 2

∫ vt

0

rm

(

w

w

)1

θ

exp

(

−δ

vθm

)

dm. (34)

Here w = wrm (0), the maximum wind speed at landfall, which we have esti-mated in Section 4.2. Therefore, the probability that the hurricane intersectsa population center is:

ρ =

∫

w

JD (w)

JcfGPD (w, σw, ξw) dw. (35)

The empirical evidence that climate change increases the number of hurri-canes is mixed, but the frequency of high wind (category 4 or 5) storms mayincrease (Bender et al. 2010). Therefore, the model takes σw as the policyvariable, with a decrease in σw interpreted as the result of climate change mit-igation. According to the empirical results, hurricane strength is thin tailed(ξw < 0). The next section derives the distribution of the aggregate damages.

6 Theoretical Results

6.1 Fat Tails

The theoretical model has several sources of random variation: hurricane windspeed, the path of the hurricane, the quantity of property in an area affectedby a hurricane, and the damage to individual properties in the area affected

23

by a hurricane. We show here that the distribution for N largely determinesthe distribution for the aggregate damages D. Because di is bounded on theunit interval, the variance of di is also bounded. Therefore, the central limittheorems imply that, regardless of the distribution of di, if N is non-stochastic(for example N = βn with 0 < β < 1), then D has a normal (thin-tailed)limiting distribution as N → ∞. Increasing the total property at risk increasesthe mean and variance of D, but the distribution remains thin tailed.

However, Section 5.6 shows that N is random. When N is random, stan-dard central limit theorems no longer hold and the distribution of D will notin general converge to a normal distribution. Instead, we will show that thelimiting distribution of D will inherit properties of the limiting distribution ofN : if N is thin (fat) tailed then D converges to a thin (fat) tailed distribution.

The definition of fat tails poses a difficult problem for modeling the distri-bution of damages from a natural disaster. As the total number of propertiesat risk is always finite, for any n the distribution of damages is bounded andtherefore fails to satisfy (2). To reconcile the apparent bound on total dam-ages with empirical tests in Section 3, which indicate total damages are fattailed, we construct a sequence of bounded distributions which converge to afat-tailed distribution as n → ∞. For any finite n, the distribution is boundedand not fat tailed, but, as n becomes large, a fat-tailed distribution becomesan increasingly accurate approximation.

We let Nn = N to emphasize that N depends on n. To see how D inheritsthe tail properties of N , assume for example, that Nn has a thin-tailed binomialdistribution: Nn ∼Binomial(n, ρ) for any n (as would be the case if eachproperty had a uniform probability of locating at each geographical point onthe coast). Then from Robbins (1948), example (ii) as n → ∞:

Dd→ N

[

nρd, nρ(

σ2d + (1− ρ) d2

)]

. (36)

Similarly, if N has a generalized inverse Gaussian distribution, then D con-verges to a heavy-tailed (i.e. not fat tailed, but with a decay rate slowerthan the normal distribution) generalized hyperbolic distribution (Haas andPigorsch 2011).

Here, conditional on the hurricane intersecting a population center, thedistribution of N is a truncated GPD.

Proposition 3. Let dj ∼ fd be iid with finite mean d and finite variance σ2d.

Let:

Nn|Nn > 0 ∼fGPD

(

N ; σn, ξ)

FGPD (n; σn, ξ), 0 ≤ Nn < n, (37)

24

where FGPD and fGPD are given by (3) and (4), respectively. Then D convergesin distribution to a normal variance mean mixture with Pareto mixing densityas n → ∞. That is:

D|Nn > 0 → qK dN + qKσdN1

2 z, (38)

where z is a standard normal and N is distributed as a GPD with parametersσn, and ξ. Further, D has tail index α = 1/ξ and mean:

E[

D|Nn > 0]

= qK dn

(

σ

1− ξ− Γ

)

, ξ < 1, Γ ≡1− FGPD (1, σ, ξ)

FGPD (1, σ, ξ)

1

1− ξ.

(39)

Proposition 3 shows that if the number of properties that experience aloss follows a GPD, then for large n, the distribution of damages is approxi-mately a mixture of a normal distribution and a GPD. The form of D given inequation (38) is known as a normal variance mean mixture, with GPD mixingdistribution.28 Proposition 3 shows that D inherits the tail index of N . Thetheory is in alignment with the empirical results of section 4, in which thepoint estimates of the tail index of D and N are similar.

From equation (39), the mean damage depends on the tail index. As seenin Section 4.1, the tail index is difficult to estimate given the lack of data in thetail. However, equation (39) may also be interpreted so that the mean damagesequal the mean damage per property times the mean amount of property ina population center. That is, by choosing σ as in equation (25), the meandamages equal qK dn/S, which may be estimated using insurance data on thetotal property under risk and the average claim size, rather than by workingwith the aggregate damage data directly.

In Proposition 3, the number of properties affected is drawn from a dis-tribution bounded by n. At any point in time, the total amount of propertyn remains finite. However, as the total property at risk grows, the boundexpands and so does the upper bound for N . Because Nn converges to a fat-tailed distribution, for large n the distribution of Nn and D will appear tohave fat tails, even though for any fixed n the distribution is bounded andtherefore not fat tailed.

The expected damage from a catastrophic storm is implicit from the dis-tribution of damages given by Proposition 3. Let D∗ denote the minimumdamage from the 1− 1/T upper quantile of the damage distribution. We then

28See Haas and Pigorsch (2011) for a discussion of normal variance mean mixtures.

25

say a storm is catastrophic if the storm results in damage large enough sothat such storms occur with with probability 1/T . For example, if f stormsoccurred per year and T = 100f , then a catastrophic storm would be a once-in-a-hundred-years storm. Higher values of D∗ imply that a catastrophic storm,which occurs once every T observations, is more damaging. Higher values ofD∗ also imply that more mass of the distribution resides in the tail (the tailis thicker). Because z and N are independent, equation (38) implies that D∗

is the solution to:

Prob (D ≥ D∗) =1

T, (40)

Prob (D ≤ D∗) = 1−1

T, (41)

1− ρ+ ρProb(

D ≤ D∗|N > 0)

= 1−1

T, (42)

FD (D) = Prob(

D ≤ D∗|N > 0)

= 1−1

ρT, (43)

∫

∞

0

∫ z(N,D∗)

−∞

fz (z) dzfN

(

N)

dN = 1−1

ρT, (44)

where fz is the standard normal density, fN is the GPD with parameters σnand ξ, and:

z(

N,D∗

)

=D∗ − qK dN

qKσdN1

2

. (45)

From equations (39) and (45), the policy variables affect both the meandamage and the damage from a T observation storm in different ways. Weexamine these effects in the next section.

6.2 Policy Implications

Four policy variables exist in the model. First, the government may set thedisaster relief policy τ . Second, the government can subsidize the cost ofcoastal property by reducing qK . Third, the government can choose the rel-ative strength of adaptations, w. Fourth, the government could, through cli-mate change regulation, attempt to reduce the mean or the mass of the upperquantiles of the storm strength distribution.

Proposition 4. Let the assumptions of Proposition 3 hold. Then an increasein the strength of adaptations (w), a decrease in the disaster relief subsidy (τ),a decrease in the property subsidy (λ), and a decrease in the scale parameter

26

of the hurricane strength distribution (σw) all cause a decrease in the damagefrom a T observation storm (D∗) and a decrease in the mean damages (E [D]).The tail index of the damage distribution is not affected by τ , λ, w, or σw.

While each policy variable affects the mean and catastrophic damages, thechannels differ substantially. First, an increase in the disaster relief subsidyadds to the total property at risk, both through the intensive and extensivemargins. In turn, an increase in the total property at risk has two effects.First, each population center has more property at risk, which causes themean damage (n/S) to increase. Second, large population centers have moreproperty at risk, so that large damages will occur more often.

To ensure that a catastrophic storm occurs only once every T observations,the amount of damage required for a storm to be considered catastrophic mustincrease. Second, an increase in the total property at risk extends the righttail of the truncated GPD. However, this effect vanishes for large n, becauseas n increases, a realization of N at the truncation point becomes less andless likely. That is, as the number of properties increases, the likelihood thata storm affects all coastal property at risk approaches zero.

As with the disaster relief subsidy, the property subsidy increases the totalproperty, along both the intensive and extensive margins. Therefore, the disas-ter relief subsidy increases the damage from all storms, including catastrophicstorms.

An increase in the strength of adaptations, w, reduces the geographic areaof damage in two ways. First, the radius of winds which are strong enough tocause damage decreases, as the wind speed at a certain range of distances fromthe eye are no longer strong enough to cause damage. Second, the time sincelandfall at which the hurricane causes no further damage decreases, becausew, the minimum wind speed which causes damage, increases. In turn, thereduction in the geographic area of damage implies that fewer hurricanes willintersect a population center and cause damage. Therefore, the unconditionalmean damage decreases. In addition, in a sample of T storms, fewer stormscause damage meaning that there will be fewer storms that cause damagegreater than a given level, D. So, D∗ must decrease to bring the unconditionalprobability of a storm causing damage greater than D∗ back to 1/T . Alterna-tively, in a 100-year sample, fewer storms cause damage, so fewer storms arecatastrophic, which decreases the likelihood that a catastrophic storm causesdamage above a given level.

We model climate change mitigation as a decrease the scale parameter ofthe wind distribution, which results in hurricanes with lower (in the sense offirst order stochastic dominance) wind speeds. As a result, the geographic

27

area of damage decreases both because the radius of the hurricane is smallerand because the hurricane dissipates more quickly. Climate change mitigationreduces the number of hurricanes that intersect a population center, reducingboth the mean damage and the damage from a catastrophic storm.

Proposition 3 indicates that disaster relief policy and the structure of theinsurance market increase the mean of D by increasing n. However, disasterrelief and the structure of the insurance market do not affect the tail index,α, which determines the tail behavior of D. Similarly, adaptation and climatechange mitigation can affect the probability that a storm causes damage, butdo not affect the tail index.

7 Conclusion

Tropical cyclones can impose significant welfare losses on coastal communities.The empirical analysis shows that hurricane damages from 1900-2012 followa fat-tailed distribution, so much so that the parameter estimates indicateinfinite variance. A fat-tailed distribution can result in additional costs: themean damages are difficult to estimate, leading to mispricing of insurance.The probability of a catastrophic loss is also difficult to estimate, leading tothe possibility that reserves will not be adequate. Past storm events are notgood predictors of the damages of future catastrophes.

A theoretical model of homeowner behavior, population centers, and hur-ricane strength is presented. The model reveals that homeowners will fullyinsure their properties from risk of damage, paying premiums to recoup alllosses not covered by the disaster-relief program. In this way, the disasterrelief program is equivalent to an insurance premium subsidy, which tendsto increase the quantity of coastal property along both the extensive and in-tensive margins. A subsidy to the cost of coastal property also increases thequantity of coastal property along both margins. These subsidies increase themean damage and the damage from a catastrophic storm. However, subsidiesdo not create a fat tail. Surprisingly, the fat tail results from the distributionof households in population centers. The empirical results agree with the pre-diction of the theoretical model that the tail index of the population centerdistribution equals the tail index of the damage distribution.

In the model, adaptations and climate change mitigation offer the possibil-ity to reduce the coastal area over which winds are strong enough to result indamage. More hurricanes miss population centers, which lowers the averagedamage and also implies storms intersect large population centers less often.

In reality, the increase in coastal property continues, implying that catas-trophic storms will become increasingly important from a welfare perspective.

28

Thus, the need for policies that can minimize the occurrence of, or damagefrom, catastrophic storms will only become more acute.

References

L. Bakkensen. Adaptation and Natural Disasters: Evidence from Global Tropi-cal Cyclone Damages and Fatalities. Yale University Working Paper, 2014.

M.A. Bender, T.R. Knutson, R.E. Tuleya, J.J. Sirutis, G.A. Vecchi, S.T. Gar-ner, and E.M. Held. Modeled Impact of Anthropogenic Warming on theFrequency of Intense Atlantic Hurricanes. Science, 327:454–458, 2010.

K. Beron, J. Murdoch, W. Thayer, and W. Vijverberg. An Analysis of theHousing Market before and after the 1989 Loma Prieta Earthquake. LandEconomics, 73:101–113, 1997.

O. Bin and S. Polasky. Effects of Flood Hazards on Property Values: EvidenceBefore and After Hurricane Floyd. Land Economics, 80(4):490–500, 2004.

E. S. Blake, C. W. Landsea, and E. J. Gibney. The Deadliest, Costliest,and Most Intense United States Tropical Cyclones from 1851-2010 (andother frequently requested hurricane facts). Technical Report NWS NHC-6, National Hurricane Center, Miami, FL, 2011.

V. Brazouskas and A. Kleefeld. Robust and efficient fitting of the general-ized Pareto distribution with actuarial applications in view. Insurance:Mathematics and Economics, 45:424–435, 2009.

R.J. Bunby. Hurricane Katrina and the Paradoxes of Government DisasterPolicy: Bringing About Wise Government Decisions for Hazardous Areas.The ANNALS of the American Academy of Political and Social Science,604:171–191, 2006.

A.C. Cebrian, M. Denuit, and P. Lambert. Generalized Pareto fit to the Societyof Actuaries’ large claims database. North American Actuarial Journal,pages 18–36, 2003.

Peter K. Clark. A Subordinated Stochastic Process Model with Finite Variancefor Speculative Prices. Econometrica, 41(1):135–55, January 1973.

Roger Cooke and Daan Nieboer. Heavy Tailed Distributions: Data, Diagnos-tics, and New Developments. Technical Report 11-19, Resources for theFuture, March 2011.

29

A.C. Davison and R.L. Smith. Models for Exceedances over High Thresholds.Journal of the Royal Statistical Society, B, 52(3):393–442, 1990.

P. de Zea Bermudez and S. Kotz. Parameter estimation of the generalizedPareto distribution – Part I. Journal of Statistical Planning and Inference,140:1353–1373, 2010.

W.H. DuMouchel. Estimating the stable index α in order to measure tailthickness: a critique. The Annals of Statistics, 11(4):1019–1031, 1983.

K. Emanuel. Increasing destructiveness of tropical cyclones over the past 30years. Nature, 436(4):686–688, 2005.

X. Gabaix. Zipf’s Law for Cities: An Explanation. The Quarterly Journal ofEconomics, 114(3):739–67, 1999.

X. Gabaix and R. Ibragimov. Rank - 1/2: A Simple Way to Improve theOLS Estimation of Tail Exponents. Journal of Business and EconomicStatistics, 29(1):24–39, 2011.

J. Gallagher. Learning about an infrequent Event: Evidence from Flood Insur-ance Take-Up in the United States. American Economic Journal: AppliedEconomics, 6(3):206–233, 2014.

S. Ghosh. A Discussion on Mean Excess Plots. Stochastic Processes and theirApplications, 120:1492–1517, 2010.

D.E. Giles, H. Feng, and W.T. Godwin. Bias-Corrected Maximum LikelihoodEstimation of the Parameters of the Generalized Pareto Distribution. Uni-versity of Victoria Econometrics Working Paper, EWP1105, 2011.

R. Gonzalez-Val, A. Ramos, F. Sanz-Gracia, and Marıa Vera-Cabello. SizeDistributions for All Cities: Which One is Best? Papers in RegionalScience, 94(1):177–96, 2015.

M. Haas and C. Pigorsch. Financial Economics, Fat-Tailed Distributions. InR. Meyers, editor, Complex Systems in Finance and Econometrics, pages308–339. Springer New York, 2011.

D. Hallstrom and V. Smith. Market Response to Hurricanes. Journal ofEnvironmental Economics and Management, 50:541–61, 2005.

G. Holland. An Analytic Model of the Wind and Pressure Profiles in Hurri-canes. Monthly Weather Review, 106:1212–1218, 1980.

30

G.J. Holland and P.J. Webster. Heightened tropical cyclone activity in theNorth Atlantic: natural variability or climate trend? Philosophical Trans-actions of The Royal Society, 365:1–22, 2007.

J.D. Holmes and W.W. Moriarty. Application of the generalized Pareto dis-tribution to extreme value analysis in wind engineering. Journal of WindEngineering and Industrial Aerodynamics, 83:1–10, 1999.

J.R.M. Hosking and J.R. Wallis. Parameter and Quantile Estimation for theGeneralized Pareto Distribution. Technometrics, 29(3):339–349, 1987.

Inc. International Code Council. 2003 International Residential Code for One-and Two-Family Dwellings. International Code Council, Inc., 2003.

W. Jenkins. Federal Emergency Management Agency: Challenges Facingthe National Flood Insurance Program. Technical Report GAO-06-174T,GAO, October 2005.

C. Jones. Pareto and Piketty: The Macroeconomics of Top Income and WealthInequality. Journal of Economic Perspectives, 29(1):29–46, 2015.

J. Kaplan, J. Dunion, N. Carrasco, and M. DeMaria. Estimating TropicalCyclone Wind Radii Utilizing an Empirical Inland Wind Decay Model.Joint Hurricane Testbed Final Report. Technical report, NOAA, April2007. http://www.nhc.noaa.gov/jht/05-07reports/final_Kaplan_

JHT07.pdf.

David L. Kelly and Zhuo Tan. Learning and Climate Feedbacks: OptimalClimate Insurance and Fat Tails. Journal of Environmental Economicsand Management, 72:98–122, 2015.

David L. Kelly, David Letson, Forrest Nelson, David Nolan, and Daniel Solıs.Evolution of Subjective Hurricane Risk Perceptions: A Bayesian Ap-proach. Journal of Economic Behavior and Organization, 81(2):644–63,2012.

D.L. Kelly, C.D. Kolstad, and G.T. Mitchell. Adjustment costs from environ-mental change. Journal of Environmental Economics and Management,50(3):468–495, 2005.

T.R. Knutson, J. Sirutis, S.T. Garner, G.A. Vecchi, and I.M. Held. Simu-lated reduction in Atlantic hurricane frequency under twenty-first-centurywarming conditions. Nature Geoscience, 436(4):686–688, 2008.

31

Carolyn Kousky and Roger Cooke. The Unholy Trinity, Fat Tails, Tail De-pendence, and Micro-Correlations. Technical Report 09-36, Resources forthe Future, November 2009.

M. Mann and K. Emanuel. Atlantic Hurricane Trends Linked to ClimateChange. Eos, Transactions, American Geophysical Union, 87(24):233–244,2006.

R. Mendelsohn, W.D. Nordhaus, and D. Shaw. The Impact of Global Warmingon Agriculture: A Ricardian Analysis. American Economic Review, 84(4):753–771, 1994.

S. Nadarajah. Generalized Pareto models with application to drought data.Environmetrics, 19:395–408, 2008.

H.F. Needham and B.D. Keim. A Storm Surge Database for the U.S. GulfCoast. International Journal of Climatology, 32(14):2108–2123, 2012.

W. Nordhaus. The Economics of Hurricanes and Implications of Global Warm-ing. Climate Change Economics, 1(1):1–20, 2010.

J. Pickands. Statistical Inference Using Extreme Order Statistics. The Annalsof Statistics, 3(1):119–131, 1975.

R.A. Pielke, Jr., J. Gratz, C.W. Landsea, D. Collins, M. Saunders, andR. Musulin. Normalized Hurricane Damages in the United States: 1900-2005. Natural Hazards Review, 9(1):19–42, 2008.

H. Robbins. The Asymptotic Distribution of the Sum of a Random Numberof Random Variables. Bulletin of the American Mathematical Society, 54(12):1151–1161, 1948.

A.B. Smith and R.W. Katz. US billion-dollar weather and climate disasters;data sources, trends, accuracy and biases. Natural Hazards, 67(2):387–410,2013.

R.L. Smith. Estimating Tails of Probability Distributions. The Annals ofStatistics, 15(3):1174–1207, 1987.

G.A. Vecchi and T.R. Knutson. On Estimates of Historical North AtlanticTropical Cyclone Activity. Journal of Climate, 21:3580–3600, 2008.

G.A. Vecchi, K.L. Swanson, and B.J. Soden. Whither Hurricane Activity?Science, 322:687–689, 2008.

32

M. Weitzman. Fat Tails and the Social Cost of Carbon. American EconomicReview, 104(5):544–6, 2014.

Appendix A Proof of Theorems

A.1 Proof of Proposition 1

The Kuhn-Tucker conditions for problem (20) require µjiAji = 0, where µji isthe multiplier which ensures property choices are non-negative and j ∈ [K, I].Eliminating the multipliers from the first order conditions for optimal propertychoice results in:

(

−u′ [ci] qK(

1− λ+ (1− τ) hd)

+ φ (i) v′ [φ (i)AK,i + AI,i])

Ak,i = 0, (46)

(−u′ [ci] qI + v′ [φ (i)AK,i + AI,i])AI,i = 0. (47)

We first show that AI,i > 0 implies AK,i = 0 for all i 6= i∗. Since AI,i > 0,µI,i = 0. Therefore, equation (47) reduces to:

u′ [ci] qI = v′ [φ (i)AK,i + AI,i] , (48)

Combining (48) and (46) results in:

(

−u′ [ci] qK(

1− λ+ (1− τ) hd)

+ u′ [ci]φ (i) qI)

AK,i = 0. (49)

Therefore either AK,i = 0 or:

φ (i) = qr(

1− λ + (1− τ) hd)

. (50)

However, since the right hand side of (21) is independent of i and φ is strictlyincreasing, equation (21) holds for at most one i, denoted i∗. Therefore, for alli 6= i∗ for which AI,i > 0, we have AK,i = 0. Reversing the argument impliesthat AK,i > 0 implies AI,i = 0 for all i 6= i∗. For the converse, we must showthat AK,i = 0 implies AI,i > 0. Suppose not, suppose AK,i = AI,i = 0. Thentotal housing is zero which cannot hold by the Inada conditions. ThereforeAK,i = 0 if and only if AI,i > 0 and the reverse.

Next from (48) and (49), the range of i for which AK,i = 0 and AI,i > 0satisfies:

u′ [ci] qK(

1− λ+ (1− τ) hd)

≥ u′ [ci]φ (i) qI , (51)

33

φ (i) ≤ qr(

1− λ+ (1− τ) hd)

. (52)

Since φ (i) is increasing, the above inequality is satisfied for all i < i∗, wherei∗ satisfies (21). Reversing the argument shows that AK,i > 0 and AI,i = 0 forall i ≥ i∗.

It remains to show that i∗ ∈ (0, 1). Since φ (0) = 0 by assumption, con-dition (52) holds with strict inequality for i = 0. Therefore i∗ > 0. Sinceφ (1) > 2qr and τ , h, and d, are all between zero and one, condition (52) isnot satisfied for i = 1. Therefore i∗ < 1. �

A.2 Proof of Proposition 2

Propositions 2.1, 2.2, and 2.5 follow from applying the implicit function theo-rem to equations (21), (22), and (23) respectively. For 2.3, note that:

qKn = qK

∫ 1

i∗(τ)

AK,i (τ) di. (53)

Applying Leibniz’s rule results in:

∂qKn

∂τ= qK

(∫ 1

i∗

∂AK,i

∂τdi− AK,i∗

∂i∗

∂τ

)

. (54)

The result then follows since Propositions 2.1 and 2.2 show that i∗ is decreasingin τ and AK,i is increasing in τ .

For 2.4, for the special case where both u and v are logarithmic, equation(22) reduces to:

qK(

1 + (1− τ) hd)

w − T − qKAK,i

(

1 + (1− τ) hd) =

1

AK,i

. (55)

Solving for AK,i results in equation (24).�

A.3 Proof of Proposition 3

The following lemma is Theorem 3 from Clark (1973), which we will applydirectly.

Lemma 5. Let di ∼ fd be iid with mean zero and unit variance and Nn be a

34

random variable independent of di such that:

Nn

n

p→ M as n → ∞. (56)

Then:

D = n−1

2

Nn∑

i=1

did→ fD, (57)

where fD is the distribution of a normal variance mean mixture. That is:D → M

1

2 z, where z is distributed as a standard normal.

The proof applies Lemma 5. First, note that M ≡ Nn/n has distribution:

Nn/n ∼ fNn

(

N(

M)) ∂Nn

∂M, (58)

=1

σNn

(

1 +ξ

nσN

(

Mn)

)

−1− 1

ξ

n, (59)

=1

σN

(

1 +ξ

σN

M

)

−1− 1

ξ

, (60)

which is independent of n. Therefore, Nn/n converges trivially to M .The next step converts the problem into one that satisfies the assumptions

of Lemma 5. Let yi =(

di − d)

/σd, then by Lemma 5, conditional on N ,

n−1

2

N∑

i=1

yid→ N

[

0, M]

, (61)

n−1

2

N∑

i=1

di − dN

d→ N

[

0, σ2dM]

. (62)

Therefore for n large:

N∑

i=1

di − dNd→ N

[

0, σ2dMn

]

, (63)

35

N∑

i=1

did→ N

[

dN , σ2dN]

, (64)

Dd→ N

[

qK dN , q2Kσ2dN]

, (65)

which is the normal variance mean mixture (38) with the parameters given inthe proposition. Since the limit of Nn is the GPD, the mixing distribution isthe GPD.

The moments and tail behavior follow from the properties of the normalvariance mean mixture.

Lemma 6. Let D be a normal variance mean mixture, i.e.:

D = µN + σN1

2 z, (66)

where z is a standard normal independent of N , and N has finite mean. Then:

1. E [D] = µE[N ].

2. The tail index of D equals the tail index of N .

We prove each result separately.

1. Since N and z are independent:

E [D] = µE[N ] + σE[

N1

2

]

E[z]. (67)

Next Jensen’s inequality implies E[

N1

2

]

≤ E[

N] 1

2

< ∞. Therefore,

since z has mean zero, the second term in (67) is zero and the resultfollows.

2. The kth moment of D is finite if and only if:

E[

Dk]

= E

[

(

µN + σN1

2 z)k]

< ∞. (68)

The binomial theorem implies:

E[

Dk]

=

k∑

i=0

[

k

i

]

E

[

(

dN)k−i (

σN1

2 z)i]

. (69)

36

A property of the GPD is that if the ith moment exists, then so do allmoments j ≤ i. The largest exponent on N in (69) occurs when i = 0.

Therefore it sufficient to show E[

Nk]

exists. But since N has tail index

equal to α, E[

Nk]

exists if and only if α > k. Therefore the kth moment

of D exists if and only α > k. Therefore, α is the tail index of D. �

A.4 Proof of Proposition 4

Equations (27) and (39) imply the unconditional mean damage is:

D = ρqK dn

(

σ

1− ξ− Γ

)

. (70)

From equation (70), the unconditional mean damage is increasing in n. Fur-ther, Proposition 2.3 shows that n is increasing in τ and λ. Therefore, E [D] isincreasing in τ and λ. The unconditional mean damage is also increasing in ρ.Further, equations (34) and (35) imply that ρ is decreasing in w. Therefore,E [D] is decreasing in w. Finally, for σw, combining equations (34) and (35)gives:

ρ =

∫

w

2

∫ vt

0

rm

(

w

w

)1

θ

exp

(

−δ

vθm

)

dmfGPD (w, σw, ξw) dw, (71)

ρ =

∫

w

2rmJc

θν

δ

(

(

w

w

)1

θ

− 1

)

fGPD (w, σw, ξw) dw =

∫

w

h (w) fGPD (w, σw, ξw) dw.

(72)Note that FGPD is decreasing in σw and therefore N1 ∼ FGPD (σw1) first orderstochastic dominates N2 ∼ FGPD (σw2) for σw1 > σw2. Therefore, since h is anincreasing function, ρ is increasing in σw. Therefore, the unconditional meandamage is increasing in w.

For the damage from a T observation storm, equation (44) defines an im-plicit function D∗ (w, τ, λ, σw). Equation (44) simplifies to:

∫

∞

0

Fz

(

D∗ − qK dN

qKσdN1

2

)

fN

(

N, σn)

dN = 1−1

ρT. (73)

Consider first the derivative with respect to n. Taking the total derivative of

37

(73) yields:

∫

∞

0

fz

(

D∗ − qK dN

qKσdN1

2

)

∂z

∂D∗

∂D∗

∂nfN

(

N, σn)

dN+

∫

∞

0

Fz

(

D∗ − qK dN

qKσdN1

2

)

∂fN∂n

dN = 0.

(74)

∂D∗

∂n=

−∫

∞

0Fz

(

D∗−qK dN

qKσdN1

2

)

∂fN∂n

dN

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

∂z∂D∗

fN

(

N)

dN

. (75)

Next, the derivative of the GPD with respect to n follows from equation (4):

∂fN

(

N, σn)

∂n=

N − σn

σ2n3

(

1 +ξ

σnN

)

−2− 1

ξ

. (76)

Therefore:

∫ ∂fN

(

N , σn)

∂ndN = −

N

σn2

(

1 +ξ

σnN

)

−1− 1

ξ

= −N

nfN

(

N)

dN . (77)

Next, the numerator of (75) simplifies using (77) and integration by parts:

−

∫

∞

0

Fz

(

D∗ − qK dN

qKσdN1

2

)

∂fN∂n

dN = −Fz

(

D∗ − qK dN

qKσdN1

2

)

N

nfN

(

N) ∣

∣

∣

∞

0+

∫

∞

0

fz

(

D∗ − qK dN

qKσdN1

2

)

∂z

∂N

N

nfN

(

N)

dN.

(78)

Simple calculations show the first term on the right hand side of (78) is zeroat both integration end points. The derivative (75) thus simplifies to:

∂D∗

∂n=

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

∂z∂N

NnfN

(

N)

dN

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

∂z∂D∗

fN

(

N)

dN

, (79)

38

∂D∗

∂n=

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

fN

(

N)

N−1

2

(

D∗ + qK dN)

12ndN

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

fN

(

N)

N−1

2dN

. (80)

Both the numerator and denominator of (80) are positive, and so the derivativeis positive. Proposition 2.3 shows that n is increasing in τ and λ. Therefore,D∗ is increasing in τ and λ.

Consider now the derivative of D∗ with respect to ρ. Following a similarlogic, equation (73) implies:

∂D∗

∂ρ=

1Tρ2

∫

∞

0fz

(

D∗−qK dN

qKσdN1

2

)

∂z∂D∗

fN

(

N)

dN

. (81)

Therefore, the derivative of D∗ with respect to ρ is positive. Since ρ is decreas-ing in w from equations (34) and (35), D∗ is decreasing in w. For σw, notethat we have shown that ρ is increasing in σw and therefore D∗ is increasingin σw.

Appendix B Figures and Tables

39

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

D

Pro

babi

lity

Den

sity

Generalized Pareto Distribution and Tail Index

α =- 1.5 (b ounded )α = ∞ (e x p onent ial )α =1 (mean= ∞)α =1.5 (var=∞)α =2.5 (th in tai l e d )

Figure 1: Generalized Pareto distribution for different values of the tail index α =1/ξ.

0 1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

D

Pro

babi

lity

Den

sity

Generalized Pareto Distribution and Mean Excess. Location=0, Scale=1

u=1 u=2 E[x|x>1] E[x|x>2]

Distan c e= M (1) =5

Distan c e= M (2) =7

α =1.5 (var=∞)

Figure 2: Mean excess for GPD with shape parameter ξ = 1/α = 2/3. In theFigure, M (1) = 5 and M (2) = 7, indicating that the mean-excess function M (u) isincreasing.

40

versus strength of adaptations

Households: buy

coastal property?

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Coastal PopulationCenters

No Yes

Coastline

Random Potential

Storm Paths

Hurricane

Effective size determined

by random storm strength

Figure 3: Illustration of the model. In the figure, seven households choose to liveon the coast, organizing into three population centers. If, for example, the hurri-cane took the west path, the hurricane would damage the two houses in the secondpopulation center.

Threshold ξ Confidence Interval Observations

45th percentile 0.946 0.593 - 1.299 133Median 0.873 0.52 - 1.227 11955th percentile 0.902 0.512 - 1.292 10860th percentile 0.806 0.42 - 1.192 9565th percentile 0.751 0.356 - 1.146 8470th percentile 0.668 0.274 - 1.061 7275th percentile 0.764 0.271 - 1.258 6080th percentile 0.618 0.141 - 1.095 48

Table 1: MLE of the shape parameter for the hurricane dam-age distribution.

41

LowerThreshold

UpperThreshold

Weighting ξ Std. Err. ConfidenceInterval

MinimumObservations

30th percentile 70th percentile obs. 0.73 0.011 0.71 - 0.75 57

30th percentile 70th percentile inverse obs. 0.71 0.011 0.69 - 0.73 57

35th percentile 70th percentile obs. 0.70 0.010 0.68 - 0.72 57

35th percentile 70th percentile inverse obs. 0.70 0.010 0.68 - 0.72 57

30th percentile 75th percentile obs. 0.72 0.008 0.70 - 0.73 49

30th percentile 75th percentile inverse obs. 0.70 0.009 0.68 - 0.71 49

35th percentile 75th percentile obs. 0.69 0.008 0.68 - 0.71 49

35th percentile 75th percentile inverse obs. 0.69 0.009 0.68 - 0.72 49

35th percentile n/a obs. 0.24 0.09 0.06 - 0.40 1

35th percentile n/a inverse obs. 0.43 0.054 0.33 - 0.54 1

Table 2: WLS estimation of the shape parameter using the mean-excess function for thehurricane damage distribution. The confidence intervals were generated by bootstrapping. Inthe second column, “n/a” corresponds to setting the maximum threshold equal to the secondlargest data point, the largest data point for which calculation of the mean excess is possible.The third column indicates that weighting is done either by the number of data points usedto calculate the mean excess, or the inverse.

42

LowerThresholdPercentile

UpperThresholdPercentile

Weighting ξ Std. Err. ConfidenceInterval

MinimumObservations

30th percentile 70th percentile obs. -0.27 0.005 -0.28 - -0.26 192

30th percentile 70th percentile inverse obs. -0.28 0.005 -0.29 - -0.27 192

35th percentile 70th percentile obs. -0.24 0.011 -0.26 - -0.22 192

35th percentile 70th inverse obs. -0.27 0.011 -0.29 - -0.25 192

30th percentile 75th obs. -0.29 0.005 -0.30 - -0.28 172

30th percentile 75th inverse obs. -0.31 0.006 -0.32 - -0.30 172

35th percentile 75th obs. -0.28 0.011 -0.30 - -0.25 172

35th percentile 75th inverse obs. -0.30 0.012 -0.33 - -0.30 172

Table 3: WLS estimation of the shape parameter using the mean-excess function for the windspeed distribution.

43

LowerThresholdPercentile

UpperThresholdPercentile

Weighting ξ Std. Err. ConfidenceInterval

MinimumObservations

35th 70th obs. 0.729 0.0007 0.728 - 0.730 934

35th 70th inverse obs. 0.726 0.0007 0.725 - 0.727 934

40th 70th obs. 0.726 0.0008 0.725 - 0.728 934

40th 70th inverse obs. 0.724 0.0008 0.722 - 0.726 934

35th 90th obs. 0.669 0.005 0.0.660 - 0.679 313

35th 90th inverse obs. 0.674 0.005 0.664 - 0.684 313

40th 90th obs. 0.660 0.005 0.650 - 0.670 313

40th 90th inverse obs. 0.668 0.006 0.657 - 0.679 313

Table 4: WLS estimation of the shape parameter using the mean-excess function for thecoastal city population distribution.

44