Vulnerability Index�

Haydar Kurbanyand Mika Katoz

November 19, 2008

Abstract

This paper develops an empirical method to measure the vulnerability ofvarious population groups during a disaster. In our model, the degree of vul-nerability depends on the nature of the disaster, i.e., its strength, durationand scope, and the household�s ability to respond and recover. Vulnerabilityindices for various socio-economic groups are computed on the basis of risk-averse public perceptions. This study provides sound analytical and empiricalguidance to decision makers regarding the most e¤ective and e¢ cient way toallocate resources among cities to minimize social and economic vulnerability.Currently, methods are lacking for assessing and ranking vulnerabilities in asystematic and integrated manner. We estimated model parameters based onthe socio-economic and loss data compiled from public and private sources afterHurricane Katrina. Our empirical method o¤ers a novel approach to quantifyand rank vulnerability of population groups during a disaster.Key Words: vulnerability, disaster loss, recovery, homeowners insurance,

Hurricane KatrinaJEL Codes: Q56, Q54, Q58, D81Paper presented at Southern Economic Association 78th Annual Meetings

2008, Washington, D.C. November 20-23, 2008

1 Introduction

In this paper, we develop an empirical method to measure vulnerability of variouspopulation groups during a disaster. We propose a vulnerbility index that can be used

�We thank Alexis Miller for excellent research assistance. This research was supported by theUnited States Department of Homeland Security through the Center for Risk and Economic Analysisof Terrorism Events (CREATE) under grant number 2007-ST-061-000001. However, any opinions,�ndings, and conclusions or recommendations in this document are those of the authors and do notnecessarily re�ect views of the United States Department of Homeland Security.

yDepartment of Economics, Howard University; hkurban@howard.eduzDepartment of Economics, Howard University; mkato@howard.edu

1

to assess vulnerability of individuals or groups in various economic status and di¤er-ent geographic locations in terms of how fast a population group recovers from lossescaused by a disaster. Two points are emphasized in the index. First, vulnerabilityshould be based not only on potential losses but also on potential ability of recovery.Second, vulnerability should also re�ect the post-recovery welfare in relation to theminimum welfare. Speci�cally, our index is made to emphasize vulnerability of in-dividuals whose welfare falls below the mimimum welfare and discount vulnerabilityof those whose welfare is above the mimimum welfare. This is somewhat similar tomaximizing the so-called risk-averse social welfare �people maximizing this type ofwelfare tend to choose a society in which the minimum level of shelter and food areguranteed. Our study, therefore, may help to provide sound analytical and empiricalguidance to emergency management agencies such as FEMA and SBA on the moste¤ective and e¢ cient way of allocating the limited resources to di¤erent groups andindividuals.Various factors a¤ect vulnerability of individuals. Not only the nature of disaster

itself, i.e., size, duration and scope, but also the individual�s ability to respond to adisaster plays an important role in explaining vulnerability. As Hurricane Katrinarevealed, the lower income groups tend to be more vulnerable because they havelimited access to private and public assets to respond to a disaster (Alwang, et al.,2001). The lower income groups are also vulnerable during the post disaster periodas the recovery can be delayed due to limited economic resources and exclusion fromsocial networks (Holzmann and Jorgensen, 2000). Social policy can reduce or elimi-nate some constraints by allocating resources according to the relative vulnerabilityof population groups during a disaster. To make the concept of vulnerability opera-tional and useful, a socially accepted minimum has to be agreed upon for each riskand outcome. Vulnerability of the poor results from their closeness to the sociallyaccepted minimum threshold level of well-being threshold. In fact, even if the lowerincome groups face similar risks, they are, ceteris paribus, more likely to fall belowthe threshold because of their inability to respond to losses in welfare. Our de�ni-tion of vulnerability also distinguishes between variability and vulnerability (Luers,et al., 2003). Even if everyone faces the same risks, some people are more vulnerablebecause of their inability to manage these risks. Although a higher income personmight face more variability in wealth as a result of a disaster, vulnerability may beuna¤ected since it is very unlikely that a higher income person will drop below theminimum level of well-being. On the other hand, a lower income household, closer tothe minimum level of well-being, can easily fall below the minimum even if losses aresmaller in magnitude.

2 De�ning the Vulnerability Index

Our interest is to create a vulnerability index that can capture not only the potentiallosses caused by an event, but also the potential recovery that can be made in the

2

Time

An Event occurs

(Hurricane,earthquake, etc.)

ImmediateLoss, L, isdetermined

Recovery Effort

(Insurance pay,public and privateassistance, etc.)

Current Stateof Wealth

W1

Recovery, R,is made

Initial Stateof Wealth

W0

Net Loss, L­R,is determined

W1 = W0 –L + R

Figure 1: Order of Events

future. Those who have ability to recover more of their loss will be considered lessvulnerable regardless of the size of their loss while those who have less ability torecover the loss will be considered more vulnerable.

2.1 The Model

We consider an individual (e.g., a household) with initial (pre-disaster) wealth W0.We set up a simple time-line of events, as shown in Figure (1), which describes theexperience of an a¤ected individual. When an event occurs, some or all of the initialwealthW0 may be a¤ected. We assume, for simplicity, that the rate of loss � dependsonly on the magnitude of an event, X. The rate of loss from the same X, however,may vary by physical and geographical conditions. For example, in case of hurricaneKatrina, coastal areas and non-coastal areas have signi�cantly di¤erent loss structuresas we show later in Section 2. Therefore, we divide the a¤ected area in smaller groupsso that each group should has a similar loss structure. Then the immediate loss L ofan individual in group i is

L = �i (X)W0; (1)

where 0 < �i (X) < 1 and �i0 (X) > 0 for any X > 0.Various private and public e¤orts will be made to recover the loss. Among all, pri-

vate insurance is most important to predict the individual�s ability to recover losses.Typically, homeowners insurance protects individual�s properties from disasters. Toreceive a higher coverage limit, an individual must pay a higher insurance premiumwith other conditions the same. The other factors, such as weather and landscape thatare speci�c to the area, also signi�cantly a¤ect the relationship between insurancepremium and the coverage limit According to the National Association of InsuranceCommissioners (NAIC), the national average premium for homeowners insurance in2005 is $764 while the top three most expensive states are Texas ($1,372), Louisiana($1,144), and Florida ($1,083), those commonly exposed to severe storms and hurri-canes. We therefore de�ne the coverage limit function by area. The coverage limit C

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of an individual in area j isC = Cj (p) ; (2)

where Cj0 > 0.Assume that the amount of insurance premium that an individual is willing to

pay is positively correlated with initial wealth W0

p = p (W0) ; (3)

where p0 (W0) > 0.If one has an insurance policy that pays the coverage limit C, the actual loss cov-

ered by this insurance is equal to either the coverage limit C or the loss L, whicheverthe smallest. If one does not have insurance, then we assume that no recovery ismade.Assume that the probability that an individual has insurance is � and it depends

on the wealth W0. Then the actual amount of recovery R is

R = � (W0)minfC;Lg, (4)

where �0 (W0) > 0.After all possible recovery is made, the post-recovery wealth W1 of an individual

in group i and area j is

W ij1 (X;W0) � W0 � L+R (5)

= W0 � �i (X)W0 + � (W0)minfCj (p (W0)) ; �i (X)W0g.

2.2 The Vulnerability Index

Our vulnerability index measures and ranks the well-being of various populationgroups impacted by disasters. Speci�cally, it computes the marginal change in after-recovery wealth when an additional stress is given to an individual in group i andin area j with initial wealth W0, i.e.,

��@W ij1 =@X

��. We also introduce an adjustmentcoe¢ cient W=W ij

1 (X;W0) to the index so that the vulnerability of individuals thatfall below some poverty line of wealth W is emphasized while the vulnerability ofindividuals that are above W is discounted.De�ne the vulnerability index for individuals in group i and area j as

V ij (X;W0) =

��@W ij1 (X;W0) =@X

���W ij1 (X;W0) =W

�� ; (6)

where � > 1 determines the degree of emphasis on vulnerability of people below thepoverty level W . A larger � implies that a policy puts higher priority on those whofall below W . Notice that this index depends only on the wind speed X and thepre-disaster wealth W0.

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3 Estimation

In this section, we attempt to actually estimate the vulnerability index function (6)that is applicable to the areas impacted by Hurricanes Katrina and Rita. We usenational, state, and county-level data to estimate functions �i (X), p (W0), Ci (p),� (W0), and W that �t to the impacted areas This helps us to predict vulnerabilityof individuals or a group of individuals with W0 in group i and area j when an eventwith magnitude X occurs.

1. Loss Rate �i (X)

We estimate loss rate �i (X) in (1). For the magnitude of an event X, we use themaximum wind speed, X = 1 (� 60 pmh), 2 (>60 pmh), 3 (>75 pmh), 4 (> 90 pmh),and 5 (> 100 pmh). We compiled the maximum wind speed data of 88 counties1

in Alabama, Mississippi and Louisiana impacted by Hurricanes Katrina, Rita andWilma. The storm maps created by NOAA is used to determine the maximum windspeed in each county that was in the path of Katrina. Table 2 in Appendix show theassignment of wind speed to each impacted county.2

Table 1: Veri�ed Losses

To estimate the loss rate, we divide the 88 impacted counties into coastal coun-ties and non-coastal counties. The loss structure of coastal counties is signi�cantlydi¤erent from that of non-coastal counties. This is so as damages in coastal countiesare not only from strong wind but also from hurricane tidal surge �ooding. We thusestimate the loss rate function for non-coastal counties �nc (X) and that for coastalcounties �c (X) separately.

1These counties were designated by the Federal Emergency Management Agency as eligible toreceive individual and public assistance as of September 14, 2005.

2For Cameron Parish, LA and Vermilion Parish, LA, we use the maximum wind speed of Hurri-cane Rita, instead of that of Hurricane Katrina, as most of their losses reported in the Federal Emer-gency Management Agency (FEMA) and the Small Business Administration (SBA) were caused byHarricane Rita.

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First, we estimate the dollar values of the full extent of housing damage in each ofthe 88 counties by using a special data set compiled by the Department of Housing andUrban Development (HUD). After Hurricane Katrina, the Federal Emergency Man-agement Agency (FEMA) and the Small Business Administration (SBA) inspectedthe disaster area and identi�ed three damage levels, �minor�, �major�and �severe�andthen estimated the dollar value of losses. 3 We summarize, in Table 1, the meanveri�ed losses for each damage level by state. We use Census 2000 data to �nd thenumber of houses in each damage level in each county. We compute the county k�sloss �gure as

Lk =3Ps=1

dksnks for k = 1; ::; 88; (7)

where s = 1 (minor), 2 (major), 3 (severe) is the damage levels de�ned by FEMA,dks is the veri�ed loss at damage level s in the state to which county k belongs fromTable 1, and nks is the number of houses in damage level s in county k from Census2000.Next, we estimate the county�s pre-disaster initial wealth W k

0 . We compute theaggregate house value of county k as

W k0 = W

k

0

3Ps=1

nks for k = 1; ::; 88; (8)

where Wk

0 is the pre-disaster mean house value of county k found in Census 2000.From Eqs. (7) and (8), the loss rate of county k can be computed by

�k � Lk

W k0

=

3Ps=1

dksnks

Wk

0

3Ps=1

nks

for k = 1; ::; 88. (9)

There are 72 non-coastal counties and 16 coastal counties. Thus we divide a¤ectedcounties into two groups as k = 1; :::; 72 (non-coastal); 73; :::; 88 (coastal) and regressthe loss rate against the wind speed level X

�k = �0 + �1Xk......for k = 1; :::; 72;

3The O¢ ce of the Federal Coordinator for Gulf Coast Rebuilding at the Department ofHomeland Security, the Federal Emergency Management Agency, the Small Business Admin-istration, and the Department of Housing and Urban Development have created a data setto assess the full extent of housing damage due to Hurricanes Katrina, Rita, and Wilma(http://www.huduser.org/publications/destech/GulfCoast_HsngDmgEst.html). FEMA inspectorsclassi�ed the damage levels as minor, major and severe. A subset of FEMA registrants with realproperty damage applied to the Small Business Administration for loans to repair their property.SBA inspectors then estimated �veri�ed loss� for units assessed by the FEMA inspector to haveeither �major damage� or �severe damage�. We used SBA median �veri�ed loss� tables, FEMAcategories and the number of occupied housing units in each category to estimate total loss for 88counties impacted by Hurricane Katrina. To estimate the aggregate damage level for each county,we multiplied the number of housing units in each damage category by the median �veri�ed losses�.

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L/W0 = ­0.222+0.1753X

L/W0 = ­0.0098+0.0195X

­0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5

Wind Speed (X)

Loss

 Rat

e (L

/W0)

CoastalNoncoastal

Figure 2: Wind Speed and Rate of Loss

and�k = �0 + �1X

k......for k = 73; :::; 88;

where Xk is the maximum wind speed in county k as reported in Table 2 in theAppendix.Figure 2 shows the average loss rates plotted against wind speeds. We �nd that:

1. the average loss rate is higher in coastal counties at any level of wind speed, and2. the marginal losses from additional wind speed is much higher in coastal countiesand thus di¤erence in losses between the two groups tends to enlarge as wind speedincreases.The estimated loss rate function for non-coastal counties is

�nc (X) = �0:0098 + 0:0195X; (10)

and that for coastal counties is

�c (X) = �0:222 + 0:1753X. (11)

2. Coverage Limit Function Cj (p)

We estimate the area (state in our case)-speci�c coverage limit function in Eq. (2) forAlabama, Mississippi, and Louisiana. However, data on coverage limit and premiumare available at the national level, not at the state level. Thus, we take two steps toderive the state-speci�c coverage limit function.First, we estimate the national coverage limit function using a linear form

Cna (p) = �+ bp. (12)

We use real insurance coverage data and average premium data compiled from Na-tional Association of Insurance Commissioners report (NAIC, 2000). The NAICreport is based on the data collected from insurance regulatory o¢ cials. Since the

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Cna(p) = 457.32p ­ 74086

Cal(p) = 412.5p ­ 74086

Cla(p) = 305.41p ­ 74086

Cms(p) = 372.09p ­ 74086

$0

$50,000

$100,000

$150,000

$200,000

$250,000

$300,000

$350,000

$400,000

$450,000

$500,000

250 450 650 850 1050 1250 1450 1650 1850

Annual Premium (USD)

Cov

erag

e Li

mit 

(US

D)

NAPLAPMAPAAP

Figure 3: Annual Premium and Coverage Limit

NAIC �gures are based on actual policy forms, they re�ect the actual nationwidecoverage limit and premiums paid. The estimated function is

Cna (p) = �74086 + 457:32p. (13)

As aforementioned, applying the national coverage limit function (13) to an individ-ual state j may be inappropriate. Insurance premium re�ects state-speci�c factorssuch as landscape and weather that determine the likelihood of disaster occurrence.Therefore, the insurance premium for identical policies can vary largely by state. Wecon�rm this point in Table 3 in the Appendix. It shows that the average premiumof the most commonly written insurance package HO-34 varies signi�cantly by stateand that states constantly a¤ected by disasters tend to have higher premiums.Second, we de�ne the national average premium as pna and the state j�s average

premium as pj and assume that state j�s homeowners pay pj=pna of national premiumfor any level of coverage limit. This assumption allows us to de�ne the state j�scoverage limit function for a given national coverage limit function (12) as

Cj (p) = a+ b

�pn

pj

�p: (14)

From Table 3 in the Appendix, the average premium for an HO-3 insurance packagein Alabama is pal = 847, that of Mississippi is pms = 939, and that of Louisiana ispla = 1144, and the nationwide average is pna = 764.Thus using the estimated national coverage limit function (13), we may derive the

coverage limit function of Alabama as

Cal (p) = �74086 + 412:5p, (15)

486.6% of homeowners insurance are this type. HO-3 is an open perils policy that covers anydirect damage to the house or other structures on the property unless it is speci�cally excluded.However the coverage for personal property is for named perils only.

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p (W 0) = 0.0018 W 0 + 256.39

0

200

400

600

800

1000

1200

1400

$0 $100,000 $200,000 $300,000 $400,000 $500,000 $600,000

Home Value (USD)

Ann

ual P

rem

ium

 (US

D)

Figure 4: Home Value and Annual Premium

that of Mississippi asCms (p) = �74086 + 372:09p, (16)

and that of Louisiana as

C la (p) = �74086 + 305:41p. (17)

3. Premium function p (W0)

We estimate the premium function p (W0) in Eq. (3) by regressing annual homeownersinsurance payments on home value data compiled from American Housing Surveys1999 National Sample. Not surprisingly, Figure 4 and Eq. (18) show that homeownerswith higher home values are willing to pay a higher annual premium. For every$10,000 increase in home value, annual insurance payments increase by $18.

p (W0) = 0:0018W0 + 256:39: (18)

4. Insured Homeowners Rate � (W0)

We estimate insured homeowners rate � (W0) in Eq. (4). The data for this variable,again, come from American Housing Surveys 1999 National Sample. We constructedthis variable by dividing the number of owner-occupied housing units with homeown-ers insurance by the total number of owner-occupied housing units in the sample.InFigure 5, di¤erent purchasing patterns are observed between homeowners with a homevalue < $100,000 and homeowners with a home value � $100,000.The rate of insured homeowners when home value is less than $100,000 is

� (W0) = 0:2327W0:12520 for 0 � W0 < 100; 000; (19)

and the rate of insured homeowners when home value is $100,000 or more is approx-imately constant at

� = 0:97 for W0 � 100; 000. (20)

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Figure 5: Home Value and Insured Homeowners

5. Poverty Line W

To estimate poverty line of an individual�s wealth W in (6), we use below povertylevel median house value as proxy for the socially accepted minimum level of wellbeing. According to American Housing Survey�s 1999 National Sample the medianhousing value of those that earned less than below poverty level income was $86,643.

W = 86643. (21)

4 Simulation

Based on the estimated loss rate functions, Eqs. (10) and (11), the coverage limitfunctions Eqs. (15)-(17), the premium function (18), the rates of insured homeowners,Eqs. (19) and (20), and the poverty line (21), we can predict vulnerability of anindividual with given initial home value W0 in group i (non-coastal or coastal) andin state j (AL, MS, or LA) when wind speed X occurs.For actual accessment of vulnerbility, � > 1, the degree of emphasis on vulnera-

bility of people who fall below the poverty level W , has to be determined. It shouldre�ects the public interest and consensus on priority rank. A larger � emphasizesvulnerbility of the poor under W0 more. We use � = 1 and � = 2 in our simulationto see the e¤ect of the choice of � on vulnerability. Figures 7, 9, and 11 show theestimated vulnerability functions for non-coastal and coastal counties in Alabama,Mississippi, and Louisiana respectively when � = 1 and Figures 7, 9, and 11 showthose when � = 2. The height of the function shows the scale of vulnerability, andthe initial home value W0 in the rage of $20,000-$300,000 and the wind speed level inthe range of 1-8 are taken on the �oor.Several important implications can be derivedfrom the simulation results.

1. Homeowners with initial wealth under $100,000 in non-coastal counties in AL,MS, and LA show some vulnerability. Vulnerabilty is higher for a lower initial

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Figure 6: Coverage Limit to Home Value Ratio

wealth. Vulnerbility of less wealthy homeowners is, however, much higher when� = 2 as shown in Figures 8, 10, and 12 than that when � = 1 as shownin Figures 7, 9, and 11 as a higher policy parameter emphasizes those underthe poverty line more. For wind speeds between 1-8, losses do not exceedthe coverage limit. Thus if a homeowner has insurance, all losses should becovered. Vulnerability is, however, higher for less wealthy individuals as therate of insured homeowners declines with decreases in home value.

2. Homeowners with initial wealth in the range of $100,000-$300,000 in non-coastalcounties in AL, MS, and LA, seem little a¤ected as they are mostly insured(97%) and their losses, when wind speed is between 1 and 8, are covered byinsurance.

3. Vulnerability of coastal counties is higher than that of non-coastal counties. Aswind speed increases, for a given home value, losses eventually exceed the homevalue and this increases vulnerability.

4. In coastal counties, vulnerability, for a given wind speed, increases when homevalue rises when � = 1, that is, less emphasis is put on vulnerability of homeown-ers under the poverty line. Welthier homeowners are assessed more vulnerbleas they tend to have larger uninsured losses. This comes from the fact that thecoverage limit to home value ratio tends to decline as home value increases asshown in Figure 6, derived from Eqs. (15)-(18). When � = 2, on the otherhand, vulnerability decreases when home value rises as the policy parameterdiscounts vulnerability for the welthier homeowners more and emphasizes thatfor the poor homeowners more.

5. In coastal counties, Louisiana is most vulnerable, Mississippi is second-mostvulnerable, and Alabama is least vulnerable among the impacted three states.This re�ects the fact that Louisiana is the most expensive state and Alabama is

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Figure 7: Vulnerability Functions for Alabama with � = 1

Figure 8: Vulnerability Functions for Alabama with � = 2

12

Figure 9: Vulnerability Functions for Mississippi with � = 1

Figure 10: Vulnerability Functions for Mississippi with � = 2

13

Figure 11: Vulnerability Functions for Louisiana with � = 1

Figure 12: Vulnerability Functions for Louisiana with � = 2

14

the least expensive state for the HO-3 type homeowners insurance, and home-owners in Louisiana tend to be underinsured. We can con�rm this point inFigure 6.

5 Conclusions

This paper developed a novel approach to measure and rank the vulnerability ofvarious population groups during and after a disaster. In our model, the degree ofvulnerability depends on the nature of the disaster, i.e., its severity, duration andscope, and the household�s ability to respond and recover. Our study contributesto the vulnerability literature by developing an index that is consistent with util-ity maximizing consumer behavior and risk-averse public perceptions. Given thatnatural catastrophes have been occurring with greater frequency and severity in thelast decade, this study provides sound analytical and empirical guidance to decisionmakers regarding the most e¤ective and e¢ cient way to allocate resources amongthe cities in order to minimize social and economic vulnerability. Currently, methodsare lacking for assessing and ranking vulnerabilities in a systematic and integratedmanner. Our model parameters were estimated based on the socioeconomic and lossdata compiled from public and private sources. As seen during Hurricane Katrina, alack of such models can lead to tremendous costs and su¤ering for vulnerable popu-lations and national economy. Our simulation indicates that less wealthy individualsin non-coastal counties are more vulnerable as there is a negative relationship be-tween home values and rate of insured homeowners. The homeowners with homevalue above $100,000 face less vulnerability because about 97 percent of homeownerspurchase insurance. For non-coastal counties we observe overall higher degrees ofvulnerability as losses exceeds the coverage limit. We observe that vulnerbility tendsto be higher for wealthier homeowners. The reason is that while the coverage limitincreases as the home value rises, it does not increase as much as the home value. Ofthe coastal counties impacted, the most vulnerable counties are in Louisiana and theleast vulnerable ones are in Alabama. This re�ects the fact that Louisiana is the mostexpensive state and Alabama is the least expensive state for the HO-3 type home-owners insurance among the three states. This implies that homeowners in Louisianatend to be underinsured.The empirical method developed and applied to Hurricane Katrina can easily be

extended to other types of disasters in the states other than Alabama, Louisiana, andMississippi. Our model parameters were estimated based on national level informa-tion on insurance coverage level, tenure, homeowners insurance premium and whetherhomeowners have insurance coverage or not. Given information on pre-disaster levelwealth, the type of disaster, its impact area and its severity, one can estimate thevulnerability for various population groups in coastal and non-coastal communities.

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Appendix: Data

Table 2: Damage in 88 counties impacted by Hurricane Katrina

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Table 3: Average Premium by State

19