MAHANAMA-DISSERTATION-2021.pdf - TTU DSpace Principal

Risk Assessment and Financial Management of Natural Disasters and Crime

by

Thilini Vasana Mahanama, M.S.

A Dissertation

in

Mathematics and Statistics

Submitted to the Graduate Facultyof Texas Tech University in

Partial Fulfillment ofthe Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Dr. W. Brent LindquistChairperson of the Committee

Dr. Dimitri Volchenkov

Dr. Svetlozar Rachev

Dr. Mark SheridanDean of the Graduate School

May, 2021

Texas Tech University, Thilini Mahanama, May 2021

ACKNOWLEDGMENTS

It is with great pleasure that I record my heartfelt indebtedness to my research ad-

visors, Dr. Dimitri Volchenkov, Dr. Svetlozar Rachev, and Dr. W. Brent Lindquist,

for their valuable counsel, guidance, and perseverance throughout my research. Their

continuous encouragement and support have been always a source of inspiration and

energy for me. I am so honored and privileged to have these incredible mentors in

my life.

My heartfelt gratitude also goes out to Dr. D.K. Mallawa Arachchi, my undergrad-

uate research supervisor, at the University of Kelaniya, Sri Lanka. His continuous

advice, guidance, and encouragement immensely helped me to pursue my postgrad-

uate studies at TTU.

I would like to extend my sincere thanks to all the professors in the Department

of Mathematics and Statistics at TTU for helping me in numerous ways during my

graduate studies. I thank the department and graduate school for giving me the

opportunity to pursue my doctoral studies and supporting me with scholarships. I

also wish to thank all my friends at TTU for making these years so memorable.

It is with great appreciation my sincere gratitude is extended to all my professors

at the Department of Mathematics and the Department of Statistics & Computer

Science at the University of Kelaniya, Sri Lanka. I am also grateful to all my teachers

for their valuable guidance and inspiration throughout my education.

I am very much indebted to my colleagues Nadeesha Jayaweera, Mr. Karunarathna

Banda, Dr. Pushpi Paranamana, Abootaleb Shirvani, and Dr. Ann Almeida for their

constant encouragement and support. Valuable experience I received from Toast-

masters International, TTU graduate writing center and library workshops greatly

acknowledged.

I am forever beholden to my parents, sisters, and relatives for their intimacy,

endless patience, and encouragement. They kept me going on and this work would

not have been possible without their input.

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CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Tornado Data Description . . . . . . . . . . . . . . . . . . . . 5

2.2 Crime Data Description . . . . . . . . . . . . . . . . . . . . . . 9

3. Learning a Statistical Manifold to Determine the Risk Covering Strate-

gies for Tornado-induced Property Losses in the United States . . . 14

3.1 Categorizing Tornado Data with respect to Physical Parameters 17

3.2 Exploratory Analysis on Determining Bilateral Coverages for

Tornado Damages . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Assessing a Distance Matrix based on the Differences between

Statistics in Tornado Scales . . . . . . . . . . . . . . . . . . . . 19

3.4 Visualizing the Distances of Property Loss Distributions be-

tween Tornado Scales using Classical Multidimensional Scaling 20

3.5 Learning an Underlying Framework for a Statistical Manifold

on Tornado Property Losses . . . . . . . . . . . . . . . . . . . 21

3.6 Upgrading the Underlying Framework to a Statistical Manifold

using a Subdivision Surface Method . . . . . . . . . . . . . . . 23

3.7 Determining a Curvature Matrix for the Statistical Manifold . 24

3.8 Determining the Risk Assessment Strategy based on the Cur-

vature Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4. Assessment of Tornado Property Losses in the United States . . . . . . 29

4.1 Classifying Tornado Data . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Non-negative Matrix Factorization Method for Classification 29

4.1.2 Classifying Tornado Data Using NMF . . . . . . . . . . . . 31

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4.2 Measuring the Dependence between Tornado Variables . . . . . 34

4.2.1 Bivariate Copula for Measuring the Non-linear Dependence

between Tornado Variables . . . . . . . . . . . . . . . . . . 34

4.2.2 Measuring the Dependence between Tornado Variables Us-

ing a Bivariate Gaussian Copula Approach . . . . . . . . . 37

4.3 Predicting Tornado Property Losses . . . . . . . . . . . . . . . 40

4.3.1 Long Short-Term Memory Neural Networks for Prediction

of Future Property Losses . . . . . . . . . . . . . . . . . . 40

4.3.2 Predicting Future Tornado-induced Damage Costs Using

LSTM and Shallow Neural Networks . . . . . . . . . . . . 42

5. A Natural Disasters Index . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Construction of the Natural Disasters Index (NDI) . . . . . . . 45

5.2 NDI Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3 NDI Risk Budgets . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.4 Evaluating the Impact of Climate Extreme Indicators on NDI

Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6. Global Index on Financial Losses due to Crimes in the United States . . 62

6.1 Financial Losses due to Crimes in the United States . . . . . . 63

6.1.1 Modeling the Multivariate Time Series of Financial Losses

due to Crimes . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1.2 Backtesting the Portfolio . . . . . . . . . . . . . . . . . . . 63

6.2 Option Prices for the Crime Portfolio . . . . . . . . . . . . . . 66

6.2.1 Defining a Model for Pricing Options . . . . . . . . . . . . 66

6.2.2 Issuing the European Option Prices for the Crime Portfolio 69

6.3 Risk Budgets for the Crime Portfolio . . . . . . . . . . . . . . 70

6.3.1 Defining Tail and Center Risk Measures . . . . . . . . . . . 71

6.3.2 Determining the Risk Budgets for the Crime Portfolio . . . 72

6.4 Performance of the Crime Portfolio for Economic Crisis . . . . 74

6.4.1 Defining Systemic Risk Measures . . . . . . . . . . . . . . 74

6.4.2 Evaluating the Performance of the Crime Portfolio for Eco-

nomic Crisis . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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7. Discussion & Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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ABSTRACT

The National Oceanic and Atmospheric Administration reports that the U.S. an-

nually sustains about 1,300 tornadoes. Currently, the Enhanced Fujita scale is used

to categorize the severity of a tornado event based on a wind speed estimate. We pro-

pose the new Tornado Property Loss scale (TPL-Scale) to classify tornadoes based on

associated damage costs. The dependence between the tornado-affected area and the

associated property losses vary strongly over time and location. The overall tornado

damage costs forecasted by a trained long short-term memory network trained on

historical data might reach $8 billion over the next five years although no systematic

increase in the number and cost of disasters is observed over time.

The approach of compensating for the property losses caused by tornadoes in

the United States is a two-fold process. First, private insurance companies cover the

tornado damage costs claimed by their clients. Second, the state distributes fundings

in the case of state emergency for local governments. We intend to recover these risk

assessment strategies for tornado-induced property losses reported in the national

tornado database. In order to that, we learn a statistical manifold on probability

distributions of property losses and define a measure of curvature on the manifold

to estimate variations of property losses with respect to changes in tornado path

lengths and path widths.

Natural disasters, such as tornadoes, floods, and wildfire pose risks to life and

property, requiring the intervention of insurance corporations. One of the most

visible consequences of changing climate is an increase in the intensity and frequency

of extreme weather events. The relative strengths of these disasters are far beyond

the habitual seasonal maxima, often resulting in subsequent increases in property

losses. Thus, insurance policies should be modified to endure increasingly volatile

catastrophic weather events. We propose a Natural Disasters Index (NDI) for the

property losses caused by natural disasters in the United States based on the “Storm

Data” published by the National Oceanic and Atmospheric Administration. The

proposed NDI is an attempt to construct a financial instrument for hedging the

intrinsic risk. The NDI is intended to forecast the degree of future risk that could

forewarn the insurers and corporations allowing them to transfer insurance risk to

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capital market investors. This index could also be modified to other regions and

countries.

Following the financial management principles in the NDI, we propose an index-

based insurance portfolio for crime in the United States by utilizing the financial

losses reported by the Federal Bureau of Investigation for property crimes and cy-

bercrimes. Our research intends to help investors envision risk exposure in our port-

folio, gauge investment risk based on their desired risk level, and hedge strategies

for potential losses due to economic crashes. Underlying the index, we hedge the

investments by issuing marketable European call and put options and providing risk

budgets (diversifying risk to each type of crime). We find that real estate, ran-

somware, and government impersonation are the main risk contributors. We then

evaluate the performance of our index to determine its resilience to economic crisis.

The unemployment rate potentially demonstrates a high systemic risk on the portfo-

lio compared to the economic factors used in this study. In conclusion, we provide a

basis for the securitization of insurance risk from certain crimes that could forewarn

investors to transfer their risk to capital market investors.

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LIST OF TABLES

2.1 The Enhanced Fujita Scale (EF-scale) [1]. . . . . . . . . . . . . . . . 64.1 Kendall’s τ and Spearman’s ρ rank correlation coefficients between

ln(Area) and ln(Loss). . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 Percent contribution to risk for standard deviation (Std) and expected

tail loss (ETL) (at 95% and 99% levels) risk budgets for the NaturalDisasters Index (NDI) . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 The left-tail systemic risk measures (CoVaR, CoES, and CoETL) onthe Natural Disasters Index (NDI) at different stress levels based onstressing the factors monthly maximum temperature (Max Temp) andthe Palmer Drought Severity Index (PDSI). . . . . . . . . . . . . . . 61

6.1 VaR Backtesting Results for ARMA(1,1)-GARCH(1,1) with Student’st and NIG innovations. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 The estimated parameters of the fitted the NIG process to the Crimeportfolio log-returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.3 The percentages of center risk (CR) and tail risk (TR) (at levels of95% and 99%) budgets for the portfolio on crimes. . . . . . . . . . . . 73

6.4 The empirical correlation coefficients of the joint densities of eacheconomic factors and the crime portfolio . . . . . . . . . . . . . . . . 76

6.5 The left tail systemic risk measures (CoVaR, CoES, and CoETL) onthe portfolio at stress levels of 10%, 5%, and 1% on the following eco-nomic factors - Unemployment Rate, Poverty Rate, Household Income 76

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LIST OF FIGURES

2.1 According to the NOAA Storm Data reports from 1950-2018, (a) thetotal tornado-induced property losses in states (measured in U.S. dol-lars adjusted for inflation in 2019) and (b) the total numbers of tor-nadoes in Tornado Alley states. Texas has the highest number ofreported tornadoes in Tornado Alley states. . . . . . . . . . . . . . . 6

2.2 The path of a tornado occurred in Raleigh, NC on April 19, 2019,according to National Weather Service, NOAA [2]. . . . . . . . . . . . 6

2.3 In the Tornado Alley states, the annual total numbers of tornadoesdetermined using the NOAA Storm Data reports between 1955 and2018. Over the years, the annual numbers of tornadoes have no com-mon trend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 In the Tornado Alley states, the annual tornado-induced propertylosses (in billions adjusted for inflation in 2019) determined using theNOAA Storm Data reports between 1955 and 2018. Over the years,the annual tornado-induced losses significantly can vary up to billionsof dollars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Fitted Gaussian distributions for tornado variables: ln(Length), ln(Width),and ln(Loss) (left to right) using the NOAA Storm Data reports be-tween 1955 and 2018. The tornado variables do not hold Gaussiandistributional assumptions. . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Kernel density plots for the property losses attributed to tornadoscales in A. Aij represents the tornadoes with path lengths in Liand path widths in Wj. . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 A visual representation of the distance matrix (D) which consists ofthe pairwise Kolmogorov-Smirnov’s distances between the distribu-tions of property losses in tornado scales. . . . . . . . . . . . . . . . . 20

3.3 The underlying framework of the statistical manifold on tornado-induced property losses. Each edge is comprised of the periodic in-terpolating cubic splines and the vertices are the tornado scales inA. The vertex, Aij, represents the tornadoes with lengths in Li andwidths in Wj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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3.4 The learned statistical manifold for the property losses of tornadoes,constructed using the reported losses between 1993 and 2018. Aijrepresents the tornadoes with lengths in Li and widths in Wj. Themanifold is constructed by implementing classical dimensional scalingand a method of subdivision surfaces. . . . . . . . . . . . . . . . . . . 23

3.5 (a) The learned statistical manifold and (b) its curvature portrait(right). Figure (b) is constructed with respect to the positions ofthe path width (Wj, j = 1, ..30) and path length (Li, i = 1, ..50)coordinates of a tornado. Yellow and blue regions in Fig (b) depictcompressed and expanded cells in Figure (a), respectively. . . . . . . 25

3.6 The curvature portrait with kernel densities of property losses of somemajor tornado scales in A. The statistics in monochromic zones showsimilar forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 The visual representation of the NMF results for tornado data between1950 and 2018. Tornado events and variables are depicted as pointsand vectors, respectively, on the two-dimensional plane constructedby the two salient components of NMF. Tornado events (points) arecolored according to the EF-scale. . . . . . . . . . . . . . . . . . . . . 32

4.2 The visual representation of the NMF results for tornado data duringthe periods 1950-1992 (left) and 1993-2018 (right). Tornado eventsand variables are depicted as points and vectors, respectively, on thetwo-dimensional plane spanned by the two salient components derivedfrom NMF. Tornado events (points) are colored according to the EF-scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 The histograms of the TPL-Scale (4.5) for the tornado data during (a)1950-1992, (b) 1993-2018, and (c) 1950-2018 are based on the propertylosses reported in NOAA Storm Data. The segment of the TPL-Scalegreater than 60 is magnified in (c); P(TPL−Scale ≥ 60) = 0.025, i.e.,there is a 2.5% chance a tornado will occur causing over ten milliondollars in damages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 The figures are constructed using the tornado data reported between1955 and 2018 in NOAA Storm Data. Figure (a) provides simplelinear regression analysis: ln(Loss) = 0.59 ln(Area) + 5.06. Figure(b) is the joint density of the generated marginal cdfs of ln(Loss) andln(Area) (F1 and F2, respectively). . . . . . . . . . . . . . . . . . . . 35

x


4.5 A sampling distribution of ln(Area): ln(Area) ∼ Γ (18.65, 0.59). Acomparison of the sampled and observed data for ln(Area) (between1955 and 2018 in NOAA Storm Data) using pdf, P-P, and Q-Q plots(left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.6 A sampling distribution of ln(Loss): ln(Loss) ∼ Γ (15.78, 0.73). Acomparison of the sampled and observed data for ln(Loss) (between1955 and 2018 in NOAA Storm Data) using pdf, P-P, and Q-Q plots(left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.7 Modeling the joint density of tornado variables, ln(Area) and ln(Loss),using a bivariate Gaussian copula associated with univariate gammadensities (4.11). Applying the generated data, we provide (a) the jointprobability density plot, (b) its contour plot, and (c) a comparison ofthe simulated and the observed data (between 1955 and 2018 in NOAAStorm Data). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.8 Bivariate Gaussian copula parameters for annual tornado data in Tor-nado Alley states, generated from NOAA Storm Data. There is nomonotonicity in the copula parameters (which provide the correlationof ln(Area) and ln(Loss)) over the years and states. . . . . . . . . . . 40

4.9 The LSTM layer architecture consists of a set of recurrently connectedLSTM blocks. The input, cell state, and output in the tth LSTM blockare denoted as xt, ct, and ht, respectively [4, 5]. . . . . . . . . . . . . 41

4.10 The structure of the tth LSTM block: The multiplicative units areinput (i), forget (f), and output (o) gates, and the cell state (g). Theinput, cell state, and output in the tth memory cell are denoted asxt, ct, and ht, respectively [4, 5]. . . . . . . . . . . . . . . . . . . . . . 42

4.11 A comparison of the observed and simulated monthly losses caused bytornadoes between 1990 and 2018 (testing data) and the errors due topredictions using a LSTM network. The simulations were generatedusing NOAA Storm Data. . . . . . . . . . . . . . . . . . . . . . . . . 43

4.12 The predictions of the monthly tornado-induced property losses in2019 (in millions) (a) using a LSTM and (b) using a shallow neu-ral network. The box plots in (b) are constructed using the 10,000simulations generated for each month based on NOAA Storm Data. . 44

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4.13 The predictions of (a) the monthly tornado-induced property lossesand (b) the cumulative monthly property losses (in billions adjustedfor inflation in 2019) between 2020 and 2025 using a LSTM network.The higher losses would be reported from March to June, and thevolume of property losses would amount up to eight billion dollars in2025. The simulations were generated using NOAA Storm Data. . . . 44

5.1 The monthly property losses (in billions adjusted for inflation in 2019)caused by drought, flood, winter storm, thunderstorm wind, hail, andtornado events between 1996 and 2018 generated using NOAA StormData. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Our proposed Natural Disasters Index (NDI) for the United States.This NDI (5.1) is constructed using the property losses of naturaldisasters reported in NOAA Storm Data between 1996 and 2018. . . . 47

5.3 The first differences of the stress testing variables, (a) Maximum Tem-perature (Max Temp) and (b) Palmer Drought Severity Index (PDSI),yield stationary time series. . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 The call option prices (5.7) for the Natural Disasters Index (NDI) attime t for a given strike price K using a GARCH(1,1) model withgeneralized hyperbolic innovations. . . . . . . . . . . . . . . . . . . . 51

5.5 The put option prices (5.8) for the Natural Disasters Index (NDI) attime t for a given strike price K using a GARCH(1,1) model withgeneralized hyperbolic innovations. . . . . . . . . . . . . . . . . . . . 52

5.6 The call and put option prices for the Natural Disasters Index (NDI)at time t for a given strike price K using a GARCH(1,1) model withgeneralized hyperbolic innovations. . . . . . . . . . . . . . . . . . . . 52

5.7 The Natural Disasters Index (NDI) implied volatilities against timeto maturity (T ) and moneyness (M = S/K, where S and K thestock and strike prices, respectively) using a GARCH(1,1) model withgeneralized hyperbolic innovations. . . . . . . . . . . . . . . . . . . . 53

5.8 The percent contribution to risk (PCTR) of the expected tail loss(ETL) risk budgets for the Natural Disasters Index (NDI) at 95%level. The legend depicts the severe weather events in ascending orderof their PCTR of ETL risk budgets at 95% level. . . . . . . . . . . . 55

5.9 The percent contribution to risk (PCTR) of the expected tail loss(ETL) risk budgets for the Natural Disasters Index (NDI) at 99%level. The legend depicts the severe weather events in ascending orderof their PCTR of ETL risk budgets at 99% level. . . . . . . . . . . . 56

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5.10 The percent contribution to risk (PCTR) of the standard deviation(Std) risk budgets for the Natural Disasters Index (NDI). The legenddepicts the severe weather events in ascending order of their PCTR ofStd risk budgets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.11 The generated joint densities of the returns of monthly maximum tem-perature (Max Temp) and the Natural Disasters Index (NDI), andthe Palmer Drought Severity Index (PDSI) and the NDI (right panel)using the fitted bivariate NIG models of the joint distributions ofindependent and identically distributed standardized residuals. Thefigures depict the simulated values and the contour plots of the jointdensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1 Call option prices against time to maturity (T , in days) and strikeprice (K, based on S0 = 100). . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Put option prices against time to maturity (T , in days) and strikeprice (K, based on S0 = 100). . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Implied volatility surface against time to maturity (T , in days) andmoneyness (M = K/S, the ratio of strike price, K, and stock price, S). 71

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CHAPTER 1

INTRODUCTION

The National Centers for Environmental Information reports the United States

has experienced 69 natural disasters with losses exceeding one billion dollars between

2015 and 2019. The accumulated loss exceeds $535 billion at an average of $107.1

billion/year. The trend of disaster frequency is expected to escalate over the years

due to changes in climate which will result in catastrophic losses [6]. These volatile

weather patterns will result in an inevitable challenge to the U.S.’s ability to sustain

human and economic development [7]. As a result, weather risk markets need to be

capable of offsetting the financial impacts of natural disasters [8, 9].

The financial losses due to natural disasters exacerbate due to changes in popula-

tion and national wealth density [10, 11]. If insurers are to retain profitability and

solvency in the event of a major catastrophe that insurers must increase their prices

for catastrophe insurance and reduce their exposure to risk [12]. Also, reinsurers un-

dergo severe financial stress in facilitating catastrophe insurance by offering tenable

reduction for risk in large catastrophic losses [13, 14, 15]. However, the substantial

losses can be alleviated using protective measures such as preparedness, mitigation,

and insurance [16, 17]. To better protect the clients, catastrophe insurance policies

should ramp-up investments in cost-effective loss reduction mechanisms by better

managing the risk.

Unequivocally, the catastrophe losses and related risks inherent create uncertainty

over the type of disaster event [13, 18]. For example, due to less coverage of insured

assets and data latency in drought and flooding events, they tend to provide uncertain

loss estimates compared to the losses of severe storm events in the United States

[18, 19]. In consequence, prioritization for mitigating the risks can be diverse and

complex.

In this dissertation, we address the risk assessment strategies for tornadoes re-

ported in NOAA Storm Data [3] in chapter 3 and 4. We consider the reported finan-

cial losses for tornadoes without considering their temporal relationships. Then, we

discuss the risk assessment strategies from the perspective of the compensations for

1


tornado property damages.

In the United States, the approach of compensating the property losses caused by

tornadoes is a two-pronged strategy. First, private insurance companies cover the


in the case of state emergency for local governments. We intend to recover the risk

assessment strategies for tornado-induced property losses reported in the database

[3]. Therefore, we learn a statistical manifold on probability distributions of property

losses to describe the processes of risk evaluations.

We categorize tornado data with respect to tornado path lengths and tornado path

widths and determine their probability distributions of property losses in sections

3.1 and 3.2. Then, we construct a distance matrix using the pairwise Kolmogorov-

Smirnov’s distances [20] in terms of property losses between tornado categories in

section 3.3. Henceforth, we learn a statistical manifold using classical multidimen-

sional scaling [21] for the distance matrix and a subdivision surface method in sec-

tions 3.4 - 3.6. Then, we define a measure of curvature on the statistical manifold to

show the existence of different risk assessment strategies for tornado-induced prop-

erty losses in sections 3.7 and 3.8. Finally, we generalize this procedure to introduce

the “Statistical Manifold Learning Algorithm” for high dimensional data.

In chapter 4, we address three problems in assessing tornado-induced property

losses. First, we introduce a new scale (known as TPL-Scale) in addition to the

Enhanced Fujita Scale [1] to classify tornadoes based on their financial impacts due

to damages using the non-negative matrix factorization analysis [22] in section 4.1.

Second, we apply a copula approach [23] for assessing property losses based on its

dependence on the area affected by tornadoes in section 4.2. Third, we predict

future tornado damage costs in section 4.3 using the shallow and the long short-term

memory neural networks. Based on our findings, the volume of tornado-induced

property losses would amount up to eight billion dollars by 2025.

The risk assessment strategies outline in chapter 3 and 4 cannot be implemented

for natural disasters other than tornadoes due to the lack of reported data in [3].

For that reason, we integrate the reported property losses in all types of natural

disasters taking the temporal ordering into account. Then, we utilize the time series

2


of financial losses for hedging the risk due to natural disasters in chapter 5.

The weather index insurance can effectively transfer spatially covariate weather

risks as it pays indemnities based on realizations of a weather index that is highly

correlated with actual losses [24]. The securitization of losses from natural disas-

ters provides a valuable novel source of diversification for investors. Catastrophe

risk bonds are a promising type of insurance-linked securities introduced to smooth

transferring of catastrophic insurance risk from insurers and corporations to capital

market investors by offering an alternative or complement of capital to the traditional

reinsurance [15]. The three types of variables that pay off in insurance-linked secu-

rities [25] are insurer-specific catastrophe losses, insurance-industry catastrophe loss

indices, and parametric indices based on the physical characteristics of catastrophic

events.

The first index-based catastrophe derivatives, CAT-futures, introduced by the

Chicago Board of Trade using the ISO-Index was ineffective due to a lack of re-

alistic models in the market [26]. Secondly, the Property Claim Services (PCS)

proposed the PCS-options based on the PCS-index and they slowed down due to

market illiquidity [27]. Then, the New York Mercantile Exchange (NYMEX) de-

signed catastrophe futures and options to enhance the transparency and liquidity

of the capital markets to the insurance sector [27]. [28] further explains alternative

risk transfer mechanisms within the context of natural catastrophe problems in the

United States.

We propose Natural Disasters Index (NDI) to address these shortcomings by cre-

ating a financial instrument for hedging the intrinsic risk induced by the property

losses caused by natural disasters in the U.S. The vital objective of the NDI is to

forecast the severity of future systemic risk attributed to natural disasters. This pro-

vides advance warnings to the insurers and corporations allowing them to transfer

insurance risk to capital market investors. Therefore, the proposed NDI is intended

to make up the shortfall between the capital and insurance markets. The NDI identi-

fies the potential risk contributions of each natural disaster and provides options and

futures in sections 5.2 and 5.3. In addition, we assess the performance of the NDI to

adverse weather events in section 5.4. Furthermore, the NDI could be modified to

3


calculate the risk in other regions or countries using a data set comparable to NOAA

Storm Data [3].

We follow the methods applied in the NDI on an ad hoc basis as a benchmark

to investigate the financial impact of crimes in the U.S. Using the financial losses

due to various types of crimes reported by the Federal Bureau of Investigation, we

propose a portfolio based on the economic impacts of these crimes in chapter 6. The

objective of this portfolio is to examine the impact of crime on insurance policies in

the United States by analyzing the financial losses associated with property crimes

and cybercrimes. Therefore, the key findings are intended to provide a basis for

the securitization of insurance risk from crimes. We investigate the financial market

implications of the portfolio using option pricing theory as well as risk budgeting

in sections 6.1 - 6.3. Furthermore, we evaluate the performance of our index with

respect to economic factors using stress testing in section 6.4.

4


CHAPTER 2

DATA DESCRIPTION

2.1 Tornado Data Description

National Oceanic and Atmospheric Administration (NOAA)’s National Centers for

Environmental Information reports severe weather events that have great impacts on

the U.S. economy. The total cost of the climate disasters since 1980 exceeds $1.775

trillion [18]. The U.S. sustains about 1,300 tornadoes and loses around fifty people

annually [3, 29]. In Figure 2.1(a), we have shown tornado property losses reported in

the U.S. in US dollars adjusted for 2019. Texas, Oklahoma, Kansas, Iowa, Missouri,

and Nebraska form a region with a disproportionately high frequency of tornadoes

known as Tornado Alley. Tornadoes in this region typically happen in late spring

and occasionally the early fall [30]. Texas alone experienced around 4,000 tornado

events since records began in 1950 [3], see Figure 2.1(b). These disastrous events

resulted in many deaths and had significant economic effects on the areas impacted.

In the United States, over 69,000 tornadoes have been reported by NOAA [3]

during the period from 1950 to 2018. For the recorded tornado events, the database

reports path length and width (measured in miles and feet, respectively) and property

losses (measured in U.S. Dollars of the given year). For example, Figure 2.2 illustrates

a path of a tornado in Raleigh, NC on April 19, 2019. Whenever appropriate, we

approximate the area affected by a tornado as the product of path length and width

(measured in square feet). We estimate property losses in U.S. dollars of 2019.

Tornado intensity, assessed by the Enhanced Fujita Scale (EF-scale) [1], assigns

each tornado a ‘rating’ based on estimated wind speed and related possible damages.

For each tornado in [3], the tornado intensity is characterized by one of six scales

[31] with an increasing degree of damage given by the EF-scale, see Table 2.1.

5


0

2,000,000,000

4,000,000,000

6,000,000,000

Total Losses (USD 2019) 1950 − 2018

(a) (b)

Figure 2.1: According to the NOAA Storm Data reports from 1950-2018, (a) thetotal tornado-induced property losses in states (measured in U.S. dollars adjustedfor inflation in 2019) and (b) the total numbers of tornadoes in Tornado Alley states.Texas has the highest number of reported tornadoes in Tornado Alley states.

Figure 2.2: The path of a tornado occurred in Raleigh, NC on April 19, 2019,according to National Weather Service, NOAA [2].

Table 2.1: The Enhanced Fujita Scale (EF-scale) [1].

EF-scale Wind Speed Estimate (mph) Damage Description

EFO 65–85 Minor

EF1 86–110 Moderate

EF2 111–135 Considerable

EF3 136–165 Severe

EF4 166–200 Devastating

EF5 ≥ 200 Incredible

6


NOAA has identified Florida and the South-Central U.S. as having a dispropor-

tionately high frequency of tornadoes [3], see Figure 2.1. Texas, Oklahoma, Kansas,

Nebraska, Iowa, and Missouri are widely known as Tornado Alley [30]. They display

high fluctuations in the annual numbers of tornadoes that occurred between 1955 and

2018, see Figure 2.3. The quadratic trendline fits (curves in red) exemplify that the

trends are not monotonic over the years. Furthermore, the annual tornado-induced

property losses in the Tornado Alley states vary up to nine orders of magnitude (i.e.

billions of dollars), see Figure 2.4.

1960 1980 2000Year

0

50

100

150

200

250

No

of T

orna

does

Texas

1960 1980 2000Year

0

20

40

60

80

100

No

of T

orna

does

Oklahoma

1960 1980 2000Year

0

20

40

60

80

No

of T

orna

does

Kansas

1960 1980 2000Year

0

20

40

60

80

No

of T

orna

does

Iowa

1960 1980 2000Year

0

20

40

60

80

100

No

of T

orna

does

Missouri

1960 1980 2000Year

0

20

40

60

80

No

of T

orna

does

Nebraska

Figure 2.3: In the Tornado Alley states, the annual total numbers of tornadoesdetermined using the NOAA Storm Data reports between 1955 and 2018. Over theyears, the annual numbers of tornadoes have no common trend.

We fit Gaussian distributions for tornado variables (i.e., path length, path width,

and property loss), see Figure 2.5. The assumptions of normality do not hold for

tornado variables as depicted in Figure 2.5 [32]. We use a copula approach for the

bivariate non-Gaussian distributed tornado affected areas and property losses [33, 34].

We thus investigate the dependence between tornado-affected areas and attributed

property losses in section 4.2.2.

7


1960 1980 2000Year

0

1

2

3

Loss

(bi

llion

s $

2019

)

Texas

1960 1980 2000Year

0

0.5

1

1.5

2

2.5

Loss

(bi

llion

s $

2019

)

Oklahoma

1960 1980 2000Year

0

0.5

1

1.5

2

2.5

Loss

(bi

llion

s $

2019

)

Kansas

1960 1980 2000Year

0

0.5

1

1.5Lo

ss (

billi

ons

$ 20

19)

Iowa

1960 1980 2000Year

0

1

2

3

4

Loss

(bi

llion

s $

2019

)

Missouri

1960 1980 2000Year

0

0.5

1

1.5

Loss

(bi

llion

s $

2019

)

Nebraska

Figure 2.4: In the Tornado Alley states, the annual tornado-induced property losses(in billions adjusted for inflation in 2019) determined using the NOAA Storm Datareports between 1955 and 2018. Over the years, the annual tornado-induced lossessignificantly can vary up to billions of dollars.

Figure 2.5: Fitted Gaussian distributions for tornado variables: ln(Length),ln(Width), and ln(Loss) (left to right) using the NOAA Storm Data reports be-tween 1955 and 2018. The tornado variables do not hold Gaussian distributionalassumptions.

8


2.2 Crime Data Description

In this section, we define the types of crimes utilized for constructing our index.

Using official data published by the Federal Bureau of Investigation (FBI), we con-

sidered financial losses caused by crimes committed in the United States between

2001 and 2019. We use the FBI’s Internet Crime Reports [35] to estimate the fi-

nancial losses attributed to cybercrimes and Uniform Crime Reports [36] to assess

the financial losses caused by property crimes (burglary, larceny-theft, and motor

vehicle theft). Using the information collected from these two reports, we calculate

the cumulative financial losses reported for the following 32 types of crimes [35, 37]:

• Advanced Fee: An individual pays money to someone in anticipation of

receiving something of greater value in return, but instead receives significantly

less than expected or nothing.

• BEA/EAC (Business Email Compromise/Email Account Compro-

mise): BEC is a scam targeting businesses working with foreign suppliers

and/or businesses regularly performing wire transfer payments. EAC is a sim-

ilar scam that targets individuals. These sophisticated scams are carried out

by fraudsters compromising email accounts through social engineering or com-

puter intrusion techniques to conduct unauthorized transfer of funds.

• Burglary: The unlawful entry of a structure to commit a felony or a theft.

Attempted forcible entry is included.

• Charity: Perpetrators set up false charities, usually following natural disas-

ters, and profit from individuals who believe they are making donations to

legitimate charitable organizations.

• Check Fraud: A category of criminal acts that involve making the unlawful

use of cheques in order to illegally acquire or borrow funds that do not exist

within the account balance or account-holder’s legal ownership.

• Civil Matter: Civil lawsuits are any disputes formally submitted to a court

that is not criminal.

9


• Confidence Fraud/Romance: A perpetrator deceives a victim into believ-

ing the perpetrator and the victim have a trust relationship, whether family,

friendly or romantic. As a result of that belief, the victim is persuaded to send

money, personal and financial information, or items of value to the perpetra-

tor or to launder money on behalf of the perpetrator. Some variations of this

scheme are romance/dating scams or the grandparent scam.

• Corporate Data Breach: A leak or spill of business data that is released

from a secure location to an untrusted environment. It may also refer to a

data breach within a corporation or business where sensitive, protected, or

confidential data is copied, transmitted, viewed, stolen, or used by an individual

unauthorized to do so.

• Credit Card Fraud: Credit card fraud is a wide-ranging term for fraud com-

mitted using a credit card or any similar payment mechanism as a fraudulent

source of funds in a transaction.

• Crimes Against Children: Anything related to the exploitation of children,

including child abuse.

• Denial of Service: A Denial of Service (DoS) attack floods a network/system

or a Telephony Denial of Service (TDoS) floods a service with multiple requests,

slowing down or interrupting service.

• Employment: Individuals believe they are legitimately employed, and lose

money or launders money/items during the course of their employment.

• Extortion: Unlawful extraction of money or property through intimidation or

undue exercise of authority. It may include threats of physical harm, criminal

prosecution, or public exposure.

• Gambling: Online gambling, also known as Internet gambling and iGambling,

is a general term for gambling using the Internet.

10


• Government Impersonation: A government official is impersonated in an

attempt to collect money.

• Harassment/Threats of Violence: Harassment occurs when a perpetrator

uses false accusations or statements of fact to intimidate a victim. Threats

of Violence refers to an expression of an intention to inflict pain, injury, or

punishment, which does not refer to the requirement of payment.

• Identity Theft: Identify theft involves a perpetrator stealing another per-

son’s personal identifying information, such as name or Social Security number,

without permission to commit fraud.

• Investment: A deceptive practice that induces investors to make purchases

on the basis of false information. These scams usually offer the victims large

returns with minimal risk. Variations of this scam include retirement schemes,

Ponzi schemes, and pyramid schemes.

• IPR Copyright: The theft and illegal use of others’ ideas, inventions, and

creative expressions, to include everything from trade secrets and proprietary

products to parts, movies, music, and software.

• Larceny Theft: The unlawful taking, carrying, leading, or riding away of

property (except motor vehicle theft) from the possession or constructive pos-

session of another.

• Lottery/Sweepstakes: Individuals are contacted about winning a lottery

or sweepstakes they never entered, or to collect on an inheritance from an

unknown relative and are asked to pay a tax or fee in order to receive their

award.

• Misrepresentation: Merchandise or services were purchased or contracted

by individuals online for which the purchasers provided payment. The goods

or services received were of measurably lesser quality or quantity than was

described by the seller.

11


• Motor Vehicle Theft: The theft or attempted theft of a motor vehicle.

A motor vehicle is self-propelled and runs on land surface and not on rails.

Motorboats, construction equipment, airplanes, and farming equipment are

specifically excluded from this category.

• Non-Payment/Non-Delivery: In non-payment situations, goods and ser-

vices are shipped, but payment is never rendered. In non-delivery situations,

payment is sent, but goods and services are never received.

• Overpayment: An individual is sent a payment/commission and is instructed

to keep a portion of the payment and send the remainder to another individual

or business.

• Personal Data Breach: A leak or spill of personal data that is released from

a secure location to an untrusted environment. It may also refer to a security

incident in which an individual’s sensitive, protected, or confidential data is

copied, transmitted, viewed, stolen, or used by an unauthorized individual.

• Phishing/Vishing/Smishing/Pharming: Unsolicited email, text messages,

and telephone calls purportedly from a legitimate company requesting personal,

financial, and/or login credentials.

• Ransomware: A type of malicious software designed to block access to a

computer system until money is paid.

• Real Estate/Rental: Fraud involving real estate, rental, or timeshare prop-

erty.

• Robbery: The taking or attempting to take anything of value from the care,

custody, or control of a person or persons by force or threat of force or violence

and/or by putting the victim in fear.

• Social Media: A complaint alleging the use of social networking or social

media (Facebook, Twitter, Instagram, chat rooms, etc.) as a vector for fraud.

Social Media does not include dating sites.

12


• Terrorism: Violent acts intended to create fear that are perpetrated for a

religious, political, or ideological goal and deliberately target or disregard the

safety of non-combatants.

Whenever necessary, we use multiple imputations with the principal component

analysis model to compute missing data [38]. Moreover, we adjust the financial

losses for U.S. dollars in 2020 using the CPI Inflation Calculator available in the

U.S. Bureau of Labor Statistics. Then, we model the time series of financial losses

due to these crime types in section 6.1.1.

13


CHAPTER 3

LEARNING A STATISTICAL MANIFOLD TO DETERMINE THE RISK

COVERING STRATEGIES FOR TORNADO-INDUCED PROPERTY LOSSES

IN THE UNITED STATES

In the United States, the approach of compensating the property losses caused by

tornadoes is a two-pronged strategy. First, private insurance companies cover the


in the case of state emergency for local governments. We intend to recover the risk

assessment strategies for tornado-induced property losses reported in the database

[3]. In order to accomplish this, we learn a statistical manifold on probability distri-

butions of property losses to describe the processes of risk evaluations.

We develop a “Statistical Manifold Learning Algorithm” to learn a statistical man-

ifold for tornado property losses reported between 1993 and 2019 [3] . First, we cat-

egorize tornado data with respect to the physical parameters (tornado path lengths

and tornado path widths) in section 3.1. In section 3.2, we determine the probability

distributions of property losses in each tornado scale. Then, we assess pairwise differ-

ences of distributions of property losses using Kolmogorov-Smirnov’s distance [20] to

construct a distance matrix in section 3.3. Then, we use classical multidimensional

scaling [21] for the distance matrix in section 3.4. As a result, we geometrize the

statistics of property losses in tornado scales on a two-dimensional manifold in sec-

tion 3.5. Henceforth, we learn a statistical manifold using the coordinates of classical

multidimensional scaling and a subdivision surface method in section 3.6.

We investigate the properties of curvature inherent to this statistical manifold in

section 3.7. First, we introduce a measure of curvature on the statistical manifold

based on the densification of cells. Then, we construct a matrix consisting of cur-

vature coefficients where the path widths and path lengths are columns and rows,

respectively. Moreover, the visual representation of the curvature matrix (curva-

ture portrait) confirms that there is no single distribution good enough to describe

all available property losses in the database. Furthermore, the curvature portrait

demonstrates a smooth connection between the points, which represent statistics of

14


property losses, on the statistical manifold. Hence, section 3.8 concludes that the

existence of different risk assessment strategies for tornado-induced property losses

reported in the database by utilizing the statistical manifold and the curvature por-

trait.

15


Statistical Manifold Learning Algorithm for Tornado Risk Assessment

Tornado Data: (Path Length, Path Width, Property Loss)n∗3

Categorize tornado data with respect to physical parameters

Samples of property losses in tornado scales

Calculate pairwise Kolmogorov-Smirnov’s distance for distributions of losses

Distance matrix

Visualize the distances using classical multidimensional scaling

Learn a framework for a statistical manifold using cubic splines

Upgrade the framework to a smooth statistical manifold using surface subdivision

Define a curvature coefficient based on the cell desifications

Determine the risk assessment strategies in data based on curvature coefficients

16


3.1 Categorizing Tornado Data with respect to Physical Parameters

In this section, we perform learning on the statistical manifold for describing the

difference between distributions of tornado-induced property losses with respect to

different physical parameters (tornado path length and path width). For that, we use

the tornado property losses (measured in USD 2019) and their physical parameters

(measured in ft) for each tornado reported between 1993 and 2018 in the storm

database published by the National Oceanic and Atmospheric Administration [3].

Then, we define the following scales of physical parameters using logarithms of path

lengths (Li, i = 1, · · · , 6) and path widths (Wj, j = 0, · · · , 3):

Li = {L | logL ∈ [ i , i+ 1)}, i = 1, · · · , 6

Wj = {W | logW ∈ [ j , j + 1)}, j = 0, · · · , 3.(3.1)

We further categorize the tornado data using the combinations of length and width

scales. Then, we examine the tornado-induced property losses for each combined

scale given below:

A = [Aij], i = 1, · · · , 6, j = 0, · · · , 3, (3.2)

where Aij represents the sample of property losses with path lengths in ith length

interval, Li, and path widths in jth width interval, Wj.

3.2 Exploratory Analysis on Determining Bilateral Coverages for Tornado

Damages

This section qualitatively analyzes the distributions of tornado-induced property

losses of tornado scales introduced in section 3.1. We provide the kernel density plots

of 24 tornado scales in Figure 3.1 and discern the inconsistent distribution shapes.

Since the densities (statistics) in A31, A41, A32, A42 scales are similar, any of

the statistics can be used to predict the losses in neighboring tornado scales. Due to

this relatively high predictability, the insurance companies are prepared to cover the

claims for such tornado property losses.

17


2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

2 4 6 8ln(Loss, $ 2019)

0

0.2

0.4

0.6

Den

sity

L6L5L4L3L2L1

W1

W3

W2

W0

A10

A13

A20

A22

A30

A33

A32

A31

A40

A43

A42

A41

A50

A53

A52

A51

A60

A61

A12

A23

A62

A63

A11 A21

Figure 3.1: Kernel density plots for the property losses attributed to tornado scalesin A. Aij represents the tornadoes with path lengths in Li and path widths in Wj.

The two tornado scales A60 and A61 have the same length scale but different

width scales. Even though the difference between their path widths is only 90 ft,

the statistics seem to be significantly different. Clearly, this is an extremity and

insurance companies fail to cover such losses. For such cases, the government intro-

duced coverage programs to fund states of emergency. This funding is distributed by

the Federal Emergency Management Agency, which is a part of in Homeland Secu-

rity. They fund the states and local governments, but they do not fund individuals

directly.

Thus, we cannot use a single probability distribution (a unique statistic) to ex-

amine all the tornado-induced property losses in the database as their distributions

are distinct variants with reference to the physical parameters. For example, the

bimodal density curves in A22, A62 and A63 might be the result of the two-fold risk

assessment strategies used in the United States: 1. private insurance 2. government

coverages. We substantiate this claim using the characteristics of the forthcoming

statistical manifold on tornado property losses. As the initial step, we quantitatively

measure the differences between distributions of tornado scales in section 3.3.

18


3.3 Assessing a Distance Matrix based on the Differences between Statistics in

Tornado Scales

In this section, we quantitatively compare the distributions of property losses for

the tornado scales introduced in section 3.1. In particular, we quantify the pair-

wise differences between the 24 distributions of property losses attributed to A using

Kolmogorov-Smirnov’s distance [20]. These measures provide the maximum dis-

tances between the empirical distribution functions of property losses of each pair in

A. For instance, the Kolmogorov-Smirnov’s distance between the property losses of

category Aij and Akl is given by

d(ij,kl) = maxx

(∣∣∣Fij(x)− Fkl(x)∣∣∣) i, k = 1, · · · , 6, j, l = 0, · · · , 3, (3.3)

where x denotes the property losses and Fij(x) is the proportion of property losses

in Aij less than or equal to x. Similarly, Fkl(x) is the proportion of property losses

in Akl less than or equal to x.

Consequently, we define a distance matrix (D) which consists of these pairwise dis-

tances, see Eq (3.4) and Figure 3.2. For example, d4,13 is the Kolmogorov-Smirnov’s

distance between the distributions of property losses in A40 and A12 tornado scales.

We utilize this distance matrix to investigate a geometrical representation to tornado

scales.

D =

A10 A20 · · · A63

d1,1 d2,1 · · · d24,1 A10

d1,2 d2,2 · · · d24,2 A20

......

. . ....

...

d1,24 d2,24 · · · d24,24 A63

(3.4)

19


Figure 3.2: A visual representation of the distance matrix (D) which consists ofthe pairwise Kolmogorov-Smirnov’s distances between the distributions of propertylosses in tornado scales.

3.4 Visualizing the Distances of Property Loss Distributions between Tornado

Scales using Classical Multidimensional Scaling

We apply classical multidimensional scaling (also known as principal coordinate

analysis) [21, 39] for obtaining a visual representation of the distance matrix (D).

In particular, we obtain a two-dimensional configuration for the tornado scales in

terms of the distances between probability distributions of property losses. The

classical multidimensional scaling provides a two-dimensional coordinate matrix for

the tornado scales by minimising the following loss function (residual sum of square):

SD (X) =

√ ∑i 6=j=1,··· ,N

(dij − ||xi − xj||)2, dij ∈ D. (3.5)

20


This maps tornado scales to a new configuration (X) such the distances in tornado

scales, dij, are well-approximated by the distances in new configuration ||xi − xj||,i.e., this quadratic optimization problem finds the best possible coordinates based

on the distances in D. The coordinate matrix (X) is calculated using the following

steps [40, 41]:

1. Use the distance matrix (D) to calculate the inner product matrix (B),

B = −1

2JDJ,

where J = I − 1n11T , I is the identity matrix, 1 is a vector of all ones and n is

the number of tornado scales (n=24).

2. Decompose B using

B = V ΛV T ,

where Λ = diag (λ1, . . . , λn) such that λ1 ≥ . . . ≥ λn ≥ 0, the diagonal matrix

of eigenvalues of B, and V = [v1, . . . ,vn], the matrix of corresponding unit

eigenvectors.

3. Extract the first and second eigenvalues Λ2 = diag (λ1, λ2) and corresponding

eigenvectors V2 = [v1,v2] .

4. The two coordinates are given by

X = [x1,x2]T = V2Λ

122 .

3.5 Learning an Underlying Framework for a Statistical Manifold on Tornado

Property Losses

In this section, we utilize the configuration of tornado scales obtained using clas-

sical multidimensional scaling to learn a statistical manifold for tornado-induced

property losses. First, we comprehend the inherent patterns related to length and

width scales (Li & Wj) in the two-dimensional configuration (with Coordinate 1 and

21


Coordinate 2). Then, we integrate these conformations by introducing a third di-

mension (say Coordinate 3) which conserves unit distance between the consecutive

contours of length scales (Li, i = 1, ..6). In Figure 3.3, we outline the contours of

length and width scales using periodic interpolating cubic splines [42] on the space

constructed using the three coordinates.

Figure 3.3: The underlying framework of the statistical manifold on tornado-inducedproperty losses. Each edge is comprised of the periodic interpolating cubic splines andthe vertices are the tornado scales in A. The vertex, Aij, represents the tornadoeswith lengths in Li and widths in Wj.

Figure 3.3 is the underlying framework for the forthcoming smooth statistical

manifold on tornado-induced property losses. In this graph, each vertex represents

the probability distribution of property losses in the corresponding tornado scale

(Aij). Clearly, this configuration of tornado scales preserves the distances between

probability distributions of property losses in A. We ameliorate this skeleton to a

smooth statistical manifold in section 3.6.

22


3.6 Upgrading the Underlying Framework to a Statistical Manifold using a

Subdivision Surface Method

The fuzzy structure in Figure 3.3 can be smoothened by enhancing the degree of

edges. We upgrade the geometry of the network in Figure 3.3 to a smooth manifold by

subdividing the ambient space of contours. In particular, we increase the cardinality

of vertices and subdivide each edge into 10 equally-spaced segments to obtain 10

supplementary vertices. Taking the new vertices into account, we add contours with

respect to width (Wj, j = 1, ..30) and length (Li, i = 1, ..50) scales using cubic

splines in Figure 3.4. As a result, we improve the coarse-grained conformational

manifold in Figure 3.3 to the smooth statistical manifold illustrated in Figure 3.4.

Figure 3.4: The learned statistical manifold for the property losses of tornadoes,constructed using the reported losses between 1993 and 2018. Aij represents thetornadoes with lengths in Li and widths in Wj. The manifold is constructed byimplementing classical dimensional scaling and a method of subdivision surfaces.

23


Figure 3.5(a) is a different orientation of Figure 3.4 that we use to demonstrate the

compactness of cells in section 3.7. In fact, each point on this manifold potentially

represents a distribution of property losses for a given path length and path width.

In section 3.7, we delineate the statistics of the manifold by defining a curvature

coefficient.

3.7 Determining a Curvature Matrix for the Statistical Manifold

Each point on the learned statistical manifold, Figure 3.4, postulates a probability

distribution of property losses with respect to the physical parameters. However,

this curved manifold seems to be too complex to interpret. Therefore, we introduce

a matrix based on the characteristics of the manifold to interpret the compensations

for property losses caused by tornadoes.

In Figure 3.5(a), the bottom leftmost (A10-A11-A21-A20) and the top rightmost

(A52-A53-A63-A62) zones consist of compressing (dense) cells. The cells in the re-

gions neighboring A22-A23-A33-A32 and A51-A52-A62-A61 illustrate relatively high

expansions. Non-trivial curvatures are present in both of these cell types. Further-

more, the sporadic tornadoes reported in the database densify the corresponding

regions (A10-A11-A21-A20 and A42-A43-A63-A62) in Figure 3.5(a). Hence, the spa-

tial characteristics of the manifold potentially detect anomalies in tornado property

losses.

The probability distributions (statistics) of property losses in compressing cells

hardly change within neighboring cells (see Figure 3.1), i.e., that most of the cell

statistics have the same form in densification. However, since the statistics tremen-

dously vary from one cell to another in expanding cells, we identify the possible

regions for extreme tornado events. This variation escalates at the border zones of

compressing and expanding regions. The statistics in expansion and transition zones

change abruptly from the conventional statistics. In such cases, distinct statistics

are essential to determine the significant variations from one cell to another. Corre-

spondingly, the predictability of tornado property losses relates to the densification

of cells in the statistical manifold, Figure 3.4.

We quantify the expansions and densifications of cells to estimate the predictability

24


of tornado property losses on the manifold, Figure 3.5(a). In particular, we determine

these deformations numerically based on the underlying curvatures of ambient spaces

in cells. Then, we define a measure of curvature (C) for a cell by comparing its area

with the mean area of the cells on the manifold as follows:

Cij = 1− Xij

E(X), i = 1, · · · , 50, j = 1, · · · , 30 (3.6)

where X denotes the area of a cell on the manifold, and Xij the area of a cell with i

and j denoting the lower leftmost two coordinates. The curvature coefficients provide

positive values for compressions and negative values for expansions in cells. Next,

we introduce a curvature matrix (C) for the statistical manifold learned in section

3.6 as follows:

C = [Cij]; i = 1, ..50, j = 1, .., 30. (3.7)

(a)

Width (Wj)

Leng

th (

Li)

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

A10 A13A12A11

A62 A63A61A60

A20

A30

A40

A50

(b)

Figure 3.5: (a) The learned statistical manifold and (b) its curvature portrait (right).Figure (b) is constructed with respect to the positions of the path width (Wj, j =1, ..30) and path length (Li, i = 1, ..50) coordinates of a tornado. Yellow and blueregions in Fig (b) depict compressed and expanded cells in Figure (a), respectively.

25


In Figure 3.5(b), we provide a visual representation of the curvature matrix with

respect to the positions of the width (Wj, j = 1, ..30) and length (Li, i = 1, ..50)

coordinates. The relative positions of some scales in A are shown in exterior locations.

In this curvature portrait, yellow and blue regions depict compressed and expanded

cells, respectively. The transitions between them (compressed and expanded cells)

are shown as green phases. Also, the legend in Figure 3.5(b) shows a color scale with

yellow for positive curvatures (compressed cells) and blue for negative curvatures

(expanded cells). In section 3.8, we describe how to utilize the curvature matrix for

assessing the risks related to tornado damage costs.

3.8 Determining the Risk Assessment Strategy based on the Curvature Matrix

Obviously, the risk assessment strategy for the damage caused by a tornado de-

pends on its severity. For typical tornadoes, the private insurance companies com-

pensate the property losses claimed by their clients. However, they fail to cover

the property losses attributed to catastrophic tornadoes. For such cases, the United

States Department of Homeland Security introduced Federal Emergency Manage-

ment Agency (FEMA) as a coverage program. When a catastrophic tornado occurs

in a state, the state governor proclaims a state of emergency. Upon the presidential

approval, FEMA distributes funds to state and local governments. Henceforth, the

local governments redistribute money to counties and municipalities but not directly

to individual victims.

The approaches of risk assessment strategies and compensations used in state

funds and private insurance companies are significantly different. In this section,

we identify these two different approaches using the statistical manifold. Then, we

analyze how the risk evaluation policies vary according to the physical parameters

of a tornado (i.e., from point to point on the manifold).

We compare the probability distribution plots of some main tornado scales with the

curvature portrait in Figure 3.6. Clearly, the probability distributions related to the

monochromic zones seem to have similar statistics. For example, the neighboring

region of A53 and A63 show a yellow zone in Figure 3.6, and their densities are

moderately similar. The statistics of either A53 or A63 potentially approximate the

26


Figure 3.6: The curvature portrait with kernel densities of property losses of somemajor tornado scales in A. The statistics in monochromic zones show similar forms.

property losses caused by a tornado with a path length in the range of 100,000-

1,000,000 ft and a path width in the range of 100-1,000 ft.

In Figure 3.6, some regions of the statistical manifold show that the small changes

in physical parameters (Li and Wj) trigger vast changes in the statistics related to

their property losses. For example, the property losses reported for the tornadoes

in the zone bounded by A51-A52-A62-A61 vary dramatically. In such scenarios, a

single statistic fails to estimate such volatilities in property losses with respect to

small changes in tornado path lengths and path widths. Therefore, the blue regions

27


on the manifold seem to demonstrate the potential tornado scales for extreme tornado

events.

The statistical manifold and its curvature portrait, Figure 3.5, provide a smooth

connection between the probability distributions of property losses in all tornado

scales of A. We described how the property losses vary with small changes of physical

parameters in section 3.7. As a result, we suggest a prospective framework for

recovering the proper risk assessment strategy for a tornado based on the statistical

manifold and its curvature matrix.

The two-dimensional statistical manifold reflects the principle of two-fold compen-

sation mechanisms for property losses in the United States. For small-scale torna-

does, the individual clients claim property losses from their insurance companies. For

losses due to catastrophic tornadoes, the government allocates funds for administra-

tive units to redistribute money to counties for compensations. We represent this

diversity of risk assessment strategies within the database using a two-dimensional

statistical manifold on property losses.

In this study, we identified the two different types of machinery to cover the prop-

erty losses caused by tornadoes by learning a statistical manifold. Even though we

determined compensations for the majority of the property losses reported in the

database [3], we failed to interpret some extremities. Furthermore, we can generalize

our statistical manifold learning algorithm for other applications with high dimen-

sional data.

28


CHAPTER 4

ASSESSMENT OF TORNADO PROPERTY LOSSES IN THE UNITED STATES

In this chapter, we address three problems in tornado-related data analysis: (i)

classifying tornado events, (ii) measuring the dependence between tornado variables,

and (iii) predicting future tornado-induced property losses based on available his-

torical data [3]. In addition to the Enhanced Fujita Scale [1] based on wind speed

estimates, we propose a novel tornado event classification accounting for the prop-

erty losses in NOAA Storm Data [3]. Underpinning the proposed classification by

the Non-negative Matrix Factorization analysis [22], we introduce a new scale for tor-

nado damage, the Tornado Property Loss scale (TPL-Scale). We also investigate the

non-linear dependence between property losses and the area affected by tornadoes

using a copula approach [23]. The resultant correlation coefficients are not mono-

tonic over time and location. As a result, we suggest forgoing the copula approach

for assessing property losses. Finally, we apply the shallow and the Long Short-Term

Memory neural networks for predicting future tornado damage costs. Based on our

findings, the volume of property losses caused by tornadoes in the U.S. would amount

to almost eight billion dollars in the next five years (by 2025).

4.1 Classifying Tornado Data

In the traditional EF-scale [1], tornadoes are classified using wind speed estimates

and related property damages. We apply the Multiplicative Update (MU) algorithm

for NMF to classify tornadoes taking the reported property losses into account. As

a result, we introduce a new tornado property loss scale (TPL-scale).

4.1.1 Non-negative Matrix Factorization Method for Classification

Data representation techniques, such as Principal Component Analysis (PCA) and

Non-negative Matrix Factorization (NMF), explore the hidden structure of a data set

[43, 44]. Notwithstanding PCA is commonplace in research, NMF outclasses PCA

on strongly correlated, non-Gaussian, multivariate data [45, 46, 47]. Furthermore,

the interpretation of principal components for non-negative data, characterized by

29


positive and negative coefficients, is vague (for example, negative property losses are

meaningless) [47, 48].

We consider the following data matrix: Vn×m = [log(Length), log(Width), log(Loss)]

(say, tornado matrix ). The tornado matrix (V ) consists of non-negative elements,

and the tornado variables (column vectors) follow non-Gaussian distributions (Fig-

ure 2.5). In order to get better results for tornado data classification, we apply the

NMF method which surpasses the potential limitations of conventional PCA [22, 47].

In NMF, the non-negative data matrix Vn×m is approximated by the product of two

non-negative matrices (Wn×r and Hr×m);

Vn×m ≈ Wn×rHr×m, (4.1)

where the rank of factorization (r) is a value less than the number of variables (m)

and the sample size (n) in the data matrix (V ) [22, 49, 50, 51]. The factor matrices

(W and H) represent latent and salient components of the data set [51]. In particular,

the jth variable in V is approximated by a linear combination of the columns in W

weighted by the elements of jth column in H [22];

V (:, j) ≈r∑

k=1

W (:, k)H(k, j) = WH(:, j), r < min(n,m), j = 1, · · · ,m.

(4.2)

The collection of all m variables results in the factorization of the data matrix

(4.1) [44, 49]. Theoretically, the m−dimensional data vectors in V project to an

r−dimensional linear subspace spanned by the columns in W (where the coordinates

are the elements of H) [50]. Thus, the linear approximation of V optimizes with the

basis W [22]. We determine the optimal W and H so that they minimize the recon-

struction error between the data matrix (V ) and its NMF image (WH) [49]. The

nonlinear optimization problem underlying NMF can be written using the Frobenius

30


norm ‖.‖F [22, 52]:

minW,H

f(W,H) =1

2‖V −WH‖2F

=1

2

n∑i=1

m∑j=1

|vij − (wh)ij|2, wij ≥ 0, hij ≥ 0.(4.3)

We apply the Multiplicative Update (MU) algorithm [22, 53] to compute factor

matrices (W and H) as it is simple to implement and the fastest per iteration. The

MU rules are as follows:

Haj ← Haj(W TV )aj

(W TWH)aj, a = 1, · · · , r, j = 1, · · · ,m

Wia ← Wia(V HT )ia

(WHHT )ia, a = 1, · · · , r, i = 1, · · · , n.

(4.4)

With a proper initialization, H and W update consecutively in each iteration

[43, 51, 52]. The iteration with the least amount of error between V and WH (4.3)

will yield the optimum W and H [46]. In general, they contain a large number

of vanishing elements leading to sparseness in outcomes that enhance accuracy and

interpretability of the results [46, 47, 49]. We classify tornado events reported by

NOAA in [3] using the MU algorithm for NMF in section 4.1.2.

4.1.2 Classifying Tornado Data Using NMF

We implemented NMF [22] to classify the reported tornado occurrences in [3]

between 1950 and 2018. For two dimensional visualization of the results, we chose

the rank coefficient, r = 2 (see the 2D plots in Figure 4.1 and 4.2). We computed the

factor matrices (W and H) using the MU algorithm (4.4) implemented in Matlab

[54]. As the iterative MU algorithm requires initializations for W and H (i.e., starting

matrices for W and H), we set 20 random replications. The three columns in H

provide the vectors for tornado variable representations on the plane constructed

31


from NMF components, see Figure 4.1;

H =

log(Length) log(Width) log(Loss)[ ]0.3564 0.2187 0.9084

0.8749 0.3479 0.3370.

Each point in Figure 4.1 (determined by a row in Wn×2 as Cartesian coordinates)

represents a tornado event reported by NOAA [3]. The EF-scale of a tornado is analo-

gized using the color scale given in the legend. This illustrates a poor classification

under the traditional EF-scale, since none of these tornado variables are included in

the EF-scale. Counterintuitively, some severe tornadoes show low property losses in

Figure 4.1 while some mild tornadoes show extremely high property losses. The EF-

scale does not exhibit a direct relation with property losses even though it includes

a qualitative measurement of tornado-induced damages (see Table 2.1). Hence, we

proposed a quantitative scale of tornado-induced property losses.

Figure 4.1: The visual representation of the NMF results for tornado data between1950 and 2018. Tornado events and variables are depicted as points and vectors,respectively, on the two-dimensional plane constructed by the two salient componentsof NMF. Tornado events (points) are colored according to the EF-scale.

32


In Figure 4.1 and 4.2, the clusters of points are orthogonal to the log(Loss) vector.

This pattern arises from the fact that before 1993, the property losses were reported

using one of eight amounts (in $): 25, 250, 2,500,· · · , 250,000,000. Figure 4.2(a)

organizes points into these categories by creating eight distinct clusters. Due to the

changes in loss assessment after 1993, the points in Figure 4.2(b) are scattered and

less uniform.

Figure 4.2: The visual representation of the NMF results for tornado data duringthe periods 1950-1992 (left) and 1993-2018 (right). Tornado events and variables aredepicted as points and vectors, respectively, on the two-dimensional plane spanned bythe two salient components derived from NMF. Tornado events (points) are coloredaccording to the EF-scale.

Tornado Property Loss Scale (TPL-Scale)

It is natural to utilize decibel scale [55] with the minimum property loss in [3] ($10)

for the TPL-scale:

TPL scale = 10 log10

(Loss

$ 10

)dB. (4.5)

A tornado event with a TPL-Scale = 50 indicates that the tornado attributed

property loss is 105 times higher than the minimal property loss ($10) reported in

NOAA [3]. Figure 4.3(a) and (b) are distinguished due to changes in the assessment

of tornado-induced losses after 1993. The TPL-Scale predicts a 2.5% chance that a

severe tornado would occur causing over ten million dollars in damage costs (i.e., the

TPL-Scale higher than 60), see to Figure 4.3(c).

33


(a) (b) (c)

Figure 4.3: The histograms of the TPL-Scale (4.5) for the tornado data during (a)1950-1992, (b) 1993-2018, and (c) 1950-2018 are based on the property losses reportedin NOAA Storm Data. The segment of the TPL-Scale greater than 60 is magnifiedin (c); P(TPL−Scale ≥ 60) = 0.025, i.e., there is a 2.5% chance a tornado will occurcausing over ten million dollars in damages.

4.2 Measuring the Dependence between Tornado Variables

The results of simple linear regression analysis, showing how property losses are

roughly proportional to the square root of a tornado’s area, have been substantially

improved in this section using a bivariate Gaussian copula approach.

4.2.1 Bivariate Copula for Measuring the Non-linear Dependence between

Tornado Variables

The tornado variables ln(Area) and ln(Loss), display a nonlinear relationship as

evidenced in Figure 4.4. Since they follow non-Gaussian distributions (see section

2.1), the Pearson linear correlation coefficient provides an inaccurate measure of their

dependence. The Kendall’s τ and Spearman’s ρ rank correlation coefficients are

nonparametric measures of nonlinear dependence [56, 57, 58]. The Kendall’s τ and

Spearman’s ρ coefficients are the average likelihood ratio dependence and quadrant

dependence, respectively [59]. However, these correlation coefficients fail to explain

the dependence structure of extreme events [60]. Therefore, we use a copula approach

to determine the dependence between strongly correlated, bivariate, non-Gaussian

distributed random variables [61, 62, 63].

A bivariate copula is a function of two variables which couples the joint cumulative

34


(a) (b)

Figure 4.4: The figures are constructed using the tornado data reported between1955 and 2018 in NOAA Storm Data. Figure (a) provides simple linear regressionanalysis: ln(Loss) = 0.59 ln(Area) + 5.06. Figure (b) is the joint density of thegenerated marginal cdfs of ln(Loss) and ln(Area) (F1 and F2, respectively).

distribution function (cdf) to the marginal cdfs [64]. According to the Sklar’s theorem

[23], there exists a bivariate copula function (C): C : [0, 1]2 → [0, 1] such that

C(F1(x1), F2(x2)) = F (x1, x2), F (x1, x2) =

∫ x1

−∞

∫ x2

−∞f(x1, x2) dx1dx2, (4.6)

where F and f are the joint cdf and pdf (probability density function), respectively,

of X1 and X2, and Fi(xi) for i = 1, 2 is the marginal cdf of Xi [65, 66, 67]. According

to the probability integral transformation [68] (i.e., the cdf of a continuous random

variable has a standard uniform distribution), a copula function has uniform margins:

Fi(xi) = ui ∼ U(0, 1) for i = 1, 2. Furthermore, since cdfs are continuous and

differentiable, (4.6) can be reformulated as

C(u1, u2) =

∫ u1

−∞

∫ u2

−∞c(u′1, u

′2) du

′1du

′2, ui = Fi(xi), i = 1, 2. (4.7)

The copula density function c(·, ·) in (4.7) contains information about the dependence

35


structure of X1 and X2 [33]. The joint pdf (f) can then be constructed as

f(x1, x2) = f1(x1) · f2(x2) · c(u1, u2), ui = Fi(xi) for i = 1, 2. (4.8)

Hence, a bivariate copula approach allows us to investigate the dependence between

X1 and X2 when their marginal distributions (f1 and f2) are properly defined [33,

62, 69].

Implementation of Copula:

We use two educated assumptions on the variables (X1 and X2): (i) the choices

of marginal distributions, f1(·) and f2(·), and (ii) the choice of the copula function,

C(·, ·) (4.7) [65, 69, 70]. We estimate the parameters of candidate distributions

of X1 and X2 using the maximum likelihood method [71]. Then, we evaluate the

goodness-of-fit using Anderson Darling or Kolmogorov-Smirnov tests to choose the

best candidate [34, 72].

There are no common principles for choosing the optimal copula function [33].

For tornado data, we selected the Gaussian copula (among Gaussian, Student-t,

Gumbel, Frank, and Clayton copulas) based on the performance of Cramer-von Mises

goodness-of-fit test [34, 72].

Bivariate Gaussian Copula:

A bivariate Gaussian copula function (C) with parameter θ is defined by

C(u1, u2; θ) = F (F−11 (u1), F−12 (u2); θ), ui = Fi(xi) for i = 1, 2 (4.9)

where

(1 θ

θ 1

)is the correlation matrix with θ ∈ (−1, 1) [73, 74, 75]. The copula

parameter (θ) determines the dependence structure in variables (i.e., the non-linear

pairwise correlation coefficient of X1 and X2) as copulas are invariant to monotonic

transformations of the margins [74, 76]. The Gaussian copula density function for a

36


bivariate Gaussian distribution is given by

c(u1, u2; θ) =1√

1− θ2exp

(−v

21θ

2 − 2θv1v2 + v22θ2

2(1− θ2)

), vi = Φ−1(ui) for i = 1, 2

(4.10)

where Φ is the cdf of the standard normal distribution [77]. We estimate the copula

parameter (θ) via the maximum likelihood method [57, 58]. The maximum correla-

tion coefficient is given by |θ| (i.e., higher the |θ|, higher the correlation) [78]. We

follow this procedure in section 4.2.2 to investigate the dependence between tornado

variables (i.e., ln(Area) and ln(Loss)).

4.2.2 Measuring the Dependence between Tornado Variables Using a Bivariate

Gaussian Copula Approach

In section 4.2.1, we examined that property losses are roughly proportional to the

square root of a tornado’s area. We investigated the dependence between ln(Area)

and ln(Loss) using a bivariate Gaussian copula approach: C(F1[ln(Area)], F2[ln(Loss)] ; θ) =

F (ln(Area), ln(Loss); θ) (4.6). Therefore, we estimated marginal pdfs (f1 and f2) and

the copula function (C) to determine the copula parameter (θ).

Estimating Marginal Distributions:

The gamma sampling distributions of ln(Area) and ln(Loss) with shape and rate

parameters are given by

ln(Area) ∼ Γ (18.65, 0.59) & ln(Loss) ∼ Γ (15.78, 0.73). (4.11)

The sampling distributions deviate for the high areas and property losses reported

in [3] because of the data cut-off phenomenon, see Figures 4.5 and 4.6. Furthermore,

Figure 4.6 depicts the quantification inherent in property losses (refer section 4.1.2).

Estimating Copula Density:

We implemented a bivariate Gaussian copula associated with univariate gamma

densities (4.11) to model the tornado data. We estimated the copula parameter as

37


0 5 10 15

Sampled cdf

0.00010.001

0.25 0.5

0.75 0.9

0.95 0.99 0.995 0.999 0.9995

0.9999

Em

piric

al c

df

0 10 20 30Sampled quantiles

0

10

20

30

Em

piric

al q

uant

iles

Figure 4.5: A sampling distribution of ln(Area): ln(Area) ∼ Γ (18.65, 0.59).A comparison of the sampled and observed data for ln(Area) (between 1955 and2018 in NOAA Storm Data) using pdf, P-P, and Q-Q plots (left to right).

0 5 10 15 20 25

Sampled cdf

0.00010.001 0.05 0.1 0.25 0.5

0.75 0.9

0.95

0.99 0.995 0.999 0.9995

0.9999

Em

piric

al c

df

0 10 20 30Sampled quantiles

0

10

20

30

Em

piric

al q

uant

iles

Figure 4.6: A sampling distribution of ln(Loss): ln(Loss) ∼ Γ (15.78, 0.73).A comparison of the sampled and observed data for ln(Loss) (between 1955 and 2018in NOAA Storm Data) using pdf, P-P, and Q-Q plots (left to right).

θ = 0.5356, using the ‘fitCopula’ R package. Figure 4.7 (a) and (b) illustrate the

joint probability density of the generated ln(Area) and ln(Loss) data. We compared

them with the reported (observed) data in [3], see Figure 4.7 and Table 4.1.

Measuring Rank Correlation Coefficients:

We estimated the Kendall’s τ and Spearman’s ρ rank correlation coefficients be-

tween ln(Area) and ln(Loss) for the NOAA data [3] and the data generated from

the bivariate Gaussian copula associated with univariate gamma densities (4.11), see

Table 4.1. The comparison of the results in Table 4.1 confirms that the bivariate

Gaussian copula is suitable for modeling tornado data [3].

38


(a) (b) (c)

Figure 4.7: Modeling the joint density of tornado variables, ln(Area) and ln(Loss),using a bivariate Gaussian copula associated with univariate gamma densities (4.11).Applying the generated data, we provide (a) the joint probability density plot, (b) itscontour plot, and (c) a comparison of the simulated and the observed data (between1955 and 2018 in NOAA Storm Data).

Table 4.1: Kendall’s τ and Spearman’s ρ rank correlation coefficients betweenln(Area) and ln(Loss).

Rank correlation coefficientType of data

Observed Generated

Kendall’s τ 0.3671 0.3591

Spearman’s ρ 0.5242 0.5169

We assessed the copula parameters (θ) for the annual tornado data for each state

in Tornado Alley. According to Figure 4.8, there is no monotonicity in copula pa-

rameters over the years and states. Thus, the dependence of ln(Area) and ln(Loss)

varies with time and geographical location. Therefore, we cannot capture the ten-

dency of property losses using this copula approach. We examined a better approach

in time series analysis to predict the losses attributed to tornadoes in the future.

39


1960 1980 2000Year

0

0.2

0.4

0.6

0.8

Cop

ula

Par

amet

er

Texas

1960 1980 2000Year

0

0.2

0.4

0.6

0.8

Cop

ula

Par

amet

er

Oklahoma

1960 1980 2000Year

0.2

0.4

0.6

0.8

1

Cop

ula

Par

amet

er

Kansas

1960 1980 2000Year

0

0.2

0.4

0.6

0.8

1

Cop

ula

Par

amet

er

Iowa

1960 1980 2000Year

0.2

0.4

0.6

0.8

Cop

ula

Par

amet

er

Missouri

1960 1980 2000Year

0.2

0.4

0.6

0.8

1

Cop

ula

Par

amet

er

Nebraska

Figure 4.8: Bivariate Gaussian copula parameters for annual tornado data in TornadoAlley states, generated from NOAA Storm Data. There is no monotonicity in thecopula parameters (which provide the correlation of ln(Area) and ln(Loss)) over theyears and states.

4.3 Predicting Tornado Property Losses

According to section 4.2.2, the calculated non-linear dependence coefficients be-

tween the property losses and areas affected by tornadoes vary with time and location.

Therefore, we use the Long Short-Term Memory (LSTM) neural networks to predict

monthly tornado-induced property losses.

4.3.1 Long Short-Term Memory Neural Networks for Prediction of Future

Property Losses

LSTM is a fast Recurrent Neural Network (RNN) with long time lags and higher

accuracy [79, 80]. We use a LSTM network composed of an input layer, a hidden layer

(LSTM layer), an intermediate fully connected layer, and an output layer [81, 82]. We

set one neuron for each input, intermediate, and output layer and then determine the

40


optimal number of neurons for the LSTM layer [82]. Furthermore, the LSTM layer

consists of a set of recurrently connected LSTM blocks (memory cells), as depicted

in Figure 4.9, and the LSTM block is further illustrated in Figure 4.10 [83, 84, 85].

Figure 4.9: The LSTM layer architecture consists of a set of recurrently connectedLSTM blocks. The input, cell state, and output in the tth LSTM block are denotedas xt, ct, and ht, respectively [4, 5].

A Constant Error Carousel (CEC) is a recurrently connected unit in a LSTM block

that determines what information to store in the memory, how long to store it, and

when to read it out, see Figure 4.10 [80, 81, 86, 87]. CEC’s activation is known as the

cell state and it is controlled by multiplicative units known as input (i), forget (f),

and output (o) gates (which analogize read, reset and write), and the cell candidate

(g), see Figure 4.10 [79, 82, 84, 86]. The gates use a sigmoid activation function,

σ(x) = (1+e−x)−1, and the cell candidate use the hyperbolic tangent, tanh, function

[4, 88].

The input, cell state, and output in the tth LSTM block are denoted xt, ct, and

ht, respectively [4]. The tth LSTM block contains the information learned (memory

gained) from the previous time step: cell state and output of (t− 1)th step (ct−1 and

ht−1) [5, 82]. At each time step, the LSTM block captures the time dependencies

to update its cell state (ct) and compute its output (ht) [5, 89, 90]. In this iterative

process, LSTM is capable of learning long-term dependencies in the data set as

it updates the memory in each block [82, 91]. We predict future tornado-induced

monthly property losses using LSTM in section 4.3.2.

41


Figure 4.10: The structure of the tth LSTM block: The multiplicative units are input(i), forget (f), and output (o) gates, and the cell state (g). The input, cell state, andoutput in the tth memory cell are denoted as xt, ct, and ht, respectively [4, 5].

4.3.2 Predicting Future Tornado-induced Damage Costs Using LSTM and Shallow

Neural Networks

We considered monthly losses attributed to tornadoes between 1950 and 2018. The

property losses due to tornadoes depend on time, geographic locations, and seasonal

effects (see Figure 2.3). We trained a LSTM [81] network to capture the long-term

dependencies in the data set and then predicted future monthly property losses.

We allow the time series data during 1950-1990 and 1991-2018 for the training and

testing datasets, respectively.

We set up the ‘Adam’ solver (Adaptive moment estimation) [92] available in Mat-

lab and trained the LSTM with 500 hidden neurons. For predictions of the future

losses, we used the ‘predictAndUpdateState’ function, which uses the previous pre-

diction as an input for the next prediction [92]. A comparison of the observed and

simulated test data reports a low root-mean-square error (RMSE = 3.7733) [93], see

Figure 4.11.

The predictions of monthly losses in 2019 are reasonable approximations of the

observations except for September and October, see Figure 4.12(a). Furthermore,

we predicted the seasonal losses in 2019 (considering the data sets of each month

separately) using shallow neural networks with ten hidden neurons. The box plots in

Figure 4.12(b) are constructed using the 10,000 simulations generated for each month.

42


1990 1994 1998 2002 2006 2010 2014 20180

5

10

15

20

ln(L

oss,

$ 2

019)

ObservedSimulated

-20

-10

0

10

20

Err

or

RMSE = 3.7733

1990 1994 1998 2002 2006 2010 2014 2018

Time

Figure 4.11: A comparison of the observed and simulated monthly losses caused bytornadoes between 1990 and 2018 (testing data) and the errors due to predictionsusing a LSTM network. The simulations were generated using NOAA Storm Data.

Then, we forecasted the monthly property losses attributed to tornadoes from 2020

to 2025, see Figure 4.13. It is apparent that the higher losses would be reported

from March to June (so-called tornado season), see Figure 4.13(a). According to our

analysis, the volume of property losses would amount up to eight billion dollars in

2025, see Figure 4.13(b).

43


Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

0

500

1000

1500

2000

Loss

(m

illio

ns $

201

9)ObservedSimulated

(a)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

0

500

1000

1500

2000

Loss

(m

illio

ns $

201

9)

Observed

(b)

Figure 4.12: The predictions of the monthly tornado-induced property losses in 2019(in millions) (a) using a LSTM and (b) using a shallow neural network. The boxplots in (b) are constructed using the 10,000 simulations generated for each monthbased on NOAA Storm Data.

2020 2021 2022 2023 2024 2025Year

0

0.2

0.4

0.6

Loss

(bi

llion

s $

2019

)

(a)

2020 2021 2022 2023 2024 2025Year

0

2

4

6

8

Cum

ulat

ive

Loss

(bi

llion

s $

2019

)

(b)

Figure 4.13: The predictions of (a) the monthly tornado-induced property losses and(b) the cumulative monthly property losses (in billions adjusted for inflation in 2019)between 2020 and 2025 using a LSTM network. The higher losses would be reportedfrom March to June, and the volume of property losses would amount up to eightbillion dollars in 2025. The simulations were generated using NOAA Storm Data.

44


CHAPTER 5

A NATURAL DISASTERS INDEX

We propose a Natural Disasters Index (NDI) for the United States using the prop-

erty losses reported in NOAA Storm Data [3] between 1996 and 2018. The NDI is

aimed to assess the level of future systemic risk caused by natural disasters. We

provide an evaluation framework for the NDI using a discrete-time generalized au-

toregressive conditional heteroskedasticity model to calculate the fair values of the

NDI options [94] in section 5.2. Then, we simulate call and put option prices using

the Monte Carlo method.

We distribute the cumulative risk attributed to our equally weighted portfolio

in order to find the risk contribution of each type of natural disaster in section

5.3. According to our assessments using standard deviation and expected tail loss

risk budgets, flood and flash flood are the main risk contributors in our portfolio.

Furthermore, we evaluate the portfolio risk of the NDI to mitigate risks using monthly

maximum temperature and the Palmer Drought Severity Index as stressors in section

5.4. We identify the stress on maximum temperature significantly impacts the NDI

compared to that of the Palmer Drought Severity Index at the highest stress level

(1%).

5.1 Construction of the Natural Disasters Index (NDI)

This section provides an exploratory data analysis for constructing an index on

financial losses caused by natural disasters. The National Oceanic and Atmospheric

Administration (NOAA) has published information on severe weather events occur-

ring in the United States between 1950 and 2018 in their “Storm Data” database [29].

We utilize the property losses caused by the following 50 types of natural disasters

from 1996-2018 to construct the index:

Avalanche, Blizzard, Coastal Flood, Cold/Wind Chill, Debris Flow, Dense

Fog, Dense Smoke, Drought, Dust Devil, Dust Storm, Excessive Heat, Ex-

treme Cold/Wind Chill, Flash Flood, Flood, Frost/Freeze, Funnel Cloud,

Freezing Fog, Hail, Heat, Heavy Rain, Heavy Snow, High Surf, High

45


Wind, Hurricane (Typhoon), Ice Storm, Lake-Effect Snow, Lakeshore

Flood, Lightning, Marine Dense Fog, Marine Heavy Freezing Spray, Ma-

rine High Wind, Marine Hurricane/Typhoon, Marine Lightning, Marine

Strong Wind, Marine Thunderstorm Wind, Rip Current, Seiche, Sleet,

Storm Surge/Tide, Strong Wind, Thunderstorm Wind, Tornado, Trop-

ical Depression, Tropical Storm, Tsunami, Volcanic Ash, Waterspout,

Wildfire, Winter Storm, Winter Weather.

The database reports the property losses incurred by natural disasters in U.S.

dollars of the given year [29]. For this study, we estimate them in U.S. dollars adjusted

for inflation in 2019. Figure 5.1 provides examples of natural disasters between 1996

and 2018 that exemplify eccentric property losses (adjusted for inflation in 2019).

2000 2005 2010 2015Year

0

0.2

0.4

0.6

0.8

1

Pro

pert

y Lo

ss (

$ bi

llion

s)

2000 2005 2010 2015Year

0

2

4

6

8

Pro

pert

y Lo

ss (

$ bi

llion

s)

2000 2005 2010 2015Year

0

0.2

0.4

0.6

0.8

1

Pro

pert

y Lo

ss (

$ bi

llion

s)

2000 2005 2010 2015Year

0

0.5

1

1.5

Pro

pert

y Lo

ss (

$ bi

llion

s)

2000 2005 2010 2015Year

0

1

2

3

4

Pro

pert

y Lo

ss (

$ bi

llion

s)

2000 2005 2010 2015Year

0

2

4

6

8

Pro

pert

y Lo

ss (

$ bi

llion

s)

Figure 5.1: The monthly property losses (in billions adjusted for inflation in 2019)caused by drought, flood, winter storm, thunderstorm wind, hail, and tornado eventsbetween 1996 and 2018 generated using NOAA Storm Data.

46


Natural Disasters Index (NDI)

To obtain an equally spaced time series, we examine the cumulative property

losses for all 50 types of natural disasters in two-week increments between 1996 and

2018. We define Lt as the total property loss at the tth biweekly period. Then, we

transform this time series Lt to a stationary time series by taking the first difference

(lag-1 difference) of L0.1t (5.1), see Figure 5.2. Thus, we propose a Natural Disasters

Index (NDI) as follows:

NDIt = L0.1t − L0.1

t−1, t = 1, · · · , T = 552. (5.1)

2000 2002 2004 2006 2008 2010 2012 2014 2016

Year

-8

-6

-4

-2

0

2

4

6

8

ND

I

Figure 5.2: Our proposed Natural Disasters Index (NDI) for the United States. ThisNDI (5.1) is constructed using the property losses of natural disasters reported inNOAA Storm Data between 1996 and 2018.

For stress testing in section 5.4, we utilize monthly maximum temperatures and

the Palmer Drought Severity Index (PDSI) used in the U.S. Climate Extremes Index

(CEI) [95, 96, 97]. We define the reported highest temperature for each month in the

U.S. as the monthly maximum temperature (measured in Fahrenheit) [98, 99]. PDSI

is a measurement of the severity of drought in a region for a given period [100, 101].

We use the monthly PDSI in the U.S. that assigns a value in [-4,4] on a decreasing

47


degree of dryness (i.e., the extremely dry condition and extremely wet condition

provides -4 and 4, respectively) [102]. Figure 5.3 depicts that the first differences of

both stress testing variables yield stationary time series.

2000 2005 2010 2015

Year

-20

-10

0

10

20

Ret

urns

of M

ax T

emp

(a)

2000 2005 2010 2015

Year

-6

-4

-2

0

2

4

6

Ret

urns

of P

DS

I

(b)

Figure 5.3: The first differences of the stress testing variables, (a) Maximum Temper-ature (Max Temp) and (b) Palmer Drought Severity Index (PDSI), yield stationarytime series.

5.2 NDI Option Prices

Standard insurance and reinsurance systems encounter difficulties in reimbursing

the extremely high losses caused by natural disasters. Insurance companies seek

more reliable approaches for hedging and transferring these types of intensive risks

to capital market investors. Catastrophe risk bonds (CAT bonds) are one of the

most important types of Insurance-Linked-Securities used to accomplish this. Our

proposed NDI is intended to assess the degree of future systemic risk caused by nat-

ural disasters. Therefore, we determine a proper model for pricing the NDI options

in this section.

Options can be used for hedging, speculating, and gauging risk. The Black-Scholes

model, binomial option pricing model, trinomial tree, Monte Carlo simulation, and

finite difference model are the conventional methods in option pricing. Recently, the

48


discrete stochastic volatility-based model was introduced to compute option prices1 and explain some well-known mispricing phenomena. Furthermore, Duan [112]

proposes the application of discrete-time Generalized AutoRegressive Conditional

Heteroskedasticity (GARCH) to price options. We extend his work by considering the

standard GARCH model with Generalized Hyperbolic (GH) innovations to compute

the fair values of the NDI options. We assume that the dynamic returns (5.1) follow

the process

Rt = logNDItNDIt−1

= r′t + λ0√at −

1

2at +

√atεt, (5.2)

where r′t and εt are the risk-less rate of return and standardized residual during

the time period t, respectively, λ0 denotes the risk premium for the NDI, and at is

the conditional variance of returns (Rt) given the information set consisting of all

linear functions of the past returns available during the time period t− 1 (Ft−1), i.e.,

at = var (Rt | Ft−1). We use the standard GARCH(1,1) to model

a2t = m+ a a2t−1 + b ε2t−1, (5.3)

where m (constant), a, and b are non-negative parameters of the model; each of these

variables is to be estimated from the data. We assume the standardized residuals (εt)

are independent and identically distributed GH (λ, α, β, δ, µ). According to [113], Rt

for given Ft−1 is distributed on real world probability space (P) as

Rt ∼ GH

(λ,

α√at,

β√at, δ√at, r

′t +mt + µ

√at

), mt = λ0

√at −

1

2at. (5.4)

The Esscher transformation given in [114] is the conventional method of identifying

an equivalent martingale measure to obtain a consistent price for options. Based on

the Esscher transformation [115], Rt for given Ft−1 is distributed on the risk-neutral

probability (Q) as follows:

Rt ∼ GH

(λ,

α√at,β√at

+ θt, δ√at, r

′t +mt + µ

√at

), (5.5)

1See [103, 104, 105, 106, 107, 108, 109, 110] and [111]

49


where θt is the solution to MGF (1 + θt) = MGF (θt) er′t , and MGF is the condi-

tional moment generating function of Rt+1 given Ft.

We generate future values of the NDI to price its call and put options using the

Monte Carlo simulations [115] as follows:

1. Fitting GARCH(1,1) with Normal Inverse Gaussian (NIG) innovations to L0.1t

and forecasting a21 (we set t = 1).

2. Beginning from t = 2, repeat the steps (a)-(c) for t = 3, 4, ..., T , where T is

time to maturity of the NDI call option.

(a) Estimating the parameter θt using MGF (1 + θt) = MGF (θt) er′t , where

MGF is the conditional moment generating function of Rt+1 given Ft on

P.

(b) Finding an equivalent distribution function for εt on Q and generate the

value of εt+1 under the assumption εt ∼ GH(λ, α, β +√atθt, δ, µ) on Q.

(c) Computing the values of Rt+1 and at+1 using (5.2) and (5.3).

3. Generating future values of L0.1t for t = 1, ...., T on Q where T is the time to

maturity. Recursively, future values of the NDI is obtained by

NDIt = R10t +NDIt−1. (5.6)

4. Repeating steps 2 and 3 for 10,000 (N) times to simulate N paths to compute

future values of the NDI.

Then, the Monte Carlo averages approximate future values of the NDI at time t

for a given strike price K to price its call and put options (C and P , respectively)

C (t, T,K) =1

Ne−r

′t(T−t)

N∑i=1

(NDI

(i)T −K

)+, (5.7)

P (t, T,K) =1

Ne−r

′t(T−t)

N∑i=1

(K −NDI(i)T

)+. (5.8)

50


We provide call and put option prices for the NDI (C and P ) at time t for a

given strike price K in Figure 5.4 and 5.5, respectively. These Figures illustrate the

relationship between time to maturity (T ), the strike price (K), and option prices.

As we expected, in Figure 5.5 the put option price for NDI (P ) increases as the

strike price increases. However, the call option price for NDI (C) increases as the

strike price decreases, see Figure 5.4. Figure 5.7 depicts the implied volatility surface

against the time to maturity (T ) and moneyness (M = S/K), where S is the stock

price. The observed volatility surface has an inverted volatility smile which is usually

seen in periods of high market stress. Options with lower strike prices have higher

implied volatilities compared to those with higher strike prices. The highest implied

volatilities of options are observed in (1.2, 1.4) of moneyness. The implied volatilities

tend to converge to a constant as the time to maturity converges to 60.

Figure 5.4: The call option prices (5.7) for the Natural Disasters Index (NDI) at timet for a given strike price K using a GARCH(1,1) model with generalized hyperbolicinnovations.

51


Figure 5.5: The put option prices (5.8) for the Natural Disasters Index (NDI) at timet for a given strike price K using a GARCH(1,1) model with generalized hyperbolicinnovations.

Figure 5.6: The call and put option prices for the Natural Disasters Index (NDI)at time t for a given strike price K using a GARCH(1,1) model with generalizedhyperbolic innovations.

52


Figure 5.7: The Natural Disasters Index (NDI) implied volatilities against time tomaturity (T ) and moneyness (M = S/K, where S and K the stock and strike prices,respectively) using a GARCH(1,1) model with generalized hyperbolic innovations.

5.3 NDI Risk Budgets

The risk budgets help investors as they provide the risk contributions of each

component in the portfolio to the aggregate portfolio risk. To accomplish this, an

investor should determine the relationships among various factors. Then, the investor

can envision the amount of risk exposure (as partly) depending on the behavior of

each component position. The primary strategies of assessing the center risk and

tail risk contributions are portfolio standard deviation (Std), Value at Risk (VaR),

and Expected Tail Loss (ETL) budgets. Some recent research applied Std and VaR

for portfolio risk budgeting 2 and ETL budgets are used in [120]. As Std and ETL

are coherent risk measures, we use them as the investment strategies for our equal-

weighted portfolio.

We delineate the marginal risk and risk contribution of each asset in the portfolio.

We define a risk measure, R(.), on the portfolio weight vector, w = (w1, w2, ..., wn)

where wi = 1n

(R(w) : Rn → R). Then, the marginal contribution to risk (MCTR)

2See [116], [117], [118], and [119].

53


of the ith asset to the total portfolio risk is

MCTRi(w) = wi∂R(w)

∂wi. (5.9)

The MCTR of the kth subset is

MCTRMk(w) =

∑i∈Mk

MCTRi(w), (5.10)

where Mk ⊆ {1, 2, ..., n} denote s subsets of portfolio assets. The percent contribu-

tion to risk (PCTR) of the ith asset to the total portfolio risk is

PCTRi(w) =MCTRi(w)∑ni=1 MCTRi(w)

. (5.11)

Since a large number of observations are involved in our analysis, we use a rolling-

method for risk budgeting. We use the first 400 data (biweekly loss returns) at each

window as in-sample-data and the last 400 data as out-of-sample data. The results

of the rolling method for finding risk contributions across the time are depicted in

Figures 5.8-5.10.

We calculate Std and ETL for risk contributions in our portfolio. Table 5.1 reports

the estimated risk allocations within the equal-weighted portfolio. According to the

results, the main center risk contributors are tornado, tropical storm, flood, ice

storm, and flash flood. However, flash flood, flood, and wildfire are the main tail risk

contributors at 95% level. Thus, flash flood and flood are the main risk contributors

in our portfolio.

5.4 Evaluating the Impact of Climate Extreme Indicators on NDI Performance

In finance, stress testing is an analysis intended to determine the strength of a

financial instrument and its resilience to the economic crisis. Stress testing is a form

of scenario analysis used by regulators to investigate the robustness of a financial

instrument is in inevitable crashes. In risk management, this helps to determine

portfolio risks and serves as a tool for hedging strategies required to mitigate against

54


Figure 5.8: The percent contribution to risk (PCTR) of the expected tail loss (ETL)risk budgets for the Natural Disasters Index (NDI) at 95% level. The legend depictsthe severe weather events in ascending order of their PCTR of ETL risk budgets at95% level.

potential losses.

We assess the performance of the NDI via stress testing using monthly maximum

temperature (Max Temp) and the Palmer Drought Severity Index (PDSI) as stressors

(refer to section 5.1). Instead of working with each factor, we use the first differences

of them as returns that yield stationary time series, see Figure 5.3.

The two series of returns inherit serial correlation and dependence according to

the results of the Ljung-Box test results (p-values are less than 0.05). Thus, to

55


Figure 5.9: The percent contribution to risk (PCTR) of the expected tail loss (ETL)risk budgets for the Natural Disasters Index (NDI) at 99% level. The legend depictsthe severe weather events in ascending order of their PCTR of ETL risk budgets at99% level.

capture linear and nonlinear dependencies in data sets, we put the series through the

ARMA(1,1)-GARCH(1,1) filter with Student-t innovations. Then, we consider the

sample innovations obtained from the aforementioned filter for our analysis.

We fit bivariate NIG models to the joint distributions of independent and identi-

cally distributed standardized residuals of each factor and the NDI (Total Loss): Max

Temp vs NDI and PSDI vs NDI. Then, we simulate 10,000 values from the models of

factors to perform the scenario analysis and to compute the systemic risk measures.

56


Figure 5.10: The percent contribution to risk (PCTR) of the standard deviation(Std) risk budgets for the Natural Disasters Index (NDI). The legend depicts thesevere weather events in ascending order of their PCTR of Std risk budgets.

Figure 5.11 shows the fitted contour plots from each model, overlaid with the 10,000

simulated values. The empirical correlation coefficients based on the observed data

suggest a weak positive relationship between the factors and the NDI (R ' 0.155).

There are various measures of systemic risk used to assess the impact of negative

events on the stress factors. Adrian and Brunnermeier [121] proposed Conditional

Value at Risk (CoVaR): the change in the value at risk of the financial system

conditional on an institution being under distress relative to its median state. [122]

improved the definition of financial distress from an institution being exactly at its

57


Figure 5.11: The generated joint densities of the returns of monthly maximumtemperature (Max Temp) and the Natural Disasters Index (NDI), and the PalmerDrought Severity Index (PDSI) and the NDI (right panel) using the fitted bivari-ate NIG models of the joint distributions of independent and identically distributedstandardized residuals. The figures depict the simulated values and the contour plotsof the joint densities.

VaR to being at most at its VaR (CoVaR). As the CoVaR is a coherent risk measure

[123], changing VaR to CoVaR allows us to consider more severe distress events, back-

test CoVaR, and improve its monotonicity concerning the dependence parameter.

The definition of CoVaR by [122] was based on the conditional distribution of a

random variable Y given a stress event for a random variable X. [124] defined an

alternative CoVaR notion in terms of copulas. They showed that conditioning on

X ≤ V aRα(X) improves the response to dependence between X and Y compared to

conditioning on X = V aRα(X). Therefore, we use the variant of CoVaR developed

by Mainik and Schaanning [124] for our study.

We define the distributions of Y and X are given by FY and FX , respectively, and

FY |X is the conditional distribution of Y given X. Then, CoVaR at level q, CoVaRq

58


(or ξq), is defined as

ξq := CoVaRq := F−1Y |X≤F−1

X (q)(q) = VaRq (Y |X ≤ VaRq(X)) , (5.12)

where VaRq (X) denotes the VaR of X at level q, which is same as the qth quantile

of X (F−1X (q)). CoVaR for the closely associated Expected Shortfall (ES) is defined

as the tail mean beyond VaR [124]:

CoESq := E (Y |Y ≤ ξq, X ≤ VaRq(X)) . (5.13)

Furthermore, Biglova et al. [125] has proposed

CoETLq := E (Y |Y ≤ VaRq(Y ), X ≤ VaRq(X)) . (5.14)

Table 5.2 reports the left-tail systemic risk measures on the NDI at different levels

based on stressing the factors (Max Temp and PDSI). At 5% and 10% stress levels,

stress on Max Temp seems to have a marginally more meaningful impact on the NDI

than the stress on PDSI. However, at the highest stress level (1%), the results show

stress on Max Temp has a greater significant impact on the NDI compared to that

of PDSI.

59


Table 5.1: Percent contribution to risk for standard deviation (Std) and expectedtail loss (ETL) (at 95% and 99% levels) risk budgets for the Natural Disasters Index(NDI)

Natural Disaster Std ETL(95) ETL(99)

Marine Lightning 0.01 0.02 0.21Marine Dense Fog 0.02 0.03 0.32Tornado 0.49 1.13 4.49Blizzard 0.49 1.96 3.53Dense Smoke 0.57 0.94 0.33Volcanic Ash 0.68 1.10 0.36Marine Hurricane Typhoon 0.78 1.24 0.61Sleet 0.79 1.24 0.54Marine Hail 0.80 1.16 0.37Winter Storm 1.05 1.92 3.40Marine Strong Wind 1.08 1.57 0.62Rip Current 1.21 1.55 0.55Funnel Cloud 1.31 1.64 0.40Seiche 1.32 1.85 0.78High Surf 1.33 2.11 3.05Avalanche 1.44 1.85 1.15Dust Devil 1.51 1.82 0.75Heavy Snow 1.58 2.86 2.59Hail 1.58 0.99 3.36Thunderstorm Wind 1.63 0.95 2.96Ice Storm 1.64 1.89 4.12Freezing Fog 1.66 2.25 0.67Dust Storm 1.66 1.45 1.72Waterspout 1.68 2.04 0.88Strong Wind 1.88 1.85 2.61Marine Thunderstorm Wind 1.89 1.54 1.32Marine High Wind 2.08 2.26 0.92Excessive Heat 2.11 2.51 0.62Heat 2.11 2.20 0.95Dense Fog 2.19 1.54 1.81Extreme Cold Wind Chill 2.27 2.38 1.06Lakeshore Flood 2.33 2.98 0.66Lightning 2.34 2.54 2.40Winter Weather 2.49 2.85 1.15Tropical Depression 2.58 2.92 1.06Storm Surge Tide 2.58 2.07 5.94Frost Freeze 2.61 3.25 0.92Lake Effect Snow 2.63 3.04 1.24Tsunami 2.63 3.49 0.67Cold Wind Chill 2.77 2.83 0.63High Wind 2.83 1.83 4.90Heavy Rain 2.84 1.24 3.39Coastal Flood 2.95 4.52 3.21Debris Flow 3.15 2.93 1.88Hurricane Typhoon 3.15 1.94 5.63Drought 3.40 2.14 2.79Tropical Storm 3.88 2.62 4.85Wildfire 4.14 3.50 3.15Flood 4.87 2.64 4.26Flash Flood 4.96 3.08 4.21

60


Table 5.2: The left-tail systemic risk measures (CoVaR, CoES, and CoETL) on theNatural Disasters Index (NDI) at different stress levels based on stressing the factorsmonthly maximum temperature (Max Temp) and the Palmer Drought Severity Index(PDSI).

Climate Extreme Indicator Stress LevelsSystemic Risk Measure

CoVaR CoES CoETL

Maximum Temperature10% -1.87 -2.53 -2.035% -2.47 -3.32 -2.471% -4.57 -5.31 -3.76

Palmer Drought Severity Index10% -1.35 -2.37 -1.555% -2.22 -4.00 -2.161% -7.86 -17.63 -4.27

61


CHAPTER 6

GLOBAL INDEX ON FINANCIAL LOSSES DUE TO CRIMES IN THE

UNITED STATES

This chapter examines the impact of crime on insurance policies in the United

States by analyzing the financial losses associated with various types of crimes as

reported by the Federal Bureau of Investigation. Taking all the available data in

uniform crime reports and internet crime reports, we model the financial losses gen-

erated by property crimes and cybercrimes. Then, we propose a portfolio based on

the economic damages due to crimes and validate it via value at risk backtesting

models. The objective of this research is to employ financial methods to construct a

reliable and dynamic aggregate index based on economic factors to provide a basis

for the securitization of insurance risk from crimes.

We hedge the investments underlying the portfolio using two methods to assess

the level of future systemic risk. First, we issue marketable financial contracts in our

portfolio, the European call and put options, to help the investors strategize buying

call options and selling put options in our portfolio based on their desired risk level.

Second, we hedge the investment by diversifying risk to each type of crime based

on tail risk and center risk measures. According to the estimated risk budgets, real

estate, ransomware, and government impersonation mainly contribute to the risk

in our portfolio. These findings will help investors to envision the amount of risk

exposure with financial planning on our portfolio.

We evaluate the performance of our index with respect to economic factors to

determine the strength of our index by investigating its resilience to the economic

crisis. The unemployment rate potentially demonstrates a high systemic risk on the

portfolio compared to the economic factors used in this study. Therefore, the key

findings are intended to provide a basis for the securitization of insurance risk from

crime. This will help insurers gauge investment risk in our portfolio based on their

desired risk level and hedge strategies for potential losses due to economic crashes.

62


6.1 Financial Losses due to Crimes in the United States

In this section, we propose an index based on the financial losses caused by various

types of crimes reported between 2001 and 2019 as a proxy to assess the level of

future systemic risk caused by crimes. First, we describe the crime data used in

this study in section 2.2. Then, we model the multivariate time series of financial

losses due to crimes in section 6.1.1. As a result, we propose a portfolio using the

annual cumulative financial losses due to property crimes and cybercrimes. Finally,

we perform backtesting for our index using value at risk models in section 6.1.2.

6.1.1 Modeling the Multivariate Time Series of Financial Losses due to Crimes

In this section, we model the financial losses due to the 32 types of crimes described

in section 6.1. In each type of crime, we transform the series of financial losses to a

stationary time series by taking the log returns:

R(i)t = logL

(i)t − logL

(i)t−1; i = 1, ..32, t = 0, .., T (6.1)

where L(i)t denotes the financial loss due to ith crime type at time t.

Then, we fit the Normal Inverse Gaussian (NIG) distribution to each log return r(i)t

series and estimate parameters using the maximum likelihood method. As a result,

we have NIG Levy processes for each type of crime. Moreover, since NIG has an

exponential form at the moment generating function, we use these dynamic returns

for option pricing in section 6.2. Then, for each NIG Levy process, we generate

10,000 scenarios to obtain independent and identically distributed data for returns.

6.1.2 Backtesting the Portfolio

In this study, we propose a portfolio based on the annual cumulative financial

losses due to all types of crimes described in section 2.2. Then, we convert this

portfolio to a stationary time series by taking their log returns. We denote rt as the

log return of the index at time t where µt is drift and σt is volatility:

rt = µt + at, t = 0, . . . , T. (6.2)

63


Then, we model the log-returns using the ARMA(1,1)-GARCH(1,1) filter to elim-

inate the serial dependence. In particular, we use ARMA(1,1) [126] to model the

drift (µt)

µt = φ0 + φ1rt−1 + θ1at−1 (6.3)

and GARCH(1,1) [127] to model the volatility (σt)

σt =atεt

σ2t = α0 + α1a

2t−1 + β1σ

2t−1

(6.4)

where φ0 and α0 are constants and φ1, θ1, α1 and β1 are parameters to be estimated.

Moreover, the sample innovations, εt, follows an arbitrary distribution with zero

mean and unit variance.

In particular, we assume Student’s t and NIG for the distributions of sample

innovations, εt. Then, we examine the performance of these two filters (ARMA(1,1)-

GARCH(1,1) with Student’s t innovations and ARMA(1,1)-GARCH(1,1) with NIG

innovations) via backtesting. Furthermore, we utilize the better model obtained in

this section to implement option pricing to our portfolio in section 6.2.

We backtest the models using Value at Risk (VaR) measures [128]. In VaR back-

testing [129], we compare the actual returns with the corresponding VaR models.

The level of difference between them helps to identify whether the VaR model is

underestimating or overestimating the risk. Moreover, if the total failures are less

than expected, then the model is considered to overestimate the VaR, and if the

actual failures are greater than expected, the model underestimates VaR.

To perform backtesting, we use the residuals of filters between 2002 and 2015 to

train the model. Then, the test window starts in 2016 and runs through the end of

the sample (2019). We perform the backtesting for the out-of-sample data at the

quantile levels of 0.01, 0.05, 0.25, 0.50, 0.75, 0.95, and 0.99. For the α quantile level,

we define VaR as follows:

VaRα(x) = − inf{x | F (x) > α, x ∈ R} (6.5)

64


Table 6.1: VaR Backtesting Results for ARMA(1,1)-GARCH(1,1) with Student’s tand NIG innovations.

Innovation VaR LevelTest Results

Traffic Light Binomial PoF CCI

Student’s t0.01 green reject reject accept0.05 green accept accept accept0.25 green reject reject accept0.50 green accept accept accept0.75 green accept accept accept0.95 green accept accept accept0.99 yellow accept accept accept

NIG0.01 green reject reject accept0.05 green reject reject accept0.25 green reject reject accept0.50 green accept accept accept0.75 yellow reject reject accept0.95 red reject reject accept0.99 red reject reject accept

where F (x) is the cumulative density function of the returns.

Table 6.1 provides the results for VaR backtesting on ARMA(1,1)-GARCH(1,1)

with Student’s t and NIG filters. First, we perform the Conditional Coverage Inde-

pendence (CCI) to test for independence [130]. According to Table 6.1, both filters

show independence on consecutive returns.

Then, we perform traffic light, binomial test, and proportion of failures (PoF)

tests as frequency tests. ARMA(1,1)-GARCH(1,1) with Student’s t model is gener-

ally acceptable in the frequency tests at most of the levels. However, ARMA(1,1)-

GARCH(1,1) with NIG model fails at all the levels except the 0.5 level.

In conclusion, ARMA(1,1)-GARCH(1,1) with Student’s t innovations outperforms

the ARMA(1,1)-GARCH(1,1) with NIG model in backtesting. Hence, we utilize

ARMA(1,1)-GARCH(1,1) with Student’s t to implement option pricing in section

6.2.

65


6.2 Option Prices for the Crime Portfolio

An option is a contract between two parties that gives one party the right, but

not the obligation, to buy or sell the underlying asset at a prespecified price within a

specific time. We provide fair prices of call and put options in our portfolio in section

6.2.2 based on the pricing model defined in section 6.2.1. Then, we investigate the

implied volatilities of our index using the Black-Scholes and Merton model. Ulti-

mately, the findings of this section are intended to help investors strategize buying

call options and selling put options of our portfolio based on their desired risk level

and predicted volatilities.

6.2.1 Defining a Model for Pricing Options

The theoretical value of an option estimates its fair value based on strike price and

time to maturity1. In pricing options, the conventional Black-Scholes Model2 assumes

the price of a financial asset follows a stochastic process based on a Brownian motion

with a normal distribution assumption. Since the asset returns are heavy-tailed in

practice, the extreme variations in prices cannot be well captured using a normal

distribution. With non-normality assumption, we implement a Levy processes for

asset returns. This provides better estimates for prices since the non-marginal vari-

ations are more likely to happen as a consequence of fat-tailed distribution-based

processes3.

Among Levy processes, the NIG process [138] is widely used for pricing options

as it allows for wider modeling of skewness and kurtosis than the Brownian motion

does. Thus, it enables us to estimate consistent option prices with different strikes

and maturities using a single set of parameters. We use the NIG process to price

options for the crime portfolio based on the estimated parameters for the returns

obtained using the maximum likelihood estimation given in Table 6.2.

The NIG process is a Brownian motion where the time change process follows

an Inverse Gaussian (IG) distribution, i.e., the NIG process is a Brownian motion

1See [103, 104, 105, 106, 107, 108, 109] and [110]2See [131] and [132]3See [133, 134, 135, 136], and [137]

66


Table 6.2: The estimated parameters of the fitted the NIG process to the Crimeportfolio log-returns

Parameters µ α β δ

Estimates -0.0014 0.4826 0.0006 0.6553

subordinated to an IG process. We define the NIG process (Xt) as a Brownian

motion (Bt) with drift (µ) and volatility (σ) as follows

Xt = µt+ σBt, t ≥ 0, µ ∈ R, σ > 0. (6.6)

At t = 1, we denote the process as X1 ∼ NIG (µ, α, β, δ) with parameters µ, α, β, δ ∈R such that α2 > β2 and the density given by

fX1 (x) =

αδK1

(α√δ2 + (x− µ)2

)π√δ2 + (x− µ)2

exp(δ√α2 − β2 + β (x− µ)

), x ∈ R. (6.7)

The characteristic function of the NIG process is derived using ϕXt (t) = E(eitXt

), t ∈

R and is given by

ϕXt (t) = exp

(iµt+ δ

(√α2 − β2 −

√α2 − (β + it)2

)). (6.8)

For pricing financial derivatives, we search for risk-neutral probability (Q) known

as Equivalent Martingale Measure (EMM). The current value of an asset is exactly

equal to the discounted expectation of the asset at the risk-free rate under Q. In

particular, we use Mean-Correcting Martingale Measure (MCMM) for Q as it is

sufficiently flexible for calibrating market data. In MCMM, the price dynamics of

the price process (St) on Q is given by

S(Q)t =

e−rtStMXt(1)

, t ≥ 0 (6.9)

67


where MXt() is the moment generating function of Xt and r is the risk free rate. We

model the risk-neutral log stock-price process for a given option pricing formula and

our market model on Q as follows

ϕlnS

(Q)t

(v) = Siv0 exp{[iv(r − lnϕX1(−i)) + lnϕX1(v)]t} (6.10)

where ϕXt(v) = E(eivXt) is the characteristic function of Xt in Eq. (6.8).

We obtain the EMM using MCMM as the pricing formula is arbitrage-free for the

European call option pricing formula under Q. First, we estimate all the parameters

involved in the process and add the drift term, Xnewt = Xold

t + mt, in such a way

that the discounted stock-price process becomes a martingale.

We define the price of a European call contract (C) with underlying risky assets

at t = 0 as

C(S0, r,K, T ) = e−rT EQ max(S(Q)T −K, 0), K, T > 0, (6.11)

for the given price process (St), time to maturity (T ), and strike price (K).

When the characteristic function of the risk-neutral log stock-price process is

known, Carr and Madan’s study [139] derives the pricing method for the Euro-

pean option valuation using the fast Fourier transform. Following that, we price the

European options contract with an underlying risky asset using the characteristics

function, Eq (6.8), and fast Fourier transform to convert the generalized Fourier

transform of the call price. For any positive constant a such that EQ(S(Q)T )

a< ∞

exists, we define the call price as

C(S0, r,K, T ) =e−rT−ak

π

∫ ∞0

e−ivkϕlnS

(Q)T

(v − i(a+ 1))

a(a+ 1)− v2 + 2i(a+ 1)vdv, a > 0, (6.12)

where k = lnK and ϕlnS

(Q)t

(v) is the characteristic function of the log-price process

under Q. By utilizing call option prices in the put–call parity formula, we calculate

the price of a put option in the crime portfolio.

68


Figure 6.1: Call option prices against time to maturity (T , in days) and strike price(K, based on S0 = 100).

6.2.2 Issuing the European Option Prices for the Crime Portfolio

In this section, we calculate the European call and put option prices for our port-

folio using the pricing model, Eq (6.12), introduced in section 6.2.1. We provide

European option prices by fixing S0 to 100 in Eq (6.12), i.e., the price of the crime

portfolio at time zero is 100 units, and the time to maturity is in days. Later, we

provide the implied volatilities of the portfolio based on the volatilities of call and

put option prices.

First, we demonstrate the relationship between call option prices, strike price (K),

and time to maturity (T ) in Figure 6.1. These calculated prices help the investors

to strategize buying the stocks in our portfolio at a predefined price (K) within a

specific time frame (T ). Second, we show put option prices in Figure 6.2 to provide

selling prices of the shares in our index. The prices of our options validate the fact

that option prices decrease as the time to maturity increases for a given strike price.

Third, we determine the implied volatilities of our portfolio using the Black-Scholes

and Merton model. This provides the expected volatility of our portfolio over the

life of the option (T ). Figure 6.3 is the implied volatility surface with respect to

moneyness (M) and time to maturity (T ). In particular, we calculate moneyness

as the ratio of the strike price (K) and stock price (S), i.e., M = K/S. Then, the

69


Figure 6.2: Put option prices against time to maturity (T , in days) and strike price(K, based on S0 = 100).

volatilities for call and put option prices are shown in the regions of the surface with

M < 1 and M > 1, respectively. The volatility surface demonstrates a volatility

smile which is usually seen in the stock market.

Figure 6.3 illustrates that implied volatility increases when the moneyness is fur-

ther out of the money or in the money, compared to at the money (M = 1). In this

case, volatility seems to be low, with a range of 0.8 and 1.2 in moneyness, compared

to the other regions in the implied volatility surface, i.e., the options with higher

premiums result in high implied volatilities. These findings help investors strategize

buying call options and selling put options in our portfolio based on their desired

risk level and predicted volatilities.

6.3 Risk Budgets for the Crime Portfolio

The investors intend to optimize portfolio performance while maintaining their

desired risk tolerance level [140]. In this section, we provide a rationale for investors

to determine the degree of variability in our portfolio. Therefore, we provide the risk

contribution related to each type of crime in section 6.3.2 using the risk measures

70


Figure 6.3: Implied volatility surface against time to maturity (T , in days) andmoneyness (M = K/S, the ratio of strike price, K, and stock price, S).

defined in section 6.3.1. As a result, we present the main risk contributors and

risk diversifiers in our portfolio. Ultimately, these risk budgets (the estimated risk

allocations) will potentially help investors with their financial planning (maximizing

the returns).

6.3.1 Defining Tail and Center Risk Measures

This section defines the risk measures that we use for assessing risk allocations in

section 6.3.2. We determine the tail risk contributors and center risk contributors

using expected tail loss and volatility risk measures, respectively [116, 120].

We use Conditional Value at Risk (CoVaR) [121, 122] for finding the tail risk

contributors in our portfolio at levels of 95% and 99%. We define the tail risk

contribution of the ith asset at α level as follows:

TRi(α) = CoVaR(i)α =

1

α

∫ α

0

V aR(i)γ (x) dγ. (6.13)

In order to find center risk contributors, we measure the volatility of asset prices

using standard deviation. Since we utilize an equally-weighted portfolio, we denote

the weight vector for 32 types of crimes as w = (w1, · · · , w32) where wi = 132

. We

define the volatility risk measure, R(w), using the covariance matrix, Σ, of asset

returns (6.1) [141]:

71


R(w) =√

w′Σw. (6.14)

Then, the center risk contribution of the ith asset is given by

CRi(w) = wi∂R(w)

∂wi. (6.15)

Having outlined the risk measures, section 6.3.2 utilizes tail and center risk con-

tributions to find the risk allocation for each type of crime defined in section 2.2.

6.3.2 Determining the Risk Budgets for the Crime Portfolio

This section assesses the risk attributed to each type of crime using the risk mea-

sures defined in section 6.3.1. First, we calculate the tail risk allocations (TR) using

Eq (6.13) to find the main tail risk contributors in our portfolio. Then, we inves-

tigate the main center risk contributors in our index using center risk allocations

(CR) computed using Eq (6.15). Finally, taking the main tail risk and center risk

contributors into account, we find the main risk contributors of our portfolio.

In Table 6.3, we provide the center risk (CR) and tail risk (TR) allocations for

each type of crime. We find the risk diversifiers in the portfolio using the negative

risk allocations shown in Table 6.3. With significantly low center and tail risk diver-

sifications, Misrepresentation and Social Media seems to be the potential main risk

diversifiers in our portfolio.

We consider the positive values outlined in Table 6.3 to identify the main risk

contributors in our portfolio. We find the main tail risk contributors using the

positive tail risk estimates at levels of 95% and 99%. At the 95% level, TR(95%),

Real Estate, Ransomware, and Government Impersonation provide a relatively higher

tail risk than the other factors. However, Real Estate, Ransomware, and Identity

Theft seem to be the main tail risk contributors at the 99% level, TR(99%).

We determine the main center risk contributors in our portfolio using the positive

center risk estimates, CR, illustrated in Table 6.3. Since Real Estate, Ransomware,

and Government Impersonation demonstrate high volatility compared to the other

types of crimes, they seem to be the main center risk contributors in our portfolio.

72


Table 6.3: The percentages of center risk (CR) and tail risk (TR) (at levels of 95%and 99%) budgets for the portfolio on crimes.

Crime Type %TR(95) %TR(99) %CR

Real Estate 13.62 12.93 11.29Ransomware 10.35 11.40 8.48Government Impersonation 9.48 8.80 8.25Identity Theft 7.96 10.23 6.74Extortion 7.72 6.90 7.19Lottery 7.09 7.37 6.26Confidence Fraud 5.56 6.64 5.48Investment 5.31 7.11 4.90Crimes Against Children 5.24 3.95 5.55Personal Data Breach 3.69 4.29 3.24Credit Card Fraud 3.58 3.16 3.96BEC/EAC 3.27 3.29 3.00Non-Payment 2.56 4.38 2.10IPR Copyright 2.02 2.04 1.95Gambling 1.97 2.55 1.40Robbery 1.80 0.67 6.77Phishing 1.48 0.97 2.01Civil Matter 1.30 -0.57 2.89Denial Of Service 1.02 -0.23 2.74Motor Vehicle Theft 1.01 1.30 0.73Check Fraud 0.98 2.19 -0.51Advanced Fee 0.75 0.43 1.04Harassment 0.74 0.10 1.14Corporate Data Breach 0.70 0.03 0.90Larceny Theft 0.50 0.56 0.39Terrorism 0.36 -0.20 2.06Burglary 0.30 0.21 0.44Employment 0.22 0.09 0.29Charity 0.17 0.18 0.54Overpayment -0.04 -0.23 0.10Social Media -0.16 -0.17 -0.31Misrepresentation -0.55 -0.41 -1.00

73


As Real Estate and Ransomware are both main tail risk and center risk contribu-

tors, they are the potential main risk contributors in our portfolio. These estimated

risk budgets and the main risk contributors will help investors to envision the amount

of risk exposure with financial planning on our portfolio.

6.4 Performance of the Crime Portfolio for Economic Crisis

We evaluate the performance of our portfolio using economic factors related to

low income as they are known to be major root causes of crime. To investigate

the robustness of the crime portfolio for inevitable economic crashes, we perform

stress testing in section 6.4.2 based on the systemic risk measures defined in section

6.4.1 The findings of this section are intended to help to determine portfolio risks

and serves as a tool for hedging strategies required to mitigate inevitable economic

crashes.

6.4.1 Defining Systemic Risk Measures

In this section, we define the systemic risk measures used for stress testing the

portfolio on crimes. We define three derived risk measures based on VaR (6.5)

denoting Y as the portfolio and X as a stress factor [94].

CoVaR is a coherent measure of tail risk in an investment portfolio. In our study,

we use a variant of CoVaR defined in terms of copulas [124]. Using the condition X ≤VaRα(X) rather than the traditional CoVaR condition, X = VaRα(X), improves the

response to dependence between X and Y . We define CoVaR at level α as

CoVaRα = VaRα (Y | X ≤ VaRα(X)) . (6.16)

The CoVaR for the closely associated expected shortfall is defined as the tail

mean beyond VaR [124]. Furthermore, we use an extension of CoVaR denoted as

Conditional Expected Shortfall (CoES). Then, we define CoES at level α as follows:

CoESα = E (Y | Y ≤ CoVaRα, X ≤ VaRα(X)) . (6.17)

Conditional Expected Tail Loss (CoETL) [125] is the average of the portfolio losses

74


when all the assets are in distress. CoETL is an appropriate risk measure to quantify

the portfolio downside risk in the presence of systemic risk. We denote CoETL at

level α as

CoETLα = E (Y | Y ≤ VaRα(Y ), X ≤ VaRα(X)) . (6.18)

We quantify the market risk of our portfolio on crime using these systemic risk

measures in section 6.4.2.

6.4.2 Evaluating the Performance of the Crime Portfolio for Economic Crisis

In this section, we evaluate how well our portfolio would perform with economic

factors related to low income. In particular, we test the impact of the Unemployment

Rate, Poverty Rate, and Median Household Income on our index. We quantify the

potential impact of these economic factors on our index using the systemic risk mea-

sures defined in section 6.4.1. Since the stress testing results indicate the investment

risk in our portfolio, the investors can utilize the outcomes to hedge strategies for

forthcoming economic crashes.

Based on backtesing results in section 6.1.2, we use the ARMA(1,1)-GARCH(1,1)

model with Student-t innovations for the log returns of our portfolio. Also, we apply

this filter to log returns of the economic factors to eliminate inherent linear and

nonlinear dependencies. Then, we fit bivariate NIG models to the joint distributions

of independent and identically distributed standardized residuals of each economic

factor and our portfolio on crime. Using these bivariate models, we generate 10,000

simulations for each joint density to perform a scenario analysis. In Table 6.4, we

provide the empirical correlation coefficients of each simulated joint density with the

corresponding economic factor. This table demonstrates weak correlations between

the economic factors and the portfolio.

We utilize the simulated joint densities to compute the systemic risk measures.

In Table 6.5, we provide the left tail systemic risk measures (CoVaR, CoES, and

CoETL) on the portfolio at stress levels of 10%, 5%, and 1% on the economic fac-

tors - Unemployment Rate, Poverty Rate, Household Income. At each level, the

75


Table 6.4: The empirical correlation coefficients of the joint densities of each economicfactors and the crime portfolio

Economic Factor Correlation Coefficient

Unemployment Rate 0.11Poverty Rate -0.24Household Income 0.17

Table 6.5: The left tail systemic risk measures (CoVaR, CoES, and CoETL) on theportfolio at stress levels of 10%, 5%, and 1% on the following economic factors -Unemployment Rate, Poverty Rate, Household Income

Economic Factors Stress LevelsLeft Tail Risk Measures

CoVaR CoES CoETL

Unemployment Rate10% -5.88 -8.85 -5.235% -9.01 -12.27 -7.061% -14.82 -16.32 -11.50

Poverty Rate10% -0.67 -1.29 -0.925% -1.31 -2.07 -1.241% -2.42 -3.15 -2.05

Household Income10% -1.45 -2.14 -1.205% -2.30 -2.91 -1.661% -3.71 -3.86 -2.46

Unemployment Rate provides the highest values for the three systemic risk mea-

sures compared to the other economic factors. Thus, among all the stressors, the

Unemployment Rate demonstrates a significantly high impact on the index. Poverty

Rate potentially has a low impact on the index according to the results of all three

systemic risk measures at all stress levels.

In conclusion, the Unemployment Rate potentially has a high impact on the fi-

nancial losses due to crimes in the United States. Hence, these findings will help

investors gauge the market risk of our portfolio for hedging strategies to alleviate

potential losses due to economic crashes.

76


CHAPTER 7

DISCUSSION & CONCLUSION

Although over 69,000 tornadoes are recorded in NOAA Storm Data [3], we con-

sidered approximately 40,000 tornado events since not all of the tornado events had

measurements for all tornado variables (length, width, and losses). In section 4.3.2,

we analyzed the natural logarithm of the monthly property losses which resulted in

constant variance over the time series. In this study, we focused on states in Tornado

Alley on an ad hoc basis as they represent a disproportionately high frequency of

tornadoes [30].

Tornadoes are rated in [3] using an intensity-based scale known as the EF-scale

[1]. Moreover, the property loss caused by a tornado unequivocally depends on the

location and value of the real estate. For example, the damage costs due to a tornado

in an urban area might be significantly high even if the tornado has a low EF-Scale.

We implemented the MU algorithm for NMF to classify tornado data in [3], and

they exhibited a conspicuous classification underpinning property losses. We have

introduced a novel scale for tornado damages based on the property losses attributed

to them, the TPL-Scale (4.5).

Using simple linear regression analysis, we found the relationship between the

property loss and area affected by a tornado: Loss ∼√Area =

√Length ·Width

(i.e., Loss approximates to the geometric mean of path length and width). Then, we

employed a bivariate Gaussian copula approach to explore the underlying non-linear

dependence between ln(Area) and ln(Loss). However, the dependence coefficients

(copula parameters) between ln(Area) and ln(Loss) strongly vary with time and

geographical location. Thus, we trained the LSTM and shallow neural networks to

capture the long-term dependencies in the data set. Finally, we predicted monthly

property losses between 2020 and 2025 using the LSTM, and the expected volume

of property losses would amount to just under eight billion dollars over the next five

years (by 2025).

The approach of compensating for the property losses caused by tornadoes is a

two-fold process in the United States. For catastrophic tornadoes, the government

77


coverage programs such as FEMA distribute fundings to local governments. For typ-

ical tornadoes, private insurance companies cover the tornado damage costs claimed

by their clients. We recovered the existence of these two risk assessment strate-

gies by learning a statistical manifold on probability distributions of property losses.

The smooth connection between the points in the manifold illustrates that there is no

single good enough distribution to describe all available property losses in [3]. There-

fore, we defined a measure of curvature on the manifold to estimate such volatilities

in property losses with respect to small changes in tornado path lengths and path

widths. However, there are few extremities in data that we failed to interpret using

the statistical manifold.

In this dissertation, chapter 3 and 4 addressed the risk assessment strategies of

tornadoes reported in the NOAA Storm Data [3]. We considered reported tornado

data as separate events and investigated the risk in terms of the property losses. Due

to the lack of reported occurrences in [3], the risk assessment methods discussed in

chapter 3 and 4 cannot be implemented for other types of natural disasters. There-

fore, we integrated the reported property losses in all 50 types of disasters in time

steps (in particular, two-week increments). Then, we utilized the temporal ordering

of financial losses for hedging the risk due to natural disasters in chapter 5.

As a result, we proposed the Natural Disasters Index, NDI (5.1), using the United

States as a model with property losses reported in NOAA Storm Data [3] between

1996 and 2018. In order to establish the NDI, we provided an evaluation framework

using three promising approaches: (1) option pricing, (2) risk budgeting, and (3)

stress testing.

We determined the fair values of the NDI options using a discrete-time GARCH

model with GH innovations and then simulated Monte Carlo averages to approxi-

mate call and put option prices (5.7),(5.8). The relationships among time to maturity,

strike price, and option prices help to construct and valuate insurance-type financial

instruments. Then, we disaggregated the cumulative risk attributed to natural dis-

asters to our equally-weighted portfolio (i.e., we investigated the risk contribution

of each type of natural disaster). The Std and ETL risk budgets for the NDI yield

that flood and flash flood are the main risk contributors in our portfolio. Finally, we

78


assessed the performance of the NDI via a stress testing analysis using Max Temp

and PDSI as stressors. We found the stress on Max Temp significantly impacts the

NDI compared to that of the PDSI at the highest stress level (1%).

The proposed NDI is an attempt to address a financial instrument for hedging the

intrinsic risk induced by the property losses caused by natural disasters in the United

States. The main objective of the NDI is to forecast the degree of future systemic

risk caused by natural disasters. This information could forewarn the insurers and

corporations allowing them to transfer insurance risk to capital market investors.

Hence the issuance of the NDI will conspicuously help to bridge the gap between

the capital and insurance markets. While the NDI is specifically constructed for the

United States, it could be modified to calculate the risk in other regions or countries

using a data set comparable to NOAA Storm Data [3].

Following the financial management principles in the NDI, we proposed construct-

ing a portfolio that outlines the financial impacts of various types of crimes in the

United States. In order to that, we modeled the financial losses of crimes reported

by the Federal Bureau of Investigation using the annual cumulative property losses

due to property crimes and cybercrimes. Then, we backtested the index using VaR

models at different levels to find a proper model for implementing in evaluation pro-

cesses. As a result, we utilized ARMA(1,1)-GARCH(1,1) with Student’s t model to

evaluate the crime portfolio.

We presented the use of our portfolio on crimes through option pricing, risk bud-

geting, and stress testing. First, we provided fair values for European call and put

option prices (Figure 6.1 and 6.2) and implied volatilities (Figure 6.3) for our portfo-

lio. Second, we found the risk attributed to each type of crime based on tail risk and

center risk measures. Third, we evaluated the performance of our index for the eco-

nomic crisis by implementing stress testing. According to the findings, in the United

States, the Unemployment Rate potentially has a higher impact on the financial

losses due to the crimes incorporated in this study compared to the Poverty Rate

and Median Household Income. The findings, estimated option prices, risk budgets,

and systemic risk outlined in the portfolio will help investors with financial planning

on our portfolio.

79


BIBLIOGRAPHY

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