MODELING CATASTROPHE
THE EXPECTED UTILITY HYPOTHESIS AND EVIDENCE FROM CALIFORNIA
EARTHQUAKE INSURANCE
A Thesis
Presented to the faculty of the Department of Economics
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF ARTS
in
Economics
by
Christopher John Newton
SUMMER
2013
© 2013
Christopher John Newton
ALL RIGHTS RESERVED
ii
MODELING CATASTROPHE
THE EXPECTED UTILITY HYPOTHESIS AND EVIDENCE FROM CALIFORNIA
EARTHQUAKE INSURANCE
A Thesis
by
Christopher John Newton
Approved by:
__________________________________, Committee Chair
Jonathan D. Kaplan, Ph.D.
__________________________________, Second Reader
Kristin Kiesel, Ph.D.
____________________________
Date
iii
Student: Christopher John Newton
I certify that this student has met the requirements for format contained in the University
format manual, and that this thesis is suitable for shelving in the Library and credit is to
be awarded for the thesis.
__________________________, Graduate Coordinator
Kristin Kiesel, Ph.D.
Department of Economics
iv
___________________
Date
Abstract
of
MODELING CATASTROPHE
THE EXPECTED UTILITY HYPOTHESIS AND EVIDENCE FROM CALIFORNIA
EARTHQUAKE INSURANCE
by
Christopher John Newton
Catastrophic events and the damages they cause have recently been paid more
attention due to their increased severity and frequency. For society to better prepare for
and ameliorate the consequences of these events, it is important to understand how people
make decisions when faced with choices involving disaster mitigation strategies. This
thesis examines the expected utility theory of insurance demand, insurance being a
widely-used disaster mitigation tool, and uses an empirical demand estimation to test the
predictions it makes concerning how the purchase of earthquake insurance varies with
price, income, and risk. Expected utility theory is one of the most widely used decision
theories of the past century, it can and has been used to model people’s choices under the
type of uncertainty catastrophic events create, and its predictive validity is still uncertain.
Three years of zip code level California Earthquake Authority (CEA) data are used
to estimate income, risk, and price elasticities for residential housing earthquake
insurance demand. In order to better deal with the disadvantages created by the time
v
invariant risk data used, several models are estimated to study the robustness of the
results across specification. A total of six models are estimated, including two pooled
models, two time fixed effects models, and two earthquake zone fixed effects models.
The first four models estimate elasticities for price, income, and risk. The two earthquake
zone fixed effects models estimate elasticities for price, and income.
The estimated elasticities are then compared to theoretical predictions of the
elasticities. Both the price and income elasticities are predicted to be either positive or
negative, the empirical results showing them negative and positive respectively. The
predicted risk elasticity is positive, and the empirical results agree with the prediction.
The empirical results find the signs of the elasticities of price, income, and risk robust
over specification, but the magnitudes change from elastic to inelastic across some of the
different models. The estimated risk elasticity changes significance depending on
specification. One concern with the analysis was in how risk was measured. Every effort
was made to characterize risk so inferences could be drawn from the analysis. In the
future, researchers should try to obtain risk data that changes over time and across entity
in order to more accurately test predictions about how risk plays a part in individual’s
insurance purchasing decisions.
_______________________, Committee Chair
Jonathan D. Kaplan
_______________________
Date
vi
ACKNOWLEDGEMENTS
First, I would like to thank Professors Jonathan Kaplan and Kristin Kiesel for
their guidance and support throughout the completion of this thesis.
Secondly, I would like to thank my family, the Berkeley Crew, Skunky Dunky,
and the Sacramento Crew. I would also like to thank those people close by and abroad I
hope to see soon, thank you.
vii
TABLE OF CONTENTS
Page
Acknowledgements .................................................................................................... vii
List of Tables ................................................................................................................ x
List of Figures ............................................................................................................. xi
Chapter
1. INTRODUCTION ………………………………………………………………... 1
2. LITERATURE REVIEW ....................................................................................... 5
2.1. Introduction ................................................................................................ 5
2.2. Expected Utility Theory............................................................................. 6
2.2.1. Theoretical Studies...................................................................... 6
2.2.2. Empirical Studies ........................................................................ 9
2.3. Demand Estimation Studies ..................................................................... 11
2.4. Theoretical and Empirical Studies ........................................................... 13
2.5. Summary .................................................................................................. 13
3. THEORY .............................................................................................................. 14
4. DATA ................................................................................................................... 19
4.1. Data Summaries and Variable Formulation ............................................. 19
4.2. Scatter Plots ............................................................................................. 28
4.3. Summary .................................................................................................. 32
viii
5. DEMAND ESTIMATION.................................................................................... 33
5.1. Estimation Models and Estimation Results ............................................. 33
5.2. Pooled Data Regression Model ................................................................ 34
5.3. Inclusion of Time Fixed Effects .............................................................. 35
5.4. Further Examination without Risk ........................................................... 38
5.5. Earthquake Zone Fixed Effects ................................................................ 40
5.5. Remarks ................................................................................................... 41
6. CONCLUSION ..................................................................................................... 43
References ................................................................................................................... 46
ix
LIST OF TABLES
Tables
Page
1.
Table 4.1. Summary Statistics for Insurance and Income Data………………. 22
2.
Table 4.2. Summary Statistics for Census Data………….………………...….. 24
3.
Table 4.3. Summary Statistics for Interpolated Census Data…………………. 24
4.
Table 4.4. Summary Statistics for Dependent and Independent Variables….... 27
5.
Table 5.1. Regression Results for Models 1 and 2……….…………………… 36
6.
Table 5.2. Regression Results for Models 3 and 4……………………….….... 38
7.
Table 5.3. Regression Results for Models 5 and 6..……………..……………. 42
x
LIST OF FIGURES
Figures
Page
1.
Figure 4.1. Scatter Plots for the 2005 Data……………….…………….….. 29
2.
Figure 4.2. Scatter Plots for the 2007 Data…………….…………….…….. 30
3.
Figure 4.3. Scatter Plots for the 2009 Data….………….……………….…. 31
4.
Figure 5.1. Estimated Zonal Effect for Model 6 By Income…….…………. 41
xi
1
Chapter 1
INTRODUCTION
The decision making processes of individuals faced with the uncertainty of low
probability-high impact events has long been studied by economists, management
scientists, and psychologists. Low probability, high consequence events can be defined
here as those events perceived both by the public and by experts as having a low
probability of occurrence. The study of this subject involves scientists employing a wide
spectrum of techniques and reporting an even broader spectrum of results. Today this
research has extra relevance due to its affiliation with modern catastrophes both natural
and man-made, garnering the continued interest of scientists, politicians, and the general
community.
Scientists use a range of tools, both theoretical and empirical, in their pursuit of
understanding people’s choices. One of these tools is the expected utility hypothesis,
which is a decision making theory described by Schoemaker (1982) as one of the most
important decision making tools developed in the last half century. The theory assumes
economic agents maximize expected utility, while at the same time adhering to several
axioms. The theory enjoys a long and diverse literature, and is applied in all types of
decisional situations. For example, Arrow (1963) shows that if actuarial fair insurance is
offered, then it is optimal to purchase full coverage. Also, the theory is discussed in
Shoemaker’s (1982) review of Bernoulli and Cramer (1738) to explain the Petersburg
2
paradox1. These empirical and theoretical studies have had mixed results in showing that
the theory can explain real world circumstances.
The expected utility hypothesis as applied to decisions specifically involving low
probability-high consequence events has a much less extensive literature associated with
it. These related studies have the same amount of varying success as the larger body of
research showing that the expected utility hypothesis is an accurate way of modeling
people’s decisions. Many, including Kunreather and Schoemaker (1979) have concluded
that an individual does not act like they are maximizing expected utility when making
decisions about these types of low probability events.
This thesis aims to study the consumer demand for catastrophic earthquake
insurance, and asks questions about people’s decision making behaviors relative to the
underlying expected utility theory assumptions. Catastrophic events and the damages
they can cause have recently been paid more attention due to their increased severity and
frequency, and this thesis uses publicly available data on earthquake insurance in
California to advance knowledge in the area of decision science as applied to catastrophic
mitigation tools. The results indicate the expected utility theory, when assuming a risk
averse utility function, correctly predicts the signs of income and risk elasticity as
positive. The empirical results are all mostly significant, with risk having possible fixed
effects issues. The theory predicts that price elasticity can have either a positive or a
negative relation with insurance demand, and the empirical results report a statistically
significant negative relationship across specifications.
1
A game considered to have an infinite expected payoff but also considered to be worth low amounts of
money
3
This thesis presents an expected utility model for the demand for insurance
developed by Lynch (1967). The model, in order to develop explicit demand expressions,
assumed people are risk averse when it comes to purchasing insurance, and assumed a
log functional form for people’s utility. This thesis makes the same assumptions. The
thesis then presents a utility maximization problem, and using first order conditions for
maximization, solves for the explicit functions of people’s demand as a function of
income, probability of event, and the price of the insurance. The equations for the
income, price, and risk elasticities are found using said first order conditions.
The thesis proceeds with a demand estimation by household over time. A total of
six models are estimated including two pooled models, two time fixed effects models,
and two earthquake zone fixed effects models, with the first four models empirically
synthesizing elasticities for price, income, and risk. In contrast, the two earthquake zone
fixed effects models empirically synthesize elasticities for only price, and income. The
different model formulations are used to check for agreement in estimated values of
independent variable across specification. The data used are California residential
earthquake insurance figures on amounts of insurance purchased, personal income
expenditures, and regional risk estimates. The data is publicly available through the
California Earthquake Authority (CEA). The signs of these elasticities can then be
compared with the theorized values, which, if they match, will add supporting evidence
to the validity of the expected utility theory’s assumptions.
All six models estimate price elasticities as negative and statistically significant.
The results continue to agree with the expected utility theory (when assuming a risk
4
averse utility function) by correctly predicting the positive sign of risk elasticity. Four
out of the six models estimate income elasticities as positive and statistically significant,
with only the zonal effects models reporting a non-statistically significant positive
elasticity. The two models estimating risk both report positive elasticities, with only the
time fixed effects model reporting a statistically significant value. Both results tentatively
correspond with the theories prediction that risk and income elasticities should always be
positive.
The remainder of this thesis is structured into four subsequent chapters. Chapter 2
presents previously conducted research in the area of decision theory, and in particular,
expected utility studies as they related to the purchase of consumer insurance. A survey
of work done in estimating different types of insurance then follows to give the reader a
solid background of the techniques, successes, and non-successes in estimating insurance
demand as a broad subject. Chapter 3 continues with a theoretical development of the
expected utility theory of catastrophic insurance demand first presented by Lynch (1967).
Theoretical predictions of the signs of income, price, and risk elasticities are developed.
The data used in the empirical estimation are presented in Chapter 4 as well as the
independent and explanatory variable formulations. Chapter 5 presents the empirical
estimation of California earthquake insurance demand and the results, along with
interpretations of the resulting variable coefficients. The thesis concludes with a
discussion of the results in Chapter 6.
5
Chapter 2
LITERATURE REVIEW
2.1. Introduction
This literature review describes past work and how this thesis adds to the current
body of knowledge concerning testing predictions of the expected utility (EU)
hypothesis. The thesis uses an earthquake insurance demand model over time to estimate
price, income, and risk elasticities and compare them with theoretical EU model
predictions. Past work on catastrophic insurance demand and the EU theory has not been
concerned with developing explicit demand functions for insurance purchases by
assuming any precise functional form of the utility function, which this thesis does.
While some studies (Lynch 1967) have assumed a functional form of the utility function
and then tested demand predictions using insurance purchase data, none have been done
using catastrophic insurance purchase data. Others, to be described below, have
performed demand estimation studies of earthquake insurance purchases but their results
were not used to test EU predictions.
This literature review first describes work related to the theoretical development
of the expected utility (EU) theory and the corresponding empirical studies supporting the
accuracy of its predictions. This thesis tests predictions derived from EU theory and the
reviewed studies also in one form or another test predictions made by the EU theory.
Some test the predictions in ways similar to this work, while others do not.
The review continues by focusing specifically on empirical insurance demand
studies and even more specifically, on earthquake insurance studies. The second part of
6
this thesis estimates several different earthquake insurance demand models, and then uses
the results of these estimations to test EU predictions. The empirical literature
summarized here provides insights for developing this thesis by exploring methods for
estimating demand models for earthquake insurance.
2.2. Expected Utility Theory
2.2.1. Theoretical Studies
This section outlines past theoretical research with regards to the expected utility
hypothesis, which is used in this thesis to explain the demand for catastrophic earthquake
insurance. Some past work has shown the EU framework does a good job explaining
observed behavior, and some have found the opposite. Though these studies do not all
use insurance demand estimation to test EU predictions, the broader concept of testing
EU predictions is what these studies have in common with this thesis.
The first contribution to the field came from Bernoulli (1738) when he
incorporated risk into the study of games. The Bernoulli/ Somers (1954) translation
elaborates on the details which include the assumption an individual perceives higher
satisfaction with higher wealth at a decreasing rate. This is another way of saying people
are assumed to be risk averse. Further, assuming that an individual’s satisfaction curve or
utility curve is concave, Somers was able to show specific insurance situations when the
expected utility after paying an insurance premium was higher than the expected utility if
not insured. This was the beginning of what would become the idea that people buy
insurance to maximize expected utility, and the framework would be used extensively. A
7
standard prediction of expected utility maximization models is that expected utility and
risk aversion are positively correlated. Though these studies by Bernoulli/ Somers (1954)
developed the language of expected utility, they did not develop explicit demand
functions to derive results from an empirical catastrophe insurance demand estimation,
which this thesis does.
Not until the work of von Neumann and Morgenstern (1953) would expected
utility be formally proved to be a rational choice. Shoemaker (1982), using this result,
explores how the EU framework contributes to a broad range of fields such as finance,
economics, psychology, and management science. Though mathematically manageable,
he describes how many empirical studies have shown the structural model of EU is not
capable of describing individual decision making.
There have been studies that have used the expected utility theory to make
predictions about insurance purchases, which is what the theoretical chapter of this thesis
does. Beenstock (1988) and Schlesinger (1981) have done work using the EU theoretical
framework that shows the purchase of insurance will increase with the size of possible
losses and the probability of these losses. Along similar lines, Beenstock (1988) shows
insurance purchases should decrease with the increasing price of the insurance. These
studies, being completely theoretical, lack real world data estimations.
An interesting result by Lee and Rice (1965) that deals with low probability
events is that people should prefer to insure low probability high loss events over high
probability low loss events for an event with the same expected value. The result
becomes clearer when analyzed through the maximization of expected utility not
8
expected value. Low probability, high loss events have a much lower expected utility
than high probability, low loss events, and therefore someone maximizing expected
utility will choose the former. Lee and Rice (1965) also use the expected utility theory
assumptions to derive theoretical predictions about low probability high impact events,
but they do not empirically test them, which this thesis does.
Another theoretical prediction of insurance demand based on an EU framework is
that insurance is an inferior good, and the demand for it should decrease with income.
Many have produced work which supports this, like Pratt (1964) who developed
methodologies to measure absolute risk aversion and argued income and the demand for
insurance are inversely related. There has also been work by Mossin (1968) showing that
if the absolute value of the risk aversion decreases with income and there is a positive
premium loading, then as income increases, ceteris paribus, deductible levels should
increase and the price a person is willing to pay for insurance will go down. Past work by
Schlesinger (1981) supports this finding but proves that it is only when assuming an
inverse relationship between absolute value of risk aversion and income. This thesis also
uses the EU theory to formulate predictions about how insurance demand varies with
income, but catastrophic insurance, not regular insurance.
The work done by Huberman (1983) and Keeton (1984) concludes that it may not
be optimal to insure completely if the potential loss an individual faces exceeds their
wealth. This type of work needs the assumption of limited liability or some social
program that aids low-income people. The assumption leads to certain mathematical
constraints that are responsible for the results. Taking a closer look at Huberman (1983),
9
there is no assumption of moral hazard and they show the optimal insurance coverage
depends on the probability distribution of potential losses, the policy holder’s wealth, and
the provisions of bankruptcy statutes. This work focuses on correcting a moral hazard
problem ex-post, or when the insured correct for a moral hazard problem after damages
have occurred.
Keeton (1984), however, focuses on different levels of an individual’s wealth and
its effect on insurance coverage. He found that only those with net worth greater than
possible damages would choose full liability insurance. All others would select less than
full liability insurance.
All of these studies have used the expected utility hypothesis to derive results for
different insurance questions. None of them have derived theoretical predictions for the
demand of catastrophic insurance, which is how this thesis adds to this literature.
2.2.2. Empirical Studies
This section outlines the past work involved with empirical tests of the EU
hypothesis. There has been varied success using the EU model to explain observed
behavior which will be described below. More research is warranted to determine if the
EU framework applies not only as a general theory, but to catastrophic insurance demand
as well. This thesis fills some of this need.
For example, Pashigian et al. (1966) used data from car deductibles to test
hypothesis of the EU model and found that most people spent too little on these insurance
products to support the EU theory. The study starts by assuming a concave utility
10
function of the Neumann-Morgenstern type and outlines the necessary mathematics. This
thesis also assumes a concave utility function of the Neumann-Morgenstern type. They
concede one can test the EU hypothesis but only if one assumes a functional form of the
utility function (in this case quadratic). They further assume only one accident per year at
most by the insured. An algebraic expression is developed for the maximized EU as a
function of the deductible. Acknowledged criticism by Pashigian et al. (1966) of this
work centers on the possibility there may have been limits set on deductibles at the time
and therefore the study may not have been actually testing the EU hypothesis.
Brookshire (1985) used data on housing prices to proxy an insurance study. This
study was intended to test the EU hypothesis by showing the price differential between
similar homes in different risk areas was a type of self-insurance. He was able to show
that the price difference was close to what the EU hypothesis would have predicted.
Beenstock (1988) shows a very direct method for testing the EU prediction by
analyzing national income. His results indicate that insurance for automobiles is a normal
good and that as income rises, more of it is purchased. Sherden (1984) analyzes
comprehensive, liability, and automobile insurance coverage. He finds that insurance
demand decreases with price and increases with loss probability, but his findings did not
support that deductible insurance decreases with wealth. Although these empirical
examples do not consider catastrophic insurance they provide a useful framework for
considering the role risk plays in such insurance decision.
11
2.3. Demand Estimation Studies
The literature outlined below gives a summary of the research on empirical
estimation of earthquake insurance demand. The most relevant studies to this thesis are
those that estimate income, price, and risk elasticities. These studies are similar to the
current thesis in that they estimate elasticities, but differ in that they are not comparing
the signs of the elasticities to theoretical predictions of elasticity signs obtained directly
from a demand function. The studies do not use California earthquake purchase data, and
are generally not testing predictions of the EU hypothesis.
Lai and Hsieh (2007) use spatial econometrics to assess the demand factors for
residential earthquake insurance against such independent variables as disposable
income, government subsidy, and region in Taiwan. The motivation for their research
was that the exposure to loss and financial impact tend to be localized to specific areas or
regions in Taiwan. Their paper uses data from the Taiwan Residential Earthquake
Insurance Pool (TREIP) a public entity created in 2002 to provide residential earthquake
insurance. They specify an econometric model using spatial autocorrelation and spatial
panel data with regional trade data. They find that income correlates positively with
insurance demand, and the number of government subsidies is negatively correlated with
the demand for TREIP insurance.
Athavale and Avila (2011) estimate the demand for earthquake insurance in the
New Madrid fault zone in Missouri. Data from the Missouri Department of Insurance
(DIFP) was used to examine the decision to purchase earthquake insurance by analyzing
data on earthquake insurance price and penetration. The double-log functional form was
12
used to specify a model with independent variables price, income, and risk where risk is
some measure of the likelihood of an earthquake. To account for the endogeneity
problem of price depending on risk, a two stage least squares regression was run. The
price was first regressed on risk, and then the demand was regressed on price. Results
indicate that homeowners acquire earthquake insurance because of risk considerations; at
higher levels of risk the demand for earthquake insurance is higher, and the price of
earthquake coverage does not provide incremental information in explaining the demand
for earthquake coverage.
Grace et al. (2003) study the demand for residential earthquake insurance. They
estimated the demand for insurance coverage in Florida and New York using a two stage
least squares regression, using data on insurance contracts, housing and demographic
data, and insurance firm characteristics. They found price elasticities that were negative
and elastic in New York and Florida, with them being higher in Florida. Income
elasticities were elastic and positive in both States as well.
A major difference between this thesis and the above summarized empirical
studies is that the data used in this thesis makes it possible to disregard the possible
endogeneity problems of price. The California Earthquake Authority sets earthquake
insurance rates every several years, and therefore price can be considered fixed by an
outside entity, not subject to free market considerations.
13
2.4. Theoretical and Empirical Studies
The study most closely related to this thesis is that of Lynch (1967). In his study,
he develops an EU theory of insurance purchases and then subsequently attempted to
empirically test this theory. He develops an expected utility maximization problem for a
three state insurance problem to derive testable hypotheses and insurance demand. He
also assumes a standard Pratt log utility function that is concave, and displays risk
aversion. Lynch then sets out to test the model via inferences about the degree of risk
aversion which should be able to be made from data on changes in insurance purchases
caused by changes in wealth, losses, and premium rates. Lynch subsequently applies the
derived demand functions to life insurance data, and asks if these functions can
accurately explain the extent and pattern of under insuring, and whether the functions can
explain the increasing ratio of aggregate face value of life insurance policies to income.
The author finds a positive result in both cases. This thesis adapts this framework
described in Lynch (1967) to the case of catastrophic insurance rather than life insurance.
2.5. Summary
We have seen above that the literature on the EU hypothesis and insurance spans
theoretical and empirical studies. This thesis extends this literature on EU and insurance
by considering the demand for catastrophic insurance within the EU framework and
applying it to California earthquake insurance. More specifically, this thesis utilizes the
analysis presented in Lynch (1962) for the case of life insurance to test hypotheses about
earthquake insurance in California.
14
Chapter 3
THEORY
This chapter presents the modified expected utility model introduced by Lynch
(1967). The model presented here is adjusted slightly by assuming a two state insurance
problem rather than a three state problem. The first order conditions are solved for
optimal coverage given the expected utility maximization objective. An explicit demand
function for coverage is then derived utilizing a representation on a Pratt (1964) utility
function. The Pratt utility function was chosen because historically it has been the only
utility function that adheres to risk averse behavior, which is the main hypothesis in the
analysis. The corresponding comparative static are derived to yield predictions about the
sign of the price, income, and risk elasticities, which are tested in Chapter 5.
The following model assumes that people are able to make insurance decisions
without background exogenous or endogenous risks being taken. Exogenous risks are
defined as being of the form of uncontrollable risks to overall wealth such as global
financial crisis, disease, political instability, and natural disasters. Endogenous risks are
defined as being moral hazard issues and asymmetric information such as an insurance
company preying on consumers who they know has insufficient information.
Let us assume an individual has initial wealth W0, and will suffer a loss αW with
probability π. The individual therefore is in a game with two possible outcomes, [π (W0αW), (1- π) (W0)]. If insurance is available the individual can instead choose to face the
game with two alternate outcomes, [π (W0- pC-αW0+ C), (1- π) (W0- pC)], given that he
15
or she can purchase insurance for a premium P, which is a linear function of both the
insurance rate p, and the amount of coverage chosen C.
P  pC
3.1
To proceed, it is necessary to assume the individual makes decisions according to
a von Neumann-Morgenstern utility function U(W), which has the important properties of
being continuous and twice differentiable, or rather, the marginal utility is positive and
decreasing in wealth (von Neumann and Morgenstern, 1953). If this is the case, an
individual will choose to purchase insurance if and only if there is a C such that the
expected utility of being insured exceeds the expected utility of refusing insurance. The
resulting inequality is of the form:
U (W0  pC  W0  C )  (1   )U (W0  pC )  U (W0  W )  (1   )U (W0 )
3.2
From this point we would like to solve an expected utility problem specified as
max E (U (W ))  U (W0  pC  W0  C )  (1   )U (W0  pC ),
3.3
subject to the following first order condition
 (1  p)U ' (W0  pC  W0  C )  p(1   )U ' (W0  pC ).
Solving this first order condition and assuming a Pratt-like utility function yields
3.4
16
U (W )  log( W )
3.5
Equation 3.5 was shown by Pratt (1964) to encompass all the necessary assumptions
gives us the relation:2
 (1  p )
p (1   )

W0  W  (1  p )C W0  pC
3.6
From this point we want to solve explicitly for C in order to have a testable expression
for demand.
`
C
(  1)pW0  ( p   )W0
p( p  1)
3.7
Comparative statics for changes in income (W0) gives the following expression:
C
p 

W0 p ( p  1)
3.8
No general predictions can be inferred about income elasticities from this result.
However, if insurance is actuarially fair (i.e., when π=p) initial wealth has no effect on
the amount of coverage purchased. If the pricing is unfair (π<p), then coverage is
inversely related to changes in initial wealth given the denominator, p(p-1), is always
negative because 0<p<1.
Comparative statics with respect to risk yields the following result:
2
Risk-averse behavior is captured by a concave Bernoulli utility function
17
C (W0 p  W0 )


p( p  1)
3.10
This result, by examination, indicates the elasticity with respect to risk is always positive,
or higher perceived risk always leads to great demand for insurance. It is straight forward
to show that the numerator is also always negative because 0<α≤1 (consumer cannot lose
more than they have).
Next, turning to the comparative static result with respect to price, one would
intuitively postulate a negative coefficient on the expression:
 (W0 p 2  W0 )  p 2 (W0  W0 )  2 pW0
C

p
p 2 ( p  1) 2
3.11
The theory again postulates nothing general about the signs of the price elasticity. The
relationship will depend on the combination of all the values and can be positive, or
negative. The denominator is always positive, and the last term in the numerator is
always negative, making the sign of the elasticity dependent on the combination of the
parameters in the first two terms of the numerator.
This chapter has presented the expected utility model developed by Lynch (1967),
adjusted slightly by assuming a two state insurance problem rather than a three state
problem. The subsequent chapters detail the empirical demand estimation for California
earthquake insurance. In the analysis different approaches are used to estimate price,
income, and risk elasticities given the difficulty in directly observing household level
18
risk. Earthquake zone risk factors are used in the analysis but do not vary over time and
thus may be correlated with other fixed factors. As such, zone fixed effects models are
also estimated and used together to draw inference about the role of risk in coverage. The
results of the estimation are then compared to the theoretical predictions described in this
chapter.
19
Chapter 4
DATA
4.1. Data Summaries and Variable Formulation
This chapter presents the data used for this thesis as well as data summaries and
variable formation methodologies. All data used is on the zip code level from California.
The major variables of interest are the amount of earthquake insurance that was
purchased, personal income per household, the price or rate of earthquake insurance, and
the perceived risk or probability of an earthquake occurrence.
The data consist of several different sets, some collected from different publicly
available sources, and some from private requests of publicly available data. The
California Department of Insurance, after direct public request from the department’s
research division, provided data on earthquake insurance coverage, and number of
earthquake insurance policies spanning the years 2005, 2007, and 2009. Each record
contains the amount and type of policies purchased for both private and CEA insurance,
and the amount of earthquake insurance coverage A, B and C in nominal dollars.
Population and household data are obtained from the California census for the years 2000
and 2010 and includes population data for each zip code as well as number of owner
occupied households per zip code and average number of residents per household.
Personal income data are obtained from the California Department of Finance from tax
records and include reported nominal income for every zip code for the years 2005, 2007,
and 2009. The risk data set is from USGS (http://earthquake.usgs.gov /hazards
20
/products/conterminous/2008/data/) and includes the probability of earthquake on a .05 x
.05 latitude and longitude gridded area for all of California.
The earthquake coverage data for the years 2005, 2007, and 2009 were the only
available years with zip code level data, and only policies sold by the CEA were
considered. This decision was made because these policies comprised more than 80% of
the policies, as well as the fact that no price data was available for non-CEA policies. The
CEA was created in 1996 by Act of the California Legislature and is privately financed
while publicly managed. Insurance companies either offer coverage through the CEA, or
offer their own policies with coverage at least equal to the CEA’s.
Earthquake insurance coverage A (structure only) is the only type of insurance
considered in this thesis due to the fact that only coverage A data was available for the
year 2007, thus narrowing the data usefulness to the common denominator of coverage
A. Only policy data on owner occupied homes were used for the study, and data for
mobile homes and renters were discarded for similar data quality issues. This data
decision is driven by the owner occupied home policies being of the majority kind, as
well as not wanting to contaminate the study by using people’s choices about insurance
that are not directly liable for the possible damages.
Both total earthquake insurance coverage, and the number of policies has
increased from 2005 to 2009. The earthquake insurance coverage increases on a much
higher rate than the number of policies, from an average of $80 million to $135 million,
where the number of policies per zip code changed from an average of 263 to 342. This
indicates people are purchasing different levels of coverage in/for/ different years, or they
21
are not always purchasing full coverage. Alternatively, this might be an indication of
changed owner-occupied demographics. The yearly totals and data summaries can be
found in Table 4.1. The maximum coverage in a zip code increases from $1.27 billion to
$1.79 billion, the minimum coverage increases from $45,000 to $81,000, and finally the
standard deviation increases from $143 thousand to $207 thousand between the years
2005 and 2009.
Data on the number of CEA earthquake insurance policies are also summarized in
Table 4.1. The average and standard deviation amount of policies by zip code increase
from 2005 to 2009. The average per zip code went from 263 to 342 policies and the
standard deviation from 404 to 459 policies. The minimum number of policies remained
at 1. The maximum amount of policies per zip code increases from 2005 to 2007 from
3,474 to 4,134, then decreases slightly from 2007 to 2009 to 3,718 policies.
Personal income data was estimated using the California adjusted gross income
(AGI) reported on tax returns for every zip code in California for the years 2005, 2007,
and 2009. The data is the total income for an entire zip code in nominal dollars, and was
not adjusted for inflation because of the short time frame and extremely low inflation in
2009. The California AGI is obtained by adjusting the Federal AGI, which consists of the
taxable income of individuals who filed a Federal income tax return. From 2005 to 2007
the average income increased from $365 million to $425 million, the maximum increased
from $4.5 billion to $5.9 billion, the minimum increased from $93 thousand to $164
thousand, and the standard deviation increased from $5.3 million to $6.5 million. From
2007 to 2009 the average income decreased from $425 million to $415 million, the
22
maximum decreased from $5.9 billion to $4.8 billion, the minimum decreased from $164
thousand to $142 thousand, and the standard deviation decreased from $6.5 billion to
$6.0 billion. Data summaries can be found in Table 4.1.
Table 4.1: Summary Statistics for Insurance and Income Data.
Year
Variable
Cov
2005
2007
2009
Description
Obs
Mean
Std. Dev.
Min
Max
Total CEA
earthquake
insurance
coverage A
(Dollars)
1,994
80,843,839.14
143,491,800.5
45,100
1,269,437,061
1,938
104,502,926.6
182,763,029.4
50,200
1,749,705,179
1,683
135,400,106.5
207,229,371.2
81,325
1,786,445,658
Aggregate
EQ
insurance
policies
1,994
263
404
1
3,474
1,938
296.07
454.00
1
4,134
1,683
342.73
458.95
1
3,718
2,372
365,414.8284
528,784.7453
93
4,483,873
2,373
425,295.6119
647,365.7275
164
5,863,104
2,391
415,318.752
603,210.0138
142
4,817,698
Pol
2005
2007
2009
Income
2005
2007
2009
Income
(Thousand
Dollars)
Population data per zip code was obtained for the years 2000 and 2010 from the
California census, and a linear trend was used to infer the years 2005, 2007, and 2009 for
every zip code in California. The data summaries for the population data can be found in
Table 4.2. From 2000 to 2010 the average population increased from 19 thousand to 21
thousand, the maximum increased from 105.3 thousand to 105.5 thousand, and the
standard deviation increased from 20 thousand to 21 thousand. The data summaries for
the interpolated population data can be found in Table 4.3.
23
The number of households owning a home and living in it per zip code was
obtained from the California census for the years 2000 and 2010. The census also
provided the average number of persons living in said home. A simple linear trend was
applied to infer the years 2005, 2007, and 2009 for both variables. From 2000 to 2010 the
average number of households increased from 6,500 to 7,100, the maximum decreased
from 33,527 to 33,342, and the standard deviation increased from 6,715 to 6,852. From
2000 to 2010 the average number of persons living in a household increased from 2.66 to
2.70, the maximum from 5.39 to 6.0, where the standard deviation decreased from 0.79 to
0.74 persons. Data summaries for number of households where the homeowners lives in
the home and number of people living in the household can be found in Table 4.2. Data
summaries for the interpolated number of households where the homeowners lives in the
home and number of people living in the household can be found in Table 4.3.
24
Table 4.2: Summary Statistics for Census Data
Year
Variable
Pop
2000
2010
House
2000
2010
Person
2000
2010
Description
Aggregate
population
on zip code
level
Number of
households
owning a
home and
occupying
Average
number of
persons in
home
Obs
Mean
Std. Dev.
Min
Max
1,756
19,288
20,642
0.0
105,275
1,763
21,116
21,332
0.0
105,549
1,756
6,550
6,715
0.0
33,572
1,764
7,129
6,852
0.0
33,342
1,756
2.66
0.80
0.0
5.39
1,764
2.70
0.75
0.0
6
Table 4.3: Summary Statistics for Interpolated Census Data
Year
Variable
Pop
2005
Description
Obs
Mean
Std. Dev.
Min
Max
Aggregate
population on
zip code level
1,654
21,215
20,920
0.0
104,380
1,938
21,522
21,109
0.0
104,847
1,683
21,829
21,352
0.0
105,315
1,654
4,051
3,902
0.0
17,697
1,654
4,088
3,924
0.0
17,986
1,654
4,126
3,958
0.0
18,275
1,654
2.76
0.63
0.0
5.37
1,654
2.75
0.64
0.0
5.36
1,654
2.75
0.66
0.0
5.69
2007
2009
House
2005
2007
Number of
households
owning a home
and occupying
2009
Person
2005
2007
2009
Average
number of
persons in
home
25
The perceived risk of an earthquake was interpreted to be the probability of an
earthquake occurring in a specific geographic area of the state, and the data were
obtained from the U.S. Geological Survey (USGS) National Seismic Hazard Maps using
the 1Hz, 10% in 50 years data set. The data consists of earthquake ground motions for
different probability levels across California, and is used in seismic provisions of
building codes, insurance contracts, and various public risk assessments. The data set is a
subset of an entire USA gridded file in 0.05 degree increments of longitude and latitude.
The same probabilities were used for each of the years 2005, 2007, and 2009. This time
invariant risk data is the source of some concern of the viability of elasticity estimate, and
motivated the inclusion of multiple model estimates.
The rate data for 2005, 2007, and 2009 was obtained by the corresponding CEA
rate manual (2001, 2006, 2009) for the proper year. The rates are calculated by the CEA
and are determined primarily be the age of the structure, the number of stories, and the
location of the structure. The rate data is in nominal dollars per $1000 of coverage. From
2005 to 2007 the average rate decreases from $4.76 to $4.49, the maximum increases
from $7.9 to $8.05, the minimum decreases from $1.60 to $1.20, and the standard
deviation decreases from $2.07 to $2.00. From 2007 to 2009 the average rate decreases
from $4.49 to $2.37, the maximum decreases from $8.05 to $4.19, the minimum
decreases from $1.20 to $0.55, and the standard deviation decreases from $2.00 to $1.03.
In general, rates have been steadily decreasing since 2005 indicating shifts in CEA policy
and underlying supply and demand issues not fully explored in this study. Table 4.3
provides descriptive statistics for the rate data.
26
The empirical portion of the study will estimate the demand for CEA earthquake
insurance as a function of price, income, and risk. The dependent variable will be
insurance demand and will be modeled as the amount of coverage demanded per
household. The coverage amount and the number of households owning a home has been
aggregated on the zip code level and will be further aggregated into earthquake zones.
The summary statistics for the variable of interest is shown in Table 4.3. From
2005 to 2007 the average coverage per household increases from $27 to $31 thousand,
the maximum increased from $76 to $89 thousand, the minimum increased from $2.8 to
$3.4 thousand, and the standard deviation increased from $19 to $21 thousand. From
2007 to 2009 the average coverage increased from $31 to $33 thousand, the maximum
increased from $89 to $90 thousand, the minimum increased from $3.4 to $6.9 thousand,
and the standard deviation increased slightly from $21.7 to $21.9 thousand.
27
Table 4.4: Summary Statistics for Dependent and Independent Variables
Year
Variable
Coverage/Household
2005
2007
2009
Description
Obs
Mean
Std. Dev.
Min
Max
Total
earthquake
insurance
coverage per
earthquake
zone
19
27,400.24
19,149.15
2,814.06
76,063.71
19
31,114.82
21,740.48
3,395.16
89,183.55
19
33,415.72
21,937.77
6,944.94
90,299.54
Disposable
income per
earthquake
zone
19
68.97
30.83
33.76
155.00
19
83.68
43.60
43.42
206.87
19
76.30
38.86
36.96
183.53
19
4.76
2.07
1.6
7.9
19
4.49
2.00
1.2
8.05
19
2.37
1.03
0.55
4.19
Income/Household
2005
2007
2009
Rate
2005
2007
2009
Rate, or
price per
earthquake
zone
In the analysis the variable for income is formulated as per-household nominal
income in a given earthquake zone. The income used is the total income of homeowners
in the zip code, which is normalized by the number of homes in the zip code. The
summary statistics for this variable can be found in Table 4.4. The price variable is the
CEA developed rate per $1000 of earthquake insurance coverage per earthquake zone.
The rate varies across earthquake zones and year. The risk variable is the average risk
measure of the probability of plate movement in an earthquake zone.
28
4.2. Scatter Plots
Figures 4.1 through 4.3 show scatter plots between coverage and income, price,
and risk for the three years 2005, 2007, and 2009. The plots reveal the same general
behavior across the three years. The price can be seen to be negatively correlated with
coverage per household while income per household (or wealth) appears positively
correlated with coverage. Risk and coverage, however, look negatively correlated, which
is counter to our intuitive expectations and the previously developed expected utility
theory.
29
Figures 4.1: Scatter plots for the 2005 data
80,000
Coverage/Household
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
2
4
6
8
10
Price
80,000
Coverage/Household
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
50
100
150
Income/Household
200
80,000
Coverage/Household
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
0.5
1
1.5
Risk
2
2.5
3
30
Coverage/Household
Figures 4.2: Scatter plots for the 2007 data
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
2
4
6
8
10
200
250
Coverage/Household
Price
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
Coverage/Household
0
50
100
150
Income/Household
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
0.5
1
1.5
Risk
2
2.5
3
31
Coverage/Household
Figures 4.3: Scatter plots for the 2009 data
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
1
2
3
4
5
Coverage/Household
Price
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
Coverage/Household
0
50
100
150
Income/Household
200
100,000
90,000
80,000
70,000
60,000
50,000
40,000
30,000
20,000
10,000
0
0
0.5
1
1.5
Risk
2
2.5
3
32
4.3. Summary
Examining the above average trends over time it can be seen, in general, that as
the price decreases and income increases, the coverage per household increases. This
trend will positively support the postulated demand function and corresponding
comparative statics results. Risk, having no time component, can’t be examined as having
any influence over time changes in demand.
There seems to be a small outlier, that being income dropping drastically from
2007 to 2009 while at the same time, the coverage increases. The drop in income may be
because of the Great Recession, which may be why the data does not support
conventional economic theory. The coverage, however, increases from 2005 to 2009 so
the trend is still positive, supporting previously presented theory. Furthermore, the Great
Recession surely had some anomalous effects on personal income, and correcting for
such effects may lead to seeing a more positive trend in income from 2007 to 2009.
This chapter has introduced the data sets used in the study and presented data
summaries for each variable of interest. Subsequent discussions have outlined their major
features. However, the analysis presented in this chapter only provides a preliminary look
at the relationships within the data. The next chapter gives a more rigor examination of
the data by presenting an econometric analysis of the demand for earthquake insurance
over time.
33
Chapter 5
DEMAND ESTIMATION
5.1. Estimation Models and Estimation Results
This chapter describes the approach used to empirically estimate demand for
California earthquake insurance over time and the subsequent results from the analysis.
As previously developed in Chapter 3, a theoretical hypothesis of the functional form of
this demand for earthquake insurance guides the analysis and allows for a suitable test of
theoretical consistency. Comparative statics derived in Chapter 3 provide hypotheses on
the signs of income, risk, and price elasticities. The analysis described below involved
estimating a series of models using pooled data and panel data techniques.
Additionally, the analysis further evaluates risk given concerns that the risk factor
measures do not change over time and may capture zonal fixed effects that might bias the
estimated effect of risk on coverage.3 To do so, two models are estimated without the risk
factor. One with time fixed effects and one without. Two additional models are estimated
as well to determine if earthquake zone fixed effects (with and without time fixed effects)
provide further insights about the role of risk in determining coverage.
The elasticity results from this analysis appear consistent with those hypothesized
earlier. The risk elasticity appears to be positive, though through the various model
specifications goes in and out of significance and may be highly correlated with other
explanatory variables. The chapter concludes with a discussion on the implications of
these results.
3
If any omitted fixed variables are correlated with the risk factor and explain differences in coverage across
zones, then the estimated coefficient is likely biased.
34
5.2. Pooled Data Regression Model
The first regression model (equation 5.1 below and hereafter referred to as Model
1) uses ordinary least-squares (OLS) to estimate elasticities with pooled data across zones
and times. The model is in log-log form, both for analytic properties and because this
functional form of the demand function leads to a direct estimation of elasticities.
ln Coverageit    1 ln Priceit   2 ln Incomeit   3 ln Risk i
5.1
The i and t subscripts in equation 5.1 above denote earthquake zone and year,
respectively. The regression results are shown in Table 5.1.
The first OLS regression finds the price coefficient to be significantly negative.
This shows that people are decreasing their purchases of earthquake insurance as the rates
set by the CEA have increased. The magnitude is below one showing an inelastic demand
with respect to price. The theory presented in Chapter 3 postulates a price elasticity that
can be either positive or negative, depending on a combination of parameters.
The risk coefficient is positive though not significant, which may be because of
the multicollinearity of risk and price. It also may be because risk is likely correlated with
factors in the error term, or factors that are the same within one risk area but different
across different risk areas. The magnitude of the elasticity is extremely small.
The income coefficient is positive and significant, showing wealth is positively
correlated with the decision. This result is could also be consistent with the theory
presented. A positive association with wealth, according to our theory, indicates that the
35
insurance being offered is more than actuarially fair, and actually being offered at a lower
price. Actuarially fair insurance is insurance offered at the price of the risk, or p=π. From
observation of equation 3.10, we can see the predicted sign of income elasticity when
p=π is positive. The magnitude suggests coverage is elastic with respect to income.
5.3. Inclusion of Time Fixed Effects
The second regression (Model 2) uses OLS and adds time fixed effects to
equation 5.1 as follows.
ln Coverage it     t  1 ln Priceit   2 ln Incomeit   3 ln Risk i
5.2
Fixed effects used here control for unobserved explanatory variables that may differ
across time and not across entities. The fixed effect assumption is that these unobserved
quantities are correlated with the independent variable. In this thesis, time fixed effects
are used to control for changes across time such as interest rates, extreme economic
fluctuations such as the financial disaster of 2008, and any other unobserved fluctuation
not taken into account with the empirical estimation model but do not vary across
earthquake zones.4
4
Earthquake zone fixed effects are not considered in this part of the analysis because the risk data does not
vary over time and their inclusion would result in perfect multicollinearity between the earthquake zone
fixed effects and the risk variable.
36
Table 5.1: Regression Results for Models 1 and 2
Variable
ln(price)
ln(income)
ln(risk)
Constant
Model 1
-0.76***
(0.153)
1.11***
(0.174)
0.07
(0.383)
6.16***
(0.756)
Model 2
-1.39***
(0.208)
1.21***
(0.157)
1.13*
(0.424)
6.28***
(0.669)
-0.258
(0.161)
-0.950***
(0.230)
0.56
24.72
0.66
22.55
2007
2009
Adjusted R squared
Overall F-statistic
Note: Robust standard errors are in parentheses. Additionally, ** and ***denote statistically significance at
the 5% and 1% levels respectively.
The pooled data model results found in Table 5.1 includes the estimates for each
independent variable. The pooled data model shows three out of the four variables are
significant. The price coefficient, income coefficient, and the constant are all statistically
significant at the 1% level. The risk coefficient in this case does not show statistical
significance. The adjusted R squared is relatively high at 0.56, showing good fit to the
model.
The time fixed effects regression resulted in all of the coefficients being
statistically significant. The income and price variables are statistically significant at the
1% level. The risk variable is statistically significant at the 5% level. Adjusted R squared
increases and shows a better fit at 0.66.
The time fixed effects model (Model 2) shows significant estimates for all three
explanatory variables, has a higher adjusted R-squared, as it controls for more
37
unobserved factors. The adjusted R squared increases from 0.56 to 0.66, a nearly 18%
increase.
The price coefficient is significantly negative. This shows that people are
decreasing their purchases of earthquake insurance as the rates set by the CEA have
increased. The magnitude changes to above one showing higher elasticity as well as a
magnitude closer in value to reported price coefficients from previous insurance studies.
The elasticity changed from -0.76 to -1.39 from the pooled regression to the fixed effects
regressions indicating that fixed effects did capture some time effects and therefore
reduced potential bias. This also displays a change from inelastic to elastic price
elasticity.
The income coefficient is positive and significant. The elasticity increases slightly
from 1.11 in the pooled data model to 1.21 in the regression with time fixed effects. The
magnitude makes the income elasticity slightly more elastic.
The risk coefficient is positive and significant. The elasticity went from a very
low inelastic number, to a larger elastic value. This is consistent with theory that the risk
is always positively correlated with the decision to buy more insurance. The elasticity
shows a large positive value. Next, the analysis estimates models without risk to observe
how this exclusion affects the price elasticity and income elasticity.
38
5.4. Further Examination without Risk
Two additional models (equations 5.3 and 5.4, hereafter denoted as Models 3 and
4) are estimated without the risk variable so we can better understand the role this
measure plays in explaining variation in coverage. Model 3 is a pooled regression using
only income and price as explanatory variables, and Model 4 is a time fixed effects
regression also excluding risk as an explanatory variable. The results are presented in
Table 5.3.
ln Coverageit    1 ln Price it   2 ln Incomeit
5.3
ln Coverageit     t  1 ln Priceit   2 ln Incomeit
5.4
Table 5.2: Regression Results for Models 3 and 4
Variable
ln(price)
ln(income)
Constant
2007
2009
Model 3
-0.72***
(0.115)
1.11***
(0.172)
6.18***
(0.736)
Model 4
-0.944***
(0.132)
1.16***
(0.165)
6.48***
(0.702)
-0.189
(0.168)
-0.590***
(0.197)
Adjusted R squared
0.568
0.658
F-statistic
37.73
22.55
p-value
5.57e-11
7.371e-12
Note: Robust standard errors are in parentheses. Additionally, ** and ***denote statistically significance at
the 5% level and 1% levels respectively.
39
The adjusted R squared of Model 3 has gone up from Model 1. The coefficients of
price and income have not changed much. The higher R squared of this model than when
compared to Model 1 suggests that the risk factor variable does not add useful
information about coverage, which is indicated by the non-statistically significant
coefficient found for risk in Model 1.
The adjusted R squared increases from Model 3 to Model 4 when adding time
fixed effects, indicating relevant variables have been added. The significance of price and
income stays the same but the magnitude of the price coefficient increases by 30%.
Moving from Model 1 to Model 2, then from Model 3 to Model 4 adjusted R
squared increased, and the significance of explanatory variables increased, leading to the
conclusion that Model 3 most likely is not the best model fit to the data. A closer look at
the results between Model 2 and Model 4 suggests that the risk factor plays a role in
describing coverage even though the adjusted R-squared does not increase measurably.
Of course, since the risk factor does not change over time it may also be capturing
unobserved fixed effects that are correlated with the time invariant risk data. Nonetheless,
when risk is included the magnitude of the estimated elasticities for income and price
change and this change is more noticeable in the own-price elasticity, which switches
from inelastic to elastic, suggesting the exclusion of the risk factor variable introduces
bias into these estimated coefficients. To further test relevance of explanatory variables
across specifications zonal fixed effects models are estimated.
40
5.5. Earthquake Zone Fixed Effects
In the next set of regression earthquake zone fixed effects are used to control for
unobserved variations across zones that are constant over time. As noted above these
fixed effects are perfectly collinear with the risk factor variable, resulting in the exclusion
of the risk factor variable in this set of models. Equations 5.5 and 5.6 (Models 5 and 6,
respectively) include the earthquake zone fixed effects model with and without risk time
fixed effects. The results are shown in Table 5.3.
ln Coverageit   i  1 ln Priceit   2 ln Incomeit
5.5
ln Coverageit   i   t  1 ln Priceit   2 ln Incomeit
5.6
The adjusted R-squared for Models 5 and 6 increase substantially when compared
to the previous models, indicating zone fixed effects play a meaningful role in explaining
coverage. The income coefficient is no longer statistically significant and extremely
small, which suggests that income may be highly correlated with earthquake zone fixed
effects. Figure 5.1 illustrates this point with respect to the estimated coefficients for the
zone fixed effects. The price coefficient is still highly significant and appears to be
inelastic.
41
Figure 5.1: Estimated Zonal Effect for Model 6 By Income
3
2.5
Coefficient
2
1.5
1
0.5
0
0
-0.5
50
100
150
200
Income
5.6. Remarks
This Chapter presented several different estimations of California earthquake
insurance demand and the results, along with interpretations of the resulting variable
coefficients. The analysis further evaluated risk given concerns that the risk factor
measures do not change over time and may capture zonal fixed effects that might bias the
estimated effect of risk on coverage. To do so, two models were estimated without the
risk factor. One with time fixed effects and one without. Two additional models were
estimated as well to determine if earthquake zone fixed effects (with and without time
fixed effects) provide further insights about the role of risk in determining coverage.
42
Table 5.3: Regression Results for Models 5 and 6.
Variable
ln(income)
ln(price)
Constant
4
5
6
7
8
11
12
13
15
18
19
20
22
23
24
25
26
27
Model 5
0.14
(0.083)
-0.29***
(0.061)
8.54***
(0.301)
-0.21***
(0.020)
0.29***
(0.021)
1.23
(0.048)
1.52***
(0.058)
0.74***
(0.027)
1.71***
(0.047)
1.62*
(0.048)
2.40**
(0.124)
1.93***
(0.075)
1.76***
(0.104)
0.83***
(0.055)
1.73**
(0.060)
1.33***
(0.070)
1.78***
(0.116)
2.21***
(0.052)
1.16
(0.045)
1.13**
(0.077)
0.87***
(0.103)
Model 6
0.03
(0.067)
-0.39**
(0.119)
9.08***
(0.367)
-0.22***
(0.022)
0.34***
(0.031)
1.20
(0.069)
1.52***
(0.066)
0.77***
(0.022)
1.72***
(0.048)
1.67*
(0.037)
2.48**
(0.102)
1.91***
(0.097)
1.62**
(0.195)
0.80***
(0.078)
1.73**
(0.066)
1.42***
(0.057)
1.90***
(0.089)
2.16
(0.083)
1.13
(0.072)
1.16*
(0.069)
0.74***
(0.188)
0.978
0.09
(0.016)
-0.05
(0.101)
0.981
2007
2009
Adjusted R squared
43
Chapter 6
CONCLUSION
This thesis set out to collect evidence from the purchase of earthquake insurance
in California and evaluate if some of the predictions of an expected utility theory of
insurance corresponded with a demand estimation made using these data. Data from three
years of insurance purchases, personal income, and risk distributions were used in
developing an empirical estimation of the demand for earthquake insurance per
household and over time. A theoretical model was presented that uses expected utility
theory, assuming a Von Neumann-Morgenstern utility function, and comparative static
predictions were derived. Secondly, a demand estimation was used to compare the
theoretical predictions with the signs of empirically estimated elasticities. This was
intended to give some evidence as to whether expected utility theory can be used to help
construct successful insurance schemes for catastrophic events.
The thesis proceeds with a demand estimation by household over time. A total of
six models including two pooled models, two time fixed effects models, and two
earthquake zone fixed effects models, with the first four models estimating elasticities for
price, income, and risk. The two earthquake zone fixed effects models estimate
elasticities for price, and income. The different model formulations were used to check
for robustness in estimated values of independent variable across specification. The signs
of the estimated elasticities were compared with theorized values, adding supporting
evidence to the validity of the expected utility theory’s assumptions. The data used are
California residential earthquake insurance figures on amounts of insurance purchased,
44
personal income expenditures, and regional risk estimates. The data is publicly available
through the California Earthquake Authority (CEA).
The study produced mixed results. The econometric analysis resulted in estimated
elasticities for price, income, and risk that were in the realms of the theoretically
predicted elasticities. Both income and price elasticities, according to the presented
theory, can be either positive or negative depending on the specific values of the
parameters. The income and price elasticities were mainly statistically significant and
were positive, and negative, respectively. The risk elasticity was always positive, though
depending on the model specification sometimes not significant. Based on several
different model specifications, however, the estimated elasticity on risk appears to be
positive, agreeing with theory and supporting the expected utility hypothesis that people,
while selecting amounts of insurance, will purchase higher amounts of insurance when
observing higher amounts of objective risk. The analysis of risk in the empirical
framework is limited, however, as the risk variable does not vary over time, and might
also capture differences across earthquake zones. In a fixed effects analysis, accounting
for unobserved differences across earthquake zones, the risk variable cannot be included
due to this data limitation.
One of the future directions this work could take is refining the data aggregation
scheme as to evaluate decisions on the zip code level. Price was formulated simply as the
rate the CEA set for each specified zone which makes for the easiest and truest measure
of price, but does aggregate the other independent variables over a large population.
Further work needs to define a more sophisticated price based on zip code that takes into
45
account housing characteristics and premiums chosen. Most importantly is finding a
measure of risk that varies over time, so as to not be measuring risk as a fixed effect. This
may or may not be feasible with earthquake data because the objective risk of an
earthquake does not vary from year to year. In the future a measurement of perceived risk
would be valuable.
46
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