Untitled - Universität St.Gallen

Transcrição

Essays on Insurance Management
— The Theory of Insurance Fraud and Empirical Analyses
of Auditing Strategies and Cat Bond Investments —
DISSERTATION
of the University of St.Gallen,
School of Management,
Economics, Law, Social Sciences
and International Affairs
to obtain the title of
Doctor of Philosophy in Management
submitted by
Katja Müller
from
Germany
Approved on the application of
Prof. Dr. Hato Schmeiser
and
Prof. Dr. Martin Eling
Dissertation no. 4164
epubli GmbH, Berlin 2013
The University of St.Gallen, School of Management, Economics, Law,
Social Sciences and International Affairs hereby consents to the printing
of the present dissertation, without hereby expressing any opinion on the
views herein expressed.
St. Gallen, May 17, 2013
The President:
Prof. Dr. Thomas Bieger
To my dear parents/ Za Mama i Tati
Valentina & Matthias
Acknowledgments
I would like to seize the opportunity to express my deepest gratitude to
a number of people who have accompanied me along the way with their
continuous support and encouragement.
To begin with, I would like to sincerely thank my supervisor, Prof.
Dr. Hato Schmeiser, for his generous support and guidance throughout
the development of this thesis and for creating an inspiring research environment at the Institute of Insurance Economics. I am also grateful
to my co-supervisor, Prof. Dr. Martin Eling, for his interest in my dissertation. Moreover, I am grateful to my co-authors, my colleagues and
my dear friends - Carin, Caroline, Joël, Tobias - for our collaborations
and discussions, and for making my time in St. Gallen an unforgettable
experience.
With all my heart, I would like to thank my parents and my brother
for their never-ending support and unconditional love and care. My
achievements would not have been possible without the generosity and
unwearying care of my dear parents, to whom I owe so much.
St. Gallen, July 2013
Katja Müller
Vorwort
Mein aufrichtiger Dank gilt einer Reihe von Personen, die mich während
der Entstehungsphase dieser Dissertation kontinuierlich unterstützt und
gefördert haben.
Zunächst möchte ich mich bei meinem Referenten und Doktorvater,
Prof. Dr. Hato Schmeiser, für seine wertvolle Unterstützung und für das
exzellente Arbeitsumfeld am Institut für Versicherungswirtschaft bedanken. Ausserdem danke ich meinem Korreferenten Prof. Dr. Martin Eling
für sein Interesse an meiner Dissertation. Ferner bin ich meinen Koautoren, Kollegen und Freunden - Carin, Caroline, Joël, Tobias - für unsere
gemeinsame Zusammenarbeit und die unvergessliche Zeit in St. Gallen
sehr dankbar.
Von ganzem Herzen möchte ich meinen Eltern und meinem Bruder
für ihre unablässige Unterstützung und ihre bedingungslose Liebe und
Zuwendung danken. Das Erreichte wäre ohne die Grosszügigkeit und
unermüdliche Fürsorge meiner lieben Eltern, denen ich so vieles zu verdanken habe, nicht möglich gewesen.
Blagodar vi ot cloto si sьrce!
St. Gallen, im Juli 2013
Katja Müller
Outline
iii
Outline
I
II
III
IV
Insurance Claims Fraud:
Optimal Auditing Strategies in Insurance
Companies
The Impact of Auditing Strategies on
Insurers’ Profitability
The Identification of Insurance Fraud:
An Empirical Analysis
What Drives Insurers’ Demand for Cat
Bond Investments? Evidence from a PanEuropean Survey
Curriculum Vitae
1
41
85
127
177
iv
Contents
Contents
Contents
iv
List of Figures
vii
List of Tables
viii
Introduction
Einführung
I
ix
xiii
Companies
1
1
Introduction
2
2
Model Framework
2.1 Optimization of Positions . . . . . . . . . . . . . . . . .
2.2 Assumptions About the Distribution of Information . . .
2.3 Analytical Results . . . . . . . . . . . . . . . . . . . . . .
6
11
12
13
3
Computational Aspects
17
4
Simulation Results
21
4.1 Reference Setting . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Sensitivity Analysis of Relevant Parameters . . . . . . . 25
5
Conclusive Remarks
30
6
Appendix
33
References
II
37
The Impact of Auditing Strategies on
Insurers’ Profitability
41
1
Introduction
42
2
Model Framework and Stakeholders’ Positions
48
2.1 Optimization Problem . . . . . . . . . . . . . . . . . . . 55
3
Optimal Auditing Strategies
55
3.1 Policyholder Claiming Scheme . . . . . . . . . . . . . . . 55
3.2 Insurance Company Auditing Strategy . . . . . . . . . . 58
Contents
4
v
3.3 Behavioral Adaptation . . . . . . . . . . . . . . . . . . .
3.4 Numerical Implementation of Iterative Optimization . .
60
63
Simulation Results
4.1 Parametrization of the Reference Setting . . . . .
4.2 Simulation Results and Sensitivity Analyses . . .
4.2.1 Development of Optimization Results Over
Several Iterations . . . . . . . . . . . . . . .
4.2.2 Sensitivity Analyses . . . . . . . . . . . . .
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72
5
Critical Discussion
77
6
Conclusion
79
References
III
81
The Identification of Insurance Fraud:
1
Introduction
2
Theory and Hypotheses Development
2.1 Development of Hypotheses . . . . . .
2.2 Data Set . . . . . . . . . . . . . . . . .
2.3 Descriptive Statistics . . . . . . . . . .
2.4 Model Derivation . . . . . . . . . . . .
85
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3
Empirical Results
103
3.1 Logistic Regression Results . . . . . . . . . . . . . . . . . 104
3.2 Special Focus on Loss Amount . . . . . . . . . . . . . . . 111
4
Conclusion and Critical Discussion
114
5
Appendix
117
References
IV
1
What Drives Insurers’ Demand for Cat
Bond Investments? Evidence from a PanEuropean Survey
Introduction
122
127
128
vi
Contents
2
The Demand for Cat Bonds
132
2.1 Current Market Size and Investor Base . . . . . . . . . . 132
2.2 Development of Hypotheses . . . . . . . . . . . . . . . . 133
3
Data and Methodology
3.1 Development of Measures . . . . . . .
3.2 Participant Recruitment . . . . . . . .
3.3 Sample Characteristics and Imputation
3.4 Exploratory Factor Analysis . . . . . .
3.5 Logistic Regression Model . . . . . . .
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137
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140
4
Empirical Results
4.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . .
4.2 Determinants of the Cat Bond Investment Decision . . .
4.3 Further Qualitative Results . . . . . . . . . . . . . . . .
142
142
150
159
5
Summary and Conclusion
165
6
Appendix
167
6.1 Aspects Encouraging Cat Bond Investments . . . . . . . 167
6.2 Aspects Opposing Cat Bond Investments . . . . . . . . . 168
6.3 Further Comments . . . . . . . . . . . . . . . . . . . . . 170
References
Curriculum Vitae
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172
177
List of Figures
vii
List of Figures
Companies
1
2
3
4
5
Acceptance
Acceptance
Acceptance
Acceptance
Acceptance
Range
Range
Range
Range
Range
from both Stakeholders’ Perspectives .
for Different Risk Aversion Parameters
for Different Fraud Amounts . . . . . .
for Different Insurance Premiums . . .
for Different Costs Per Audit . . . . .
22
26
26
28
29
Impact of Auditing Strategies on Insurers’
Profitability
6
7
8
9
10
11
12
13
14
Overview of the Processes Associated with the Filing
and Handling of Insurance Claims . . . . . . . . . . . . .
Interaction between Insurance Company and Policyholders
Development of the Optimal Auditing Strategy . . . . . .
Development of the Number of Audits and the Number
of Fraudulent Claims . . . . . . . . . . . . . . . . . . . . .
Development of the Insurance Company’s Net Present
Value and the Policyholders’ Gain in Utility . . . . . . . .
Auditing Range and Objective Quantities Depending
on Cost Per Audit . . . . . . . . . . . . . . . . . . . . . .
on Relative Fraud Amount . . . . . . . . . . . . . . . . . .
on Policyholder’s Initial Threshold Value . . . . . . . . . .
Auditing Range Including an Additional Auditing Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
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Empirical Analysis of Insurance Fraud
15
Contribution of Loss Amount to Predicting the Likelihood of Fraud . . . . . . . . . . . . . . . . . . . . . . . . . 112
viii
List of Tables
List of Tables
Companies
1
Input Parameters for the Reference Setting . . . . . . . .
21
Impact of Auditing Strategies on Insurers’
Profitability
2
Input Parameters for the Reference Setting . . . . . . . .
68
Empirical Analysis of Insurance Fraud
3
4
5
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10
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Descriptive Statistics for the Sample Composition I . . .
Descriptive Statistics for the Sample Composition II . .
Logistic Regression Results Model 1 . . . . . . . . . . .
Likelihood Ratio Tests for the Models . . . . . . . . . .
Classification Table for Full Model . . . . . . . . . . . .
Explanatory Variables Used in the Models . . . . . . . .
Variance Inflation Factors for All Explanatory Variables
Descriptive Statistics for the Whole Sample I . . . . . .
Descriptive Statistics for the Whole Sample II . . . . . .
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99
101
104
105
107
108
110
111
117
118
119
120
Drivers of Cat Bond Investments
15
16
17
18
19
20
21
22
23
24
Sample Composition . . . . . . . . . . . . . . . . . . . . . 144
Descriptive Statistics for the Company Sizes . . . . . . . . 147
Mann-Whitney U Test . . . . . . . . . . . . . . . . . . . . 148
Potential Determinants of the Investment Decision . . . . 149
Rotated Factor Loadings Matrix with Additional Statistics 152
Logistic Regression with all Potential Determinants . . . . 155
Classification Table for Model with all Potential Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Logistic Regression with Significant Determinants . . . . . 158
Classification Table for Model with Significant Determinants159
Open Questions . . . . . . . . . . . . . . . . . . . . . . . . 160
Introduction
ix
Introduction
The phenomenon of insurance fraud is usually met with an ambivalent
attitude as few individuals perceive insurance fraud as a white-collar
crime. At the same time, however, many are appalled by the ruthless actions undertaken to scam excessive indemnity payments at the expense
of the entire policyholder population. From the insurance company perspective, fraud represents a major challenge which has existed for many
years. It can be observed in all classes of insurance and results in a large
amount of excess payments each year. In this doctoral thesis, two out of
the overall four research papers focus on one particular aspect of insurance economics: developing theoretical models to derive approaches on
how to tackle different challenges related to insurance fraud.
The first paper, “Insurance Claims Fraud: Optimal Auditing Strategies in Insurance Companies”, presents a theoretical model framework
for determining conditions under which both the insurance company and
the policyholder are willing to enter into a relationship. Assuming that
fraud is indeed present, the aim is to derive an agreement range consisting of all auditing and defrauding strategy combinations that all
stakeholders are inclined to accept. Hereby, the behavioral strategies
are characterized by the probability of performing an audit or engaging
in fraudulent activities respectively. Our findings show that the number of all valid constructs strongly depends on parameters like insurance
premiums or the cost per audit. For a final statement with regard to optimality, however, it is necessary to take the participants’ market power
into consideration.
The following paper, entitled “The Impact of Auditing Strategies on
Insurers’ Profitability” alters the aspect of agreement and addresses the
question of how to optimally configure the auditing strategy from the
insurance company perspective. While the aim is to maximize the insurer’s objective quantity, we still take the policyholder perspective into
consideration to ensure a willingness to adhere to the insurance relationship. Our results evolve around optimal auditing strategies in the form
x
Introduction
of ranges triggered by the magnitude of a filed claim. Furthermore, we
allow for both stakeholders to adapt their behavior by picking up market
signals. One of our key findings shows that, against all expectations, it
is actually not favorable to try and anticipate the supposedly prevalent
auditing strategy from the policyholder point of view.
In connection with our studies, we were given the opportunity to
conduct numerous interviews with industry experts from different fraud
investigation divisions. In summary, they all agree on the severity of this
matter and collectively call for necessary actions to be taken. This dissertation provides a thorough insight into the characteristics and challenges
associated with insurance fraud and can help to implement appropriate
measures for handling it more effectively in insurance companies.
While the first part of the dissertation evolves around theoretical
frameworks, we set our focus on the analysis of insurance markets in the
second part. Founded by the increasing availability of data throughout
the last decades, empirical research has advanced to become a soughtafter approach in the field of insurance economics. It complements theorectical models and can help to promote a deeper understanding for the
underlying processes. With regard to behavioral patterns and decisionmaking, empirical research provides a wide range of methods for unveiling potential indicators and drivers. Based on this line of reasoning, the
remaining two research papers of this dissertation pertain to the area of
empirical research in the context of insurance markets.
The first paper of this dissertation in the context of market analyses
is entitled “The Identification of Insurance Fraud - An Empirical Analysis”. It is an empirical analysis of potential fraud indicators with a focus
on the automobile sector. Based on a comprehensive data sample from
a Swiss insurance company, we identify characteristics which allow for
a distinction between legitimate and illegitimate incoming claims. The
set of significant determinants comprises variables on the policyholder,
vehicle, policy and loss level. Based on our findings, we are able to draw
a profile which identifies fraud-prone policyholders as middle-aged indi-
Introduction
xi
viduals having a good driving record and owning high-valued vehicles.
It seems worth mentioning that the option to cheat on one’s insurance
company is solely taken into consideration in the case of small- and
medium-sized losses. These results indicate that individuals try to anticipate the supposed auditing strategy to hopefully remain undetected.
Finally, the second paper in the center of empiricism “What Drives
Insurers’ Demand for Cat Bond Investments? Evidence from a PanEuropean Survey” addresses the issue of low demand in cat bond investments from the insurance and reinsurance company perspective in
Europe. Despite the asset class’ attractive risk-return profile and diversification potential, insurers account for merely ten percent of the current
demand in the market, raising the question as to how this observation
may be explained. Our comprehensive study reveals that the firm’s experience and expertise related to cat bonds, their perceived fit with the
asset and liability management goals as well as the prevailing regulatory
regime have a significant impact on the companies’ decision whether or
not to invest in this particular asset class. The results seem to be of high
relevance, particularly for practitioners and regulators, and can support
further growth of this still relatively small segment of the capital markets.
Founded on a broad data base, the last two research papers of this
thesis uncover main indicators for fraudulent behaviour and drivers in
the decision-making process of insurance companies related to cat bonds.
The results are not only of interest to practitioners, but also provide new
insights which may help to advance theoretical models.
Einführung
xiii
Einführung
Ansichten zum Thema Versicherungsbetrug sind stark divergierender Natur. Nicht wenige sehen darin einen Kavaliersdelikt oder eine Gelegenheit, vergangene Prämienzahlungen zusätzlich zu kompensieren. Demgegenüber wird die Rücksichtslosigkeit von vielen verurteilt - insbesondere
basierend auf der Erkenntnis, dass die einhergehenden Zahlungen von
der Gesamtheit aller Versicherungsnehmer getragen werden. Aus Sicht
der Unternehmen ist Versicherungsbetrug eine der zentralen Herausforderungen. Betrugsversuche sind in fast allen Versicherungssparten zu
finden und haben jedes Jahr erhebliche Zahlungen zur Folge. In der vorliegenden Dissertation sind zwei der insgesamt vier Forschungsarbeiten
dieser Thematik gewidmet. Unser Bestreben ist es, mit Hilfe theoretischer Modellrahmen, Empfehlungen zum Umgang mit Betrugsfällen und
-versuchen herzuleiten.
Die erste Forschungsarbeit mit dem Titel “Insurance Claims Fraud:
Optimal Auditing Strategies in Insurance Companies” hat zum Ziel, Konditionen herzuleiten und zu analysieren, unter denen sowohl Unternehmen als auch Versicherungsnehmer grundsätzlich bereit sind, ein Vertragsverhältnis miteinander einzugehen. Hierzu betrachten wir aus Sicht
des Versicherungsunternehmens die Wahrscheinlichkeit, ein Prüfverfahren
einzuleiten, während bei den Versicherungsnehmern die Betrugswahrscheinlichkeit einbezogen wird. Unter der Annahme, dass Betrug tatsächlich stattfindet, werden diejenigen Verhaltenskombinationen ermittelt,
die beide Teilnehmer jeweils gewillt sind zu akzeptieren. Unsere Ergebnisse zeigen, dass die Ausprägung dieses Einigungsbereiches insbesondere
von der Wahl der Parameter Prämienhöhe und Prüfkosten abhängt. Eine abschliessende Bewertung mit Blick auf eine Gleichgewichtssituation
ist lediglich unter Hinzunahme der Markmacht der Parteien möglich.
In der zweiten Arbeit “The Impact of Auditing Strategies on Insurers’
Profitability” verabschieden wir uns von der Zielsetzung einer Einigung,
und untersuchen die Konfiguration optimaler Kontrollstrategien. Dazu
versetzen wir uns in die Lage des Versicherungsunternehmens, unter
xiv
Einführung
Berücksichtigung der Teilnahmebedingungen des Versicherten. Die resultierenden Prüfstrategien indizieren, ob ausgehend von der Schadenhöhe
eine Überprüfung eingeleitet oder die Meldung direkt abgewickelt werden
soll. Zusätzlich erweitern wir unser Modell um die Option einer kontinuierlichen Verhaltensanpassung beider Parteien. Es stellt sich heraus, dass
es für den Versicherungsnehmer entgegen der Intuition unvorteilhaft ist,
das vermutete Kontrollverhalten des Versicherers zu antizipieren.
Im Zuge unserer Forschung hatten wir mehrfach die Gelegenheit, Interviews mit Versicherungsexperten aus dem Bereich der Betrugsbekämpfung zu führen. Zusammenfassend kann gesagt werden, dass angesichts
der umfassenden Präsenz und der wirtschaftlichen Implikationen, adäquate Lösungen für den effektiven Umgang mit Versicherungsbetrug immer
wichtiger werden. Der erste Teil dieser Dissertation gewährt einen umfassenden Einblick sowohl in die Thematik als auch die einhergehenden
Herausforderungen und legt Vorschläge für geeignete Massnahmen dar.
In Ergänzung zu den modelltheoretischen Ansätzen der ersten beiden
Forschungsarbeiten ist der zweite Teil der Dissertation der empirischen
Analyse von Versicherungsmärkten gewidmet. Im Zuge der zunehmenden Verfügbarkeit von Daten und Informationen in den vergangenen
Jahrzehnten sind empirische Analysen zu einem zentralen Element der
Forschung im Bereich Versicherungswirtschaft avanciert. Sie sind als Mittel zur Ergänzung und Verifizierung theoretischer Modelle zu sehen, und
können dazu beitragen, zusätzliche Erkenntnisse über zugrundeliegende
Prozesse zu gewinnen. Insbesondere mit Blick auf die Erforschung von
Verhaltensmustern und Entscheidungsprozessen bietet die Empirie ein
breites Spektrum von Verfahren und Methoden an.
Bei der ersten empirischen Forschungsarbeit mit dem Titel “The Identification of Insurance Fraud - An Empirical Analysis” handelt es sich um
eine Analyse von Betrugsindikatoren. Ausgehend von einem umfassenden
Datensatz geprüfter Schadenmeldungen aus der Motorfahrzeugversicherung eines grossen Schweizer Versicherers werden Faktoren ermittelt, die
eine eindeutige Abgrenzung zwischen ehrlichen und betrügerischen Mel-
Einführung
xv
dungen ermöglichen. Die potenziellen Kriterien sind auf Ebene des Versicherten selbst, dem Motorfahrzeug, der Police sowie der Schadenmeldung
angesiedelt. Unsere Ergebnisse porträtieren betrugswillige Versicherungsnehmer als Personen mittleren Alters mit einer überdurchschnittlichen
Anzahl von Schadenfreiheitsjahren und in Besitz von hochpreisigen Fahrzeugen. Insbesondere scheinen sie Betrugsversuche lediglich im Falle von
kleinen und mittleren Schäden in Erwägung zu ziehen. Diese Erkenntnisse legen die Vermutung nahe, dass Versicherungsnehmer in der Tat
Vermutungen über die bestehenden Kontrollmechanismen anstellen und
versuchen, diese zu umgehen.
Die zweite Arbeit im Fokus der Empirie widmet sich dem Investitionsverhalten von Versicherungsunternehmen in Bezug auf Katastrophenanleihen. Unter dem Titel “What Drives Insurers’ Demand for Cat Bond
Investments? Evidence from a Pan-European Survey” werden potentielle Entscheidungsfaktoren für oder gegen diese Anlageklasse ermittelt.
Trotz ihres attraktiven Risiko-Rendite-Profils und Diversifikationspotentials kommen Versicherer lediglich für zehn Prozent der gesamten Nachfrage im Markt auf. Dies wirft die Frage nach möglichen Erklärungen
auf. Zu diesem Zweck wurde eine detaillierte Umfrage mit zahlreichen europäischen Erst- und Zweitversicherungsunternehmen durchgeführt. Die
statistische Auswertung der Angaben zeigt, dass die Faktoren “Erfahrung und Expertise” in Bezug auf Katastrophenbonds, deren “Kompatibilität mit den Assetmanagementzielen” des Unternehmens sowie die
“bestehenden Regulierungsrahmenwerke” einen signifikanten Einfluss auf
die Investmententscheidung haben. Unsere Ergebnisse sind für Praktiker
und Regulierer von besonderer Relevanz, und können dazu beitragen, aktuelle Investitionsbarrieren zu überwinden.
Ausgehend von einer umfassenden Datengrundlage aus zwei verschiedenen Bereichen des Versicherungsmanagement werden in den letzten
beiden Forschungsarbeiten Indikatoren für Betrugsverhalten sowie treibende Faktoren bei der Investitionsentscheidung von Versicherungsunternehmen ermittelt und analysiert. Die Ergebnisse sollten dazu geeignet
xvi
Einführung
sein, neue Impulse für die Weiterentwicklung von theoretischen Modellen
zu geben.
1
Part I
in Insurance Companies
Abstract
Insurance claims fraud is one of the major concerns in the insurance industry. According to many estimates, excess payments due to fraudulent
claims account for a large percentage of the total payments affecting all
classes of insurance. In this study, we develop a model framework based
on a costly state verification setting in which, while policyholders observe the amount of loss privately, the insurance company can decide to
audit incoming claims at some cost. In particular, optimization problems are formulated from both stakeholders’ positions considering that
for each of them willing to sign an insurance contract, certain participation constraints need to be fulfilled. Besides deriving analytical solutions
regarding optimal fraud and auditing strategies, we provide a numerical
approach based on Monte Carlo methods. The simulation results illustrate the acceptance range that consists of all valid fraud and auditing
probability combinations both stakeholders are willing to tolerate. We
discuss the impact of different valid probability combinations on the insurance company’s and policyholder’s objective quantities respectively
and analyze the sensitivity of the acceptance range with respect to different input parameters. 1
1 K. Müller, H. Schmeiser, and J. Wagner. Insurance Claims Fraud: Optimal Auditing Strategies in Insurance Companies. Working Papers on Risk Management and
Insurance, 2011.
This paper has been presented at the Annual Meeting of the Western Risk and Insurance Association in January 2012 and at the Jahrestagung des Deutschen Vereins
für Versicherungswissenschaft in March 2012.
2
1
I Theory of Insurance Fraud
Introduction
Insurance claims fraud has been one of the major concerns in the insurance industry for a long time and has attracted much attention in
both scientific and firm-level environments. There exist many research
papers and studies on the detection and deterrence of fraudulent activities and the optimal design of insurance contracts (see, e.g., Viaene
and Dedene (2004), Picard (2009), Dionne, Giuliano, and Picard (2009),
GDV (2011)). Despite all efforts, the Association of British Insurers
(2012) reports the uncovering of 139’000 dishonest claims in the UK in
2011 alone, adding up to almost 1 billion in illegitimate loss reports;
the estimated number of unrevealed cases is assumed to be substantially
higher. In this study, we show that with the goal of minimizing insurance companies’ costs, the complete elimination of fraudulent activities
is not always desirable. We derive acceptance ranges that comprise all
valid fraud and auditing probability combinations under which contract
conditions remain attractive enough for both insurer and policyholder
to adhere to the insurance relationship. The actual strategies are chosen
based on the respective market power.
Insurance claims fraud is a multi-layered phenomenon. While it is
often associated with criminal activities, only a minority of illegitimate
claims is said to contain outright fraud (see, e.g., Viaene and Dedene
(2004), Tennyson (2008)). This observation is probably due to the fact
that for a case to be declared criminal fraud it needs to be proven that it
is ”a willful act of obtaining money or value from an insurer under false
pretenses or material misrepresentations” (Derrig (2002)). The more
common and frequent type of insurance claims fraud is referred to as
soft fraud. Even though there exists no clear definition of the term, it
is generally associated with the attempt to exaggerate the magnitude
of an otherwise legitimate claim (see, e.g., Weisberg and Derrig (1991),
Viaene and Dedene (2004), Tennyson (2008)). This form of inflation is
also called build-up. In the context of our study, we use the term insurance claims fraud in the sense of soft fraud or build-up, i.e., fraud-prone
1 Introduction
3
policyholders claiming loss amounts exceeding their actual value.
The appearance of insurance claims fraud is based on information being asymmetrically distributed between the policyholder and the corresponding insurance company (see, e.g., Derrig (2002)). Since individuals
observe the amount of loss privately after the time of occurrence, they
may decide to misrepresent the magnitude. In a costly state verification environment as applied by researchers (see, e.g., Townsend (1979),
Mookherjee and Png (1989), Bond and Crocker (1997) and Picard and
Fagart (1999)), the insurer has the opportunity to perform verification
processes to determine the truthfulness of an incoming claim. Any detected engagement in fraudulent activities can be charged with a penalty
payment. However, since this auditing comes at some cost, the insurance
company has to weigh the benefits against the accompanying expenses.
Another component that needs to be considered in this calculation is the
policyholder perspective. Viaene and Dedene (2004) found that policyholders who had negative experiences throughout the insurance relationship such as delayed indemnifications or underpayment were more likely
to engage in fraudulent activities.
While costly state verification is based on the insurance company
being able to detect attempts of misrepresentation, the costly state falsification approach focuses on the policyholder incurring some cost to
manipulate the magnitude of loss such that it becomes unverifiable. The
general setting was introduced by Lacker and Weinberg (1989) and transferred to the specific features of the insurance environment by Crocker
and Morgan (1998) and Crocker and Tennyson (2002).
A different approach in the fight against insurance claims fraud was
taken by Dionne and Vanasse (1992) and Moreno, Vázquez, and Watt
(2006). Instead of engaging in cost-intensive auditing, the insurance company raises the insurance premium whenever the policyholder filed some
claim in the previous period. This strategy is often applied in the automobile insurance sector where contract renewals are a common standard.
As a result, illegitimate claims when no insured loss occurred may be prevented.
4
In this study, we consider an alternative approach on the subject
of claims handling, especially when fraud is present. For this purpose,
we focus on the parties’ behavioral strategies, i.e., from the insurance
company perspective, we consider the verification scheme. It is characterized by the probability of performing an audit, whereas from the
policyholder point of view, the defrauding strategy is taken into account
as represented by the probability of engaging in fraudulent activities.
The aim is to identify and analyze all conditions under which the insurance relationship is attractive for the insurance company and the
policyholder, i.e., both stakeholders are willing to adhere to the insurance contract. Particularly, for any fraud strategy, i.e, the probability of
filing an inflated loss amount, we determine the set containing all valid
corresponding auditing probabilities, given some constant cost per audit and vice versa. With regard to optimality, we can make predictions
based on the stakeholders’ respective market power.
Previous research has focused on deriving optimal contracts such that
at equilibrium, the policyholders have an incentive to always report their
losses truthfully (see, e.g., Townsend (1979), Picard and Fagart (1999)).
However, the question arises as to how the two stakeholders representing
two opposing groups of interest can be brought together in the first place.
Which behavioral patterns, i.e., defrauding and auditing probabilities, is
the respective other willing to accept without being worse off than in the
situation when no insurance contract was signed prior to the occurrence
of loss? From the insurance company perspective, one can assume that,
given some fixed cost per audit, it is not appealing to enter an insurance
relationship with individuals who engage in build-up on a large scale in
terms of frequency and severity. From the policyholder point of view
on the other side, it appears to be intuitive to assume that delayed
or reduced indemnification due to extensive auditing might curtail the
attractiveness of insurance.
Obtaining the resulting set of all acceptable fraud and auditing probability combinations, we make a crucial observation. In the context
of cost-minimizing insurance companies, a mutual acceptance between
both stakeholders can be reached for any fraudulent behavior the policyholder might exhibit. In particular, even in the case when the loss
1 Introduction
5
amount is inflated in most of the incoming claims, there may exist auditing strategies such that the insurer is willing to adhere to the insurance
relationship and even be able to optimize its objective quantity. This
result underlines the expectation expressed by Watt (2003) showing that
for cost-minimizing insurance companies it is not necessarily desirable
to undercut all fraudulent activities.
Based on the results of the acceptance range, we analyze all valid
fraud and auditing probability combinations with respect to their optimality for the stakeholders’ respective objective quantities. As expected,
we are able to show that the best possible outcome can never be achieved
for both the insurance company and the policyholder at the same time.
Which one out of all valid behavioral strategy combinations they settle
on, depends on their respective market power.
We want to emphasize that the acceptance range is not to be understood as a cooperative agreement that both parties decide upon. It
intends to demonstrate the range of all possibilities attractive enough so
that both insurer and policyholder are willing to maintain the insurance
relationship.
The model derived in this study is based on the costly state verification environment considering the insurance company’s net present value
of future cash flows and the policyholder’s expected utility of his terminal wealth position. To make sure that both stakeholders are willing to
sign an insurance contract, we include participation constraints. We derive and analyze some analytical solutions to the optimization problems.
Due to the complexity of the model, however, it is not always possible
to obtain closed-form solutions. Therefore, we present a numerical approach using Monte Carlo methods. The simulation results and their
implications for both stakeholders’ optimal strategies are discussed and
illustrated graphically.
The remainder of this study is organized as follows: we start by
presenting the model framework and first analytical results in Section 2.
Thereafter, the numerical approach and the corresponding program are
6
introduced in Section 3. In Section 4, we discuss the simulation results
before concluding in Section 5.
2
Model Framework
An individual with initial wealth W0 is offered the possibility to sign an
insurance contract with a fixed premium P due by the time of inception
of insurance coverage in t = t0 . At the same time, he faces some uncertain loss θ of stochastic amount which, by the time of occurrence, is
observed privately. In case he signed the insurance contract earlier, the
policyholder can then chose to file a claim of some size θ̂. In the case
of honest behavior, the amount of that claim will equal the actual loss,
i.e., θ̂ = θ. If the policyholder decides to commit fraud, he reports some
finite θ̂ > θ. The probability of the policyholder choosing to report a
fraudulent claim is denoted by p.
The insurer on the other hand has no information about the actual
occurred loss. He therefore audits incoming claims with some probability
q and at the constant cost of k per audited claim. Depending on whether
auditing took place or not and the outcome in case of an audit, a payment
R is made from the insurance company to the policyholder. Considering
the different possible combinations of fraud and auditing probabilities p
and q, the payment R can be defined as follows:
R(θ, θ̂) = (1 − p) θ + p [(1 − q) θ̂ + q (θ − B)],
(1)
with B being the penalty payment deducted from the claim amount θ.
Equation (1) can be interpreted as an indemnity payment if R is
positive, whereas a negative R represents the payment made from the
insured to the insurance company in case of detected fraud when B > θ.
There are several possible cases: if the reported loss is not checked, the
insured receives the payment of θ̂. In the case of auditing, the payment
depends on whether the policyholder committed fraud or not. Proven
honesty leads to a payment of θ = θ̂. If a misrepresentation of loss is
determined, the policyholder faces some penalty B. In practice, B is
mostly chosen such that θ − B = 0.
2 Model Framework
7
In this setting, we take audits to be perfect, i.e., if a fraudulent claim
is made, it will surely be detected in the case of auditing.
In the following, we introduce the setting as well as the objective
quantity and the participation constraint from the insurance company
perspective. The same is done from the policyholder point of view.
Based on this information, we state the resulting optimization problems
for both stakeholders in Section 2.1. Assumptions about the distribution
of information among the policyholder and the insurance company are
given in Section 2.2 before presenting analytical results in Section 2.3.
Insurance Company: Cash Flow, Net Present Value,
Participation Constraint
In the framework introduced above, we observe the future cash flows
from the insurance company perspective at the time of insurance inception in t = t0 and the time of loss realization and settling in t = t1 and
analyze their resulting present value.
In the case of an insurance contract coming into existence, the insurance company receives the premium payment P in t = t0 . An incoming
claim in t = t1 that is audited with probability q and at some given cost
k(> 0) per analyzed claim, results in −R(θ, θ̂) − qk.
The insurance company’s net present value of its future incoming and
outgoing cash flows is denoted by
N P V = P − E(R(θ, θ̂)) − qk, 2
(2)
where R(θ, θ̂) denotes the indemnity payment as defined in (1).
Condition 1 The insurance company is willing to participate in an insurance contract if its net present value is positive. Hence, one obtains
the following participation constraint:
2 We consider the expected value of future cash flows discounted with the risk-free
interest rate rf = 0.
8
N P V ≥ 0.
(3)
Applying Equation (2), participation constraint (3) can be formulated as
P ≥ E(R(θ, θ̂)) + qk.
(4)
Apparently, the expression on the right-hand side represents a lower
bound for the premium payments the insurance company is willing to
accept. Its value depends on the expected value of the indemnity payments that will be made and a loading that reflects the auditing effort.
Policyholder: Wealth Position, Utility Function,
From the policyholder perspective, we analyze his wealth position and
the corresponding expected utility at the time of inception of insurance
cover t = t0 and the time of loss realization and claiming t = t1 for the
framework introduced above.
An individual initially owns some wealth W0 . Its consecutive development depends on whether he signs an insurance contract prior to the
occurrence of loss or not. In a situation without an insurance contract,
the individual holds the unchanged wealth position:
W0B = W0
(5)
at time t = t0 . At the time of loss occurrence in t = t1 , this amount
decreases to:
W1B = W0B − θ = W0 − θ.
(6)
The decision to sign an insurance contract is accompanied by the
payment of an insurance premium P . Consequently, when signing the
contract at time t = t0 , the individual owns the wealth position:
2 Model Framework
9
W0A = W0 − P.
(7)
Assuming a loss θ of some stochastic level occurs and therefore a
claim is filed at time t = t1 , the policyholder’s wealth at that point in
time is denoted by:
W1A = W0A − θ + R(θ, θ̂) = W0 − P − θ + R(θ, θ̂)
(8)
with R(θ, θ̂) as defined in (1).
We assume the policyholder’s utility being described by a standard
mean-variance utility function of his individual wealth. For a given
wealth position W and the risk aversion parameter a(≥ 0) of the individual, this utility function is given by
U (W ) = E(W ) −
a
Var(W ),
2
(9)
where E(W ) denotes the expected value of the stochastic variable W .
In the case where no insurance contract was signed prior to the occurrence of loss, using Equation (6) and definition (9) the final utility is
written as:
a
Var(W0 − θ)
2
a
= W0 − E(θ) − Var(θ).
2
U (W1B ) = E(W0 − θ) −
(10)
For the setting in which an insurance contract was signed by applying
the definition in (9) to Equation (8), we obtain:
a
Var(W0 − P − θ + R(θ, θ̂))
2
a
= W0 − P − E(θ − R(θ, θ̂)) − Var(θ − R(θ, θ̂)).
(11)
2
U (W1A ) = E(W0 − P − θ + R(θ, θ̂)) −
Comparing Equations (10) and (11), one sees the difference in influencing factors for the final expected utility for each situation. In a
10
setting without the existence of an insurance contract, the final expected
utility U (W1B ) solely depends on the extend of the actual loss θ and the
policyholder’s risk aversion parameter a. However in a situation in which
insurance coverage exists, the value of the corresponding expected utility
U (W1A ) is not only influenced by θ, P and a. In addition, the policyholder’s fraud strategy p and the insurer’s auditing strategy q have an
impact on that value due to the payment of R (see (1)). Moreover, if the
insured decides to commit fraud, the size of θ̂ that he chooses to claim
is relevant as well as the enforced penalty payment B (see (1)) in case
the fraudulent claim gets detected.
Condition 2 The individual’s decision to get insurance coverage in the
first place depends on whether his utility by the time of loss occurrence
is greater with having insurance than without it. In other words:
U (W1A ) ≥ U (W1B ).
(12)
Using (11) and (10) this participation constraint (12) can be written
as:
−P + E(R(θ, θ̂)) −
a
a
Var(θ − R(θ, θ̂)) ≥ − Var(θ)
2
2
a
⇐⇒ P − E(R(θ, θ̂)) ≤ − Var(R(θ, θ̂)) + a Cov(θ, R(θ, θ̂)).
2
(13)
Based on the representation
P ≤ E(R(θ, θ̂)) −
a
Var(R(θ, θ̂)) + a Cov (θ, R(θ, θ̂)),
2
the inequality in (13) can be interpreted as an upper bound for the
insurance premium the potential policyholder is willing to pay for his
insurance coverage. It depends on the utility of the payment R and the
covariance between actual loss θ and R. Furthermore, the individual’s
risk aversion parameter a has an influence on his willingness to pay.
2.1 Optimization of Positions
2.1
11
Optimization of Positions
So far, the model framework and the participation constraints for both
the policyholder and the insurance company have been presented. Based
on this information, we state the corresponding optimization problems.
The insurance company is aiming to maximize the net present value
of the incoming and outgoing future payments with respect to its audit strategy such that both stakeholders are still willing to participate,
i.e., Equations (12) and (3) hold. Again, it is assumed that the other
parameters are given. This objective function can be written as:
Insurance Company’s Optimization Problem
Find the optimal audit strategy q s.t. N P V is maximized
(14)
and Equations (12), (3) hold.
At the same time, the policyholder’s aim is to maximize his final
expected utility with respect to his fraud strategy such that both participation constraints (12) and (3) hold, i.e., an insurance contract exists.
It is assumed that all the other parameters are given. We will denote
this optimization problem by:
Policyholder’s Optimization Problem
Find the optimal fraud behavior p s.t. U (W1A ) is maximized
(15)
and Equations (12), (3) hold.
Both stakeholders try to optimize their own respective position. Our
aim is to analyze these conflicting objectives and participation constraints
from both the insured’s and insurer’s perspective and find a common acceptance range for the resulting fraud and auditing strategies.
12
2.2
Assumptions About the Distribution of
Information
Before presenting the results of our analytical analyses, we summarize
the assumptions regarding the distribution of information among the
stakeholders.
Insurance Company Perspective
The choice of the insurance company’s feasible auditing strategies depends on the policyholder’s prevalent defrauding behavior, i.e., the potential fraud amount and the probability of an incoming claim to be inflated. The insurance company is assumed to have full information about
the distribution of the reported losses θ̂ due to having observed incoming
claims to date. In particular, this information can be specified for each
insurance segment or even loss type. Furthermore, we expect that it has
an adequate estimate for the distribution of the actual losses θ based on
the outcomes of previous auditing processes. Consequently, the insurer is
able to deduce the deviation from the magnitude that is to be expected
for the particular loss type α := θ̂/θ, i.e., the potential fraud amount
in case fraudulent behavior occurs. Since the optimal auditing strategy
also depends on the second component of the policyholder’s defrauding
strategy, the prevalent fraud probability p, the insurance company has
to estimate this value as well. For this purpose, whole catalogs consisting of criteria, so-called red flags, have been derived and implemented
aiming to estimate the probability of a claim being illegitimate as accurately as possible (see, e.g., Belhadji, Dionne, and Tarkhani (2000) and
Bermúdez, Pérez, Ayuso, Gómez, and Vázquez (2008)). Such indicators
can be targeted at the individuals’ characteristics itself like gender, nationality or place of residence as well as the attributes associated with
the loss event. Combining this information, one is able to obtain a precise predictor for the fraud probability p (see, e.g., Dionne, Giuliano,
and Picard (2009)). It then chooses the corresponding optimal audit
probability that maximizes its N P V in response.
2.3 Analytical Results
13
Policyholder Perspective
We assume that fraud-prone policyholders do not have sufficient information about the insurance company’s auditing process itself, i.e., he or
she does not know the exact criteria for a claim to undergo verification.
As a consequence, the individual cannot manipulate the claim in a way
such that the insurer would not be able to identify the fraud attempt.
This assumption is essential and not unrealistic. Waiving it would make
auditing of any kind redundant since the insurance company would never
be able to detect loss inflation or other kinds of manipulation regardless
of how the verification process is designed.
For the purpose of our analysis, we assume the policyholder to be
able to estimate the probability of being audited by the insurance company when submitting a claim. This assumption is not in conflict with
the one made before. The knowledge of the probability of one’s claim
being audited does not imply an ability to manipulate the verifiability.
Rather, it provides the policyholder with the possibility to become aware
of which fraud behavior is advantageous in this particular situation and
maybe choose the optimal one.
In the context of our study, we consider one observation period and
determine all potentially feasible behavioral strategies from both the insurance company and the policyholder perspectives. Based on their decisions, however, other behavioral strategies may become more favorable
for one or the other party in the consecutive periods. Therefore, the potential behavioral options need to be reconsidered by both stakeholders
at the beginning of each observation period.
2.3
Analytical Results
In the course of this subsection, we derive analytical results for the presented optimization problems assuming different conditions. The proofs
can be found in the Appendix.
In the first proposition, we derive optimal fraud and auditing strategies p and q for a special setting of the model framework. The crucial
14
assumption in this case is concerning the policyholder’s risk aversion
parameter a that is set a = 0, i.e., we assume the policyholder to be riskneutral. This implies optimizing the insured’s objective function from a
present value perspective.
Proposition 1 For a = 0 and θ, θ̂ such that 0 ≤ θ < θ̂, the optimal
fraud and auditing strategies from both stakeholders perspectives’ are p =
1 and q = 0. This results in P = E(θ̂).
This proposition implies that under the given assumptions, the insurance company should waive auditing incoming claims and allow fraudulent behavior instead. In return, the expected amount of fraud will be
added to the insurance premium. Furthermore, the proposition confirms
a characteristic behavior that risk-neutral policyholders show. They are
assumed to have no interest in insuring a potential loss at a premium
which exceeds its expected value.3 Since in this specific setting, all policyholders claim the fraudulent amount θ̂ at all times, the premium cannot
be set higher than the expected value of θ̂. On the other hand, for the
insurance company to be willing to participate in the insurance relationship, this premium cannot deceed this value. Hence, the insurance
premium equals exactly E(θ̂).
In the remainder of this subsection, optimal fraud and auditing strategies will be derived for the policyholder and insurance company respectively in a more general setting. First of all, the policyholder is assumed
to be risk-averse, i.e., the risk aversion parameter a is strictly positive,
a > 0. This means his objective function is actually given as an expected
utility function, i.e., the variance of the difference between indemnity
payment R(θ, θ̂) and actual loss θ, denoted by Var(θ − R(θ, θ̂)), has an
impact on the final result.
Furthermore, whenever the policyholder decides to make a fraudulent
claim, he reports θ̂ = αθ for some given finite α ≥ 1 to the insurance
company. This setting implies that the relative amount of fraud is constant. We deem it likely to assume that fraud-prone policyholders take
the actual loss amount into consideration when trying to inflate it, i.e.,
3 See,
e.g., Kirstein (2000).
2.3 Analytical Results
15
they regard the relative fraud amount as a percentage surcharge. In
this way, the filed claim does not deviate substantially from the loss
amounts that can be expected related to the corresponding loss type.
Consequently, the inflated claim is not perceived as illegitimate by the
insurance company which makes it less probable to undergo verification.
This assumption is in line with the observations stated by Viaene and
Dedene (2004). They found that in the context of soft fraud, the excess
amounts tend to be relatively small.
We will derive optimal fraud and auditing strategies, namely popt
and q opt , for the setting introduced above. Other than in Proposition 1,
the potential policyholder is assumed to be risk-averse.
Proposition 2 Assume p, q to be in the acceptance range, i.e., an insurance contract exists. For a > 0, B = θ, θ̂ = αθ with some given
α ≥ 1, the respective optimal strategies popt , q opt are given by:
(i) Insurance company perspective
Let some p be given. In order for the net present value N P V to be
maximized, choose
q
where p∗ :=
opt
=
(
as large as possible if
p > p∗
as small as possible if
p ≤ p∗
,
(16)
k
.
αE(θ)
(ii) Policyholder perspective
Let some q be given. In order for the final expected utility U (W1A )
to be maximized, choose
popt

as large as possible s.t.
=
as small as possible s.t.
−E(θ)
ap(1−α(1−q)) Var(θ)
−E(θ)
≥1
≤0
. (17)
16
For the case 0 <
be made.
−E(θ)
< 1 no general statement can
Proposition 2(i ) looks at the optimization problem from the insurance company perspective. It states the optimal auditing strategy with
respect to a given fraud probability. The insurance company has two
general strategies to choose from. It can either decide to audit the incoming claims with the maximal probability possible, i.e., such that the
participation constraints of both policyholder and insurance company
hold true, or the auditing probability can be chosen as small as possible.
This decision depends on an estimate of the policyholder’s behavior p.
k
, the insurBased on whether it exceeds or deceeds the threshold αE(θ)
ance company opts for a high or low auditing probability respectively.
According to Proposition 2(i ), the exceed of the threshold is influenced
by the costs per audit k. The lower these are, given some fixed α and
θ, the more likely it is for the fraud probability to exceed the resulting
threshold. In this case, it becomes optimal for the insurance company
to verify the incoming claims with a high probability. The opposite relationship holds true for the expected loss amount θ and the degree of
fraud that is represented by α. The higher their values are, the lower
the threshold becomes and the more likely it is for the estimated fraud
probability to exceed the latter. For the insurance company this implies
auditing the incoming claims with the highest probability possible as
well. For an illustration of the results obtained in Proposition 2(i ) see
Figure 1(a) and the discussions in Section 4.1.
Proposition 2(ii ) considers the policyholder point of view in this optimization problem. In this case, the decision whether to chose the fraud
probability as large or small as possible given a certain auditing strategy,
is not as clear as in the previous situation described in Proposition 2(i )
especially since there are situations for which no forecast can be made.
Furthermore, difficulties arise when trying to interpret the impact of single model parameters on the value of the threshold that determines the
optimal auditing behavior in the known cases. However, see Figure 1(b)
and the discussions in Section 4.1 for an illustration of the optimal fraud
3 Computational Aspects
17
probability from the policyholder perspective.
The challenges that occur with finding a closed-form analytical solution to the introduced maximization problem emphasize the need for
a numerical approach. In Section 3, we therefore present a method
for deriving the acceptance range for both policyholder and insurance
company. Furthermore, the impact of valid p − q combinations on the
objective quantities U (W1A ) and N P V is analyzed and illustrated.
3
Computational Aspects
As discussed in the previous section, simple analytical solutions to the optimization problem cannot be derived for all general settings. Moreover,
the results may be hard to interpret both graphically and economically.
In this section, we will approach these challenges by using numerical
methods and Monte Carlo simulation. The aim is to compute the acceptance range with respect to the fraud and auditing strategies for various
parameterizations of the model. After having introduced the procedure,
the results of the simulations will be analyzed and presented graphically.
Monte Carlo Simulation and Numerical Methods
We use the Monte Carlo technique to find the optimal acceptance range
regarding the fraud and auditing strategies of the policyholder and the
insurance company respectively. The main idea behind this approach is
to generate a sufficiently large number of realizations N of the random
variable θ. Furthermore, we consider all fraud and auditing probabilities
1
for l = 0, 1, ..., M where M denotes
p and q that are represented by l · M
the number of discretization points on the interval [0, 1]. Based on these
assumptions, the resulting indemnity payments R, the policyholder’s
wealth positions with and without having signed the insurance contract
W1A and W1B and the insurance company’s value V are calculated for
each outcome of the simulation and each fraud and auditing probability
combination. Using Equations (1), (6) and (8) for R, W1B and W1A
respectively, this can written as follows:
18
R[n, i, j] = (1 − p[i])θ[n] + p[i]((1 − q[j])αθ + q[j](θ[n] − B[n])) (18)
W1B [n, i, j] = W0 − θ[n]
(19)
W1A [n, i, j]
(20)
= W0 − P θ[n] + R[n, i, j],
where θ[n] denotes the nth realization of the random variable θ and p[i]
and q[j] are the considered fraud and auditing probability represented
1
1
by i M
and j M
for i, j = 0, 1, ..., M respectively. Consequently, the term
[n, i, j] indicates for which combination of loss realization and fraud and
auditing probabilities the quantities R, W1B and W1A are evaluated.
The next step to determining the acceptance range is to derive the
objective quantities, i.e., the policyholder’s final utility depending on
whether he signed the insurance contract prior to the loss or not and the
insurance company’s present value based on the corresponding wealth
and value positions calculated before. For this purpose, we use arithmetic averaging with respect to the realizations of the random variable
θ for each possible combination of p and q. Regarding the individual’s
final utility when having decided against insurance coverage, we use the
following formula, derived from Equation (10):
a
(21)
U (W1B )[i, j] = µ̂n (W1B [n, i, j]) − σ̂n2 (W1B [n, i, j]),
2
where µ̂n denotes the estimator for the expected value with respect
to all realizations n = 1, ..., N and σ̂n2 the estimator for the variance with
respect to all realizations n = 1, ..., N . The same procedure applies for
the case when an insurance contract was signed, this time using Equation
(11):
a
U (W1A )[i, j] = µ̂n (W1A [n, i, j]) − σ̂n2 (W1A [n, i, j]).
(22)
2
From the insurance company point of view, the net present value of
its future incoming and outgoing cash flows depending on the fraud and
auditing probability can be derived as follows, based on Equation (2):
N P V [i, j] = P − µ̂n (R[n, i, j]) − q[j]k.
(23)
3 Computational Aspects
19
We are now in the position to check for the participation constraints
of both the policyholder and the insurance company. Only if these hold
true, an insurance contract comes into existence and merely in this case,
the optimization problems are well defined. The idea here is to systematically analyze the participation constraints given in Equations (12) and
(3) for each combination of fraud and auditing probabilities. In case
these are verified, we consider the corresponding p − q combination as
valid. At the end of this procedure, we obtain the acceptance range.
The actual aim is to find the optimal strategies p and q such that the
objective quantities, i.e., the policyholder’s final wealth position U (W1A )
and the present value of the insurance company’s future incoming and
outgoing cash flows N P V , are maximized from each of the participants’
perspectives. For these to be determined, we calculate the results for
U (W1A ) and N P V evaluated with respect to the valid p − q combinations respectively. Once the maximal values have been found, we can
retrace the corresponding fraud and auditing probabilities under which
the maximum was attained. This procedure is performed separately for
the two participants.
Choice of Parameters
We analyze the implementation of the model for different parameterizations. The aim here is to study the influence of certain model parameters
on the acceptance range regarding the valid fraud and auditing probabilities.
We make assumptions concerning the distribution of the loss variable
θ, the policyholder’s initial wealth position W0 and the penalty payment
B that remain fixed throughout the whole analysis. For instance, the
policyholder’s wealth position is set to W0 = 0. Since his participation
constraint that is given by Equation (12) is independent of this parameter, our choice will not have any influence on whether he signs the
insurance contract or not. Furthermore, we assume the random variable
θ to follow a log-normal distribution. This assumption is commonly used
20
as mentioned in Marlin (1984) since it guarantees positive values for the
realizations of the random variable. In particular, the expected value is
set µ = 1 and the variance σ 2 = 0.4. Regarding the penalty payment
B, we take it to be of the same value as the corresponding realization
of the loss θ such that in the case of detected fraudulent behavior the
indemnity payment is 0. Additionally, we will not consider exogenously
given penalties. Viaene and Dedene (2004) claim that in practice insurance companies tend to negotiate with allegedly suspicious policyholders
since substantial legal evidence is needed to prosecute insurance claim
fraud successfully.
In this subsection, we analyze the influence of the policyholder’s risk
aversion a, the amount of fraud that is represented by α, the insurance
premium P and the cost per audit k on the acceptance range respectively. For this purpose, we use the ceteris paribus assumption in the
analysis, i.e., we study the change in the acceptance range caused by
one isolated factor while keeping all the others constant. Unless noted
otherwise, the policyholder is taken to be risk averse. Hence, to start
with, his risk aversion parameter a is set 6. Furthermore, we first assume
that in the case of fraudulent behavior the policyholder always decides
to claim an amount that is 20% higher than the actual loss. According
to Derrig, Johnston, and Sprinkel (2006), this value seems reasonable.
In an auto injury insurance claim study from 2002, they revealed that
the average payments that were made related to bodily injury claims
added up to approximately $7,872 if no buildup or fraud was detected,
whereas in cases where fraudulent behavior appeared, the amount rose
up to $9,559 on average. The last assumption that we have to make concerns the insurance premium. It can be split up into the fair premium
and an appropriate loading factor. The fair premium corresponds to the
expected loss. Hence, having set the expected value of the loss variable θ
to µ = 1, it implies a fair premium of 1 as well. However, the loss ratio in
the automobile insurance in many industrialized countries over the last
years averaged out to approximately 70%.4 Using this observation and
the assumption of µ = 1, we set the fair premium to 1.4. Furthermore,
since the insurance company faces additional costs due to the auditing
4 See,
e.g., U.S., German or Swiss market supervisory data.
4 Simulation Results
21
process with positive probability, it will add a corresponding loading factor to the fair premium. However, as mentioned in Cummins and Mahul
(2004), the loading factor cannot be chosen too big since potential policyholders would not sign the insurance contract under such conditions.
For the purpose of starting our analysis, we will assume the total insurance premium to be P = 1.45. The last parameter whose influence on
the acceptance range will be analyzed is the cost per audit k. It is set
k = 0.1 which corresponds to 10% of the expected value of the loss θ.
For the purpose of our analysis and in order to keep focused, we will
disregard costs other than the ones due to auditing.
Table 1 sums up the choices for the input parameters for the reference
setting as introduced. In the course of this study, we base our simulations
and studies on these values.
Input parameter
Initial wealth position
Insurance premium
Occurred loss
Fraud amount
Risk aversion parameter
Auditing cost
Penalty payment
Reference level
W0
P
θ
α
a
k
B
0
1.45
lnN (1, 0.4)
1.2
6
0.1
realization of θ
Table 1: Input Parameters for the Reference Setting
Unless otherwise noted, the simulation results are based on N =
100, 000 realizations of the loss variable θ and M = 50 discretization
points in the interval [0, 1].
4
Simulation Results
This section contains the results based on the numerical simulation.
First, we discuss the reference setting and the impacts on the objective quantities and the corresponding optimal strategies. Furthermore,
a sensitivity analysis of the relevant parameters is performed.
22
4.1
Reference Setting
1.0
Auditing probability q
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(a) Insurance Company Perspective
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Before discussing the effects of different parameterizations regarding the
policyholder’s risk aversion, the amount of fraud, the insurance premium
and the cost per audit on the acceptance range, we will first illustrate
the results given the input parameters as summarized in Table 1.
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(b) Policyholder Perspective
Figure 1: Acceptance Range from both Stakeholders’ Perspectives
All parameters are chosen as presented in Table 1. p − q combinations which result in the
highest third of N P V and U (W1A ) respectively are displayed in the darkest color, the ones
which result in the lowest third of N P V and U (W1A ) respectively in the lightest color and
the remaining ones in a medium color.
Figure 1 shows the acceptance range from both the policyholder and
the insurance company perspective based on the values for the input parameters that were presented above. Each point in the graphic represents
a valid fraud and auditing probability combination.
To illustrate the dimension of the objective quantities U (W1A ) and
N P V that result from the current parameter choice and a certain p − q
combination, the points in Figure 1 are displayed in different colors according to the value. For this purpose, given that the input parameters
are fixed, the p − q combinations that lead to the lowest third of outcomes are presented in the lightest color, whereas those combinations
that result in the highest third of outcomes are shown in the darkest
color. The remaining points are displayed in a medium color. This implies that the darker the color of a point, the higher is the relative value
4.1 Reference Setting
23
of the corresponding U (W1A ) or N P V .
Insurance Company Perspective
From the insurance company point of view, we are interested in deriving
all feasible and, in particular, the optimal verification strategies characterized by the probability of auditing q when the prevalent defrauding
probability p is known.
For this purpose, let some constant fraud behavior that is characterized by p be given. The choice regarding the optimal corresponding
auditing strategy q from the insurance company perspective depends on
the value of the fraud probability p. As already proven in Proposition
2, there exists a threshold p∗ that determines whether it is optimal to
audit the incoming claims with the highest probability possible or the
lowest valid probability, i.e., the highest and lowest q respectively contained in the acceptance range. Considering the choice of the input
parameters for the reference setting, the value of this threshold is given
by p∗ = k/αE(θ) = 0.083. This implies that in the case p > 0.083, it is
best for the insurance company to audit the incoming claims with the
highest valid probability whereas if p ≤ 0.083, the optimal strategy is
to chose q as small as possible. These relationships can be observed in
Figure 1(a).
Another interesting observation can be made when considering p = 1.
In this specific setting, the fraud-prone policyholders decide to inflate
their claims by 20% each time they incur an insured loss. Intuition would
tell us that such an extensive case of build-up cannot be acceptable from
the insurance company point of view, i.e., it would not be possible to
find feasible auditing strategies in this context. However, we are able
to observe the opposite in our analyses due to the circumstance that
the insurer in our reference setting has to incur relatively low costs to
detect fraudulent attempts. Additionally, in the case of proven build-up,
no indemnity payments are made to the policyholder, i.e., neither the
excess nor the loss amounts are paid out. As a consequence, it is possible
to have the savings from detected fraud outweigh the additional costs
24
from indemnifying inflated losses such that the net present value N P V
is positive. Thereby, the best result from the insurance company point
of view is achieved when performing audits with the highest feasible
probability q.
Policyholder Perspective
Similarly to the case above, we determine all acceptable and especially
optimal defrauding strategies p from the policyholder perspective, given
that they have knowledge of the current insurance company’s auditing
scheme q. The former are defined by the probability of filing an inflated
loss amount.
Hence, we assume the insurance company to be committed to some
constant auditing strategy q. From the policyholder perspective, it is
always optimal to correspond with reporting fraudulent claims at the
highest valid probability p. Figure 1(b) supports this result.
This finding appears to be rather intuitive. The premise in this context is the insurance company having committed itself to some constant
verification scheme expressed by some constant probability q. However,
this implies that the share of incoming claims that does not have to
undergo the auditing process remains constant as well. In this case, it
is advisable for the policyholder population to increase the probability
p of exaggerating their loss amounts, i.e., the share of build-up among
the claims that are indemnified instantly rises as well leading to higher
payouts for the individuals.
As indicated in Section 1, Figure 1 illustrates that in the setting of our
model framework, it is impossible to find a feasible p−q combination that
maximizes the objective quantities of both stakeholders at the same time.
The prevalent behavioral strategies result in an optimum of either the
insurance company’s net present value N P V or the policyholder’s utility
U (W1A ) of having signed an insurance contract prior to the occurrence of
loss. Which one of these events will be observed depends on the market
power of the respective parties. Assuming a highly competitive market,
it is likely for those defrauding and auditing probability combinations to
be applied that maximize the policyholders’ objective while the insurer
is still willing to adhere to the insurance relationship. However, if the
4.2 Sensitivity Analysis of Relevant Parameters
25
insurance company is in the position of possessing the position of power,
other probability combinations become of interest since the insurer will
be able to maximize its own objective while making sure to keep contract
conditions attractive enough for its policyholder population.
Furthermore, it seems possible for defrauding and auditing strategies
that have once been optimal for the respective stakeholder to become
unattractive in the consecutive observation period. Consequently, the
insurance company and the policyholders need to identify all acceptable
probability combinations p − q at the beginning of each period, and
possibly realign their strategies on this basis.
4.2
Sensitivity Analysis of Relevant Parameters
In the remainder of this section, we present and discuss the resulting
acceptance ranges, i.e., all valid p − q combinations based on different
choices regarding the input parameters of risk aversion a, fraud amount
α, insurance premium P and cost per audit k. Since the effects of the different valid p − q combinations on the policyholder’s final utility position
U (W1A ) and on the insurance company’s present value N P V has been
presented and analyzed, we restrict ourselves to showing the acceptance
range itself without the impacts on the objective quantities.
Influence of Policyholder’s Risk Aversion
In this subsection, we will analyze the impact of different risk aversion
parameters on the acceptance range of the fraud and auditing probabilities. For this purpose, we chose different values for a while keeping all
the other input parameters as given in Table 1. In particular, Figure 2
shows the acceptance range for the risk aversion parameters a = 5 and
a = 10.
Comparing the two graphics for the acceptance range, we find that
the upper bound shifts in an upward direction when increasing the policyholder’s risk aversion parameter. This implies that the higher the
risk aversion of the policyholder is, the broader the acceptance range
becomes assuming all the other input parameters to be constant.
26
1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(a) Acceptance Range: a = 5
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(b) Acceptance Range: a = 10
Figure 2: Acceptance Range for Different Risk Aversion Parameters a
The remaining parameters are chosen as presented in Table 1.
In other words, the more risk averse the policyholder is, the higher
the auditing probability q can be chosen for each fraud strategy p while
the policyholder is still willing to participate in the insurance contract.
Influence of Fraud Amount
1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Fraud probability p
0.8
(a) Acceptance Range: α = 1.1
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
In Figure 3, the acceptance range is displayed for the fraud amounts
α = 1.1 and α = 1.8 respectively, i.e., in the case of fraudulent behavior,
the claimed loss is given by θ̂ = 1.1·θ or θ̂ = 1.8·θ. Again, the remaining
input parameters are chosen as displayed in Table 1.
0.2
0.4
0.6
Fraud probability p
0.8
(b) Acceptance Range: α = 1.8
Figure 3: Acceptance Range for Different Fraud Amounts α
1.0
27
Comparing the graphics for the different choices of α, we find that the
upper bound of the acceptance range as well as part of the lower bound
shift in an upward direction when increasing the fraud amount. To be
more precise: while the number of valid p − q combinations with high
auditing probabilities q increases for all fraud strategies p, the change
in the lower bound occurs only in the area of high fraud probabilities
p. Summing up these effects, we can state that the higher the amount
of fraud, the wider the acceptance range becomes. However, a change
from α = 1.2 in the reference setting to α = 1.1 results in marginal
modifications within the acceptance range.
This outcome can be interpreted in the following way: the higher the
amount of fraud α per claim, the more likely it is for the policyholder to
accept higher auditing probabilities q, given that his own fraud probability p is fixed. In these cases, even though the auditing activity increased,
the gain in final utility U (W1A ) due to excessive claiming is still positive, despite the higher chance of being convicted and imposed with a
penalty payment. On the other hand, it becomes unattractive from the
insurance company perspective to audit the incoming claims with a low
probability q when the amount of fraud is increased, assuming a high
fixed fraud behavior p. Such a strategy would imply that the majority
of fraudulent claims remained undetected, which consequently leads to
an increase in outgoing cash flows due to excessive fraud amounts. This
increase, however, is not covered by incoming positions like insurance
premiums or penalty payments. Therefore, if the fraud amount α goes
up, p − q combinations with higher values for q become acceptable to
both stakeholders, whereas no insurance contract will come into existence
with individuals who are expected to commit excessive fraud frequently.
Influence of Insurance Premium
Insurance premiums are another common way to influence the willingness of both the potential policyholder and the insurance company to participate in an insurance contract. In Figure 4, the acceptance range is presented for two different values of the insurance premium, i.e., P = 1.35
and P = 1.55 while the other input parameters are chosen as in the
28
1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Fraud probability p
0.8
(a) Acceptance Range: P = 1.35
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
reference setting.
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(b) Acceptance Range: P = 1.55
Figure 4: Acceptance Range for Different Insurance Premiums P
A comparison of the acceptance ranges when choosing P = 1.35 and
P = 1.55 respectively shows that the upper bound shifts in a downward
direction when increasing the value of the insurance premium. This
means that the higher the insurance premium is, the smaller the acceptance range gets while keeping the remaining input parameters unchanged.
In other words, the lower the insurance premium P is, the more
willing the policyholder is to accept higher audit probabilities q when
keeping his own fraud probability p constant. However, if the insurance
premium is set too high, i.e., it exceeds the expected loss amount by
far, potential policyholders will have no benefit from signing such an
insurance contract.
The effect of shrinking acceptance ranges due to high insurance premiums can be weakened by offering such contracts to potential policyholders whose risk aversion is assumed to be high as well. As we have
seen in Figure 2, the increase in risk aversion has the opposite effect on
the acceptance range as the choice of the insurance premium.
It needs to be pointed out that the insurance premium seems to have
a significant impact on the acceptance range. Even though the values of
P have been varied only marginally throughout the analysis, i.e., ≈ ±7%
29
of the reference level, the resulting number and positions of the valid p−q
combinations differ markedly.
Influence of Cost Per Audit
1.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
Fraud probability p
0.8
(a) Acceptance Range: k = 0.01
1.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
The last input parameter that can be adjusted easily is the cost per
audit k. Its value can give an indication of what type of auditing is being performed by the insurance company. Auditing procedures in which
standard techniques are applied require minor costs, whereas investigative processes that are initiated to verify major claims result in high
costs.
Figure 5 displays the acceptance ranges when the cost per audit is
chosen to be k = 0.01 and k = 1.0 respectively while keeping the remaining input parameters as in the reference setting.
0.2
0.4
0.6
Fraud probability p
0.8
1.0
(b) Acceptance Range: k = 1.0
Figure 5: Acceptance Range for Different Costs Per Audit k
When comparing the graphic where the cost per audit is set k = 0.01
to the one with k = 1.0, we find that the upper boundary of the acceptance range shifts in a downward direction in the case of low fraud
probabilities p while there appears to be no change in the remaining
valid p − q combinations. This implies that the higher the cost per audit,
the smaller the acceptance range becomes when keeping the other input
parameters constant. However, only marginal changes within the acceptance range can be observed when choosing k = 0.01 instead of k = 0.1
30
as given in the reference setting.
This observation can be explained in the following way: the higher
the cost per auditing process, the less willing the insurance company
becomes to verify those incoming claims for which a low fraud probability
is assumed. Such a strategy would lead to high expenses for the insurer
that are not likely to be covered. The policyholder rarely commits fraud
and even in case he does, the additional amount claimed is not excessive.
Therefore, relatively high auditing costs and comparably low expenses
resulting from undetected fraudulent claims are opposing each other.
As a consequence, no insurance contracts will come into existence with
policyholders whose fraud probability p and amount α are expected to
be low while the cost per audit k is set at a high level.
A way to avoid this effect is to adjust the effort put into the auditing
process to each specific case. Depending on the type of loss and the corresponding amount claimed, the insurance company can decide whether
to apply a basic procedure at low cost or an extensive process that leads
to high expenses.
As indicated by the very extreme choice of the parameters, i.e., in
the first case k = 0.01 corresponds to 1% of the expected loss and in the
second one k = 1.0 equals the expected loss, the cost per audit k does
not have a significant influence on the acceptance range. However, the
results imply that extensive auditing in the form of high values for q is
not sustainable for the insurance company if the cost per audit k is high.
5
Conclusive Remarks
In this study, we build and analyze a model framework that depicts the
handling of insurance claims fraud based on a costly state verification
approach. We present analytical solutions as well as numerical methods
for solving the resulting optimization problems that take both the insurance company and the policyholder perspectives into account. Our focus
is set on deriving an acceptance range consisting of all valid fraud and
auditing probability combinations and analyzing their optimality regard-
5 Conclusive Remarks
31
ing both stakeholders’ objective quantities respectively. In addition, we
discuss the impact of different relevant input parameters on the size of
the acceptance range. Furthermore, we are able to calculate a threshold
value for incoming claims based on which the insurance company can
decide whether to perform auditing or not.
One of our main findings is the derivation of optimal auditing and
fraud strategies from the stakeholders’ perspectives. Especially from
the insurance company point of view, it seems intuitive: the optimal
answer to low fraud probabilities is to perform auditing with a small
probability as well, whereas medium and high fraud probabilities require
the largest valid audit probability to maximize the net present value.
An interesting observation in this regard is that the insurance company
benefits from the existence of insurance contracts (almost) regardless of
the policyholders’ defrauding strategy. This finding demonstrates that in
the context of cost-minimizing insurers, it is not essential to completely
prevent all defrauding attempts ventured by the policyholder population.
Based on our numerical approach, we present and analyze the acceptance range for different parameterizations as well as the optimality
of different auditing and fraud probability combinations regarding the
stakeholders’ respective objective quantities. While a relatively high risk
aversion, a high relative amount of fraud and low insurance premiums
result in broadening the acceptance range, the latter becomes smaller
whenever the value of these input parameters is chosen the opposite way.
We also find that the cost per audit merely influences the number of
valid fraud and auditing probability combinations. Furthermore, the
simulation results support and illustrate our analytical findings regarding optimal fraud and auditing strategies.
The model that we present in this study can be extended for future
research. On the one hand, another type of auditing could be introduced
that while less costly than the perfect one, detects fraud only with some
probability less than one. On the other hand, insurance premiums could
depend on the auditing probability since more strict auditing policies
require a longer period to process incoming claims and policyholders
32
might not be willing to pay the original insurance premium due to possible delays in indemnity payments. Another topic for further research
is to back test the results derived in this study with insurance company
data and profiling experience.
6 Appendix
6
33
Appendix
In the Appendix, we state the propositions and the corollary presented
in the main part of the paper once again and provide their proofs respectively.
Proposition 1 For a = 0 and θ, θ̂ such that 0 ≤ θ < θ̂, the optimal
fraud and auditing strategies from both stakeholders perspectives are
p = 1 and q = 0. This results in P = E(θ̂).
Proof of Proposition 1
By setting a = 0, the policyholder’s participation constraint given in
Equation (13) can be written as
P ≤ E(R(θ, θ̂)).
(24)
Since both participation constraints have to be met for an insurance
contract to come into existence, (4) and (24) result in
E(R(θ, θ̂)) + qk ≤ P ≤ E(R(θ, θ̂)) ⇐⇒ q = 0 ∀k > 0,
(25)
i.e., for any k > 0, q = 0 is the only solution.
In this case, the policyholder’s objective function (15) can be written
as U (W1A ) = W0 − P + p[E(θ̂) − E(θ)]. Due to the assumption of θ < θ̂,
it attains its maximum at p = 1.
Furthermore, setting q = 0 and p = 1 in Equation (1), we get
E(R(θ, θ̂)) = E(θ̂). At the same time, one can conclude from (25) that
P = E(R(θ, θ̂)). This leads to P = E(θ̂).
Proposition 2 Assume p, q to be in the acceptance range, i.e., an
insurance contract exists. For a > 0, B = θ, θ̂ = αθ with some α ≥ 1,
the respective optimal strategies popt , q opt are given by:
(i) Insurance company perspective
34
Let some p be given. In order for the net present value N P V to
be maximized, choose
q
where p∗ :=
opt
=
(
as large as possible if
p > p∗
as small as possible if p ≤ p∗
,
(26)
k
αE(θ) .
(ii) Policyholder perspective
Let some q be given. In order for the final expected utility U (W1A )
to be maximized, choose
popt

as large as possible s.t.
=
as small as possible s.t.
For 0 <
−E(θ)
−E(θ)
−E(θ)
≥1
≤0
. (27)
< 1 no general statement can be made.
Proof of Proposition 2
(i) Using Equations (1), (2) and the assumptions B = θ, θ̂ = αθ with
α ≥ 1, we get
N P V = P − (1 − p)E(θ) − p[(1 − q)E(θ̂) + qE(θ − B)] − qk
= P − (1 − p)E(θ) − p(1 − q)αE(θ) − qk
= P − E(θ) + p(1 − α)E(θ) + q[αpE(θ) − k].
(28)
Deriving (28) with respect to q leads to
∂
N P V = αpE(θ) − k,
∂q
(29)
which can be distinguished into two cases with respect to its sign.
6 Appendix
35
k
holds, the N P V as
(a) If for the given fraud strategy p > αE(θ)
defined in (2) has a positive slope with respect to the parameter q. Consequently, the optimal auditing strategy q opt has to
be chosen as large as possible in order to maximize the value
of N P V .
k
(b) If the given fraud strategy p is given such that p ≤ αE(θ)
holds,
the N P V has a negative slope with respect to the parameter
q. Hence, the optimal auditing strategy q opt has to be chosen
as small as possible for the N P V to be maximized.
(ii) Applying the assumptions a 6= 0, B = θ, θ̂ = αθ with α ≥ 1 to
Equations (1) and (11), we obtain
U (W1A ) =W0 − P − E(θ) + (1 − p)E(θ) + p[(1 − q)E(θ̂) + qE(θ − B)]
a
− Var[−θ + (1 − p)θ + p(1 − q)θ̂ + pq(θ − B)]
2
=W0 − P − E(θ) + (1 − p)E(θ) + p(1 − q)E(θ̂)
a
− Var[−θ + (1 − p)θ + p(1 − q)θ̂]
2
=W0 − P − pE(θ) + pαE(θ) − pqαE(θ)
a
− Var(−pθ + pαθ − pqαθ)
2
=W0 − P − p(1 − α + qα)E(θ)
a
− p2 (1 − α + qα)2 Var(θ).
(30)
2
Deriving (30) with respect to p results in
∂
U (W1A ) = −(1 − α + qα)E(θ) − ap(1 − α + qα)2 Var(θ). (31)
∂p
Based on (31), three cases can be identified:
−E(θ)
(a) For ap(1−α(1−q))
Var(θ) ≥ 1, the policyholder can choose any
fraud strategy p ∈ [0, 1], especially any p in the acceptance
36
−E(θ)
range, such that p ≤ a(1−α(1−q))
Var(θ) . Applying this inequal∂
ity to Equation (31), we obtain ∂p
U (W1A ) ≥ 0. From this can
be concluded that U (W1A ) has a positive slope. Consequently,
the optimal fraud strategy popt has to be chosen as large as
possible in order to maximize the value of U (W1A ).
−E(θ)
(b) Similarly, for ap(1−α(1−q))
Var(θ) ≤ 0, the policyholder can
choose any fraud strategy p ∈ [0, 1], especially any p in the
−E(θ)
acceptance range, such that p ≥ a(1−α(1−q))
Var(θ) . For Equa∂
A
tion (31) this implies that ∂p U (W1 ) ≤ 0. This means that in
this case U (W1A ) has a negative slope and hence, the optimal
fraud strategy popt needs to be chosen as small as possible for
U (W1A ) to be maximized.
−E(θ)
(c) For 0 < a(1−α(1−q))
Var(θ) < 1, no general statement about
the corresponding optimal fraud strategy popt can be made.
References
37
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66(2):893–919.
Picard, P. and M.-C. Fagart, 1999, Optimal Insurance Under Random
Auditing, Geneva Papers on Risk and Insurance Theory, 24(1):29–54.
Tennyson, S., 2008, Moral, Social, and Economic Dimensions of Insurance Claims Fraud, Social Research, 74(4):1181–1204.
Townsend, R., 1979, Optimal Contracts and Competitive Markets with
Costly State Verification, Journal of Economic Theory, 21(2):265–293.
Viaene, S. and G. Dedene, 2004, Insurance Fraud: Issues and Challenges,
Geneva Papers on Risk and Insurance - Issues and Practice, 29(2):313–
333.
References
39
Watt, R., 2003, Curtailing Ex-Post Fraud in Risk Sharing Arrangements,
European Journal of Law and Economics, 16(2):247–263.
Weisberg, H. and R. Derrig, 1991, Fraud and Automobile Insurance: A
Report on Bodily Injury Liability Claims in Massachusetts, Journal
of Insurance Regulation, 9(4):497–541.
41
Part II
The Impact of Auditing
Strategies on Insurers’
Profitability
Abstract
Insurance claims fraud is a major concern in the insurance industry. According to estimates, excess payments due to fraudulent claims account
for somewhere between 15 and 25 percent of all claim payments, affecting all classes of insurance. In this paper, we develop a model framework
based on a costly state verification setting in which - while policyholders
observe the amount of loss privately - the insurance company can decide
to audit incoming claims at some cost. The aim is to derive optimal
auditing strategies from the insurance company perspective while maintaining contract attractiveness to policyholders who are willing to adhere
to the insurance relationship. We present and analyze an auditing strategy which is triggered by the filed claims amount and also includes an
option for each stakeholder to adapt its behavior based on a signal specific to its position. The impact of the optimal auditing strategy on the
insurer’s profitability is examined. Finally, practical implementations
are discussed. 5
5 K. Müller, H. Schmeiser, and J. Wagner. The Impact of Auditing Strategies on
Insurers’ Profitability. Working Papers on Risk Management and Insurance, 2012.
This paper has been presented at the International Congress on Insurance: Mathematics and Economics in June 2012, the European Group of Risk and Insurance
Economists Seminar in September 2012 and the Annual Meeting of the Western Risk
and Insurance Association in January 2013.
It is currently in the second round of the review process at The Journal of Risk and
Insurance and has been resubmitted in November 2012.
42
1
II Theory of Insurance Fraud
Introduction
Insurance claims fraud is one of the major industry concerns. It occurs
in all classes of insurance and accounts for a substantial portion of the
indemnity payments each year, yet due its nature, is difficult to estimate
in total value. The Insurance Research Council (2008) estimated the
excess payments due to fraudulent claims in 2007 as somewhere between
$4.8 and $6.8 Mrd in the auto injury insurance sector in the U.S. alone,
corresponding to 13 to 18 percent of total payments. These fraudulent
activities, while undertaken by some individuals, have an impact on the
policyholder population as a whole through higher insurance premiums
(see Tennyson (2008)).
Previous research has explained the existence of fraud due to information being asymmetrically distributed between policyholder and the
corresponding insurance company (see, e.g., Derrig (2002)). Since insureds may hold private information about the actual amount of the
loss suffered, there exists the possibility of misrepresentation. Based
on the approach of costly state verification (see, e.g., Townsend (1979),
Mookherjee and Png (1989), Bond and Crocker (1997), Picard and Fagart (1999), Picard (2000)), the insurance company can consequently
choose to audit incoming claims in order to determine their truthfulness. In cases when fraudulent activities haven been proven, a penalty
payment can be imposed on the policyholder. However, these verification processes incur costs. From the insurance company perspective this
implies that the costs for auditing have to be traded off against the
savings resulting from detected fraud. One of the costs may be on policyholder attitude, and therefore additionally, the policyholders’ point of
view needs to be considered as well. According to Viaene and Dedene
(2004), individuals are more likely to develop an opportunistic attitude
towards insurance fraud after having gained negative experiences with
their insurance companies. Underpaid claims or the endurance of long
waiting periods for indemnity payments may encourage fraudulent behavior in the future.
1 Introduction
43
An opposing strategy to minimize the occurrence of insurance fraud
comprises the implementation of bonus-malus systems. Dionne and
Vanasse (1992) and Moreno, Vázquez, and Watt (2006) develop model
frameworks where, instead of performing costly verification processes,
the policyholder’s insurance premium is increased whenever he or she
files a claim in the previous period. This approach is applicable in the
case of insurance contract renewals; however, in some countries policyholders may avoid this penalization mechanism by switching their insurance company (see, e.g., Dionne and Ghali (2005)).
An additional strand of research dealing with the occurrence and
handling of insurance claims fraud is based on the costly state falsification approach introduced by Lacker and Weinberg (1989) and adapted
to the insurance setting by Crocker and Morgan (1998) and Crocker and
Tennyson (2002). While the premise with regard to the asymmetric distribution of information remains unchanged, this time the policyholder
engages in costly manipulations such that a verification of the claims
becomes impossible.
The term fraud has a rather negative connotation implying the engagement in illegal activities such as staged accidents. The range of
actions, however, which is colloquially subsumed under this notion, is by
far broader (see, e.g., Picard (2001), Derrig (2002), Tennyson (2008)). In
general, a distinction is made between ex-ante and ex-post moral hazard,
referring to the timing of fraudulent behavior. The former occurs by the
time of insurance purchase, e.g., in case the potential policyholder fails
to provide relevant information which might have resulted in unfavorable
contract conditions or even in a rejection on part of the insurance company (see, e.g., Picard (2001)). The term ex-post moral hazard implies
the engagement in fraudulent activities by the time of claims filing, i.e.,
after a potential loss has been suffered. In this context, there are several possible distinctions which can be made. A common one is to differ
between criminal fraud, also called hard fraud, and soft fraud which is situated in an ethical gray area (see, e.g., Tennyson (2008)). Derrig (2002)
defines criminal fraud as ”the willful act of obtaining money or value
from an insurer under false pretenses or material misrepresentations”.
44
While it is assumed that only a small number of claims contain outright
fraud, the more frequent and more costly type of fraudulent behavior
falls within the term of soft fraud (see, e.g., Weisberg and Derrig (1991),
Viaene and Dedene (2004), Tennyson (2008)). Even though there does
not exist an explicit definition of the term, it is associated with the misrepresentation of the loss magnitude after its occurrence, i.e., claimants
exaggerate the amount to obtain higher indemnity payments. A notion
often used in this context is build-up. In this paper, we will refer to
fraud as soft fraud or build-up, i.e., we assume that some policyholders
inflate the magnitude of loss after its occurrence if it appears profitable.
In addition to the policyholders, also third parties like service providers might be involved in fraudulent activities to exaggerate the loss
amount (see, e.g., Tennyson (2008), Dulleck and Kerschbamer (2006)
Derrig and Zicko (2002)). Prominent examples are known from automobile insurance. Derrig and Zicko (2002) found that repair shop are
likely to have developed insight about the insurance companies’ prevalent auditing strategies due to repeated experiences in repairing cars for
insured damages. As a consequence, there exists the possibility to adjust
the defrauding strategy. Such actions are either undertaken without the
knowledge of the policyholder, or performed by mutual agreement between the policyholder and the corresponding repair shop. The service
providers’ incentive to engage in fraudulent activities against the insurance company may stem from the hope to gain the policyholders’ favor.
As a result, the latter might request their services again at a later point
in time. For the purpose of our study, we will assume that all fraudulent
activities - even if committed by repair shops - to be advantageous for
the policyholders only.
Considering its large prevalence, effective measures in dealing with
the phenomenon of insurance claims fraud need to be found. As already
mentioned, insurance companies have the possibility to perform audits.
Many insurers have dedicated teams to identify and combat fraudulent
claims. In interviews with fraud managers, Morley, Ball, and Ormerod
(2006) find that verification processes are initiated whenever investigators detect anomalies or inconsistencies in the circumstances of the loss
1 Introduction
45
event or unusual behavior from the claimant. This approach seems to be
suitable for filed claims that exhibit obvious signs indicating potential
fraudulent behavior and/or the ones which are of significant magnitude.
Apparently, high-magnitude claims are be subject to verification right
away. The majority of claims, however, appear to be legitimate at first
sight, seeking low to medium indemnity payments, which do not trigger
immediate investigation. Consequently, the key question is how to deal
with the fraudulent cases that appear legitimate, and which account for
the majority of inappropriate indemnity (see, e.g., Derrig (2002)).
We therefore set our focus on developing audit strategies which help
to improve the efficiency of the claims settlement process with regard to
inconspicuous claims. The key element in our model framework - which
pursues the costly state verification approach - is the use of threshold
values which indicate whether an incoming claim should be verified or
not. This approach allows the insurer to determine the optimal auditing
strategy based on the magnitude of the filed claim. For this purpose,
we make use of information on policyholder claiming and, more importantly, defrauding behavior which insurance companies are expected to
hold due to previous experiences and verification processes. In particular, the share of fraud-prone policyholders among the population has
an impact on the actual threshold values for auditing. Interviews with
experts as well as previous research (see, e.g., Belhadji, Dionne, and
Tarkhani (2000), Bermúdez, Pérez, Ayuso, Gómez, and Vázquez (2008))
have shown that numerous criteria exist which help indicate the potential fraud behavior of an individual. Such factors may include gender,
nationality or place of residence. Additionally, investigations in the past
have shown that certain accidents are more prone to fraud than others.
For example, wind screen and glass damages as well as thefts inside the
passenger compartment are known to be cases where policyholders are
likely to report exaggerated claim amounts. Combining this information
on individual characteristics with fraud signals presented by Dionne, Giuliano, and Picard (2009)), one obtains accurate estimators for the share
of defrauders among the claimants.
46
In what follows, a model framework is presented based on a costly
state verification setting in which - while policyholders observe the amount
of loss privately - the insurance company can decide to audit incoming
claims at some cost. Our goal is to derive optimal auditing strategies
from the insurance company perspective indicating which of the incoming claims are subject to verification. In any case, we take into account
that contract conditions must still be attractive to policyholders such
that they are willing to adhere to the insurance relationship. A key
element of our study is the option of both stakeholders adapting their
behavior respectively based on different signals is taken into account.
Subsequently, the impact of the optimal auditing strategy on the insurers’ profitability is analyzed.
Unlike other strands of research in this area (e.g., Townsend (1979),
Picard and Fagart (1999)), which often set up the problem of optimizing
insurance contracts in a way such that policyholders always report loss
amounts truthfully in equilibrium, we do not develop an auditing scheme
which completely prevents or deters insurance claims fraud. From a
macroeconomic point of view, both the approach and the result are
desirable. From a single company perspective, however, this does not
necessarily hold true. Similarly as Dionne, Giuliano, and Picard (2009),
our aim is to minimize the insurer’s overall costs from fraud, in general
including paying some portion of fraudulent claims, as well as incurring
expenses to detect and address fraud, and in experiencing the reputational effects of investigations themselves. In fact, we can show that for
low costs per audit process the insurance company’s net present value
increases in the presence of fraud compared to the case where no fraud
exists. Our findings are in line with the assumption made by Watt
(2003).
Allowing for the existence of some fraudulent activities accounts for
the widespread attitude to consider insurance as an investment which
is expected to yield a return. As evidence, a study from GDV (2011)
reported that more than 20% of the Germans consider insurance fraud
to be a ”gentlemen’s offence” which is committed by almost everyone at
least once. This point of view can be found among all socio-demographic
1 Introduction
47
groups. Duffield and Grabosky (2001) compile different approaches individuals use to justify fraud. Among others, insurance fraud is assumed
to cause no significant harm; insurers are accepted targets which can
afford it; build-up is a way to recover past premium payments (see also
Miyazaki (2008)). Given this line of reasoning, we assume therefore that
there will always exist policyholders trying to defraud in terms of inflating their magnitude of loss. In this context, auditing processes can help
to minimize the share of fraudulent activities among the policyholder
population. But a complete eradication seems unlikely.
Including the option to change one’s strategy based on signals provides another insightful result. We can show that there exist situations
where - while the possibility to adapt one’s behavior might be desirable for the insurance company - this option is disadvantageous from
the policyholder perspective. This observation stands in contrast to the
widespread opinion that the adaptation of the defrauding strategy, especially based on signals from service providers, would be favorable from
the individuals point of view.
We assume that while the policyholders and the service providers,
which might be involved, may obtain signals and information based on
which they change the defrauding strategy, they do not know the exact
auditing threshold values nor do they have enough information to derive
them themselves. This is a crucial assumption in our model which also
seems to be realistic. Leaving out this assumption, any auditing strategy
would be redundant since the policyholder would know how to adapt his
fraud behavior in a way to avoid being caught. From the insurance company perspective, it would not make sense to verify any incoming claim
in this case.
The remainder of this paper is organized as follows: We start by
presenting the model framework and optimization problem in Section 2.
Section 3 constitutes the introduction of the policyholder’s and insurer’s
respective strategies as well as the behavioral adaptation process. The
corresponding numerical results are presented in Section 4. In Section 5,
we analyze practical implications, before we conclude in Section 6.
48
2
Model Framework and Stakeholders’
Positions
We consider an insurance company with a fixed number of policyholders.
The latter are assumed to differ from one another only in their willingness
to defraud, i.e., the population consists of both honest individuals who
never commit fraud and those who defraud if it appears to be profitable.
We denote the share of fraud-prone policyholders among the population
with p. Depending on the loss amount and the prevalent defrauding
strategy, the share of claims which actually contain build-up might be
lower.
Since we assume all individuals to belong to the existing policyholder
population, they all pay an insurance premium P at the beginning of a
period, i.e., in t = t0 . We assume this premium to equal the given market
premium which is demanded by other existing insurance companies as
well. At the same time, with probability 0 ≤ π ≤ 1, they face some
uncertain loss θ of stochastic amount which, by the time of occurrence,
is observed privately. In case a policyholder suffers a loss, he chooses
to file a claim of some size Θ̂(θ) during the period (t0 , t1 ). In case of
honest behavior, the amount of the claim will equal the actual loss, i.e.,
θ̂ = θ. If the individual decides to defraud, he reports some finite θ̂ > θ.
Equation (32) summarizes all values the policyholder can report to the
insurance company in the claiming scheme Θ̂:
θ̂ = Θ̂(θ)



= 0,
= θ,


> θ,
no loss occurred
loss but no fraud .
(32)
loss and fraud
In the course of this paper, we will use the notation
F = {θ̂|θ̂ > θ}
(33)
to denote the set consisting of all fraudulent claims filed by the policyholders. The elements in F are characterized by the individuals’ defrauding strategy. We will analyze different scenarios for the fraud behavior
in the course of this paper.
2 Model Framework and Stakeholders’ Positions
49
The insurance company however has no information about the true
loss amount. It therefore has the opportunity to verify the truthfulness
of incoming claims. However, this comes at some cost per audit k. In
this model framework, we take the audit process to be perfect, i.e., fraud
is detected with probability of one anytime auditing is performed. As
a consequence of detected fraud, the insurance company will reject any
indemnification.6 Figure 6 illustrates the interaction of the different
processes introduced in this model framework.
Following Figure 6, the payment of an indemnification depends on
several aspects. On the one hand, the policyholder needs to have suffered
an insured loss7 during the course of the observation period. Otherwise,
he would not have a reason to file a claim. In the case when the claimant
decides to defraud, his indemnification is dependent on whether auditing
takes place or not: If the reported loss is not verified, he receives the
payment of θ̂. If the filed claim undergoes an auditing process, however,
the attempt to defraud will be revealed and any indemnification payment
will be rejected, i.e., both excess and actual loss amount are denied. In
case the policyholder belongs to the group of honest individuals, his loss
θ will be indemnified no matter whether auditing took place or not.
Two key elements in the model framework which we have not elaborated on so far, are the behavioral strategies of both stakeholders: The
amount of the reported claim in case of dishonesty is defined by the
policyholders’ defrauding behavior, while the decision whether or not to
induce a verification process is dependent on the insurance company’s
auditing strategy. We present and analyze different examples for the
respective strategies as well as behavioral adaptations in Sections 3 and
4.
6 In practice, gradations with regard to the indemnification are possible, i.e., the
insurer might decide to pay the full loss amount or parts of it.
7 Remember that we are considering only the situation where fraud will be committed through claim build-up and not through planned fraud.
50
Insurance Company
Claims Processing
Policyholder
Claims Filing
Original
Claim
Submitted
Verification
at cost k
No Fraud θ̂ = θ
Result
Claim
No Audit
(No) Indemnification
θ̂ ≥ θ
Audit ?
θ
Fraud θ̂ > θ
Claiming depends on
policyholders’ behavior
Selection of claims for auditing
depends on insurer’s strategy
Figure 6: Overview of the Processes Associated with the Filing and Handling of Insurance Claims over the
Course of One Period
An indemnification may or may not be paid out by the end of the period depending on the previous events.
Premium Payment
No
Claim
51
Insurance Company: Contribution Margin and
From the insurance company perspective, we observe the future cash
flows at the beginning of a period in t = t0 and at the time of loss
realization and settling at the end of that period in t = t1 and analyze the
resulting contribution margin per contract CM in (t0 , t1 ). The insurance
company receives a premium payment P from each policyholder in t =
t0 . As already presented in the beginning of Section 2, the outflows of
each period in t = t1 depend on whether claims have been filed by the
policyholders, and if so, whether they were audited or not and the result
of the eventual verification process (see Figure 6).
A key element in this model framework is the insurance company’s
auditing strategy. It indicates which of the incoming claims are subject
to verification and which of them are indemnified without verification.
We use the following general notation to denote the auditing strategy
A = {θ̂|θ̂ is audited}.
(34)
We distinguish four scenarios which lead to different values in the
contribution margin for a single contract CM (P, k, θ̂, A, F). Using the
notations introduced in Equations (33) and (34), we define


P




P − θ̂
CM (P, k, θ̂, A, F ) =

P − θ̂ − k




P −k
, no loss occurred, i.e., θ̂ = 0
, no audit, i.e., θ̂ ∈ Ac
, audit, no fraud, i.e., θ̂ ∈ A ∩ F c
, (35)
, audit, fraud, i.e., θ̂ ∈ A ∩ F
which can also be the stated as:
CM (P, k, θ̂, A, F) = P − θ̂ · 1Ac (θ̂) − θ̂ · 1A∩F c (θ̂) − k · 1A (θ̂)
h
i
= P − θ̂ 1 − 1A∩F (θ̂) − k · 1A (θ̂),
(36)
52
where 1X (y) denotes the indicator function, i.e., it takes the value 1 if
y is in the set X or 0 if y is not in X.
Hence, in this context 1A (θ̂) represents the number of claims which
were subject to verification whereas 1Ac (θ̂) states the number of claims
which were indemnified without auditing. 1A∩F (θ̂) counts the number
of cases when fraudulent claims underwent an auditing process, i.e.,
the number of cases when the attempt to defraud was unveiled while
1A∩F c (θ̂) returns the number of cases when honest claims where verified.
Overall, the insurance company is interested in its whole policyholder
population and hence considers the average contribution margin per contract E(CM ), i.e., the net present value of future incoming and outgoing
cash flows N P V . We define :
N P V = N P V (P, k, θ̂, A, F) = E(CM (P, k, θ̂, A, F)),
(37)
where E(Y ) denotes the expected value of the stochastic variable Y .
Hence, we analyze the insurance company’s net present value as the expected value of future cash flows discounted at the risk-free rate rf = 0.
This assumption rf = 0 holds throughout the paper.
Based on Equation (2), we can formulate the insurance company’s
participation constraint.
Condition 3 The insurance company is willing to maintain the insurance relationship with its policyholder population if the net present value
per contract is non-negative:
N P V ≥ 0.
(38)
Policyholder: Expected Utility and Participation
Constraint
From the policyholder perspective, we analyze their wealth positions
and the corresponding average expected utility at the end of the period
t = t1 in the framework introduced above. We assume each individual
53
initially to hold the same wealth position W 8 and to take out the same
insurance contract. At the beginning of the observation period in t = t0 ,
the payment of the insurance premium P is due. Consequently, each
policyholder is endowed with the wealth position W0I = W − P , where
the superscript I indicates the existence of an insurance contract.
The consecutive development of this wealth position W0I depends on
whether the policyholders suffer an insured loss and if so, whether they
choose to report their loss truthfully or not and whether their fraud is
being revealed in case of dishonesty or not. Taking the occurrence of loss
into account, the wealth position at the end of the observation period in
t = t1 is given by
h
i
W1I = W − P − θ + θ̂ 1 − 1A∩F (θ̂) ,
(39)
where W is invested riskless with rf = 0. In the case when no insured loss
has occurred throughout the observation period, Equation (39) simplifies
to W1I = W0I = W − P .
In order to be able to formulate a participation constraint from the
policyholder perspective, we consider the development of their wealth
position throughout the observation period when not having signed an
insurance contract prior to the occurrence of loss. In t = t0 , the individuals would not have to make a payment in the amount of the insurance premium. Hence, the corresponding wealth position is given by
W0N = W , where the superscript N implies the absence of an insurance
relationship. If some loss θ occurs up until t = t1 , this amount decreases
to W1N = W0N −θ = W −θ, whereas, without loss during the observation
period, we have W1N = W0N = W .
We assume the policyholders’ expected utility to be described by
a standard mean-variance utility function of the corresponding wealth
position. The degree of risk aversion is expressed by the parameter
8 Since the actual value of the initial wealth position will have no impact on the
decision whether to maintain the insurance relationship or not, the policyholders
might even be endowed with different wealth positions. We assume the latter to be
invested safely with the risk-free rate rf = 0.
54
a (> 0). Generally, for a given stochastic wealth position Z, its expected
utility for the individual is given by U (Z) = E(Z) − a2 Var(Z), where
Var(Z) denotes the variance of the stochastic variable Z.
Using the equations above and considering that the probability of
loss occurrence is denoted by π, the final expected utility in case no
insurance contract exists can be written as
a
(40)
U (W1N ) = W − πE(θ) − π 2 Var(θ).
2
Similarly, for the setting where an insurance relationship between
policyholders and insurance company already exists, the final expected
utility is given by
h
i
U (W1I ) = W − P − E πθ − θ̂ 1 − 1A∩F (θ̂)
h
i
a
− Var πθ − θ̂ 1 − 1A∩F (θ̂) .
2
(41)
Comparing Equation (41) with Equation (40) results in the policyholders’ participation constraint. For this purpose, we introduce the
notion of the gain in expected utility from having signed an insurance
contract
∆U = U (W1I ) − U (W1N ).
(42)
Condition 4 The policyholders are willing to maintain the insurance
relationship with the insurance company if their final expected utility is
greater with having insurance than without it, i.e.,
∆U ≥ 0.
(43)
We want to point out that the policyholders’ participation constraint
functions the same in the case of a multi-period model, i.e., they do not
quit the insurance relationship unless their gain in expected utility from
having insurance coverage ∆U is negative. In particular, the insurance
company is in the position to optimize its position while the policyholders
would choose to cancel their insurance contracts only if ∆U < 0.
2.1 Optimization Problem
2.1
55
Optimization Problem
Summing up the information we have presented so far with regard to
the model framework as well as the insurance company’s and the policyholders’ participation constraints, we can formulate the resulting optimization problem.
The insurance company is aiming to derive an auditing strategy A
such that its net present value of future cash flows N P V is maximized.
At the same time, it needs to be made sure that the stakeholders are
willing to maintain their insurance relationships, i.e., Equations (38) and
(43) hold. Formally, this objective is given by


max N P V (P, k, θ̂, F, A)

 A
NPV ≥ 0



∆U ≥ 0
.
(44)
We make the assumption that the former charges the given market
premium P . In particular, this implies that an adaptation of the insurance premium as part of the fraud handling is not feasible. Furthermore,
we assume no interaction with the number of acquired contracts.
3
Section 2 constitutes the introduction of the model framework as well
as the insurance company’s and policyholders’ behavioral strategies in
general. In this section, we present a case for the individuals’ defrauding
behavior as well as a suitable approach to deriving the resulting optimal
auditing strategy. Furthermore, the process of behavioral adaptation is
included.
3.1
Policyholder Claiming Scheme
As discussed above, we consider insurance fraud in the form of build-up,
i.e., in the case of occurrence of loss some of the policyholders among the
56
population decide to file an exaggerated fraudulent claim if it appears
to be profitable.
In the context of our study, we assume that the fraud amount by
inflating the magnitude of a loss not be a decision variable for the policyholders. In case an individual decides to engage in fraudulent activities,
he or she will consider the build-up as a percental ”surcharge” on the actual loss amount instead of filing some value which deviates significantly
from the actual loss amount. This approach increases the likelihood of a
fraudulent claim to be perceived as legitimate by the insurance company,
and not to be audited as a consequence. This assumption is in line with
Viaene and Dedene (2004) who find that policyholders involved in soft
fraud typically tend to file claims containing small fraud amounts. Further evidence can be found in the behavioral economic theory literature.
Individuals weigh the consequences accompanied by losses stronger than
the ones from a gain of the same size (see, e.g., Kahneman and Tversky
(1979), Kerr (2012)). Applied to the context of our model framework this
implies that the loss from being caught committing soft fraud (i.e., indemnification is waived completely) is perceived as a higher burden than
the potential profit from a successful build-up attempt. Additionally, regarding build-up as a surcharge on the loss suffered gives policyholders
the opportunity to ”back down and claim the appropriate amount” if investigated by the insurance company rather than in the case of outright
fraud (see, e.g., Emerson (1992)).
From the perspective of the repair shops, which might also be involved in fraudulent activities, extensive inflations of the loss amount
seem to be unlikely as well. According to Hubbard (2002), service
providers in general have a reputational incentive to act in their clients
favor since the latter tend to return more often if they are satisfied with
previous services. These findings are in line with statements made by
experts from an insurance company whose experiences have shown that
repair shops orientate themselves on the actual loss amount when trying to charge too much for certain services. This way even exaggerated
claims seem legitimate and are less likely to undergo an auditing process
resulting in direct indemnification.
3.1 Policyholder Claiming Scheme
57
As part of the defrauding strategy, we assume that whenever those
policyholders have suffered a loss during the observation period, they
report back a multiple of the actual loss amount. We denote this constant multiplicative factor by α = θ̂/θ. However, it is well known that
insurance companies do perform audits in order to verify the truthfulness of incoming claims. Consequently, fraud-prone policyholders can
be expected to adjust their fraud strategy accordingly. We assume that
∗
all of these individuals have an inner threshold value for defrauding θ̂ph
.
With regard to their fraud strategy this threshold value implies that they
∗
apply their fraud strategy α up to the preset threshold value θ̂ph
. To
be more precise, if the amount of the actual loss θ is smaller than the
∗
∗
threshold θ̂ph
, the minimum of αθ and θ̂ph
is reported to the insurance
company. However, if the amount of the loss suffered already exceeds
that threshold, the policyholder claims this amount truthfully. We can
write this strategy as



0
∗
θ̂ = Θ̂(θ, α, θ̂ph ) = θ


min{αθ, θ̂∗ }
ph
, no loss occurred
∗ .
, no fraud or θ > θ̂ph
, fraud and θ <
(45)
∗
θ̂ph
The set consisting of all fraudulent claims associated with this strat∗
egy Θ̂(θ, α, θ̂ph
) is denoted by Fα,θ̂∗ .
ph
The question arises as to how policyholders determine an adequate
threshold value for defrauding. Here we refer to the data presented in
Section 1 regarding survey reports indicating that policyholders may
use third parties, such as repair shops, to act as their partners in fraud.
These repair shops are likely to have developed insight about the value
of θ̂∗ due to repeated experiences in repairing cars for insured damages.
Based on this information they can make proper assumptions concerning
the insurance companies’ prevalent auditing strategies. There are two
potential ways repair shops can proceed: provide too much service or bill
too much for a particular service (see Dulleck and Kerschbamer (2006)).
A particular phenomenon is to include the cost to fix existing damages
58
when repairing the ones which were caused by a current insured event.
Such actions might often be undertaken even without the knowledge of
the policyholder. On the one hand, the latter might not have the incentive to carefully check the work performed by repair shops since the
bill is passed on to the insurance company (see Tennyson (2008)). On
the other hand, the policyholders might lack the necessary know-how to
do so (see Dulleck and Kerschbamer (2006)). Policyholders, however, in
particular the ones who are willing to defraud if it appears to be advantageous, may also seek information themselves to adapt their claiming
behavior for purposes of defrauding the insurer. They may seek out
others for ideas of successful fraud or perhaps be approached by third
parties to participate in defrauding activities. Staying in the field of
auto insurance, fraudulent activities may be performed by mutual agreement between the policyholder and the corresponding repair shop who
share the excess insured payments with one another. In particular, there
exists the possibility to adjust the policyholders’ defrauding strategy as
presented in Equation (45). Recall that we make the crucial assumption
that neither the policyholders nor a third party other than the insurance
company itself holds exact information concerning the prevalent auditing strategy. The contrary case would make the principal of auditing
redundant.
3.2
Insurance Company Auditing Strategy
From the insurance company perspective, this fraud strategy implies
the necessity to adjust its auditing strategy A. After having performed
audits for at least one period, the company should have gained enough
information on policyholder fraud behavior to do so. Revealed fraud
can serve as an especially helpful information base for improving the
existing auditing strategy. Assuming that a sufficiently high number of
verification processes has been performed (which has revealed sufficiently
many defrauding attempts), the insurer will note that fraudulent claims
do not exceed a certain threshold value. Hence, it is unnecessary to
audit claims above that value, allowing insurers instead to verify medium
sized claims. Consequently, in this case the aim is to derive the optimal
3.2 Insurance Company Auditing Strategy
59
auditing range AR . The upper bound of this range would have to be
∗
the policyholders’ persistent inner threshold value θ̂ph
. Unfortunately,
the insurance company does not hold this information. However, an
adequate estimate for this value is the maximum fraud amount which
was detected during the last period. We denote this value by θ̂max and
formally define it as
θ̂max = max{θ̂ · 1A∩F }.
(46)
Setting the upper bound of the auditing range AR to θ̂max practically
implies that no incoming claim above that value will be verified. However, since one cannot be absolutely sure whether the actual maximum
of all fraudulent claims has been determined, i.e., whether or not fraud is
being committed beyond θ̂max , it is reasonable to include a safety margin
s > 0. Consequently, the updated upper bound of the auditing range is
set (1 + s) · θ̂max .
The second parameter that characterizes the audit range AR is its
∗
lower bound which is expressed by θ̂R
. Summing up, we can formulate
this auditing strategy as
∗
∗
AR = AR (θ̂R
, s, θ̂max ) = {θ̂|θ̂R
≤ θ̂ ≤ (1 + s) · θ̂max }.
(47)
We can specify the optimization problem given in Equation (44) with
regard to this new setting. The aim here is to derive the optimal lower
∗
bound θ̂R
of the auditing range such that the insurance company’s net
present value N P V is maximized:


max N P V (P, k, θ̂, Fα,θ̂∗ , AR )


ph
 θ̂∗
R
NPV ≥ 0




∆U ≥ 0.
(48)
As already mentioned above, the auditing strategy AR requires the
availability of information with regard to the prevalent defrauding behavior, i.e., θ̂max needs to be determined. Consequently, AR is applicable
after the first observation period the earliest when enough audits have
60
been performed to specify the value of θ̂max . However, this implies the
need for a different verification scheme for the very first observation period. In our case, we assume that the insurance company will revert to
an initial strategy for auditing during the first period. This one is char∗
acterized by a threshold value θ̂init
, i.e., all incoming claims which exceed
this threshold are subject to verification whereas claims whose amount
is below this indicator will be indemnified right away. We denote this
specific auditing strategy by Ainit and define it formally as
∗
∗
Ainit = Ainit (θ̂init
) = {θ̂|θ̂init
≤ θ̂}.
(49)
Then, when having gathered enough information throughout the first
observation period, the insurer switches to the auditing range AR in the
consecutive periods.
3.3
Behavioral Adaptation
So far, the policyholders in the population who are likely and willing to
defraud were assumed to adhere to a constant fraud strategy. They chose
a constant multiplicative factor α and/or the same threshold value for
∗
defrauding θ̂ph
over the course of several periods. This constancy results
from the assumption that the policyholder population does not obtain
any information on the insurance company’s prevalent verification process. Hence, there was no basis to give occasion for an adjustment of
their behavior. That also had an impact on the insurance company’s
corresponding optimal auditing strategy. As soon as the optimal verification process with respect to the prevalent defrauding behavior was
found, i.e., right after the first observation period, there was no need for
the insurance company to perform any adjustments to it. Summarizing,
no one of the participants in the insurance relationship had to change
their behavior.
However, it is realistic to assume that there might exist signals indicating whether the insurance company changes its auditing strategy
in the subsequent period or not. Especially in the case of automobile
insurance, signaling is issued by (authorized) repair shops. Since dealing
3.3 Behavioral Adaptation
61
with a large number of insured events, they are able to estimate changes
in the auditing behavior of different insurance companies. In these cases,
the policyholders who are prone to defrauding would be given a chance
to change their fraud behavior and react to the new verification scheme.
In case the insurance company announces strengthened controls in order
to combat insurance claims fraud, the fraudulent part of the policyholder
population would choose to act more carefully in terms of their defrauding strategy, i.e., lower the multiplicative factor α and/or their threshold
∗
value for defrauding θ̂ph
. In the opposite case, knowing that the insurance company will relax their auditing scheme, it can be assumed that
attempts are made to exaggerate the actual loss amount θ even more and
∗
obtain higher indemnification payments, i.e., by increasing α and/or θ̂ph
.
Figure 7 gives an overview of the interaction between insurance company and (defrauding) policyholders and the resulting adjustment processes of the respective behavioral strategies.
Both stakeholders define their respective initial strategy at the beginning of the very first observation period in t = t0 . For the insurance
company, this is its initial auditing strategy denoted by Ainit . Since by
that point in time, no information regarding the policyholders’ defrauding strategy is available, it will be characterized by an initial threshold
∗
value for auditing θ̂init
. The policyholders themselves choose a claiming
9
scheme Θ̂init . In particular, the initial fraud strategy is defined by the
∗
multiplicative factor α and an initial threshold value for defrauding θ̂ph,0
(see Equation (45)). These two strategies are applied throughout the first
observation period [t0 , t1 ]. At its end, in t = t1 , all information with regard to the actual distribution of the claimed losses throughout that
period as well as indications on the policyholders’ defrauding scheme,
i.e., the maximum value of detected fraud θ̂max,1 in [t0 , t1 ], are available to the insurance company.10 Based on this information, the insurer
determines its optimal auditing strategy AR,1 for the first observation
9 The function Θ̂ includes the defrauding strategy for the dishonest policyholders
(see Equation (45)).
10 We assume that a sufficiently high number of audits has been performed during
the course of the first period which has resulted in the detection of fraudulent claims,
i.e., θ̂max,1 is observed.
62
ex-post optimization
applied in [t0 , t1 ]
IC
AR,1
Ainit
ex-post optimization
θ̂max,1
Θ̂init
t = t0
I(AR,1 )
Θ̂(I(AR,1 ))
t = t1
θ̂max,2
I(AR,2 )
Θ̂(I(AR,2 ))
...
t = t2
Figure 7: Interaction between Insurance Company (IC) and Policyholders (PH) over the Course of the First
Two Periods in an Insurance Relationship
PH
...
AR,2
3.4 Numerical Implementation of Iterative Optimization
63
period ex-post. The corresponding optimization problem is defined in
Equation (48). This adjusted auditing scheme will then be applied in the
second observation period [t1 , t2 ]. At the same time, some signal I(AR,1 )
concerning the adjustment of the verification process is communicated
to the policyholder population. They themselves now have the opportunity to adapt their behavior accordingly. While the honest policyholders
adhere to reporting the actual loss amount when having suffered an insured loss, the ones who are willing to defraud adjust their threshold
∗
value for defrauding to θ̂ph,1
. This new claiming scheme will be denoted
by Θ̂(I(AR,1 )) and is applied in the second observation period [t1 , t2 ].
However, the change in the defrauding behavior will be registered by the
insurance company in the form of a different maximum value of detected
fraud θ̂max,2 in the course of that second period. This new piece of information on the policyholders’ behavior induces once again an ex-post
optimization of the prevalent auditing strategy to AR,2 at the end of
this very period in t = t2 . Again, the adjusted verification scheme AR,2
is applied in the following observation period after having provided the
policyholder population with a signal I(AR,2 ) concerning the change in
auditing. The interaction and adaptation processes can be repeated in
the same fashion over the course of several periods.
It needs to be emphasized that while we focus on the derivation of
the optimal auditing strategies AR,n , we also assure that all participants
are willing to maintain the insurance relationship, i.e., all participation
constraints as defined in Equations (38) and (43) need to hold when
applying the optimal verification process.
3.4
Numerical Implementation of Iterative
Optimization
In this subsection, we present the iterative approach regarding the optimization of the auditing range AR,n in the nth period from the insurance
company perspective when interaction and hence adaptation is observ-
64
able.11
Step 1: Adjustment of claiming scheme We first consider the
∗
policyholders’ claiming scheme Θ̂n (θn , α, θ̂ph,n
) is applied at the beginning of each iteration. It is characterized by the defrauders’ strategy
indicated by a constant multiplicative factor α and a threshold value
∗
for defrauding θ̂ph,n
which is adjusted each period based on the signal
I(AR,n−1 ). In accordance to Equation (45), we obtain for the policyholders’ claiming scheme of the nth period



0
∗
θ̂n = Θ̂n (θn , α, θ̂ph,n ) = θn



∗
min{αθn , θ̂ph,n
}
, no loss occurred
∗
, (50)
, no fraud or θn > θ̂ph,n
∗
, fraud and θn < θ̂ph,n
where θn represents a realization of the loss variable in the nth period.
The set consisting of all fraudulent claims associated with this strategy
is denoted by Fα,θ̂∗ .
ph,n
Step 2: Determining maximum value of detected fraud The
next step to determining the optimal auditing strategy AR,n is the identification of the maximum fraud value which was actually detected in
the nth iteration, i.e., period. Apparently, it depends on the auditing
strategy AR,n−1 which was derived to be optimal in the (n − 1)th period
and is then applied in the nth period. Using the notation presented in
Equation (36), the maximum value of detected fraud in the nth period
can be defined as
θ̂max,n = max{θ̂n · 1AR,n−1 ∩Fα,θ̂∗
ph,n
(θ̂n )}.
(51)
Step 3: Ex-post optimization of auditing strategy The value of
θ̂max,n forms the basis for the actual optimization process. Considering
11 The subscripts n in the course of this subsection indicate the nth iteration process,
i.e., the respective quantities for the nth observation period. Hereby, we consider all
n ≥ 2. The special case of n = 1 corresponds to the initial period.
3.4 Numerical Implementation of Iterative Optimization
65
Equation (47), the insurance company’s auditing strategy for the nth
period is given by
n
o
∗
AR,n = θ̂n |θ̂R,n
≤ θ̂n ≤ (1 + s) · θ̂max,n ,
(52)
with s being the safety margin.
∗
The aim is now to find the lower bound θ̂R,n
of this audit range
such that the insurance company’s net present value of future cash flows
N P V is maximized and the stakeholders’ participation constraints hold:


max N P V (P, k, θ̂, Fα,θ̂∗ , AR,n )

 θ̂∗
ph,n

R,n
N P Vn ≥ 0




∆Un ≥ 0,
(53)
where ∆Un denotes the policyholders’ gain in expected utility from having insurance coverage in the nth period.
Step 4: Communication of signal After having found the op∗
timum value for the lower bound θ̂R,n
of the auditing range, i.e., the
optimal auditing strategy AR,n , at the end of observation period n, a
signal I(AR,n ) is communicated to the policyholder population informing about the adjustment of the prevalent verification scheme. For this
purpose, we define this signal as follows:
I(AR,n ) =
∗
θ̂R,n
∗
θ̂R,n−1
+
θ̂max,n
θ̂max,n−1
!
/2.
(54)
This signal is used to adapt the claiming scheme Θ̂n+1 at the beginning of period n + 1. In this context, the signal I is an average of
two ratios: The first one represents the change of the upper bound from
period n − 1 to n whereas the second one constitutes the change of the
maximum value of detected fraud from period n − 1 to n.
66
4
Simulation Results
In Sections 2 and 3, we have presented a model framework and potential optimal auditing strategies when interaction between policyholder
and insurance company and consequently behavioral adaptation over
the course of several observation periods is possible. In this section, we
present and analyze the corresponding numerical solutions for the optimal auditing strategy from the insurance company perspective.
4.1
Parametrization of the Reference Setting
In our simulations, we consider a policyholder population consisting of
M = 2′ 500′ 000 individuals. An assumed probability of loss occurrence
of π = 0.2 leads to N = 500′ 000 loss realizations of θ per observation
period. Hereby, the latter follow a log-normal distribution. This assumption is commonly used as mentioned in Marlin (1984) since it guarantees
positive values for the realizations of the random variable. In particular,
the expected value E(θ) is set µ = 1 and the variance V ar(θ) = σ 2 = 0.4.
All individuals among the population are assumed to be risk averse. For
instance, their risk aversion parameter a is considered to be 6. Furthermore, the policyholders’ initial wealth position is set W = 0. At the
same time, the individuals have to pay an insurance premium P at the
beginning of each observation period. The latter can be split up into
the fair premium and an appropriate loading factor. The fair premium
corresponds to the expected loss. Hence, having set the expected value
of the loss variable θ to µ = 1 and considering the probability of suffering
a loss to be π = 0.2, this implies a fair premium of 0.2. However, since
the insurance company faces additional costs, it will add a corresponding loading factor to the fair premium. As mentioned in Cummins and
Mahul (2004), the loading factor can not be too large since potential
policyholders would not sign the insurance contract under such conditions. For our analyses, we will assume the total insurance premium
to be P = 0.3. Furthermore, the cost per audit is set k = 0.05 which
corresponds to 16.67% of the insurance premium P . For the purpose of
our analyses, we will disregard costs other than the ones due to auditing.
4.2 Simulation Results and Sensitivity Analyses
67
The share of fraud-prone policyholders among the population is assigned
the value p = 0.2, i.e., 20% of all policyholders who suffer an insured loss
may exaggerate that amount if it is in accordance with their defrauding
strategy. Their defrauding strategy is accompanied by the choice of the
relative fraud amount α and/or an appropriate threshold value for de∗
frauding θ̂ph
. To start with, we take α to be 2, i.e., the policyholders
who decide to defraud report back an amount twice as high as the actual
loss amount. However, in case the individuals have a threshold value for
defrauding as described in Section 3.1, they never claim more than that
∗
∗
value θ̂ph
. For the purpose of our analyses, we assume θ̂ph
= 1.1 which
is 10% higher than the expected value of the loss variable. Finally, the
insurance company needs to decide on the parameters concerning its auditing strategy A. During the very first period, it opts for a verification
process Ainit which is characterized by a threshold value for auditing
∗
θ̂init
, we set its initial value to 1 which corresponds to the expected value
of the loss variable θ. As already explained at the end of Section 3.2,
information with regard to the policyholders’ defrauding behavior needs
to be gathered first before being able to apply this verification scheme
∗
AR . Hence, we initially set θ̂init
= 1, and determine the paramters θ̂max
∗
and θ̂R based on the information obtained based on the first program
run. With regard to the upper bound of the auditing range, the value
for the safety margin s is assumed to be 0.1, i.e., the upper bound of the
auditing range is 10% higher than maximum of detected fraud θ̂max . The
resulting auditing range AR is then applied in a consecutive program run.
Table 2 sums up the choices for the input parameters for the reference
setting as introduced above. In the course of this and the following
sections, we base our simulations and studies on these values.
4.2
Simulation Results and Sensitivity Analyses
The remainder of this section constitutes the presentation and discussion of the simulation results. For the simulations, we adhere to the
parameter choice presented in Table 2 unless noted otherwise.
68
Input parameter
Total number of policyholders
Number of loss realizations
Loss distribution
Insurance premium
Share of fraud-prone policyholders
Relative fraud amount
Initial threshold value for auditing
Safety margin
Policyholder’s initial threshold
Auditing cost
Risk aversion parameter
Reference level
M
N
θ
P
p
α
∗
θ̂init
s
∗
θ̂ph,0
k
a
2’500’000
500’000
lnN (1, 0.4)
0.3
0.2
2
1
0.1
1.1
0.05
6
Table 2: Input Parameters for the Reference Setting
4.2.1
Development of Optimization Results Over Several
Iterations
In this subsection, we present and discuss the development of the optimal
auditing range AR over the course of several iterations. Furthermore, we
analyze its impact on quantities like the number of performed audits, the
amount of fraudulent claims, the net present value N P V and the gain
in utility ∆U . To get a better insight of the effects, we consider both the
parametrization of the reference setting with costs per audit k = 0.05
as well as the case when the costs per audit are raised to k = 0.3 and
compare the results.
Development of Optimal Auditing Range AR
Comparing Figure 8(a) with Figure 8(b), we can see that higher costs per
audit k result in a slightly broader optimal auditing range AR , i.e., the
share of values which may be verified becomes greater. At the same time,
the optimal auditing range shifts in an upward direction, i.e., the value
of the claims which may be subject to auditing becomes higher. For the
insurance company this implies that, in case of high expenditures per
case, they should focus their investigations on those claim which exhibit
high saving potential whenever an engagement in fraudulent activities is
detected.
1.0
1.5
2.0
θ*R
θmax
θ*ph
2
4
6
number of periods
8
10
(a) Development of Optimal Auditing
Strategy, k = 0.05
0.0
0.5
1.0
1.5
2.0
θ*R
θmax
θ*ph
0.5
0.0
69
2.5
2.5
2
4
6
number of periods
8
10
(b) Development of Optimal Auditing
Strategy, k = 0.3
Figure 8: Development of the Optimal Auditing Strategy throughout
the Course of several Iterations
The development is displayed for two different choices of cost per audit k respectively. The
remaining parameters are chosen as presented in Table 2.
From the policyholder perspective, we find that higher costs per audit
∗
k lead to a higher threshold of defrauding θ̂ph
, i.e., the value up to which
policyholders take build-ups into consideration increases. This observation shows that the cost per audit does not only have an impact on the
insurance companies auditing strategy but indirectly also on the policyholders’ behavior, in particular on their defrauding strategy. Since the
insurer signals an upward shift in his verification behavior, policyholders
(and the corresponding service providers) are left with the impression
that inflating a loss amount up to some value is more likely to remain
undetected than in the previous period. In return, they then raise their
threshold value for defrauding.
Development of Number of Performed Audits and Fraudulent
Claims
We measure both the number of performed audits and the number of
fraudulent claims in relation to the total number of filed claims.
Figure 9(a) confirms the intuition that higher costs per audit k result
in a lower share of incoming claims which are subject to verification,
i.e., fewer auditing processes are performed. As mentioned above, this
70
number of fraud (in %)
5
10
15
number of audits (in %)
10
20
30
40
50
20
k=0.05
k=0.3
0
0
k=0.05
k=0.3
2
4
6
number of periods
8
10
(a) Development of the number of
audits performed by the insurance
company when applying the optimal
auditing strategy for the respective
period.
2
4
6
number of periods
8
10
(b) Development of the number of
fraudulent claims when the insurance
company applies the optimal auditing
strategy for the respective period.
Figure 9: Development of Number of Audits and the Number of Fraudulent Claims over the Course of Several Iterations
Both quantities are measured in relation to the total number of losses, i.e., filed claims. The
development is illustrated for two different choices of the cost per audit k. The remaining
parameters are chosen as presented in Table 2.
restriction in the number of verification processes leads the insurance
companies to focus on those claims which, in case of detected fraud,
have a higher saving potential, i.e., higher valued claims.
From the policyholder point of view, the number of fraudulent claims
consequently increases (see Figure 9(b)). This finding is in line with our
results presented in Figure 8. Due to receiving a signal indicating an
upward shift, the policyholders themselves raise their threshold value
for defrauding. As a consequence of this elevation, the number of losses
below the threshold increases resulting in more cases where actions are
taken to inflate the claim amount.
It needs to be noted that even though 20% of the policyholder population are willing to exaggerate their loss amount, the real numbers lie
below that value (see Figure 9(b)). This phenomenon can be explained
∗
with the existence of the threshold value for defrauding θ̂ph
up to which
fraud is actually taken into consideration. The actual share of buildup among all claims may depend on different factors. As already seen
in Figure 8, higher costs per audit k result in an increased defrauding
∗
threshold θ̂ph
. This, however, implies a higher likelihood of the actual
71
loss amount being below the threshold value. As a consequence, the
amount of fraud may increase up to the maximum of 20%.
0.6
Development of Net Present Value N P V and Gain in
Utility ∆U
0.2
0.4
NPV k=0.05
NPV k=0.3
∆U k=0.05
∆U k=0.3
2
4
6
number of periods
8
10
Figure 10: Development of the Insurance Company’s Net Present Value
and the Policyholders’ Gain in Utility over the Course of Several Iterations
For each period, the optimal auditing strategy is applied. The development is illustrated
for two different choices of the cost per audit k. The remaining parameters are chosen as
presented in Table 2.
Looking at Figure 10, it strikes attention that both N P V and ∆U
are positive for both chosen values of the cost per audit k. Especially
with regard to gain in utility ∆U , this implies that the policyholders
among the population are willing to adhere to the insurance relationship
(see Equation (43)). This observation proves that the derived optimal
auditing schemes are feasible from both stakeholders’ perspectives.
From the insurance company point of view, higher costs per audit
k lead to a lower net present value N P V . This phenomenon may be
explained by the fact that higher auditing costs result in fewer auditing
processes (see Figure 9(a)) but in a higher share of fraudulent claims
(see Figure 9(b)). As a consequence, more exaggerated claim amounts
remain undetected resulting in higher, unjustified expenses from the insurance company perspective.
72
Another intriguing result can be taken from Figure 10 recalling the
case where behavioral adaption is not possible. Since in that simplified
setting signals are not exchanged, no changes in both the auditing and
the claiming strategy would be possible, resulting in a stable solution
after period two. Comparing the values from the second observation
period with the consecutive ones in Figure 10, we see that, given the
current setting, from the policyholder point of view, the gain in utility
in the second observation period is considerably higher than the ones
in all the consecutive observation periods, implying that the option to
change ones strategy is disadvantageous to them. This result is very
enlightening since it contradicts the widespread opinion that adapting
the defrauding strategy based on signals especially from third parties
like service providers is favorable from the individuals perspective.
4.2.2
Sensitivity Analyses
The remainder of this section constitutes the presentation and discussion
the impact of relevant input parameters have on the insurance company’s
optimal auditing strategy AR and the resulting effects on the net present
value N P V . In particular, the influence of the cost per audit k, the
relative fraud amount α and the policyholder’s initial threshold value
∗
θ̂ph
will be analyzed respectively.
For this purpose, we consider the final values of the optimal auditing range AR , net present value N P V and gain in utility ∆U after 12
iterations each when stable results are achieved.
Cost Per Audit
We take k ∈ [0.05, 0.5] and illustrate the results for the auditing range as
well as the corresponding values for the insurance company’s net present
value N P V in Figure 11 keeping the remaining parameters as presented
in Table 2.
Figure 11(a) shows that a change in the cost per audit k merely
has an impact on the width of the auditing range. However, the values
themselves which trigger the verification process shift upwards for higher
values of this specific input parameter. The higher the cost per audit k
∆U
NPV
0.6
2.0
0.4
1.5
θ*R
θmax
0.1
0.2
0.3
cost per audit k
0.4
(a) Auditing Range Depending on
Cost Per Audit k
0.5
0.0
0.2
1.0
0.5
0.0
73
0.8
0.1
0.2
0.3
cost per audit k
0.4
0.5
(b) Corresponding Values for ∆U and
NP V
Figure 11: Auditing Range and the Corresponding Objective Quantities
from Insurance Company and Policyholder Depending on the Cost Per
Audit k
is, the higher are the claim amounts which will be subject to auditing.
These findings are in line with the ones presented in Figure 8. The lower
the costs per audit are, the more verification processes the insurance
company can perform assuming a given budget. This allows the insurer
to review a larger number of incoming claims and consequently enhances
the probability of revealing the fraudulent ones, in particular those which
are close to the policyholders’ threshold value. Such an approach enables
the insurance company to adjust its auditing strategy optimally to the
prevalent defrauding behavior at an early stage. As a consequence, potential escalations with regard to the fraud strategies, i.e., expansion of
the policyholder’s individual threshold value, can be prevented. In this
context, note that the upper bound of the audit range in Figure 11(a)
which is defined by the maximum values of detected fraud, is decreasing
when lowering the costs per audit k.
From the policyholder perspective, we observe that the gain in utility
∆U is always positive, implying that the auditing strategies discussed
above are feasible.
74
Relative Fraud Amount
0.4
1.5
θ*R
θmax
0.5
1.0
1.5
2.0
relative fraud amount α
(a) Auditing Range Depending
Relative Fraud Amount α
2.5
0.0
0.2
1.0
0.5
0.0
∆U
NPV
0.6
2.0
0.8
Considering α ∈ [0.25, 2.5], we illustrate the effects on the optimal auditing strategy AR in Figure 12 and discuss them afterwards. Again, the
remaining input parameters are chosen as given in Table 2.
0.5
1.0
1.5
2.0
2.5
relative fraud amount α
NP V
from Insurance Company and Policyholder Depending on the Relative
Fraud Amount α
As can be seen in Figure 12(a), the relative fraud amount α has a considerable impact on the insurance company’s optimal auditing strategy.
The width of the auditing range increases (slightly) for greater values of
this input parameter. At the same time, the claim values which indicate
the necessity of verification shift in an upward direction. As a result,
the relative fraud amount α also has an impact on both stakeholders’
objective quantities. Figure 12(b) illustrates that the insurance company’s net present value N P V is increasing when raising the values of
this input parameter. The reason for this observation is that in case of a
successful verification process, i.e., the detection of fraudulent behavior,
the relative fraud amount α will become known. The higher the relative
fraud amount α is - while assuming the loss distribution itself has not
changed - the more profitable it is from the insurance company perspective to audit claims which demand high indemnity payments. Hence,
the relative fraud amount α has a direct influence on the upper bound
of the auditing range. This observation gets even clearer when keeping
75
in mind that the latter is determined by the maximum of all detected
fraudulent claims during one period.
Furthermore, Figure 12(b) shows that the insurance company profits from raising and widening the auditing range whenever the relative
fraud amount α is increased. In this current scenario, the relative fraud
amount α is raised while keeping the probability for fraudulent behavior
p constant, i.e., fraud is not committed more often but more severely.
Since the loss distribution remains unchanged, this implies that fraudulent claims are more likely to be the higher valued ones. From the
insurance company perspective this means that auditing becomes more
profitable when shifting its auditing range into this area. Since in the
case of detected fraudulent behavior no indemnity payments to the policyholder are made, this auditing strategy has a positive effect on the
insurance company’s net present value N P V .
Again, we see that the policyholder’s gain in utility ∆U is positive
throughout all observation periods guaranteeing that the corresponding
auditing strategies are feasible.
Policyholder’s Initial Threshold
The final input parameter whose influence on the insurance company’s
optimal auditing strategy we aim to analyze is the policyholders’ initial
∗
threshold value θ̂ph
. This values serves as an upper bound for the potential fraud amount. As already introduced in Section 3.1, policyholders
who decide to commit fraud exaggerate their loss amount by some con∗
stant factor α up to that threshold value θ̂ph
. For the purpose of our
∗
sensitivity analysis, we consider θ̂ph ∈ [1.05, 1.5] and present the results
in Figure 13.
Figure 13 illustrates that the choice of the policyholders’ threshold
∗
value for defrauding θ̂ph
has a significant impact on the optimal auditing
∗
range AR . Higher values of θ̂ph
imply an increasing discrepancy with re∗
gard to the insurance company’s initial threshold for auditing θ̂init
which
is kept at a constant value of 1.0. As a result, the optimal auditing range
∗
AR becomes broader for increasing values of θ̂ph
. In particular, the upper
bound of the auditing range is continuously increasing while the lower
bound remains almost constant. The explanation for this phenomenon
76
0.4
1.5
θ*R
θmax
1.1
1.2
1.3
1.4
threshold for defrauding θ*ph
1.5
(a) Auditing Range Depending on
∗
Policyholder’s Initial Threshold θ̂ph
0.0
0.2
1.0
0.5
0.0
∆U
NPV
0.6
2.0
0.8
1.1
1.2
1.3
1.4
threshold for defrauding θ*ph
1.5
NP V
from Insurance Company and Policyholder Depending on the Policy∗
holder’s Initial Threshold Value θ̂ph
∗
is that a raise in the policyholders’ defrauding threshold θ̂ph
results in
an increase in the share of fraudulent claims among all filed claims. In
particular, the amount of exaggerated claims among the higher-valued
ones will increase. Since the upper bound of the optimal auditing range
is determined as the maximum of detected fraud θ̂max and some safety
margin s, its value will become higher as well.
This explanation can also be used for understanding the resulting
marginal increase in the insurance company’s net present value N P V .
∗
Since higher values of θ̂ph
lead to an increased likelihood of a loss amount
being below this threshold, the number of cases where fraud-prone policyholders engage in build-up rises, especially in the high-value segment.
Whether this development has an impact on the insurance company’s
net present value depends on the prevalent auditing strategy AR and
its detection success. Interpreting Figure 13(b), the number of detected
fraudulent activities increases compared to the number of unjustifiably
paid out claims resulting in fewer indemnification which in turn leads to
a slightly higher net present value.
Like in the previous analyses, the gain in utility ∆U from having
signed an insurance contract prior to the occurrence of loss is always
positive implying that all participation constraints are met.
5 Critical Discussion
5
77
Critical Discussion
The insurance company’s optimal auditing strategies derived in this paper are characterized by two threshold values indicating the range of
claimed values which should be subject to verification. All the other incoming claims are indemnified without further particular proof of their
truthfulness. In particular, the presented approach results in examining claims especially from the medium segment leaving out small and
high-valued ones. This strategy is based on the assumption that policyholders avoid any engagement in fraudulent activities whenever the
actual loss amount is above some personal threshold since they fear the
probability of being caught to be particularly high in this segment. As
a consequence, theoretically there is no need to verify incoming claims
of higher magnitude.
From a practical point of view, however, this approach appears to
be incomplete. It seems unimaginable that insurance companies indemnify loss amounts which are far above the corresponding expected value
without further examination of their legitimization. For this purpose,
we once more extend our model framework to accommodate this aspect.
∗
We therefore introduce an additional threshold value for auditing, θ̂high
.
In addition to verifying all incoming claims whose value fall withing the
auditing range AR , the insurance company also audits those which are
∗
above the new threshold value θ̂high
.
In order to depict the impact of adding an additional threshold value
∗
to the existent auditing range AR , we consider θ̂high
= 1.5 which equals
one and a half the expected value of the loss amount θ. The remaining
parameters are chosen as in the reference setting. Figure 14 illustrates
the results.
As can be seen from Figure 14, in the particular setting of our model
framework, an additional auditing threshold identifying high-valued claims
for verification does not generate benefit for the insurance company. The
∗
optimal auditing range AR including the additional threshold θ̂high
results in a net present value of N P V = 0.118 after the tenth observation
78
θ*R
θmax
θ*ph
0.5
0.0
0.2
1.0
1.5
0.4
2.0
0.6
2.5
2
4
6
number of periods
8
NPV
∆U
10
(a) Auditing Range with Additional
∗
Auditing Threshold θ̂high
= 1.5
2
4
6
number of periods
8
10
NP V
Figure 14: Auditing Range Including an Additional Auditing Threshold
and the Corresponding Objective Quantities
∗
The additional auditing threshold θ̂high
= 1.5 is displayed as well as the corresponding
objective quantities from insurance company and policyholder perspective. The remaining
parameters are chosen as presented in Table 2.
period, whereas the optimal auditing range AR alone in N P V = 0.120,
i.e., we obtain a change of 2%. From the policyholder perspective, the
∗
introduction of the new threshold value θ̂high
has no impact. The gain
in utility ∆U remains positive.
This observation can be explained by the underlying defrauding behavior of the policyholders. Since fraud-prone individuals do no inflate
loss amounts in this segment, the insurer performs costly auditing processes without ever detecting any fraudulent activities, i.e., additional
costs arise without ever leading to savings due to refusal of indemnification. As a consequence, the insurance company’s net present value N P V
decreases when introducing the additional threshold value for auditing
high-valued claims. From the policyholder perspective, no changes in
the gain in utility arise since the new auditing scheme has no impact on
the indemnification payments.
In practice, however, this observation does not necessarily have to
hold true. On the one hand, policyholders and/or service providers
might indeed engage in fraudulent activities in case of high-valued losses
or even if no insured loss occurred at all. As an example, Emerson
(1992) recapitulates the case ”State vs. Book” in which the policyholder
6 Conclusion
79
exaggerated the value of his stolen luxury class automobile by 20%. On
the other hand, insurance companies might profit from performing verification processes even in those cases when no fraudulent activities are
detected. Their investigations might have a deterrent effect discouraging
policyholders and service providers to dare fraud attempts in the future.
The corresponding monetary benefits, however, are almost impossible to
measure (see, e.g., Viaene and Dedene (2004)).
6
Conclusion
In this paper, we develop a model framework which depicts an optimal
auditing scheme with regard to inflated insurance claims. The key element in this context is the auditing range which - triggered by the filed
amount - selects those claims which should be subject to verification.
Its actual configuration is chosen in way that maximizes the insurance
company’s position while at the same time maintaining contract attractiveness such that policyholders are willing to adhere to the insurance
relationship. In addition, we incorporate the possibility for each stakeholder to adapt its behavioral strategy over the course of several periods.
By this means, we take into consideration that changes in the policyholder defrauding behavior have a crucial impact on the optimal corresponding auditing strategy and vice versa. Insurance companies may
use their experiences from previous verification processes as a source of
information whereas policyholders often rely on third parties like service
providers.
One of our main findings is the derivation of the optimal auditing
scheme characterized by a range whose exact boundaries we are able
to calculate. We come to the conclusion that given some constant cost
per audit it is optimal to verify the truthfulness of claims from the midvalue segment. In particular, it is not reasonable from the insurance
company point of view to examine small claims since the accompanying
costs outweigh the savings potential in case of detected fraud. Not verifying high-valued claims results from the assumption that fraud-prone
policyholders do not inflate the magnitude of their losses above some
80
personal threshold value since they fear that the likelihood of getting
caught in this segment to be above-average. Omitting this assumption,
however, may require the introduction of an additional threshold value
for auditing.
Furthermore, we are able to show that while the option to adapt one’s
strategy might be favorable from the insurance company perspective, it
actually has a negative impact on the policyholders’ position compared
to the situation where no signals are exchanged based on which one
could change its behavior. This result is astonishing since it disproves
the common believe that adapting the defrauding strategy with the help
of signal from service providers would be advantageous from the policyholder point of view.
Using a numerical approach based on Monte Carlo simulations, we
are able to illustrate and analyze the impact of different parametrizations
on the optimal auditing range. High costs per audit as well as a high
relative fraud amount result in an upward shift of the auditing range,
whereas an increase in the policyholders’ defrauding threshold leads to
broadening the range. With regard to the insurance company’s objective
quantity, the relative fraud amount has a particularly strong impact on
its result.
References
81
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85
Part III
The Identification of
Insurance Fraud:
Abstract
Fraud is a major concern in the insurance industry. Time after time,
spectacular incidents become public of individuals trying to scam tremendous indemnifications from their insurance companies. The majority of
claims, however, particularly those seeking low to medium indemnification, exhibit no obvious signs of fraudulent activity thereby leading the
insurer to believe they were legitimate. In this study, we therefore focus on determining the characteristics that make an accurate distinction
between fraudulent and legitimate claims possible. In addition to identifying dishonest cases more systematically, applying a criteria catalog
would enable an efficient use of the limited resources with which fraud
investigation divisions are usually endowed. The basis of our analysis
is established by a comprehensive data set of automobile claims from a
large Swiss insurance company collected throughout the years of 2004
to 2011. The results of the logistic regression analyses reveal different
relevant determinants on the policyholder, vehicle, policy and loss level.
Contrary to common assumptions, it is most often individuals with a
flawless driving record possessing high-valued cars who decide to defraud their insurance company. In extension, we place special focus on
how the amount of loss affects an individual’s likelihood of engaging in
fraudulent activities.12
12 K. Müller. The Identification of Insurance Fraud - An Empirical Analysis. Working Papers on Risk Management and Insurance, 2013.
This paper has been presented at the 2013 Annual Conference of the Asia-Pacific
Risk and Insurance Association in July 2013.
86
1
III Empirical Analyses
Introduction
Insurance fraud has been a key concern in the industry ever since. To
date, particularly astonishing incidents have regularly made headlines
involving tremendous illegitimate indemnifications from insurance companies. These cases, however, may just be the tip of the iceberg. According to a report by the Association of British Insurers (2012), 15
fraud attempts are being detected each hour of every day, summing up
to 139,000 cases worth nearly 1 billion GBP in the United Kingdom in
the year of 2011. Even though insurance companies and related organizations take numerous measures to combat this wide-spread phenomenon,
due to its secretive nature, a major part of fraud goes undetected, resulting in an estimated total of another 2 billion of excess payments each
year in the United Kingdom (see Association of British Insurers (2012)).
In light of its prevalence and economic extent, several insurance companies established their own investigative units to uncover insurance
fraud. Being equipped with limited budgets, however, they are forced to
verify only those claims which exhibit a comparatively high probability
of containing fraud and a relatively high saving potential rather than analyzing every single incoming claim. A recent survey conducted by Coalition Against Insurance Fraud (2012) among 74 mostly property/casualty
insurers revealed that 88% of the respondents employ technologies to support their investigators, two of the most common being “automated red
flags” and “scoring capabilities”.
Nevertheless, many insurers are just beginning to discover the necessity of establishing fraud investigation divisions within their own company. Interviews with experts in this field have revealed that, in particular, smaller insurance companies may not deem it worthwhile to invest in
costly software, still relying on their intuition when it comes to detecting
fraudulent claims.
Our aim is, hence, to identify the determinants that would make
it possible to draw conclusions on the likelihood of a claim seeking unfounded indemnification. Based on such a catalog of criteria, insurance
companies would be able to use their limited resources to reveal defraud-
1 Introduction
87
ing attempts more effectively. In addition, honest policyholders may
also benefit from an improved auditing scheme and hit ratio. Processing
times would likely shorten, thereby resulting in reduced waiting periods
for indemnification.
With the aim of detecting insurance fraud by engaging in auditing
processes, the insurance companies’ strategy can be assigned to the category of costly state verification (see, e.g., Townsend (1979), Mookherjee
and Png (1989), Bond and Crocker (1997), Picard and Fagart (1999),
Dionne, Giuliano, and Picard (2009)). The latter is based on the assumption of information regarding the (allegedly) insured event being
distributed asymmetrically between the policyholder and the respective
insurance company. It is, therefore, possible that the policyholder may
misrepresent facts and figures in order to obtain a higher or even unjustified indemnification. To confront and discourage any defrauding
attempts, insurance companies perform verification processes to determine the truthfulness of incoming claims and may then choose to impose
penalties. Since audits, however, incur costs and the respective divisions
have limited funds at their disposal, a choice must be made as to which
of the incoming claims to test. An important consideration in this context is the weighing of incurred costs against potential savings related
to detected fraud attempts.
An alternative approach in the handling of insurance fraud is subsumed under the term costly state falsification (see, e.g., Crocker and
Morgan (1998), Crocker and Tennyson (2002)). Other than in the firstmentioned one, the idea behind this approach is for the policyholder
to be able to manipulate a claim at monetary expense such that the
fraud attempt becomes undetectable. In this case, auditing proves to be
obsolete, leaving the insurer with the potential option of indemnifying
all incoming claims without further verification while at the same time
raising the premium payments. This approach would be in line with the
findings of Clarke (1989) and Morley, Ball, and Ormerod (2006), who revealed that insurance companies were concerned with reputational risks
as a consequence of excessive auditing. Such an approach could create
a negative image in public perception, as a result of which individuals
88
may be tempted to switch to one of the company’s competitors.
For the purpose of our study, we will use the term “fraud” or “fraud
attempt” as a collective term for all those cases within our data sample
for which the respective insurance company has found sufficient evidence
to categorize them as such. The phenomenon of insurance fraud having
many facets, there is a variety of forms that may be observed in this context (see, e.g., Picard (2001), Crocker and Tennyson (2002), Tennyson
(2008)).
Based on the severity of the offense, a common distinction is made
between soft fraud and criminal/ hard fraud. According to Derrig (2002),
criminal fraud is defined as the “willful act of obtaining money or value
from an insurer under false pretenses or material misrepresentations”.
Expert interviews as well as previous research (see, e.g., Weisberg and
Derrig (1991), Viaene and Dedene (2004), Tennyson (2008)), however,
have revealed that the majority of defrauding attempts is situated in
an ethical gray area rather than containing outright fraud. Even in the
absence of a definition, the term soft fraud is related to attempts to
inflate the claims amount after the occurrence of an insured event in
order to obtain higher indemnification.
Aside from policyholders, other potential actors associated with the
occurrence of insurance fraud include insurance brokers, intermediaries
and service providers (see, e.g., Dulleck and Kerschbamer (2006) and
International Association of Insurance Supervisors (2011)). Whether it
is charging excessive prices or providing unnecessary services and treatments, such activities can be performed either with or without the knowledge of the respective policyholder aiming to obtain additional payments
from the insurance company (see Tennyson (2008)).
The aim of our study is to identify the determinants that help to accurately distinguish between legitimate and illegitimate incoming claims.
Hereby, we take into account characteristics regarding the policyholder
himself, the insured vehicle, the signed policy and the loss event itself.
1 Introduction
89
Previous literature has analyzed potential indicators that predict
the likelihood of fraud by employing discrete choice models.13 For the
specifics of the US insurance market, Tennyson and Salsas-Forn (2002)
as well as Derrig, Johnston, and Sprinkel (2006) analyze the phenomenon
of insurance fraud related to automobile personal injuries requiring medical treatment. While the latter present some exemplary measures to
handle fraud attempts, Tennyson and Salsas-Forn (2002) find that auditing processes contain both a detection and a deterrence component.
Furthermore, Belhadji, Dionne, and Tarkhani (2000) identify fraud indicators to determine their actual impact on the fraud probability of a
claim using a representative data set from Canadian insurance companies. A slightly different path in this context is followed Dionne et al.
(2009). Using the scoring approach, they derive a red flag strategy indicating which of the suspicious claims should be referred to an external
investigative units. The result is an optimal auditing strategy in the face
of a cost-minimizing insurance company.
Apart from that, Artı́s, Ayuso, and Guillén (1999), Artı́s, Ayuso,
and Guillén (2002), Caudill, Ayuso, and Guillén (2005), Pinquet, Ayuso,
and Guillén (2007) and Bermúdez, Pérez, Ayuso, Gómez, and Vázquez
(2008) address potential issues which may surface in relation to the data
sample itself. These include selection biases based on the insurers’ own
criteria for selecting claims to undergo auditing in the first place (see
Pinquet et al. (2007)) and oversampling of fraudulent claims in the data
set (see Artı́s et al. (1999) and Bermúdez et al. (2008)). Furthermore,
Artı́s et al. (2002) and Caudill et al. (2005) account for misrepresentation of honest claims, i.e., cases that the insurance company mistakenly
considers as legitimate.
With this paper, we aim to extend the existing studies on the identification of insurance fraud. Based on the literature, we develop a number
of hypotheses to gain new insights into the drivers of fraudulent behavior.
Furthermore, we utilize of a comprehensive data set from the automobile
13 For different approaches to determining fraud indicators see, e.g., Derrig, Weisberg, and Chen (1994) and Brockett, Derrig, Golden, Levine, and Alpert (2002). Ai,
Brockett, Golden, and Guillén (2013) use such indicators to determine the overall
fraud rate in a population of filed claims.
90
insurance market in Switzerland. To the best of our knowledge, such an
analysis on indicators predicting the existence of insurance fraud has not
been performed for the Swiss market to date.
The data sample we acquired for our analysis is comprised of audited claims from a major Swiss insurance company. The audits were
performed throughout the time period between 2004 and 2011 within
their automobile devision. Potential fraud indicators are available on
the policyholder, vehicle, policy and loss level. By applying logistic regression methods, we determine which characteristics have a significant
impact on the occurrence of fraud and could therefore be used to trigger
auditing processes.
One particular interesting result refers to the impact of the insured
loss amount on the policyholder’s decision to engage in fraudulent activities. We are able to show that the option to defraud one’s insurance
company is solely taken into consideration for comparably small loss
amounts, proving that behavioral adaption in the context of insurance
fraud does take place.
Particularly from a practical perspective, the identification of factors
revealing the probability of defrauding attempts is crucial. Being able to
assess the fraud potential of an incoming claim is an essential step in the
claims settlement process. Since the resources that are set aside to combat insurance fraud are limited, it is of great importance to distinguish
between those claims for which verification is deemed sensible and those
which should be paid out right away. This paper’s derived catalog of
criteria can serve as a basis for implementing auditing strategies to handle defrauding attempts more effectively. The extent to which insurance
companies make use of this information, however, depends particularly
on their available budgets.
The remainder of this paper is structured as follows: Section 2 sets
forth ten hypotheses with regard to potential fraud indicators and their
respective effect on the likelihood of committing fraud. We then provide
a comprehensive overview of our data sample using descriptive measures,
before presenting our theoretical model. The results of the logistic re-
2 Theory and Hypotheses Development
91
gressions are reported and discussed in Section 3. Finally, in Section 4,
we summarize our findings and provide an outlook for future research.
2
Theory and Hypotheses Development
2.1
Development of Hypotheses
In the following, we develop several hypotheses with regard to the determinants that may serve as potential fraud indicators revealing the
probability of an incoming claim being untruthful. Using data from
auto insurance policies of a Swiss insurance company, we take into consideration characteristics on the policyholder, vehicle, policy and loss
level.
Previous literature has already analyzed suitable indicators in the
context of insurance claims fraud. In our study, we pick up the presented research results to examine whether or not the fraudulent claims
in our data sample exhibit the same characteristics. In addition, several
additional hypotheses are introduced which, to the best of our knowledge, have not yet been tested empirically.
Fraud Indicators Based on Policyholder Characteristics
Policyholder Age According to a representative population survey
commissioned by the German Insurance Association GDV (2011), there
is a wide-spread perception among all age groups that defrauding one’s
insurance company would generally be easy. A closer look, however,
reveals this attitude to be slightly more prevalent among younger policyholders than older ones. Similarly, a study published by the Insurance
Fraud Bureau (2012) reveals that while 8% of all survey participants
stated their willingness to participate in a staged accident for financial
profit, this number increases to 14% among young people. One reason
behind this attitude may be that financial benefits from successful fraud
attempts carry more weight for younger policyholders than for older ones
due to their respective average assets. These elaborations are also in line
with the findings of Artı́s et al. (2002) who show in their data sample
92
that younger drivers are more likely to try to defraud their insurance
company. Therefore, we hypothesize:
H1 : The younger the policyholders are, the more likely they
are to engage in fraudulent activities.
Fraud Indicators Based on Vehicle Characteristics
Vehicle age In connection with characteristics related to the insured
vehicle itself, its age may be of interest to predicting the probability of
a claim being fraudulent. Artı́s et al. (2002) were able to prove this link,
empirically showing that older vehicles are more likely to be involved in
fraudulent activities since policyholders may perceive its cash value as a
form of additional funds when purchasing a new car. Following this line
of reasoning, one can assume fraud in this context not only to occur in
the form of build-up, but also as seeking indemnification for uninsured
events in order to gain financial benefits. We include this aspect in our
study, and hypothesize:
H2 : The older vehicles are, the more likely they are to be
involved in insurance claims fraud.
Vehicle type Additionally, the vehicle’s class may be associated with
a particular probability of being involved in fraudulent activities. In our
data set, we can distinguish between regular passenger cars, transporters
and motorcycles. Insurers have long been known to take the vehicle class
into account when pricing the policy since it serves as an indicator for
driving behavior and related accident frequency. Therefore, we include
this variable in our analysis and postulate:
H3 : The class of an insured vehicle has a significant impact
on the probability of filing a fraudulent claim.
Vehicle value Another characteristic related to the vehicle’s characteristics is its value, which is composed of its catalog price and the value
of any accessories, such as audio systems, car phones or air conditioning.
In particular, these additions, whether fitted already by manufacturer or
2.1 Development of Hypotheses
93
at some later point, have the potential to substantially increase the insured vehicle’s value the consequence being higher insurance premiums.
As policyholders then have financial incentives to engage in fraudulent
activities, we aim to verify the following hypothesis within our data sample:
H4 : The higher the value of an insured vehicle, the more
likely defrauding attempts become.
Leasing More and more individuals are choosing to lease their automobiles instead of purchasing them. A recent representative study
in Switzerland commissioned by comparis.ch (2011), the leading Swiss
Internet comparison service, revealed that the share of leased vehicles
accounts for 14% of the overall private automobile market. This number
rises even up to 23% with regard to the share of leased cars among all
new private ones. With an average price of 42,328 CHF, leased cars are,
on average, slightly more expensive than those paid for in cash costing
40,091 CHF. Leasing contracts usually provide the lessee with the right
to purchase the then-used vehicle at the end of contract. Since the price
is generally determined already by the time of signing the leasing agreement, it is in the lessee’s interest to obtain the car in its best possible
condition. This, however, may incentivize individuals to misuse their insurance coverage, to eliminate defects of any kind at the expense of the
insurance company. As a consequence, one could expect the magnitude
of claims to be disproportionally high for leased vehicles. Therefore, we
hypothesize:
H5 : Leased vehicles are more likely to be engaged in fraudulent activities than purchased ones.
Fraud Indicators Based on Policy Characteristics
Loss-free An individual’s perception of insurance in general may
serve as an incentive in the context of fraud. Surveys have discovered
that policyholders perceive build-up in particular as a way to obtain
a compensation for former premium payments without having made
a claim (see, e.g., International Association of Insurance Supervisors
94
(2011), Miyazaki (2008), Duffield and Grabosky (2001)). This attitude
adopts the common idea of treating insurance as an investment which
has to eventually pay off. Consequently, we expect policyholders who
have been in an insurance relationship for several periods without filing
a claim to use the opportunity to inflate the amount of an insured loss
by the time of its occurrence. We therefore postulate:
H6 : The longer the insurance relationship exists while remaining loss-free, the more likely defrauding attempts become.
Records In the context of automobile insurance, most insurance
companies offer their policyholders bonus-malus policies providing them
an incentive not to file claims for all kinds of minor losses and at the
same time rewarding them for accident-free driving records (see, e.g.,
Moreno, Vázquez, and Watt (2006)). We believe that bonus-malus policies may be an obstacle in filing a claim to begin with, particularly for
small damages. Since, however, it implies negative consequences for the
policyholder in the form of increased premium payments for the consecutive period, this penalty may at the same time provide an incentive
to obtain additional payments from a claim in order to compensate for
additional future expenses. This kind of attitude is expected to be particularly observable among individuals having a bad driving record since
they already are likely to be at the highest premium level. In these cases,
Artı́s et al. (1999) argue that the claimants may feel like they have “nothing to lose” anyway. Based on their data sample, Artı́s et al. (1999) are
able to show that the number of previous claims indeed has an impact
on the likelihood of a fraud attempt. We therefore aim to verify the
following:
H7 : The higher the number of previous claims, the higher the
likelihood of a claim containing fraud.
Fraud Indicators Based on Loss Characteristics
Type of damage In discussions with experts from several fraud
investigation divisions, attention was drawn to the different types of
95
damages for which policyholders file claims. Particular focus was placed
on loss events whose magnitude may easily be manipulated by either
“overprovision” or “overcharging” (see Tennyson (2008)) as well as to
damages that are allegedly difficult to verify and, hence may encourage defrauding attempts. These include, among others, glass breakage
and collisions. Therefore, we include this variable in our analysis and
postulate:
H8 : Types of damages which are deemed to be difficult to
verify (e.g., glass breakage and collisions) are more likely to
contain fraud than those which are deemed easily verifiable.
Loss amount In filing a fraudulent claim, its magnitude is of particular importance. We are convinced that policyholders do have a presentiment of the existence of auditing and hence take it into consideration
when engaging in fraudulent activities. With claims of high magnitude
being supposedly one of the targets under investigation, we expect fraud
prone policyholders to contemplate such actions solely in cases of smallervalued loss events and in the form of a percental surcharge on the actual
loss amount. This approach would additionally leave the option to excuse any incorrect claims as a mistake if audited by the insurance company(see, e.g., Emerson (1992)). Hence, we hypothesize the following:
H9 : Smaller-valued claims are more likely to contain some
kind of fraud than higher valued ones.
Delay Previous studies have shown (see, e.g., Artı́s et al. (2002),
Dionne et al. (2009)) that the longer the lag between the accident and
the filing of a report to the insurance company, the higher the likelihood
of the respective claim containing some kind of fraud. The reason behind this observation is assumed to be that policyholders take this time
to elaborate on the alleged story that they are trying to sell to their insurance company. We therefore include this aspect in our analysis, and
postulate:
H10 : Greater delay in reporting an event to the insurance
company increases the probability of fraud attempts being undertaken.
96
2.2
Data Set
Our data set is constituted of personal-, vehicle-, policy- and loss-related
information on the population in the automobile insurance division of
a Swiss insurance company. It is comprised of all claims filed between
the years of 2004 and 2011, summing up to a total of 1, 429, 896 claims
seeking for almost 2.5 bn CHF in indemnification. Throughout this time
period, 7, 407 (0.52 percent) of those claims were examined by the company’s fraud investigation division. The indemnification payments for
these cases summed up to a total of more than 60 mn CHF. Among the
7, 407 audited claims, 402 (5.43 percent) were identified as fraudulent,
mainly exhibiting signs of build-up. Consequently, the majority of these
claims received some partial indemnification, only 1.49 percent of them
were denied any payment.
As indicated previously, we make use of the word “fraud” as a collective term for all cases that were categorized as such by the insurance
company. This, however, does not imply that every single one of these
cases is an offense in the criminal-law sense. Judging from the high
amount of partial indemnifications, the majority of audited claims seems
to have exaggerated the actual loss amount rather than completely forging an insured loss event. Hence, these cases would fall into the category
of soft and not criminal fraud. Nevertheless, in our study we choose
not to make a distinction regarding the extent to which the individuals
defrauded the insurance company.
Furthermore, we do not differentiate between the potential actors in
the context of insurance fraud. However, besides the policyholders themselves, third parties like repair shops may also be involved in fraudulent
activities. On the one hand, the initiative may be taken by the insured
hoping for previous damages, unrelated to the current accident, to get
repaired. On the other hand, the repair shops may be the ones to inflate
the loss amount, either by charging overly high prices or providing unnecessary services (see Tennyson (2008)). These actions can be undertaken
with or without the knowledge of the policyholder.
2.2 Data Set
97
Data Selection
The insurance company’s decision as to whether an incoming claiming
has to undergo verification or not was based on personal evaluation of the
incoming cases. The investigation division consists primarily of employees with a police background having broad experience with fraudulent
activities in the insurance context. A predefined set of fraud indicators, however, that may serve as hints for the probability of fraud being
present in a claim, had not existed during the time period between 2004
and 2011.
Nevertheless, it can be expected that the investigators did not proceed arbitrarily. Being aware of the limited resources at their disposal,
they sought to focus on those claims that appeared to have a high
probability of being illegitimate and that exhibited a high saving potential. Even in the absence of a predefined set of selection criteria, they
likely chose the claims for auditing accordingly. These criteria, however,
whether chosen deliberately or not, may influence the composition of
our data sample of audited claims and therefore impact the results of
the regression analyses.
For this purpose, we report measures on sample composition for the
sample of all filed claims as well as the subsamples of audited and not
audited claims. The results can be found in Tables 13 and 14 in the Appendix. According to the results, investigators seem to have selected disproportionally young policyholders who drive either older or high-valued
vehicles. They exhibited flawless driving records, however by the time
of loss occurrence, seeking comparably high indemnification. Regarding
the type of damage, cases reporting the theft of the insured vehicle seem
to have been the target of investigations.
Selection bias being probably present to some extent, we nevertheless
perform our analyses to identify potential fraud indicators. Insurance
companies being driven by the need to minimize their cost and time
consumption, it seems unrealistic to expect anyone to perform auditing
on a completely random basis. We will therefore not be able to acquire
a data sample that is free of all selection biases.
98
2.3
Descriptive Statistics
In this section, we present descriptive measures to provide insight into
the full data samples of audited claims as well as the subsamples of
fraudulent and legitimate ones. Tables 3 and 4 give an overview of all
variables used in our analysis.
The first column in Table 3 shows the mean and standard deviation
for a number of policyholder-, vehicle-, policy- as well as loss-related
characteristics for the overall data set. Policyholders whose claims had
to undergo verification were on average just over 39 years old. The
vehicles involved in the loss events were a little over 7 years of age, being
worth more than 48, 000 CHF including accessories. While the claimants
had remained loss-free for over 4 consecutive years, their driving records
were comprised of 3 previous loss events. On average, it took insured
individuals almost 16 days to file a claim after the loss event occurred,
seeking over 8, 700 CHF of indemnification.
The second and third columns in Table 3 specify this information for
the subsamples of fraudulent and legitimate claims in order to uncover
potential differences between these two groups. To identify whether any
discrepancies are the result of significant differences between the subsamples or simply arise randomly, we perform a two-sample t-test for
the equality of the means (see p-values). While policyholders proven
to have engaged in fraudulent activities were on average over 40 years
old, honest ones were nearly two years younger. With the corresponding
p-value of the t-test being 0.0111, i.e., less than 0.05, this observation
may be a hint that the policyholder’s age serves as an indicator of the
existence of fraud. Regarding their vehicles, it is striking that those participate in a defrauding attempt were approximately one year younger
but almost 4000 CHF more expensive than those in the opposing group.
Again, taking the results of the t-test into account, these variables might
allow us to draw conclusions on the probability of fraud. Furthermore,
in terms of driving behavior, there seem to be significant differences
between the two subsamples. Claimants belonging to the group of defrauders have remained loss-free for almost one year longer and, at the
same time, were involved in fewer accidents during the whole duration
Defrauders
mean
s.d.
N=7407
N=402
Non-defrauders
mean
s.d.
p-value
N=7005
Policyholder age
39.18
13.87
40.78
12.52
39.09
13.94
0.0111
Vehicle age
Vehicle Value (CHF)
7.39
48,313
5.75
59,250
6.54
51,929
4.68
39,809
7.43
48,105
5.80
60,171
0.0004
0.0707
No. consec. loss-free years
No. previous records
4.26
3.18
2.33
7.55
4.84
2.11
2.53
1.84
4.22
3.24
2.31
7.75
< 0.0001
< 0.0001
Loss amount (CHF)
Delay in filing claim (days)
8,711
15.90
16,996
43.61
5,379
13.23
12,088
35.39
8,847
16.06
17,153
44.05
< 0.0001
0.1462
2.3 Descriptive Statistics
Audited Claims
mean
s.d.
Table 3: Descriptive Statistics for the Sample Composition
This table reports the mean and standard deviation (s.d.) of different characteristics related to policyholder, vehicle, policy and loss with
regard to the full sample of audited claims. This information is narrowed down particularly for the two subsamples of proven fraud attempts
(i.e., defrauders) and legitimate claims (i.e., non-defrauders). Furthermore, the last two columns provide the results of a two-sample t-test.
99
100
of their insurance relationship. Surprisingly, however, by the time of the
loss occurrence, fraud-prone policyholders claimed loss events totaling
to a little more than half the cost of that of their honest counter parts.
Lastly, we observe that the delay in filing a claim appears to be irrelevant
when predicting the probability of fraud.
Table 4 provides further information on the composition of the data
set. Similarly to Table 3, we report the number and percentages of
characteristics on the policyholder, vehicle, policy and loss level for the
data set of audited claims in column one, and specify this information
for the subsamples of fraudulent and legitimate claims in columns two
and three respectively.
Comprising 58.90 percent of the whole population, Swiss citizens accounted for the majority of all policyholders. This number drops slightly
(to 44.53 percent) among the subsample of fraud-prone claimants. While
the greater portion of the overall policyholder population (72.09 percent)
had their place of residence in the German-speaking part of Switzerland,
only 22.82 percent indicated that their place of residence was among the
French-speaking cantons of Switzerland and merely 5.09 percent in the
Italian-speaking part.14 These numbers do not seem to change considerably when comparing the subsamples of fraudulent and honest individuals. With respect to vehicle-related characteristics, we report the
vehicle type and whether the latter was leased or not. A majority of
about 71 percent of all policyholders had insurance coverage for a regular passenger car. This number rises by almost seven percentage points
among the subsample of defrauding claimants. The opposite holds true
for motorcyclists among the population. While their share among all
claimants sums to 23.36 percent, they only account for 16.04 percent of
all detected fraud attempts. Transporters form the smallest part of all
vehicle types comprising around 5.30 percent of the overall data sample
as well as within the two subsamples. While less than 19 percent of all insured vehicles were leased, they account for about 30 percent of all cases
14 Based on the prevalent quantity, the Swiss cantons are allocated as follows: Ticino
to the Italian-speaking part, Geneva, Vaud, Neuchatel, Jura and Fribourg to the
French-speaking part and the remaining cantons to the German-speaking part.
101
Audited Claims
No.
Percent
Defrauders
No.
Percent
Non-defrauders
No.
Percent
Policyholder related characteristics
Citizenship
Swiss
other
4363
3044
58.90
41.10
179
223
44.53
55.47
4184
2821
59.73
40.27
Total
7407
100.00
402
100.00
7005
100.00
German-speaking part
French-speaking part
Italian-speaking part
5313
1682
375
72.09
22.82
5.09
282
83
37
70.15
20.65
9.20
5031
1599
338
72.20
22.95
4.85
Total
7370
100.00
402
100.00
6968
100.00
Car
Transport
Motorcycle
3525
262
1145
71.34
5.30
23.36
210
15
43
78.36
5.60
16.04
3315
247
1111
70.94
5.29
23.77
Total
4941
100.00
268
100.00
4673
100.00
Leased
Not leased
1404
6003
18.96
81.04
123
279
30.60
69.40
1281
5724
18.29
81.71
Total
7407
100.00
402
100.00
7005
100.00
Included
Not included
2991
4416
40.38
59.62
203
199
50.50
49.50
2788
4217
39.80
60.20
Total
7407
100.00
402
100.00
7005
100.00
Theft
Glass
Collision
Others
2437
1130
1368
2472
32.90
15.25
18.71
33.37
100
25
124
153
24.86
6.22
30.85
38.06
2337
1105
1244
2319
33.36
15.77
17.76
33.10
Total
7407
100.00
402
100.00
7005
100.00
Area of residence
Vehicle related characteristics
Vehicle type
Leasing
Policy related characteristics
Bonus protection clause
Loss related characteristics
Type of damage
Table 4: Descriptive Statistics for the Sample Composition
This table describes the sample composition using different categorical variables on the
policyholder, vehicle, policy and loss level. Besides providing an overview of the complete
data sample of audited claims, the information is further differentiated with regard to
fraudulent (i.e., defrauders) and legitimate (i.e., non-defrauders) claims.
102
proven to have engaged in fraudulent activities. Additionally, we provide information as to whether the policyholders had included a bonus
protection clause in their contracts or not. This holds true for about 40
percent of the whole population, and over 50 percent among the subsample of detected defrauders. Finally, with respect to the claimed loss type,
we distinguish between theft of the vehicle, glass breakage, collision and
other damages.15 Almost one third (32.90 percent) of all audited claims
had reported the theft of the insured vehicle, whereas glass breakage and
collision accounted for approximately 15 and 19 percent of all incidents,
respectively. The shares of theft and glass breakage, however, drop notably by eight percentage points each within the subsample of fraudulent
claims, while the portion of cases including collisions rises by more than
twelve percentage points.
2.4
Model Derivation
The aim of our study is to identify the impact a set of explanatory
(predictor) variables has on a dichotomous (binary) dependent variable,
i.e., taking on solely one of the two values - one and zero (fraud and no
fraud). We are hence envisaging the possibility of employing the logistic
regression model.
In this relation, let us consider the following linear regression model:
yi = β0 + β1 xi1 + β2 xi2 + . . . + βm xim + ǫi = xi β + ǫi
(55)
where yi denotes the outcome of the dependent variable for the ith claim,
i.e., fraud or no fraud, and xim represents the value of the mth explanatory variable for the ith claim. Furthermore, βm specifies the regression
coefficients to be estimated, with β0 being the intercept, and the random variables ǫi indicate the error terms. Having a system of linear
equations, one may also abbreviate by using matrix notation. Hereby,
xi is a column vector with each row containing the values of the explanatory variables for the ith claim, and β a column vector with the
15 The sub-category “others” comprises, among others, damages caused by hail,
martens and other wild life, parking damages and theft of valuable left in the vehicle.
3 Empirical Results
103
corresponding regression coefficients.
In contrast to linear regression models, however, the logistic regression does not pursue the estimation of the dependent variable’s outcome
yi itself, but rather its probability of occurrence πi which is defined as
πi = Prob(yi = 1) = E(yi ) since yi is dichotomous, i.e., Bernoulli distributed. Applying this to Equation (55), we obtain
πi = xi β,
(56)
since E(ǫi ) = 0 for all i. This equation is commonly referred to as linear
probability model.
Unfortunately, applying the ordinary least squares (OLS) method
for estimating the regression coefficients βi , as usually done in relation
to linear regression models, would give rise to a series of problems in
this context. It may predict values outside the permitted range of [0, 1],
and is not able to capture heteroscedasticity and non-normality of error
terms arising with dichotomous dependent variables (see, e.g., Pohlmann
and Leitner (2003)). In addition to all this, utilizing OLS may produce
nonsensical predictions for the estimation results. These obstacles can
be overcome by drawing on the logistic model, which makes use of the
logistic function Prob(z) = 1/(1 + exp(−z)). With Equation 56, this
results in
πi = Prob(xi β) =
1
.
1 + exp(−xi β)
(57)
In this case, the regression coefficients βi , also referred to as logit coefficients, are derived by means of maximum likelihood estimation (MLE).
Its aim is to determine those parameter values βi , which make the observation of the collected data yi and xi the most likely.
3
Empirical Results
In this section, we present and discuss the results of the logistic regression
model introduced previously. Hereby, we begin with the set of explanatory variables with regard to policyholder characteristics and extend this
104
initial model stepwise by adding the variables on the vehicle, policy and
loss level in each iteration, respectively.
In order to compare the different models against each another and ultimately to assess how well the final model actually fits the observations,
we present different measures of model adequacy.
3.1
Logistic Regression Results
Model 1: Policyholder Characteristics
The first model considers only the influence that the policyholder
characteristics introduced in Tables 3 and 4 have on the decision to defraud the insurance company or not. Results of the logistic regression
are reported in Table 5.
N = 7002
Constant
Policyholder age
citizenship:other
area of residence:fr
area of residence:it
βi
exp(βi )
s.e.
p-value
sig.
–3.5406
0.0098
0.6718
–0.1945
0.4960
0.0290
1.0098
1.9578
0.8232
1.6421
0.1798
0.0038
0.1067
0.1338
0.1926
< 0.0001
0.0104
< 0.0001
0.1461
0.0100
***
*
***
*
Table 5: Logistic Regression with Determinants Related to Policyholder
Characteristics (Model 1)
Results for the logistic regression of the dependent variable (fraud/no fraud) with three
explanatory variables on the policyholder level (plus constant). The regression coefficients
βi indicate the contribution of each explanatory variable on the logit, exp(βi ) the corresponding effect on the odds ratio, and s.e. represents the standard error of the respective
determinant. Significance levels (sig.): *** = 0.1 percent, ** = 1 percent, * = 5 percent,
. = 10 percent.
As already indicated by the result of the two-sample t-test in Table 3,
the claimant’s age has a significant effect on the likelihood of engaging
in fraudulent activities (p < .05). The respective regression coefficient
being positive, older policyholders are more fraud prone than younger
ones (β = .01). This confirms our assumption that the policyholder age
is indeed an indicator for detecting fraud, however, it contradicts H1 .
3.1 Logistic Regression Results
105
Furthermore, we find both citizenship and area of residence to be
statistically significant. In particular, claimants not having the Swiss
citizenship appear to be more involved in dishonest activities than their
counter-group (β = .67, p < .0001). With regard to the categorical
variable area of residence, we use all policyholders living in the Germanspeaking part of Switzerland as a reference group. While individuals
from the French-speaking cantons do not exhibit significantly different
defrauding behavior (β = −.19, p > .1) compared to the Germanspeaking ones, claimants from the Italian-speaking part have a higher
probability of engaging in fraudulent activities (β = .50, p < .05).
Model 2: Policyholder and Vehicle Characteristics
In addition to the policyholder characteristics, the second model
takes the variables on the vehicle level into account (see Tables 3 and
4). The results of the corresponding logistic regression can be found in
Table 6.
N = 5156
Constant
policyholder age
citizenship:other
vehicle age
vehicle type:transporter
vehicle type:motorcycle
Vehicle value
leasing:no
βi
exp(βi )
s.e.
p-value
sig.
–2.772
0.0059
0.5955
–0.1285
0.3551
–0.0296
–0.0852
–0.3243
0.0001
–0.4962
0.0625
1.0059
1.8139
0.8794
1.4263
0.9708
0.9183
0.7230
1.0000
0.6088
0.2694
0.0047
0.1255
0.1580
0.2204
0.0201
0.3231
0.0208
0.0001
0.1512
< 0.0001
0.2060
< 0.0001
0.4159
0.1071
0.1424
0.7920
0.1196
0.9459
0.0010
***
***
**
Table 6: Logistic Regression with Determinants Related to Policyholder
and Vehicle Characteristics (Model 2)
Results for the logistic regression of the dependent variable (fraud/no fraud) with four
explanatory variables on the vehicle level in addition to the three policyholder characteristics (plus constant). The regression coefficients βi indicate the contribution of each
explanatory variable on the logit, exp(βi ) the corresponding effect on the odds ratio, and
s.e. represents the standard error of the respective determinant. Significance levels (sig.):
*** = 0.1 percent, ** = 1 percent, * = 5 percent, . = 10 percent.
106
Examining the regression coefficients and p-values for the policyholder characteristics, we find that result from Model 1 regarding the
impact of citizenship is confirmed.
Furthermore, we see that, in this model set up, the vehicle’s age may
not serve as an indicator for fraud being prevalent in a claim (β = −.03.
p > .1). With regard to vehicle type, the reference group is comprised
of all cases where a regular passenger car was involved in the loss event.
In comparison, neither transporters (β = −.09, p > .1) nor motorcyclists (β = −.32, p > .1) exhibited a significantly different defrauding
behavior than drivers of passenger cars. The criterion of whether the vehicle was leased or not seems to be a good indicator of fraud (β = −.50,
p < .005). With the regression coefficient being negative, we can conclude that owners of non-leased vehicles are less tempted to defraud the
insurance company. This finding supports the assumption stated in hypothesis H5 . Surprisingly, however, the vehicle’s value does not appear
to be statistically significant for the existence of fraud (p > .1).
To evaluate whether extending the initial Model 1 by the variables
on the vehicle level improves the fit, we conduct a likelihood ratio test.
The results are reported in Table 9. According to the corresponding values (χ2 = 24.7, p < .0001), Model 2’s fit proves to be significantly better.
Model 3: Policyholder, Vehicle and Policy Characteristics
The next step in extending our model is to additionally include all
variables on the policy level. Table 7 shows the corresponding results.
With regard to the parameters already included in Models 1 and 2,
we see that our previous results are confirmed. The explanatory variables
on the policyholder and vehicle level remain significant with respect to
the prediction of the likelihood of fraud.
Both the number of consecutive loss-free years and the number of
previous records seem to be statistically significant in predicting the
probability of fraud. In particular, claimants who remained loss-free for
a longer time period are more likely to get involved in fraudulent activi-
N = 5156
Constant
policyholder age
citizenship:other
vehicle age
vehicle type:motorcycle
vehicle value
leasing:no
No. consecutive loss-free years
bonus protection clause
107
βi
exp(βi )
s.e.
p-value
sig.
–2.6110
0.0063
0.5163
–0.1862
0.3675
–0.0151
–0.0529
–0.5520
0.0001
–0.5370
0.0664
–0.2506
0.3362
0.0735
1.0063
1.6758
0.8301
1.4441
0.9850
0.9485
0.5758
1.0000
0.5845
1.0687
0.7783
1.3996
0.3580
0.0048
0.1272
0.1584
0.2243
0.0184
0.3254
0.2117
0.0001
0.1505
0.0280
0.0422
0.1360
< 0.0001
0.1859
< 0.0001
0.2399
0.1014
0.4097
0.8709
0.0091
0.6818
0.0003
0.0178
< 0.0001
0.0134
***
***
**
***
*
***
*
Table 7: Logistic Regression with Determinants Related to Policyholder,
Vehicle and Policy Characteristics (Model 3)
explanatory variables on the policy level in addition to the three policyholder and four
vehicle characteristics (plus constant). The regression coefficients βi indicate the contribution of each explanatory variable on the logit, exp(βi ) the corresponding effect on the odds
ratio, and s.e. represents the standard error of the respective determinant. Significance
levels (sig.): *** = 0.1 percent, ** = 1 percent, * = 5 percent, . = 10 percent.
ties once they experience an insured loss event (β = .07, p < .01). This
result provides proof for hypothesis H6 . Regarding driving records, we
find that the less previous claims a policyholder has had the higher the
likelihood of defrauding the insurance company (β = −.25, p < .0001).
On the one hand side, this observation is consistent with the result from
the number of consecutive loss-free years. On the other hand side, it
seems counter-intuitive since a high driving record is associated with individuals having a bad and/or insecure driving behavior and therefore is
deemed to be a predictor for the likelihood of fraud. Another indicator
for the presence of fraud in a claim is the information whether a bonus
protection clause was included in the insurance contract or not. According to the results of the logistic regression, policyholders who included
this option in their contracts were involved in fraudulent activities more
often than their counter-group (β = .34, p < .05).
108
The results of the likelihood ratio test of Model 3 against Model 2,
presented in Table 9, confirm that the addition of the variables on the
policy level does help to significantly increase the predictive accuracy
(χ2 = 29.10, p < .0001).
Model 4: Policyholder, Vehicle, Policy and Loss
Characteristics
Our final Model 4 reflects the effect of all variables on the predictability of fraud being present in a claim. The results are presented in Table 8.
N = 4863
Constant
Policyholder age
citizenship:other
vehicle age
vehicle type: motorcycle
vehicle value
leasing:no
no. consecutive loss-free years
no. previous records
type of damage:glas
type of damage:collision
type of damage:other
loss amount
delay in filing claim
βi
exp(βi )
s.e.
p-value
sig.
–2.2030
0.0097
0.4635
0.1347
0.6636
–0.0214
0.2390
–0.6660
0.0001
–0.7682
–0.0011
–0.2206
0.0065
–2.3110
0.4653
–0.1971
–0.0001
–0.0077
0.1105
1.0097
1.5896
1.1442
1.9418
0.9788
1.2700
0.5138
1.0000
0.4638
0.9989
0.8020
1.0065
0.0992
1.5925
0.8211
0.9999
0.9923
0.4343
0.0054
0.1519
0.1879
0.2555
0.0233
0.3505
0.2768
0.0001
0.1774
0.0337
0.0460
0.1661
0.5363
0.2096
0.2125
0.0001
0.0035
< 0.0001
0.0793
0.0023
0.4734
0.0094
0.3585
0.4954
0.0161
0.0006
< 0.0001
0.9728
< 0.0001
0.9688
< 0.0001
0.0264
0.3535
< 0.0001
0.0284
***
.
**
**
*
***
***
***
***
*
***
*
Table 8: Logistic Regression with Determinants Related to Policyholder,
Vehicle, Policy and Loss Characteristics (Model 4)
explanatory variables on the loss level in addition to the three policyholder, four vehicle
and three policy characteristics (plus constant). The regression coefficients βi indicate the
contribution of each explanatory variable on the logit, exp(βi ) the corresponding effect
on the odds ratio, and s.e. represents the standard error of the respective determinant.
Significance levels (sig.): *** = 0.1 percent, ** = 1 percent, * = 5 percent, . = 10 percent.
Starting with the loss characteristics, we find the magnitude of loss
amount to be highly significant for filing fraudulent claims (p < .0001).
109
The regression coefficient being negative implies that the smaller the loss
amount, the more likely the existence of fraud. This observation was already indicated by the results from the two-sample t-test in Table 3 and
provides proof for our predication stated in hypothesis H9 . The outcome
for the delay in filing a claim, however, seems surprising. Even though
its effect on the dependent variable is significant (p < .05), the negative
sign of β = −.01 suggests that the shorter the time lag between the
occurrence of loss and the report to the insurance company, the higher
the likelihood for fraud. This is directly contrary to our assumption expressed in H10 . A possible explanation may be that, like in the case with
the loss amount, claimants do suspect the insurance company to control
for the delay in filing a claim and hence not only take the magnitude of
loss into consideration when defrauding, but also manipulate the date of
loss occurrence whenever it is possible. Once more, the aim is to feign
realistic scenarios in order to not get audited and moreover detected.
On the policyholder level, the final model confirms some of the effects
already predicted in Model 1: Older claimants have a higher probability
of cheating on their insurance company. This observation, however, contradicts our assumption in hypothesis H1 . Put in the context of driving
behavior and premium payments to date, a potential explanation may
be that it is actually the older and thus (usually) more experienced policyholders who remain loss-free throughout long periods of time. Having
paid insurance premiums over the course of many years, they may consider themselves long-standing customers who expect good will in form
of generous indemnification in trade for their loyalty.
Regarding the variables related to the vehicle itself, the fully extended
model once again confirms previous results: Both the vehicle value and
the information whether the vehicle is leased or not are very good indicators for fraudulent claims. The proven effects support our assumptions
expressed in hypotheses H4 and H5 respectively. Furthermore, we are
able to show that the vehicle class has an impact on the likelihood of
fraud, motorcyclists cheating less on the insurance company than drivers
of regular passenger cars (see H3 ). Solely, hypothesis H2 does not hold
true. Against our prediction, the age of the insured vehicle does not
110
seem to have an impact on the likelihood to engage in fraudulent activities (β = −.02, p > .1).
Also, from the policy perspective, the results from Model 3 prove
to be true: Exhibiting a small number of previous claims increases the
likelihood of inflating the loss magnitude once an insured event occurs.
In comparison with Model 3, the fully extended Model 4 leads to a
significantly better fit according to the likelihood ratio test in Table 9
(χ2 = 106.55, p < .0001), indicating that the fully extended Model 4 is
to be preferred over the less evolved ones.
Model
Model
Model
Model
1
2
3
4
∆χ2
∆df
χ2
df
p-value
sig.
1651.1
1626.4
1597.3
1490.8
4858
4854
4850
4845
24.69
29.10
106.55
4
4
5
< 0.0001
< 0.0001
< 0.0001
***
***
***
Table 9: Likelihood Ratio Tests for the Models
Results of the pairwise likelihood ratio tests between the consecutive Models 1 to 4. Each
extension of the previous model leads to a significant improvement in fit as indicated
by the increasing values for χ2 and the corresponding p-values. Significance levels (sig.):
*** = 0.1 percent, ** = 1 percent, * = 5 percent, . = 10 percent.
Having decided on the final model for identifying fraud indicators,
i.e., Model 4, we conclude this section by assessing its adequacy.
To check for potential problems with regard to the multicollinearity
of independent variables within our data set, we determine their variance
inflation factors, which can be found in greater detail in Table 12 in the
Appendix. The fact that their values do not exceed the critical threshold
value, the highest being 1.59 for the variable vehicle type, indicates that
it is reasonable to assume our explanatory variables to be uncorrelated.
Table 10 displays the classification table for the full logistic regression
model using all explanatory variables in our data set. The results confirm
the good predictive accuracy of our model. In particular, with regard to
the number of fraudulent claims, we are able to predict 73.76 percent correctly.This number, however, decreases slightly to 67.97 percent with respect to predicting the cases of legitimate claims. These figures indicate
3.2 Special Focus on Loss Amount
111
that our model is slightly more suitable for detecting fraudulent claims
than honest ones. Apparently, many of the characteristics that help to
identify fraudulent claims are also present among legitimate claims. On
the one hand, the small number of detected fraudulent claims may be
to blame. Only 402 out of the 7,407 audited claims (5.43 percent) were
classified as fraudulent, making these cases rare events. On the other
hand, our data sample was restricted to those criteria which are solely
assumed to be helpful with respect to fraud detection. In light of this,
there may exist other explanatory variables besides those included in
our data sample, which may improve the distinction between honest and
dishonest claims. These may include many determinants already known
to the insurance company, but also some that have not been gathered
yet.
Observed
Predicted
Fraud No Fraud
% Correct
Fraud
149
53
73.76
No Fraud
1493
3168
67.97
Table 10: Classification Table for Full Model
This classification table illustrates the predictive accuracy of the logistic regression model
in Table 8 by showing how many of the observed values for the dependent variable (fraud/
no fraud) are correctly predicted. The full model correctly predicts 73.76 percent of the
fraud attempts and 67.97 percent of the legitimate claims.
3.2
Special Focus on Loss Amount
As already revealed in the course of this section, fraud-prone policyholders do take the loss amount into consideration when deciding whether
to actually engage in fraudulent activities or not. More precisely, the
results of the logistic regression model as presented in Table 8 indicate
that the two are inversely proportional to each other, i.e., actions to obtain higher indemnification tend to be undertaken when the insured loss
amount is comparably small.
In this subsection, we focus on this particular fraud indicator and
discuss its effect on the decision to cheat one’s insurance company. For
112
this purpose, Figure 15 illustrates the link between loss amount and the
likelihood of fraud being present in an incoming claim. It must be noted
that, for the purpose of this analysis, we consider solely the loss amount
as a factor for predicting the existence of fraud. While taking other
significant exploratory variables into account would certainly increase
the overall accuracy in detecting defrauding attempts (see Table 10), it
would not impact the link between the loss amount’s magnitude and the
likelihood of fraud.
1.0
0.8
0.6
0.4
0.2
0.0
20000
60000
100000
140000
Loss Amount
Figure 15: Contribution of Loss Amount to Predicting the Likelihood of
Fraud
This figure illustrates both the actual magnitudes of loss events for honest and dishonest
claims within our data sample as well as the loss amount’s overall effect on the likelihood
of fraud. While the blue line indicates the predicted probability of fraud depending on the
magnitude of loss, each of the black points represents the loss amount of an actual claim
from the data set, the corresponding y-value hinting fraud (1) or no fraud (0).
In Figure 15, we include the audited cases from our data set exhibiting a loss amount up to 150000 CHF. Each of the points in the figure
depicts one claim within our data sample in terms of the corresponding
loss amount and the information as to whether fraud was detected or
not. Secondly, we add the predicted effect of the magnitude of a loss
3.2 Special Focus on Loss Amount
113
event on the likelihood to commit fraud, represented by the blue line in
Figure 15.
This demonstates that, with regard to the honest claims, the loss
amounts vary greatly across a wide range of magnitudes, reaching values
of 150000 CHF and higher. For cases proven to be illegitimate, however,
the contrary holds true. Here, the loss amounts seem concentrated in the
range of up to 20000 CHF, occasionally going up as far as 40000 CHF.
This observation is reflected by the predicted effect on the likelihood of
fraud being involved in a claim. The blue line indicates that, while for
small magnitudes,the loss amount accounts for almost ten percent of the
predictability of fraud being present, this value drops rapidly to zero
percent for loss amounts higher than 50000 CHF.
This leads us to conclude that defrauding attempts are not considered an option if the insured loss amount exceeds some threshold value.
Primarily for relatively small magnitudes, some individuals may try to
obtain higher indemnification payments from their insurance companies.
This observation provides proof for behavioral adaptation in the context of insurance fraud. Fraud-prone policyholders can be expected to
adjust their defrauding schemes in such a way that their attempts remain
undetected. Previous research has already shown that individuals fear
negative consequences in the form of losses as a result of their actions
more than they would appreciate a gain of the same size (see, e.g., Kahneman and Tversky (1979), Kerr (2012)). Hence, the overall objective
is deemed to be perceived as presenting a legitimate claim rather than
solely “maximizing” indemnity payments. Additional evidence for this
link can be found, e.g., in Viaene, Derrig, Baesens, and Dedene (2002)
and Tennyson (2008) who state that while only very few claims contain
outright fraud, the majority of defrauding attempts is detected in cases
seeking low to medium amounts of indemnification.
114
4
Conclusion and Critical Discussion
With fraud being identified as one of the central challenges in the industry to date and in the future, many insurance companies have established
their own investigation divisions in the recent years. Nevertheless, many
still rely largely on intuition when it comes to detecting wrongful claims.
In our study, we identify criteria that allow for an accurate distinction
between fraud-prone and honest policyholders and, by this means, predict the existence of fraud in a filed claim. Such a catalog of variables
allows for a systematic approach to the combat against fraud, hopefully
resulting in a higher detection rate. Moreover, it enables a targeted utilization of the limited resources that investigation divisions have at their
disposal.
Our analysis is based on a sample of claims data comprised of 7,407
audited loss events in an automobile insurance division. The data was
collected from a large Swiss insurance company between 2004 and 2011.
The target variable being dichotomous, we employ logistic regression
models to determine significant predictors of the presence of fraud in
claims. The fit and adequacy of the different models, and particularly
the final one, are assessed with the help of different measures. The analysis is rounded off with an in-depth examination of the effect of the loss
amount on the likelihood of engaging in fraudulent activities.
The results of our logistic regression analyses portray fraud-prone
policyholders as middle-aged individuals who prefer to drive high-valued
cars and having signed a leasing contract more often than their honest counterparts. With regard to their driving behavior, individuals
engaging in fraudulent activities prove to be rather experienced and
safe drivers. They are characterized by having a low number of claims
throughout their entire insurance relationship. Particularly noteworthy
is the fact that they tend to file fraudulent claims for comparably small
loss amounts, probably with the intention of remaining undetected. In
a similar manner, they try to not attract attention by filing claims too
late after the insured loss event occurred.
4 Conclusion and Critical Discussion
115
Another central result of this study is the empirical documentation
of the link between loss amount and the probability of resulting fraud.
As previous research has already suggested, the magnitude of an insured
loss event has an inverse effect on the likelihood of taking fraudulent activities into consideration. The main reason for this observation can be
policyholders’ anticipation of auditing strategies to remain undetected
and collect on the higher indemnification.
Our findings could be highly relevant to fraud investigators and underwriters alike. The information regarding fraud indicators can be utilized to perform auditing more effectively. Investigators would be given
the opportunity to focus specifically on those claims which are deemed
to have a high likelihood of being dishonest. Furthermore, the knowledge that some individuals are more prone to inflating loss amounts than
others may be useful for other aspects of risk management as well. With
regard to the pricing of insurance policies, the information on whether
an individual should be counted among the fraud-prone or honest policyholders may be a relevant differentiation criterion. As part of the
risk selection process, insurance companies may even decide not to provide coverage for individuals who can be expected to defraud the company to a large extent or who are unwilling to pay the corresponding
insurance premium. These options are particularly interesting, since
limited resources will prevent investigation units from verifying all incoming claims, even those which exhibit sufficient signs of the presence
of fraud.
As with all studies, the current study has some limitations which may
establish a basis for future research. Relying on the insurance company’s
decision as to which incoming claims to audit and which to indemnify
right away, may have biased our view on potential fraud indicators. On
the one hand, determinants which are identified based on a preselected
sample predict the overall likelihood of fraud correctly. However, if they
were already among the company’s selection criteria, they may tend to
overestimate the actual probability of its occurrence being amenable to
the self-fulfilling prophecy. On the other hand, by disregarding part of
116
the dishonest cases due to omission error, we may have been unable to
capture additional fraud indicators, possibly even more suitable ones for
predicting the existence of fraud in a claim.
5 Appendix
5
117
Appendix
Variable
Definition
Policyholder Characteristics
Policyholder Age
Citizenship
area of residence
Age of policyholder by the time of loss
occurrence
Policyholder’s citizenship
(equals Swiss or other)
Policyholder’s area of residence within
Switzerland (equals ge, fr or it)
Vehicle Characteristics
Vehicle age
vehicle type
Vehicle value
Leasing
Vehicle age at the time of loss occurrence
Type of vehicle
(equals car, transport or motorcycle)
Value of vehicle including accessories
(in CHF)
Vehicle is leased (equals 1, otherwise 0)
Policy Characteristics
no. consec. loss-free years
Number of consecutive years without
filing a claim
Total number of claims filed to date
Policy includes a bonus protection clause
(equals 1, otherwise 0)
Loss Characteristics
Type of damage
loss amount
Type of damage indemnification is seeked
for
(equals theft, glas, collision or others)
Estimated loss amount of the claim
(in CHF)
Time lag between occurrence of loss and
filing claim to insurance company in days
Table 11: Explanatory Variables Used in the Models
An overview of all variables and their respective definitions contained in our data set. We
distinguish between variables on the policyholder, vehicle, policy and loss level. Based on
this information, we perform logistic regression to determine potential indicators for the
presence of fraud in a claim.
118
vif
Policyholder age
citizenship
area of residence
vehicle age
vehicle type
vehicle value
leasing
no. consecutive loss-free years
type of damage
loss amount
1.076
1.058
1.154
1.345
1.587
1.255
1.324
1.154
1.170
1.260
1.513
1.202
1.022
Table 12: Variance Inflation Factors for All Explanatory Variables Used
in the Analyses
The variance inflation factors are employed to detect potential multicollinearity with regard to the explanatory variables. All corresponding values remaining below the critical
threshold value - with 1.587 for the variable vehicle type being the maximum - this can be
ruled out.
5 Appendix
Filed Claims
mean
s.d.
N=1,429,896
Audited Claims
mean
s.d.
N=7407
Not Audited Claims
mean
s.d.
p-value
N=1,422,489
Policyholder age
44.89
14.99
39.18
13.87
44.92
14.99
< 0.0001
Vehicle age
Vehicle Value (CHF)
6.21
44,948
5.12
38,076
7.39
48,313
5.75
59,250
6.21
44,931
5.12
37,934
< 0.0001
< 0.0001
No. consec. loss-free years
3.45
4.67
2.16
33.02
4.26
3.18
2.33
7.55
3.44
4.68
2.15
33.10
< 0.0001
< 0.0001
Loss amount (CHF)
Delay in filing claim (days)
1,775
19.17
3,760
40.23
8,711
15.90
16,996
43.61
1,740
19.19
3,535
40.21
< 0.0001
< 0.0001
Table 13: Descriptive Statistics for the Whole Sample Composition
This table reports the mean and standard deviation (s.d.) of different characteristics related to policyholder, vehicle, policy and loss with
regard to the full sample of filed claims. This information is narrowed down particularly for the two subsamples of audited claims and not
audited claims. Furthermore, the last two columns provide the results of a two-sample t-test.
119
Audited Claims
No.
Percent
Not Audited Claims
No.
Percent
120
Filed Claims
No.
Percent
Policyholder related characteristics
Citizenship
Swiss
other
1,055,362
371,131
73.98
26.02
4,363
3,044
58.90
41.10
1,050,999
368,087
74.06
25.94
Total
1,426,493
100
7,407
100
1,419,086
100
Area of residence
German-speaking part
French-speaking part
Italian-speaking part
1,001,286
338,016
83,340
70.38
23.76
5.86
5,313
1,682
375
72.09
22.82
5.09
995,973
336,334
82,965
70.37
23.76
5.86
Total
1,422,642
100
7370
100
1,415,272
100
980,258
52,930
29,588
92.24
4.98
2.78
3,525
262
1,154
71.34
5.30
23.36
976,733
52,668
28,434
92.33
4.98
2.69
Total
1,062,776
100
4,941
100
1,057,835
100
Leasing
Leased
Not leased
340,247
1,089,649
23.80
76.20
1,404
6,003
18.96
81.04
338,843
1,083,646
23.82
76.18
Total
1,429,896
100.00
7,407
100.00
1,422,489
100.00
Vehicle related characteristics
Table 14: Descriptive Statistics for the Whole Sample Composition
This table describes the sample composition using different categorical variables on the policyholder, vehicle, policy and loss level.
In addition, the information is further differentiated with regard to audited and not audited claims.
Vehicle type
Car
Transport
Motorcycle
5 Appendix
Filed Claims
No.
Percent
Audited Claims
No.
Percent
Not Audited Claims
No.
Percent
Policy related characteristics
Bonus protection clause
Included
Not included
Total
746,440
683,456
52.20
47.80
2,991
4,416
40.38
59.62
743,449
679,040
52.26
47.74
1,429,896
100.00
7,407
100.00
1,422,489
100.00
9,049
367,832
328,867
721,376
0.63
25.77
23.04
50.55
2,437
1,130
1,368
2,472
32.90
15.26
18.47
33.37
6,612
366,702
327,499
718,904
0.47
25.83
23.07
50.64
1,427,124
100.00
7,407
100.00
1,419,717
100.00
Loss related characteristics
Type of damage
Theft
Glass
Collision
Others
Total
Table 14: Descriptive Statistics for the Whole Sample Composition – continued
121
122
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127
Part IV
What Drives Insurers’
Demand for Cat Bond
Investments? Evidence from a
Pan-European Survey
Abstract
Despite the fact that insurance and reinsurance companies should be
familiar with the risk characteristics of the cat bond asset class, they
jointly account for less than 10 percent of the current investor demand
in the market. In order to be able to develop explanations for this observation, a deeper insight into the respective decision-making process
is needed. Accordingly, our main research goal in this paper is to identify major determinants of the cat bond investment decision of insurers.
For this purpose, we conducted a comprehensive survey among senior
executives in the European insurance industry. Evaluating the corresponding data set by means of exploratory factor analysis and logistic
regression methodology, we are able to show that the firm’s expertise and
experience with regard to cat bond investments, the perceived fit of the
asset class with its asset and liability management philosophy, as well
as the prevailing regulatory regime are significant drivers of an insurer’s
propensity to invest. These statistical findings are supported by further
qualitative survey results and additional information from structured interviews with the investment managers of four large dedicated cat bond
funds.16
16 A. Braun, K. Müller, and H. Schmeiser. What Drives Insurers’ Demand for Cat
Bond Investments? Evidence from a Pan-European Survey. Working Papers on Risk
Management and Insurance, 2012.
This paper has been accepted for publication at The Geneva Papers on Risk and
Insurance.
128
1
IV Empirical Analyses
Introduction
For almost two decades, insurance and reinsurance companies have been
employing insurance-linked securities (ILS) and derivatives to hedge against peak losses in the capital markets. The undoubtedly most successful of these alternative risk transfer measures is the catastrophe (cat)
bond, an instrument that allows natural disaster risk to be traded over
the counter. As is typical for securitizations, cat bonds are issued out of a
special purpose vehicle (SPV), which then holds the principal paid by investors in the form of highly rated collateral.17 The sponsoring company
enters into a reinsurance contract (or cat swap) with the SPV and, in
case a catastrophe occurs and causes losses in excess of the preset threshold, it is reimbursed with the proceeds of the collateral while investors
lose all or part of their principal. To determine whether a payment under the embedded reinsurance contract is due, cat bond structures can
feature a variety of different trigger mechanisms.18 Up until the trigger event or maturity, investors are compensated for bearing the natural
disaster risk through regular coupons that typically consist of a floating interest rate such as LIBOR, plus a risk-adjusted spread (see, e.g.,
Braun, 2012). Due to their comparatively high yields and rather low correlations with traditional asset classes, cat bonds have repeatedly been
described as an appealing investment opportunity (see, e.g., Litzenberger
and Beaglehole, 1996, Schoechlin, 2002, Cummins and Weiss, 2009). Yet,
the current investor base for this kind of asset is largely dominated by
money managers and a few specialized investment funds (see, e.g., Swiss
Re, 2009). This raises questions about the determinants of the institutional demand for cat bonds and, in turn, limits to the future growth
potential of this still relatively small segment of the capital markets. A
particularly puzzling phenomenon in this context is the fact that these
17 Note
that until early 2009, cat bonds used to be protected against collateral losses
by means of a total return swap (TRS). However, in the aftermath of the financial crisis four transactions ended up in distress, since the default of their swap counterparty
Lehman Brothers coincided with a severe impairment of the collateral assets. Consequently, the TRS feature has been removed in more recent transactions. Instead,
credit risk is meant to be largely eliminated through stricter collateral arrangements
(see, e.g., Towers Watson, 2010).
18 A description of the different trigger types for cat bonds can, e.g., be found in
Swiss Re (2006).
1 Introduction
129
instruments seem to play a negligible role in the asset management of
insurance companies, which, in contrast to other institutional investors,
such as banks and pension funds, should be very familiar with the inherent risks.
Several authors have discussed potential catalysts and impediments
for the evolution of the catastrophe risk markets, considering both traditional reinsurance and securitization. Froot (1999), for example, suggests
that securitization can help to improve the efficiency of the distribution
of natural hazard losses, and postulates five key success factors for cat
bond issues. Furthermore, Niehaus (2002) explores market imperfections
that hamper the optimal sharing of natural disaster risk via reinsurance
contracts and cat bonds. In his opinion, the unresolved question of pricing these instruments in a portfolio context has a very important impact
on the demand, since it determines whether or not investors believe that
the asset class truly exhibits zero-beta characteristics. Similarly, Froot
(2001) develops a number of supply-and-demand-related explanations
for the fact that the empirically observed amount of reinsurance and cat
bond transactions is considerably lower than suggested by risk management theory. He finds the market power of reinsurance companies and
impediments to the inflow of financing from the capital markets to be
the most likely reasons for this phenomenon. Another related paper has
been written by Gibson et al. (2007), who examine why capital-marketbased risk transfer solutions have failed to replace traditional reinsurance
as the primary means for sharing catastrophe risk. They conclude that
reinsurance should be preferred in situations where information from the
capital markets is costly to acquire and largely redundant. In addition,
Cummins and Trainar (2009) consider the advantages and disadvantages
of reinsurance and securitization from a risk management perspective. In
this context, they note that cat bonds primarily attract investors due to
their still relatively high yields, their low correlation with traditional asset classes, the fact that they are collateralized, and the lower complexity
as well as better alignment of interest between investors and ceding companies compared to other types of asset-backed securities (ABS). Apart
from that, Ibragimov et al. (2009) derive a model that serves to ex-
130
plain the limited supply of protection against catastrophe risk offered
by insurers and reinsurers. In their view, the typical heavy-tailed loss
distributions associated with natural disasters imply a substantial reduction in diversification benefits that may lead to a situation where
individual firms do not have an incentive to offer coverage. Finally, Lakdawalla and Zanjani (2011) illustrate that the full collateralization of cat
bonds hinders deeper market penetration, since diversification benefits
in the context of traditional reinsurance portfolios allow for a more efficient deployment of capital. In their view, this impediment can only be
surpassed if securitizations offer substantially lower friction costs than
reinsurance contracts.
Another strand of related literature is directly concerned with factors
that determine the supply of cat bonds by sponsors and the demand for
these instruments by investors. Bantwal and Kunreuther (2000) employ
behavioral economic aspects such as ambiguity aversion, myopic loss
aversion, and the fixed costs of education in order to explain institutional asset managers’ reluctance to invest in cat bonds. They suggest
that, to increase demand and promote further market growth, issuers
should aim for a larger degree of security standardization, reduce pricing uncertainty, and strive to enhance investor expertise with regard to
the asset class. Moreover, within his comprehensive overview of alternative risk transfer instruments, Cummins (2005) describes cat bonds as a
valuable means of portfolio diversification for investors, and emphasizes
that more standardized and transparent transactions as well as the development of a public secondary market would help to realize the full
potential of the asset class. Cummins (2008) additionally mentions the
difficulty involved in obtaining transactional information as an obstacle
for further growth. A wide range of issues that hinder the expansion
of the ILS markets is also discussed in a study by the World Economic
Forum (2008). Amongst others, the considered problems for sponsors
comprise basis risk, the instruments’ accounting and regulatory treatment, inconsistent rating methodologies, insufficient data quality and
disclosure, the costs of structuring a securitization deal, and the relatively low level of experience with securitization in the insurance indus-
1 Introduction
131
try. Investors, on the contrary, are said to be concerned about the lack
of standardization, the limited liquidity and secondary market trading
activity, the nontransparent nature of certain trigger mechanisms, and
the complexity involved in ILS valuation. Similar issues are identified by
Cummins and Weiss (2009) as well as Bouriaux and MacMinn (2009),
who also discuss major demand drivers, such as the risk-return profile
and diversification benefits of the asset class and the latest advances in
risk modeling methodology. Barrieu and Loubergé (2009), in contrast,
claim that the common arguments for the supposedly disappointing development of the cat bond market to date, such as the lack of investor
familiarity with the instrument, parameter uncertainty, and the trade-off
between moral hazard and basis risk, are not convincing. Instead, they
suspect downside risk aversion in combination with ambiguity surrounding the correlation between natural disaster losses and capital market
crash scenarios to be responsible for the limited demand. Finally, Hagendorff et al. (2011) draw on the model framework proposed by Merton
(1974), in order to demonstrate that the risk reduction benefits of cat
bonds are confined to sponsors with a high default probability or a large
exposure to natural disaster risk. They point to this lack of universal
applicability as an explanation for the underwhelming development of
the market to date.
Despite the relatively large body of literature on supply, demand,
and growth drivers in the markets for catastrophe risk, to the best of our
knowledge, relatively little is known about the motivation of insurance
companies to act as cat bond investors. In particular, no analysis of the
determinants of their respective investment decision has been conducted
to date. Therefore, with this paper we aim to address an important research gap by identifying and analyzing the main drivers and obstacles
regarding the demand for cat bond investments in the insurance industry.
For this purpose, we developed a comprehensive questionnaire that has
been distributed to senior executives of almost 500 European insurance
companies. Moreover, to complement the statistical data and qualitative information provided by the survey participants and provide additional evidence for the robustness of our results, we have also conducted
132
structured interviews with the investment managers of the four largest
dedicated cat bond funds that together absorb almost 80 percent of the
outstanding volume. Our findings should provide a thorough insight into
the decision-making process that underlies cat bond investments of insurance companies and can thus help to address pressing issues, reduce
investment barriers, and support further growth of the cat bond market.
The remainder of this paper is structured as follows. In Section 2,
we provide facts on the current size and investor base of the cat bond
market and develop eight hypotheses concerning the determinants of an
insurance company’s decision to invest in this asset class. Furthermore,
Section 3 contains a description of our survey as well as a brief introduction to the statistical techniques of exploratory factor analysis and logistic regression that are used to evaluate the resulting data set. Section 4
represents the main part of the paper, including descriptive statistics,
the derivation of our key empirical findings, and the discussion of the
qualitative results from the open survey questions and expert interviews.
Finally, in Section 5 we draw our conclusion and propose ways to tackle
the major barriers that currently seem to prevent insurance companies
from engaging in cat bond investments on a larger scale.
2
2.1
The Demand for Cat Bonds
Current Market Size and Investor Base
Although, in its early days, the cat bond market suffered from low capacities and a lack of investor interest, it has undergone a major development since the beginning of the 1990s. In the 10 years between 1997
and 2007, issuance volume has increased more than sevenfold from less
than USD 1 billion to over USD 7 billion (see Cummins and Weiss, 2009).
Furthermore, Guy Carpenter (2011) estimates that the outstanding risk
capital for nonlife cat bonds summed up to more than USD 10 billion in
2011. Apart from the market size, a significant evolution could also be
observed in the investor base of this asset class. According to Swiss Re
(2009), primary insurers and reinsurers together purchased a total of 55
133
percent of the cat bond volume issued in 1999. The remaining demand
came from money managers (30 percent) as well as hedge funds, banks,
and dedicated cat bond funds (5 percent each). By 2009, however, the
market structure had changed dramatically, with dedicated cat bond
funds (46 percent), money managers (23 percent), and hedge funds (14
percent) now providing the vast majority of risk capital. At the same
time, the combined share of insurance and reinsurance companies had
fallen to a mere 8 percent. To explain this low level of demand compared
to other types of institutional investors, the driving factors behind the
investment decision of asset managers in the insurance sector need to be
revealed.
2.2
Development of Hypotheses
In the following, we develop a total of eight hypotheses that served us
as guidance with regard to the design of the questionnaire as well as
the schedule for the expert interviews. Although some of the postulated
determinants have already been mentioned in earlier articles, they have
not yet been empirically tested with an explicit focus on insurance companies as investors.
Larger insurers generally have more financial and professional resources at their disposal than small or medium-sized companies. Consequently, they may, for example, afford to put the necessary cat-modeling
and data evaluation technology in place, hire additional asset management specialists, or establish a dedicated investment team that focuses
on the analysis of ILS. This implies that the sheer size of an insurance
company could have an influence on its ability to gain access to the cat
bond market, leading us to hypothesize the following:
H1 : Larger insurance companies are more likely to invest in
cat bonds.
In addition, due to the ever-growing importance of modern risk management techniques and processes, insurers make their investment decisions in accordance with preset strategic asset and liability management
134
(ALM) goals. Thus, they will tend to avoid assets that are perceived to
be at odds with their ALM philosophy, while focusing on investments
that they find to be overall attractive and to fit well into the firm’s
portfolio. We therefore state the following hypothesis:
H2 : The better the perceived fit of cat bonds with the strategic
ALM goals of an insurer, the more likely the company is to
invest.
Asset managers in the insurance industry continuously search for
and analyze potential investment opportunities. Due to the complexity
of today’s capital markets, organizations need to exhibit a lot of inhouse expertise and experience if they want to be able to structure and
maintain portfolios of a wide range of assets. In particular, with respect
to niche markets such as cat bonds, an experienced and well-attuned
asset management is a crucial factor for investment success. Based on
these considerations, we postulate:
H3 : More expertise and experience with regard to the cat bond
asset class increase an insurer’s propensity to invest.
It has been repeatedly emphasized that cat bonds exhibit an attractive risk-return profile. However, potential investors need to perceive
and value this benefit as such, in order to become interested in the asset
class. Hence, we hypothesize:
H4 : Insurance companies that perceive the risk-return potential of cat bonds to be attractive are more likely to invest.
Another typical characteristic of cat bonds is their low correlation
with other asset classes. This circumstance provides them with a considerable diversification potential. For this particular benefit to play a role
with regard to the investment decision, however, the firms need to notice
and acknowledge it. This observation leads to the following hypothesis:
H5 : The propensity of insurers to invest in cat bonds rises
with the degree to which they perceive the asset class’s diversification potential.
135
Furthermore, it can be expected that insurers are more likely to consider investments in cat bonds if they perceive them as rather liquid and
standardized assets, which are associated with low administration costs.
We summarize these and similar aspects in a factor called perceived administrative complexity and hypothesize the following:
H6 : The less administrative complexity insurance companies
associate with cat bonds, the more likely they are to invest in
this asset class.
Whether an asset class represents an adequate investment choice generally depends on the evaluation of data and information. The decisionmaking process of insurance companies can be facilitated if, for example,
transaction data, pricing information, and historical performance figures
for the respective asset classes are readily available. Hence, we derive
the following hypothesis:
H7 : The more they perceive relevant data and information
on the asset class to be readily available, the more likely insurance companies are to invest in cat bonds.
For the protection of the policyholders’ interests, the assets of an
insurance company are classified as either tied (restricted) or free. The
free assets reflect the firm’s equity capital and thus typically account for
a comparatively small percentage of the total portfolio. A much larger
fraction, in contrast, is represented by the tied assets, which are meant
to cover the firm’s technical provisions at all times. Consequently, they
need to adhere to strict requirements with regard to investment types,
diversification, and risk management.
As will be discussed in Section 4, all companies in our sample are either subject to EU or Swiss regulation. Explicit lists of asset categories
that can be employed to cover the technical provisions of insurers within
the European Union are included in Article 21 of the third nonlife insurance Directive of the European Council (Directive 92/49/EEC) and
Article 23 of the Directive of the European Parliament and the Council
136
concerning life assurance (Directive 2002/83/EC). These legal acts comprise a general category termed “debt securities, bonds, and other money
and capital market instruments”, which, taking into account their fixed
income format, also seems applicable to cat bonds. However, in accordance with this EU legislation, individual member states may also establish more detailed guidelines with regard to the characteristics of acceptable investments. In Germany and Austria, for example, the act on the
supervision of insurance undertakings (in German: “Versicherungsaufsichtsgesetz”, VAG) empowers the government and the national regulatory authority, respectively, to enact provisions that contain binding
conditions for the tied assets.19 These legal acts, called “Anlageverordnung” (AnlV) in Germany and “Kapitalanlageverordnung” (KAVO) in
Austria, do not explicitly rule out cat bond investments. In addition,
they contain an opening clause for asset classes that are not included in
their predefined lists.20 Since the requirements with regard to the tied assets in other European countries are also based on the above-mentioned
EU directives, they can be assumed to be quite similar. Hence, we do
not expect explicit regulatory constraints with regard to the cat bond
asset class for insurance companies in EU member states.
Furthermore, in Switzerland the act on the supervision of insurance
undertakings states that the Swiss Federal Council may enact provisions
that govern tied assets and leave it up to the Swiss Financial Market Supervisory Authority (FINMA) to determine additional details.21 The respective guidelines have been incorporated into Article 79(1) of the “Aufsichtsverordnung” (AVO) and are further substantiated in FINMA’s circular letter on the investment of tied assets (see FINMA, 2008). Clause
II.D.a of this document, which includes the general principles with regard
to eligible investments, states that the tied assets must not include insurance risk, and clause III.C.b.bb explicitly forbids the purchase of ILS.
Hence, Swiss insurers may only consider cat bond investments within
their free assets. This implies that their asset management departments
19 See
paragraph 54(2) VAG Germany and paragraph 78(3) VAG Austria.
opening clauses can be found in paragraph 2(2) AnlV and paragraph 2(1)
No. 9 KAVO.
21 See Article 20 VAG Switzerland.
20 These
3 Data and Methodology
137
have less options to employ the instrument than those of their European
counterparts. The consideration of this legal background results in our
last hypothesis:
H8 : Due to binding regulatory constraints, Swiss insurers are
less likely to invest in cat bonds.
In this context it should be noted that regulatory constraints in the
broader sense could also arise due to the capital requirements associated
with cat bond investments under the Swiss Solvency Test (SST) or Solvency II. More specifically, if an insurer perceives the resulting capital
charges to be ambiguous or inappropriate, it could tend to avoid the asset
class. However, under both regulatory frameworks cat bond investments
are virtually treated in the same way as the catastrophe risk exposure
incurred through traditional (re)insurance contracts. Hence, we do not
expect a significant impact on the investment decision of insurance companies and refrain from formulating an explicit hypothesis. Similarly, we
deem it unlikely that the accounting treatment of the instrument turns
out to be relevant in this regard. Nevertheless, our survey included a
set of questions to measure the firm’s perception of the capital requirements and accounting treatment of cat bond investments. Based on the
corresponding information, the correctness of these expectations will be
confirmed.
3
3.1
Data and Methodology
Development of Measures
Before the development of our questionnaire, we conducted several interviews with ILS experts from a Swiss reinsurance company. The aim
was to identify and better understand potential factors driving the demand for cat bond investments. Based on the results, we then devised
measures and scales for the determinants and generated an initial set of
items.22 After having prepared a first draft of the questionnaire, we again
22 To the best of our knowledge, there are no existing scales in the literature that
would fit the context of our study.
138
consulted industry professionals and risk management academics to obtain in-depth feedback concerning wording and completeness, based on
which we implemented final changes by rephrasing, including, or deleting
certain items.
3.2
Participant Recruitment
To recruit relevant participants for our web-based survey, we made use of
the key informant technique. The aim was to contact senior executives,
preferably CEOs and CFOs, since they should be well informed about the
companies’ strategic investment decisions as well as the corresponding
reasons behind them. Hence, we collected the addresses of a total of 490
insurance and reinsurance companies from Austria, France, Germany,
Italy, the Netherlands, Sweden, Switzerland, the UK, Finland, Portugal,
Belgium, and Greece. Based on the gathered information, we invited
the senior executives per regular mail to take part in our survey. In
a first stage, the questionnaire was available online for three weeks in
February 2012. Subsequently, two reminders were sent out per e-mail to
those persons whose e-mail addresses were available. This happened at
an interval of two to three weeks so that the companies had a total of
nine weeks’ time to answer the questionnaire. While survey participation
in general was anonymous, the respondents were offered to be sent the
results of our study if they chose to include their contact information.
3.3
Sample Characteristics and Imputation
Overall, 64 companies reacted to the invitation, which corresponds to a
response rate of 13.1 percent. On average, it took the participants 11
minutes to complete the survey. However, a number of firms terminated
the questionnaire too early to be included in our analysis. We therefore
had to adequately impute missing values as far as possible or remove all
cases for which essential items were missing and imputation was impossible. Our final data set is based on the responses of 44 participants,
who completed the essential parts of the questionnaire, such as the company background, and indicated whether or not they have invested in
3.4 Exploratory Factor Analysis
139
cat bonds in the past and/or plan to do so in the future.23 Of these 44
participants, 36 provided their views on the potential determinants that
have an impact on the investment decision. This subsample will be used
for the inference statistical analysis.
3.4
Exploratory Factor Analysis
Apart from the company size and constraints due to the regulatory
regime, the potential determinants of the cat bond investment decision
hypothesized in Section 2.2 are likely to be latent variables, i.e., multifaceted constructs that are not directly observable. In order to capture
those, we have included a total of 41 items (observed variables) in the
questionnaire, which were measured on six-point Likert scales.24 Potential biases due to individuals who are too inexperienced with the subject
to provide a reliable opinion are minimized, since we additionally allowed the respondents to choose a “do not know” button for each item.
The derivation of the data points for each potential determinant will be
conducted by means of exploratory factor analysis (EFA), a statistical
methodology that explains the covariance (correlation) structure of observed random variables in terms of a smaller number of latent variables.
The following is an analytical representation of the general EFA model:25
X = Λξ + δ,
(58)
where X is the vector of observed variables (items), Λ represents the
factor loadings matrix, ξ is the vector of latent variables (factors), and
23 Fourteen of the participants provided us with information about their positions in
the company. According to their statements, five Chief Investments Officers and two
other members of the executive boards answered our questionnaire. Furthermore, two
survey participants are directors whereas another five claimed they were appointed
either Head of Asset Management or Senior Risk Manager. Thus, we believe that we
actually did approach the key informants we wanted to.
24 Likert scales are a common way to capture an individual’s level of agreement or
disagreement with a specific statement (see Likert, 1932). An even number of points
means that a neutral answer is not possible. Instead, the respondent is forced to
choose a positive or negative stance.
25 The notations in this subsection have been adopted from Jöreskog (1967).
140
δ stands for the vector of unique factors (residuals). Applying matrix
algebra, one can derive the covariance matrix Σ implied by the model:
Σ = ΛΦΛ′ + Ψδ ,
(59)
with Φ being the covariance matrix of the factors and Ψδ being the covariance matrix of the residuals. The parameters (factor loadings and
residual variances) for the EFA model are determined by means of maximum likelihood estimation (MLE), so that the model-implied covariance
(correlation) matrix fits its empirically observed counterpart as closely
as possible.26 Based on the derived factor loadings, it is possible to
ˆ For this purpose, we will choose the
compute factor score estimates ξ.
regression method, which employs the sample covariance matrix Σ̂ and
the estimated factor loadings matrix Λ̂ as follows:
ξˆ = Λ̂′ Σ̂−1 X.
(60)
These factor scores can then be used as a measure for the hypothesized latent determinants of the cat bond investment decision in further
analyses.
3.5
Logistic Regression Model
To address our research question, we require a statistical methodology
that determines the influence of a set of metric explanatory variables
xi on a binary (dichotomous) dependent variable y, i.e., one that takes
on only two values: one and zero (investor and non-investor). In this
situation, linear regression is unsuitable, since it may also produce outcomes of less than zero and greater than one, and its basic assumptions
of homoscedastic and normally distributed residuals are violated (see
Wooldridge, 2008). Given these drawbacks of the linear approach, we
decide to resort to a binary response model to predict the probability
that the dependent variable y assumes a value of one, given the realizations of a number of independent variables xi (i = 1...n). More specifi26 Note that the standard EFA model requires the observed variables to be normally
distributed.
3.5 Logistic Regression Model
141
cally, we employ the common logit model, which is based on the logistic
function:27
1
.28
(61)
p(z) =
1 + exp(−z)
The logistic function translates any real number z into a value p(z)
between zero and one.29 In order to derive the logistic regression model,
z is assumed to be a latent variable called the logit that can be expressed
as a linear combination of the regressors xi (i = 1...n):
z = β0 + X’β + ǫ = β0 + β1 x1 + β2 x2 + ... + βn xn + ǫ,
(62)
where β0 denotes the intercept, the βi are the regression (logit) coefficients, and the error term ǫ is assumed to be independent of the xi .30
The logit equals the natural logarithm of a magnitude termed the odds
ratio (OR), which equals the probability of the dependent variable taking on the value one divided by the probability of it taking on the value
zero:
OR =
p(y = 1|X)
= exp(β0 + β1 x1 + β2 x2 + ... + βn xn + ǫ).
p(y = 0|X)
(63)
Being defined on [0, +∞), OR is the key measure of effect strength
in the logistic regression model. If it equals one, both outcomes of the
dependent variable are equally likely. The further it deviates from one,
the stronger the (positive or negative) link between the dependent variable and the regressors.
27 The
following derivation of the logit model is based on Wooldridge (2008).
expression also represents the cumulative distribution function for a logistically distributed random variable.
29 In contrast to logistic regression, discriminant analysis, which is another binary
response model, also generates values above one and below zero. Apart from that it
requires the explanatory variables to be normally distributed (see Press and Wilson,
1978). Therefore, it is not considered in the context of our analysis.
30 In matrix notation, X’ is the random vector of explanatory variables and β
represents the vector of regression coefficients.
28 This
142
Combining Equations (61) and (62), the relationship between the
explanatory variables xi and the response probability p(y = 1|X) can be
expressed as follows:31
p(y = 1|X) = 1/(1 + exp(−β0 − β1 x1 − β2 x2 − ... − βn xn − ǫ)). (64)
Hence, although the observed values of the dependent variable are
binary, the logit model actually describes a continuous variable, i.e., the
probability of y assuming a value of one. The βi are determined by means
of MLE. More specifically, based on the data for y and the explanatory
variables xi , an iterative procedure serves to choose the logit coefficients
that are most likely associated with the observed values.
4
4.1
Empirical Results
Descriptive Statistics
In this section, we present descriptive statistics to characterize the composition of our sample. The first column of Table 15 shows the number
(and percentages) of insurance companies categorized by country, business model (insurer, reinsurer), business line (life, nonlife, multiline), and
geographic investment scope (global, regional). In addition, we indicate
whether or not the firms act as cat bond sponsors, invest in securitizations in general, and belong to a larger insurance group. Accounting for
31.82 percent of the survey respondents, Swiss insurers are slightly overweighted in the data set. Apart from that, however, the participants are
relatively evenly spread across European countries. Furthermore, almost
80 percent of the covered firms are primary insurers. This is a positive indication for the representativeness of the sample, since reinsurers are also
much rarer in the reference population (i.e., European insurance companies). Similarly, a majority of about 70 percent run a multiline insurance
business, comprising both life and nonlife divisions.32 With respect to
31 Note that z, OR, and p(y = 1|X) essentially provide the same information. A
probability of 0.5 corresponds to a logit of 0 and an OR of 1. Each of these values
implies that both outcomes of y exhibit the same likelihood.
32 Four of the firms in this category responded that they additionally run a health
insurance line.
143
their geographic investment scope, around 60 percent replied that they
exclusively focus on regional (national or European) assets and markets,
while only 40 percent consider themselves to be global investors. Hence,
we seem to have a well-balanced mix of small, medium-sized, and large
companies. Moreover, more than 80 percent of the respondents stated
that they do not act as cat bond sponsors. Again, this implies that the
data should be representative, since only a few large primary insurers
and reinsurers in Europe with the resources and capabilities to sponsor and structure cat bond transactions have actually employed this risk
transfer instrument to date. Finally, half of the firms in the sample invest
in at least one other type of securitization, such as ABS, collateralized
debt obligations (CDOs), or covered bonds, and the vast majority of the
responding entities are part of a group.
In the second and third column of Table 15, we have split the sample
into those firms that do and those that do not invest in cat bonds.33 As
could be expected, only about 30 percent (13/44) of the respondent firms
actually invest in cat bonds. Taking into account the low percentage of
insurers among the current investor base of the cat bond asset class
as discussed in Section 2.1, the large fraction of 70.45 percent (31/44)
non-investors is another cue for the representativeness of our sample. Interestingly, the majority of investors (61.54 percent) come from Austria,
Germany, or Italy. At least half of the survey participants from these
countries have disclosed themselves as cat bond investors. Furthermore,
most non-investors in the sample are based in Switzerland (38.71 percent), and over 90 percent of the non-investors are primary insurers. In
contrast, more than 40 percent of the investors are reinsurers or, to put
it differently, two-thirds (6/9) of the reinsurers in the sample do invest in
cat bonds. This might be some initial evidence for hypotheses 1, 2, and
3, since the firm’s size, the expertise and experience with cat bonds, as
well as the fit of the asset class with the firm’s strategic ALM are characteristics that can be expected to vary considerably between the average
primary insurer and reinsurer. Similarly, the fact that almost all non33 We define cat bond investors as those insurance companies, which stated that
they will continue or begin to hold cat bonds in their asset portfolios in the future.
144
Full Sample
No. Percent
Investors
No. Percent
Non-investors
No.
Percent
Austria
France
Germany
Italy
Netherlands
Sweden
Switzerland
UK
Finland
Portugal
Belgium
Greece
4
1
6
5
2
3
14
2
3
2
1
1
9.09
2.27
13.64
11.36
4.55
6.82
31.82
4.55
6.82
4.55
2.27
2.27
2
0
3
3
0
1
2
1
1
0
0
0
15.38
0.00
23.08
23.08
0.00
7.69
15.38
7.69
7.69
0.00
0.00
0.00
2
1
3
2
2
2
12
1
2
2
1
1
6.45
3.23
9.68
6.45
6.45
6.45
38.71
3.23
6.45
6.45
3.23
3.23
Total
44
100.00
13
100.00
31
100.00
Primary Insurers
Reinsurers
35
9
79.55
20.45
7
6
53.85
46.15
28
3
90.32
9.68
Total
44
100.00
13
100.00
31
100.00
Life Business
Nonlife Business
Multiline Business
4
9
31
9.09
20.45
70.45
1
4
8
7.69
30.77
61.54
3
5
23
9.68
16.13
74.19
Total
44
100.00
13
100.00
31
100.00
Global Investor
Regional Investor
18
26
40.91
59.09
7
6
53.85
46.15
11
20
35.48
64.52
Total
44
100.00
13
100.00
31
100.00
Cat Bond Sponsor
No Cat Bond Sponsor
7
37
15.91
84.09
6
7
46.15
53.85
1
30
3.23
96.77
Total
44
100.00
13
100.00
31
100.00
Securitization Investor
No Securitization Investor
20
24
45.45
54.55
11
2
84.62
15.38
9
22
29.03
70.97
Total
44
100.00
13
100.00
31
100.00
Group
Single Entity
35
9
79.55
20.45
9
4
69.23
30.77
26
5
83.87
16.13
Total
44
100.00
13
100.00
31
100.00
Table 15: Sample Composition
This table shows the composition of the sample of 44 firms that has been generated from
the respondents of a survey among 490 European insurers and reinsurers. The data is categorized by country, business model, business line, geographic investment scope, sponsoring
activity, other securitization investments as well as organizational structure. In addition,
each category is further differentiated into cat bond investors and non-investors.
145
investors (96.77 percent) do not act as cat bond sponsors and a majority
of them have a regional investment focus (64.52 percent) supports these
hypotheses. The reason for this is that only a few large and experienced
reinsurance companies dominate the cat bond sponsoring business, and
insurers with a geographically limited asset management scope are likely
to be smaller and less experienced with rather exotic asset classes such
as cat bonds. In addition, judging by the higher share of pure nonlife
insurers among the investors, it seems that the propensity to purchase
cat bonds might somehow be related to the risk expertise that an insurance company accumulates through its business lines. Lastly, we observe
that almost 85 percent of the cat bond investors in our sample also hold
other securitizations, while more than 70 percent of the non-investors
do not. Thus, the experience with and the general affinity to invest in
securitized assets could also exert an influence on insurance companies’
demand for cat bonds.
Table 16 contains mean, median, standard deviation, minimum, and
maximum for the number of employees, the balance sheet size, and the
premium volume of the firms in our sample. These variables have been
included in the survey as proxies for the company size. Again, we additionally distinguish between investors and non-investors. When examining the respective figures, we notice that for all three variables the
medians are smaller than the means, i.e., the distributions seem to exhibit some degree of positive skewness (long right tail). Moreover, the
minimum and maximum values indicate that the sample covers a range
of very differently sized companies. The smallest insurer, for example,
employs only 14 staff members, while the largest workforce amounts to
60,000 people.34 Similar observations can be made for the balance sheet
sizes and premium volumes. A simple comparison of the sample means
of these three size proxies between investors and non-investors reveals
that, on average, the former are larger. However, since this discrepancy could be simply caused by the random draw through our survey,
one needs to rely on statistical inference in order to make a statement
about the underlying population of European insurance companies. The
34 Note
that all entities with less than 4,000 employees are part of insurance groups.
146
most common procedure in this regard is the two-sample t-test for the
equality of means. The typical prerequisites for this test are normally
distributed data, equal sample sizes, and equal sample variances (see,
e.g., Sawilowsky and Blair, 1992). The former condition can be checked
by means of the Kolmogorov-Smirnov (K-S) goodness-of-fit test, the results of which are displayed in the last two columns of Table 16. Since
we reject the null hypothesis of normality in all cases but one (see pvalues), the aforementioned two-sample t-test will not be applied. Instead, we decide to conduct the more robust Mann-Whitney U test to
assess whether the differences in the means between investors and noninvestors are statistically significant. This nonparametric test is based
on the null hypothesis that the two samples under consideration have
been drawn from the same distribution. The results are shown in Table 17. For all three size proxies (number of employees, balance sheet
size, premium volume) the p-values exceed 0.1. To put it differently, this
is a first indication that company size might not matter with regard to
the cat bond investment decision of insurance companies.
Finally, to get an initial impression of the importance of some potential determinants discussed in Section 2.2, we have asked the insurance
companies that participated in the survey to state whether or not certain factors had an influence on their cat bond investment decision.35
Table 18 shows the respective results.36
Interestingly, the aspects expertise and experience, risk-return (and
correlation) profile, administrative complexity, and data availability are
considered relevant by the majority of investors, while they do not seem
to be of importance to most non-investors. Constraints due to capital
35 As explained in Section 3.4, most of the determinants are latent factors, which
have been measured by means of observed variables or so-called items. Within the
questionnaire, each determinant was represented by a whole battery of such items.
The relevant/irrelevant questions to which we refer in this paragraph preceded those
item batteries. The only exception is the factor “perceived fit with the company’s
strategic ALM” (hypothesis 2), for which a relevance/irrelevance question has not
been posed. The reason is that the respective item battery was included in a more
general section of the questionnaire that captured further details with regard to the
firm’s past, present, and future cat bond investments.
36 Note that the number of respondents N has dropped from 44 to 36, since eight
firms have not provided information with regard to the item batteries for the latent
variables at all. Hence, imputation was not possible and the respective cases have
been excluded from the following analyses due to missing data.
Mean
Median
S. D.
Min
Max
K-S Stat.
p-value
No. of Employees
Balance Sheet (mn EUR)
Premium Volume (mn EUR)
6,667.80
38,975.24
8,529.05
1,225.00
10,492.93
1,775.00
13,365.14
78,202.53
18,971.43
14.00
0.18
0.11
60,000.00
400,000.00
107,900.00
0.3224
0.3249
0.3265
0.0002
0.0002
0.0002
***
***
***
Investors
No. of Employees
9,102.85
50,347.32
8,580.91
1,250.00
22,500.00
3,735.00
17,035.41
84,651.67
11,197.83
87.00
360.00
460.00
47,000.00
240,000.00
32,600.00
0.3644
0.3732
0.2774
0.0633
0.0392
0.2697
*
**
Non-investors
No. of Employees
5,646.65
34,206.30
8,507.31
1,200.00
9,296.00
1,308.00
11,674.88
76,287.88
21,580.55
14.00
0.18
0.11
60,000.00
400,000.00
107,900.00
0.3147
0.3334
0.3766
0.0043
0.0020
0.0002
***
***
***
Full Sample
Table 16: Descriptive Statistics for the Company Sizes
This table contains the mean, median, standard deviation, minimum, and maximum for the number of employees, the balance sheet size,
and the premium volume of the insurance companies in the sample. These three variables serve as proxies for the firm size. In addition,
location and dispersion statistics have been provided separately for the subsamples of cat bond investors and non-investors. The last three
columns show the results of a Kolmogorov-Smirnov (K-S) test of the null hypothesis that the variables are normally distributed.
147
148
No. of Employees
N
Mann-Whitney U
Standard Error
p-value
Balance Sheet Size (mn EUR)
44
218.0000
39.6440
0.8400
N
Mann-Whitney U
Standard Error
p-value
44
237.5000
39.6820
0.4880
N
Mann-Whitney U
Standard Error
p-value
44
249.0000
39.6850
0.3260
Table 17: Mann-Whitney U Test
Results of a Mann-Whitney U test to assess whether the differences in the average number of employees, balance sheet size, and premium
volume between investors and non-investors are statistically significant. The test is based on the null hypothesis that the two samples under
consideration have been drawn from the same distribution.
relevant
Full Sample
irrelevant
sum
relevant
Investors
irrelevant
sum
Non-investors
relevant irrelevant
sum
Expertise/Experience
in percent
15
41.67
21
58.33
36
100.00
8
66.67
4
33.33
12
100.00
7
29.17
17
70.83
24
100.00
Risk/Return/Correlation
in percent
16
44.44
20
55.56
36
100.00
10
83.33
2
16.67
12
100.00
6
25.00
18
75.00
24
100.00
Administrative Complexity
in percent
16
44.44
20
55.56
36
100.00
9
75.00
3
25.00
12
100.00
7
29.17
17
70.83
24
100.00
Data Availability
in percent
12
33.33
24
66.67
36
100.00
8
66.67
4
33.33
12
100.00
4
16.67
20
83.33
24
100.00
Regulatory Constraints
in percent
10
27.78
26
72.22
36
100.00
3
25.00
9
75.00
12
100.00
7
29.17
17
70.83
24
100.00
Accounting Issues
in percent
7
19.44
29
80.56
36
100.00
4
19.05
17
80.95
21
100.00
3
12.50
21
87.50
24
100.00
Determinant
Table 18: Potential Determinants of the Investment Decision
Numbers and percentages of the insurance companies that stated whether or not a certain factor has influenced their decision to invest in
cat bonds. Due to the exclusion of cases with missing data, the overall sample size drops to N = 36.
149
150
charges and accounting issues, in contrast, have been declared to be irrelevant by both groups. This finding supports our expectation expressed in
Section 2.2 that these two aspects do not affect an insurance company’s
propensity to invest in cat bonds.
4.2
Determinants of the Cat Bond Investment
Decision
As explained in Section 3.4, before estimating the logistic regression
model, we run an EFA to extract a set of latent constructs from the observed variables in our sample. For the resulting factor structure to be
meaningful and properly interpretable, the correlations between items
that are associated with the same factor should be high, while those
between items that represent different factors should be low. This idea
underlies the Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy,
which indicates whether a data set is suitable for an EFA. Being defined
between zero and one, KMO values below 0.5 imply that EFA should not
be applied, whereas KMO values in excess of 0.8 mean that the sample
is particularly well suited for the analysis (see Kaiser, 1974). Following
an iterative process guided by the KMO measure, we identified 15 of
the 41 items as problematic and removed them from our sample. The
remaining 26 items lead to a solid KMO value of 0.7041. Furthermore,
in order to check the EFA precondition of normality, the K-S goodnessof-fit test has been employed. Apart from very few exceptions, we do not
find significant deviations from the normal distribution, implying that
the items can be used in a factor analysis.37 Subsequently, we conduct
Bartlett’s test of sphericity and find the correlations of the items to be
statistically significant on the 1 percent level with a χ2 test statistic of
1,106.16 and 325 degrees of freedom. Since the EFA procedure relies on
the correlation (covariance) matrix of the observed variables to derive
the latent constructs, this test result is another indication for the suitability of our data.
37 Due to the standard nature of the K-S statistic and the multitude of variables,
we decided not to report these results.
4.2 Determinants of the Cat Bond Investment Decision
151
For the initial factor extraction we choose the principal components
analysis. This procedure produces an orthogonal factor structure (i.e.,
the pairwise factor correlations are zero) by repeatedly searching for linear combinations of the items that account for the largest fraction of the
still unexplained variance, until the number of extracted factors equals
the number of items.38 The usual result is that, on the one hand, the
majority of items strongly load on the first few factors and, on the other
hand, substantial cross-loadings of items with more than one factor arise.
Consequently, it is common to rotate the extracted factor structure in
order to generate a more coherent pattern of loadings that considerably
improves interpretability.39 For this purpose, we resort to the orthogonal varimax rotation approach, which repositions the axes of the factor
space such as to maximize the variance of the squared loadings per factor. Moreover, since EFA does not provide a theoretical foundation for
the number of factors to be retained, our hypotheses from Section 2.2
serve as the main guidance for the dimensionality of the model. In addition, we employ the Kaiser criterion, which states that only factors with
eigenvalues in excess of one should be retained.40
Table 19 summarizes the results of the EFA.41 As can be inferred
from the rotated factor loadings matrix, we decided in favor of a fivefactor model that is compatible with our hypotheses. This solution is
also supported by the Kaiser criterion (see eigenvalues).42 In sum, the
five factors account for more than 80 percent of the variation in the
38 Taken
together, the principal components explain the total variance of all items.
rotated factor structures, most items tend to load relatively high on one factor,
while exhibiting rather weak cross-loadings with others.
40 Being defined as the sum of squared factor loadings across all items, eigenvalues
represent the amount of the total variance explained by a factor. As EFA is commonly
performed with standardized variables, each item exhibits an eigenvalue of one. Thus,
intuitively the Kaiser criterion requires factors to explain at least as much variance
as individual items.
41 In order to enhance the readability and interpretability of this table, factor loadings below 0.4000 have been suppressed.
42 Note that there is actually a sixth factor, which exhibits an eigenvalue just slightly
above one. However, it contributes a mere 3.85 percent of explained variance and is
neither supported by our theory, nor clearly associated with a specific item battery.
In addition, no item exhibits a loading in excess of 0.6 with regard to this factor.
Hence, we chose not to include it in the final factor model.
39 In
Item
Factor 1
0.4792
Our firm ...
... is very experienced in the asset class
... has a strong cat bond expertise
... fully understands the typical risks
... can handle the modeling/valuation/risk mgmt
... possesses cat bond portfolio mgmt skills
... understands the accounting treatment
... understands the regulatory treatment
... commands the necessary overall resources
0.8706
0.8930
0.8171
0.9292
0.8915
0.8479
0.7481
0.9126
The following information is readily available:
– Transaction data
– Pricing information
– Historical performance figures
– Loss experience
– Deal documents
– Overall cat bond data availability
Factor 4
Factor 5
0.8440
0.9165
0.6470
0.4125
0.9066
0.8346
0.5494
0.7486
0.8976
0.9489
0.5020
0.6568
0.5616
0.8083
0.6631
0.8532
0.7821
0.4289
8.9929
34.5879
34.5879
0.9653
6.7188
25.8416
60.4295
0.9318
Table 19: Rotated Factor Loadings Matrix with Additional Statistics
2.1114
8.1209
68.5504
0.9152
1.9237
7.3990
75.9494
0.8761
1.7278
6.6455
82.5949
0.8819
Factor loadings resulting from an EFA for 26 items that have been measured on Likert scales. In order to enhance the readability and interpretability of this table,
factor loadings below 0.4000 have been suppressed. In sum, the five factors explain 82.5949 percent of the total variance.
0.8025
0.7539
0.8458
0.7713
0.4828
Cat bonds ...
... are standardized
... are liquid
... are associated with low administration costs
... expose investors exclusively to insurance risk
... are not associated with credit risk
Eigenvalues
Explained Variance (percent)
Cumulative Explained Variance
Cronbach’s α
Factor 3
152
Cat bonds ...
... fit well in our portfolio
... are compatible with our strategic ALM goals
... are an attractive asset class
The cat bond asset class exhibits ...
... a strong historical performance
... an attractive return potential
... a high relative value
... an appealing overall risk-return profile
Factor 2
153
data. Furthermore, apart from two exceptions, all factor loadings that
belong to one item battery exceed 0.60, underlining a strong influence
of the common factors on the observed variables through which they
have been measured.43 This is also reflected by the low number of crossloadings above 0.4 as well as the high values for Cronbach’s alpha, which
measures the internal consistency of each factor. Thus, the 26 items that
have been included in the EFA lead to a meaningful factor structure. In
line with our arguments in Section 2.2, we interpret the five factors as
follows:
- Factor 1: expertise and experience with regard to the cat bond
asset class (hypothesis 3)
- Factor 2: perceived availability of data and information on the cat
bond asset class (hypothesis 7)
- Factor 3: perceived attractiveness of the risk-return profile (hypothesis 4)
- Factor 4: perceived administrative complexity (hypothesis 6)
- Factor 5: perceived fit with the insurance company’s strategic
ALM goals (hypothesis 2)
This EFA output enables us to test five of our eight hypotheses within
a logistic regression analysis. In order to control for the remaining three
determinants postulated in Section 2.2, we need to draw on further independent variables. More specifically, 2 of the 15 items that were not
suited for the EFA have been averaged to form an additional factor, reflecting hypothesis 4 (diversification benefits of cat bonds). In addition,
we include the number of employees as a proxy for the firm size (hypothesis 1).44 Finally, to account for hypothesis 8, we coded a dummy
variable for the regulatory constraints, which equals one if the company
is subject to Swiss investment rules and zero otherwise.
43 The factor loadings can be interpreted as correlation coefficients between indicator variables and factors.
44 We have also estimated alternative model specifications based on the balance
sheet size and the premium volume to ensure that the reported significance level for
the size factor is robust with regard to the employed proxy.
154
The results for a logistic regression model, comprising these eight
independent variables, are shown in Table 20. Unreported collinearity
diagnostics such as the tolerance and the variance inflation factor (VIF)
indicate that multicollinearity is not an issue. The coefficients βi reflect
the magnitude of the effect of each independent variable on the logit,
exp(βi ) represents the corresponding impact on the OR, and s.e. is the
standard error of the respective parameter. The Wald statistic is employed to test the significance of the individual logit coefficients. Examining the corresponding p-values and significance levels, we notice that
the coefficients of the regressors “expertise and experience”, “perceived
fit with the firm’s strategic ALM”, and “regulatory constraints” turn
out to be statistically significant.45 Consistent with the corresponding
hypotheses, the first two variables exhibit a positive impact, while the
last factor reduces the logit and the investment probability. All other factors, on the contrary, appear to be irrelevant with regard to the insurers’
decision to add cat bonds to their portfolios.
Table 20 also contains the typical goodness-of-fit measures for logistic
regression models. −2LL0 and −2LLm equal minus two times the loglikelihood value for a null model that includes only a constant and minus
two times the log-likelihood value for the considered model, respectively.
The higher the value of −2LLm , the worse the actual model fit. Since
−2LLm (also known as deviance) is χ2 -distributed with N −k−1 degrees
of freedom (k equals the number of explanatory variables), we can use
it to test the null hypothesis of a perfect model fit that, due to the p45 Note that we also conducted analyses with the 13 spare items, which are neither
used in the EFA nor reflect a specific hypothesis. Five of these items cover more detailed aspects with regard to accounting treatment and solvency capital requirements,
another five represent specific transactional characteristics such as TRS features and
collateral arrangements, and one asks for the respondents’ overall risk perception.
Unreported logistic regression results indicate that none of these variables adds any
explanatory power to the model. Furthermore, additional qualitative characteristics
such as the company type (primary insurer vs reinsurer) or business line (life vs
nonlife) have been tested via dummy variables. In this regard, we find that being a
reinsurance company exhibits a significant positive impact on the investment decision.
However, this can be simply explained by the fact that the characteristic “reinsurer”
is highly predictable through a combination of the factors expertise/experience as well
as perceived fit with the strategic ALM, and is thus already covered by our model.
Constant
Company Size
Expertise and Experience
Perceived Fit with Strategic ALM
Perceived Data Availability
Perceived Administrative Complexity
Perceived Risk-Return Profile
Perceived Diversification Benefits
Goodness of Fit
−2LL0 (null model)
−2LLm (considered model)
LR (likelihood ratio test)
HL (Hosmer-Lemeshow test)
βi
exp(βi )
s.e.
Wald
p-value
0.3867
–0.0002
3.5537
3.0297
0.4424
–0.8880
0.1325
0.5492
–4.9872
1.4721
0.9998
34.9413
20.6905
1.5565
0.4115
1.1416
1.7319
0.0068
0.9034
0.0001
1.6473
1.4511
0.6264
0.8645
0.6774
0.6929
2.5702
0.1832
1.6369
4.6537
4.3591
0.4989
1.0550
0.0382
0.6283
3.7652
0.6686
0.2008
0.0310
0.0368
0.4800
0.3044
0.8449
0.4280
0.0523
χ2
df
p-value
45.8290
20.3100
25.5190
10.2841
34
26
8
7
0.0847
0.7766
0.0013
0.1730
sig.
**
**
*
Pseudo R2 -measures
Cox and Snell
Nagelkerke
McFadden
0.5078
0.7053
0.5568
Table 20: Logistic Regression with all Potential Determinants
Results for a logistic regression of the dichotomous dependent variable (investor/non-investor) on eight explanatory variables (plus constant).
The coefficients βi indicate the magnitude of the effect of each independent variable on the logit, exp(βi ) represents the corresponding impact
on the OR, and s.e. is the standard error of the respective parameter. The Wald statistic is employed to test the significance of the logit
coefficients. Goodness of fit (based on the χ2 distribution): −2LL0 = minus two times the log-likelihood value for the null model (includes
only a constant); −2LLm = minus two times the log-likelihood value for the considered model (H0 : perfect model fit); LR (likelihood ratio)
equals the difference between −2LL0 and −2LLm (H0 : all logit coefficients of the considered model are zero); HL = Hosmer-Lemeshow
statistic (H0 : the observed and predicted event rates do not differ in each category of the dependent variable). Pseudo R2 -measures are
defined between zero and one with values in excess of 0.4 indicating a good model fit. Significance levels: *** = 1 percent, ** = 5 percent,
* = 10 percent.
N = 36
155
156
value of 0.7766, cannot be rejected.46 A closely related statistic is the
likelihood ratio, LR, which equals the difference between −2LL0 and
−2LLm , thus providing a means for the assessment of the fit of the
considered model relative to the null model. It is also χ2 -distributed
with degrees of freedom equal to the difference in the degrees of freedom
of −2LL0 and −2LLm and forms the basis for the likelihood ratio test.
In our case, the respective null hypothesis that all logit coefficients are
jointly zero can be rejected on the 1 percent significance level (p-value:
0.0013). Therefore, adding the tested regressors leads to a significant
improvement in the model fit compared to the null model. Furthermore,
we conduct the Hosmer-Lemeshow test. Based on the corresponding
variable HL, which is χ2 -distributed with seven degrees of freedom, it is
not possible to reject the null hypothesis that the observed and predicted
event rates are equal for each category of the dependent variable. The
overall good model fit indicated by these statistics is further underlined
by the fact that the values of the pseudo R2 -measures by Cox and Snell,
Nagelkerke, and McFadden are all above 0.4.47 Finally, turning to the
classification table (Table 21), we find that the model correctly predicts
83.33 percent of the investors, 95.83 percent of the non-investors, and
91.67 percent of all firms.48
Since all but three of the tested independent variables turned out to
be insignificant, we should be able to remove them without losing much
explanatory power. The results for such a reduced model, merely comprising the regressors “expertise and experience”, “perceived fit with
the firm’s strategic ALM goals”, and “regulatory constraints”, can be
found in Table 22. Again, the coefficients for these variables are statisti46 We are aware that the −2LL
m statistic is sensitive to the distribution of the cases
among the categories of the dependent variable. If the sample is very unbalanced in
this regard, it may provide a too optimistic assessment of the model fit.
47 Although they cannot be interpreted exactly in the same way, pseudo R2 measures have been developed to mimic the well-known R2 of the linear regression
analysis (see, e.g., Wooldridge, 2008). They equal zero if the independent variables
exhibit no explanatory power at all. Values above 0.4 indicate a good model fit.
48 These figures are based on a cutoff value of 0.5, i.e., all firms for which the
probability of investing as implied by the model exceeds 0.5 are classified as investors.
Observed
Investor
Non-investor
Predicted
Investor Non-investor
10
1
157
% Correct
2
23
83.33
95.83
Overall
91.67
Table 21: Classification Table for Model with all Potential Determinants
This classification table can be employed to evaluate the predictive accuracy of the logistic
regression model in Table 20. It shows how many of the observed values for the dichotomous
dependent variable (investor/non-investor) are correctly predicted. The figures are based
on a cutoff value of 0.5, i.e., all firms for which the probability of investing as implied
by the model exceeds 0.5 are classified as investors. The model correctly predicts 83.33
percent of the investors, 95.83 percent of the non-investors, and 91.67 percent of all cases.
cally significant.49 Moreover, as expected, the goodness-of-fit statistics
and pseudo R2 -measures have hardly changed. Similarly, the figures reported in the classification table for this new model (Table 23) suggest
a strong predictive accuracy. More specifically, although it includes five
independent variables less than the previous model, it still correctly predicts 83.33 percent of the investors, 83.33 percent of the non-investors,
and 83.33 percent of all firms in the sample.
To sum up, through our quantitative results, we are able to provide
evidence for hypotheses 2, 3, and 8. The remaining hypotheses, however, cannot be confirmed. Consequently, the expertise and experience
of insurance companies concerning cat bonds, the extent to which they
perceive a fit of the asset class with their strategic ALM goals, and the
prevailing regulatory regime seem to be the key determinants of the investment decision. To assess the importance of these factors relative to
each other, one needs to consider the respective effect strengths. According to the logit coefficient of 2.4740, the strongest impulse for an
investment in cat bonds emanates from the expertise/experience of a
company. With a corresponding value of 2.2814, however, the percep49 Peduzzi et al. (1996) propose that as a rule of thumb, logistic regression models
require a minimum of 10 events per explanatory variable to avoid biased regression
coefficients and misestimation of the standard errors. Since the 36 participants in
our sample comprise only 12 investors, we have additionally tested each of the three
determinants in a separate model, thus increasing this ratio above the critical level.
In doing so, we ensure the robustness of our results.
158
N = 36
Expertise and Experience
Perceived Fit with Strategic ALM
Goodness of Fit
−2LL0 (null model)
−2LLm (considered model)
LR (likelihood ratio test)
HL (Hosmer-Lemeshow test)
βi
exp(βi )
s.e.
Wald
p-value
sig.
2.4740
2.2814
–3.3251
11.8694
9.7899
0.0360
0.9830
0.9887
1.4184
6.3336
5.3239
5.4955
0.0118
0.0210
0.0191
**
**
**
χ2
df
p-value
49.9070
24.9120
24.9950
4.2433
35
32
2
7
0.0490
0.8096
0.0000
0.7514
Pseudo R2 -Measures
Cox and Snell
Nagelkerke
McFadden
0.5006
0.6674
0.5008
Results for a logistic regression of the dichotomous dependent variable (investor/non-investor) on three explanatory variables (without
a constant). The coefficients βi indicate the magnitude of the effect of each independent variable on the logit, exp(βi ) represents the
corresponding impact on the OR, and s.e. is the standard error of the respective parameter. The Wald statistic is employed to test the
significance of the logit coefficients. Goodness of fit (based on the χ2 distribution): −2LL0 = minus two times the log-likelihood value for
the null model (includes only a constant); −2LLm = minus two times the log-likelihood value for the considered model (H0 : perfect model
fit); LR (likelihood ratio) equals the difference between −2LL0 and −2LLm (H0 : all logit coefficients of the considered model are zero);
HL = Hosmer-Lemeshow statistic (H0 : the observed and predicted event rates do not differ in each category of the dependent variable).
Pseudo R2 -measures are defined between zero and one with values in excess of 0.4 indicating a good model fit. Significance levels: *** = 1
percent, ** = 5 percent, * = 10 percent.
Table 22: Logistic Regression with Significant Determinants
4.3 Further Qualitative Results
Observed
Investor
Non-investor
159
Predicted
Investor Non-investor
20
2
% Correct
4
10
83.33
83.33
Overall
83.33
Table 23: Classification Table for Model with Significant Determinants
This classification table can be employed to evaluate the predictive accuracy of the logistic
regression model in Table 22. It shows how many of the observed values for the dichotomous
dependent variable (investor/non-investor) are correctly predicted. The figures are based
on a cutoff value of 0.5, i.e., all firms for which the probability of investing as implied
by the model exceeds 0.5 are classified as investors. The model correctly predicts 83.33
percent of the investors, 83.33 percent of the non-investors, and 83.33 percent of all cases.
tion that the asset class is in line with the strategic ALM goals has a
similarly large positive impact on the likelihood to invest. In contrast
to that, the binding regulatory constraints with regard to the tied assets
faced by Swiss companies strongly oppose these factors (logit coefficient:
–3.3251), causing a significant reduction of the investment probability.
Thus, Swiss insurers seem to be a lot less likely to invest in cat bonds
than EU-based firms, even if they exhibit the same values with regard
to the two aforementioned factors.
4.3
Further Qualitative Results
Open Survey Questions
To complement our inference statistics, we additionally included five
open questions in the questionnaire. Thereby, the participants were given
the opportunity to express opinions and ideas with regard to different
aspects of their cat bond investment decision. Overall, we obtained
38 responses to open questions from 24 different key informants. A
comprehensive list of quotations has been included in the Appendix.
For reasons of efficient reporting, we have grouped the answers in this
section based on their key messages. The respective results are shown in
Table 24. A total of 14 statements contain aspects that encourage the
firms to invest in cat bonds. Six of these, i.e., 15.79 percent of all comments, are centered around the attractive risk-return profile of cat bonds
160
Full Sample
No. Percent
6
6
2
15.79
15.79
5.26
Aspects opposing cat bond investments
”They do not fit with our asset and liability management.”
”We have not undertaken a particular effort due to regulatory constraints.”
”Missing know-how.”
Other reasons
9
6
4
2
23.68
15.79
10.53
5.26
Further comments
3
7.89
Table 24: Open Questions
This table gives an overview of the responses to the open questions in the survey. 24 participants have commented on different aspects
regarding their decision to invest or not to invest in cat bonds. The answers are categorized by aspects encouraging and aspects opposing
cat bond investments. The percentage figures are based on a total of 38 responses to open questions.
Aspects encouraging cat bond investments
”Cat bonds offer attractive returns/have a low fundamental correlation with other asset classes.”
”It can be better to write cat bond business than to use the conventional market.”
”We are obtaining knowledge of the cat bond market to use these instruments in the future.”
161
as well as the low correlation with other asset classes. Since these factors were not found to be statistically significant in the logistic regression
analysis of Section 4.2, we need to assume that they are merely a decisive
factor for certain insurers but not for the majority of firms. In another
six responses (15.79 percent), the respective participants point out that
their their companies view cat bond investments as a means for market
expansion and to complement their traditional insurance business. This
is quite an interesting consideration, which has not been hypothesized
ex ante and was thus missing in our questionnaire design. Hence, for the
time being, we need to take into account that this might be an additional
determinant while leaving the confirmation of its statistical significance
for future research. Finally, two answers (5.26 percent) name the intension to acquire know-how about the cat bond market as the main reason
for the investment decision. This provides further support for the factor
expertise and experience, which has been identified as one of the main
drivers in the previous section.
Moreover, 21 answers brought forward reasons not to purchase cat
bonds. Since the majority of survey participants are non-investors (see
Table 15), it could be expected that the negative statements outnumber
the arguments in favor of an investment in this asset class. Consistent
with our statistical results, a lacking fit of cat bonds with regard to
the ALM considerations of the company is stressed in nine comments
(23.68 percent). Furthermore, in six (15.79 percent) responses the key
informants note that their firm refrains from cat bond investments due
to regulatory constraints. Particularly the FINMA guidelines regarding “tied assets” are referred to several times, thus confirming our logistic regression results for H8 .50 Apart from these aspects, four answers
(10.53 percent) revolve around the missing expertise and experience that
would be needed to make adequate investment decisions with regard to
cat bonds. Again, this supports our previous findings with regard to
H3 . Finally, the responses to the open survey questions also comprise a
50 A penalization in terms of the Solvency II capital charges for cat risk, in contrast,
is only criticized in a single response.
162
totally new aspect counted under “other reasons”.51 One company does
not seem to show any interest in cat bond investments because they want
to avoid losses due to perils that their stakeholders do not expect to be
part of their exposure. The fact that earthquakes in Japan would not
be associated with a purely European insurance company is mentioned
as an example in this context.
Most answers to the open survey questions help to underline and
further illustrate the results of our empirical analysis. Indeed, it appears
that a large number of the participants made comments that can be
associated with one of the three significant determinants identified in
Section 4.2. In addition to that, however, we were able to gather new
information with regard to the investment decision that could not be
captured by the preset items in the questionnaire. The aspects revealed
in this regard should be taken into account in future empirical research
on this topic.
Interviews with Managers of Dedicated Cat Bond Funds
In addition to the survey, we conducted structured interviews with the investment managers of four large and influential dedicated cat bond funds.
Together, their assets under management amount to approximately USD
7.73 billion, which represented 76.19 percent of the outstanding cat bond
volume in 2011 (see Guy Carpenter, 2011). Thus, our interview partners
possess information about a large part of the market and profound knowledge of their clients’ investment decisions as well as the reasons behind
them. This characterizes them as key informants for our study. The
interviews were carried out in March and April 2012. All participants
received the same set of questions and were asked to answer either in
form of a telephone interview or in writing. The generally low level of
cat bond investments in the insurance industry (see Section 2.1) is also
reflected by the investor base of the considered funds. While two of them
do not have insurers among their current investors at all, the other two
51 The second statement in the category “other reasons” refers to the low degree
of liquidity of cat bonds. This consideration is covered by factor 4 of the empirical
analysis (perceived administrative complexity).
163
pointed out that less than 1 percent of their total client money comes
from the insurance industry. Several reasons for this phenomenon have
been provided by our interview partners.
One of them emphasizes that the regulatory constraints in Switzerland are the main factor behind the limited interest in cat bond investments shown by the local insurance industry. According to his clients,
FINMA insists on a strict separation of business lines. Companies that
are exclusively regulated as life or health insurers are not permitted to
invest in cat bonds at all, since this would be economically equivalent
to the underwriting of insurance contracts in the property and liability
sector. Nonlife insurers, in contrast, for which natural disaster risk is
part of their core business, do not face such an explicit restriction with
regard to the asset class. However, their potential for cat bond investments is still considerably constrained due to the tied asset investment
rules set by FINMA. This statement provides further support for H8 .
Other explanations revolve around the determinant expertise and experience. Our interviewees find this aspect to be especially relevant in
times of market turbulence. Under such circumstances as during the
financial crisis in 2008, those insurance firms without an in-depth knowledge about the cat bond market are the first to abandon their engagement as investors. It has also been stated that there are a number of
reinsurers as well as large primary insurers with considerable cat bond
expertise and experience. When willing to invest, those firms only consider direct investments. Smaller insurers, on the other hand, often lack
know-how with regard to the structures, the market, or certain catastrophe risks in general and thus, if at all, access the asset class through
dedicated cat bond funds. We view these arguments as an additional
confirmation of H3 .
Furthermore, the question whether the risk-return and correlation
profiles of cat bonds have a certain impact on the investment decision
was affirmed during three of the interviews. This is in line with the statements of some of the survey participants. Hence, those insurers that
164
actually invest in cat bonds might view the beneficial characteristics of
the asset class to be great enough to overcome some of the other factors
that exhibit a negative impact on their investment decision. Nevertheless, the corresponding determinants did not turn out to be statistically
significant in our empirical analysis, implying that one should be cautious about a generalization of these opinions. Therefore, in the absence
of further evidence, H4 and H5 can still not be ultimately confirmed.
Similarly, three interview partners state that an improvement in the
availability of data and information on the cat bond asset class would be
likely to have a positive influence on the demand. In this context, they
point to a potentially severe conflict of interest. Since the asset management of those insurance companies that act as cat bond investors needs
to assess prospective transactions, it desires as much publicly available
information as possible. The sponsor, in contrast, is interested in a high
level of discretion, particularly for indemnity deals, in order to avoid
that its competitors gain too much insight into its underwriting activities. Consequently, insurers seem to believe that they can get better
data on the actual exposure when they insure the risk rather than relying on offering documents of cat bond transactions. Moreover, it has
been mentioned that substantial changes in the risk models of the major
analytics firms (RMS, EQECAT, AIR) negatively affect the interest in
the asset class, since investors generally tend to avoid cat bonds that are
perceived to hide a considerable amount of model risk. Although these
are interesting new insights, the insignificance of the respective factor
within the empirical analysis prevents the confirmation of H7 .
Finally, the interviews revealed two perspectives concerning the cat
bond investment decision of insurance companies that had not been anticipated in Section 2.2. Interestingly, these are consistent with the new
aspects that we identified based on the answers to the open survey questions. The first point, which has been mentioned by two of the interviewees, can be described as political reasons for the decision not to buy
cat bonds. More specifically, although stakeholders might benefit from
this asset class through enhanced diversification, the management of in-
5 Summary and Conclusion
165
surance companies can be reluctant to accept exposures outside of their
core market due to the associated career risks. In doing so, they aim
to avoid the responsibility for losses due to natural hazards in locations
where their company is not even represented with its insurance operations. Apart from that, our interview partners indicated that some firms
have embraced cat bonds as a complement to their conventional business.
While some insurers exclusively approach the topic from an asset management perspective, others consider cat bond investments as a relative
value trade with regard to insurance products. If the pricing of a cat
bond issue is more attractive than that of a corresponding traditional
contract, these companies will switch to the former in order to benefit
from its superior risk-return trade-off.
5
Summary and Conclusion
Although they are familiar with the risks inherent in cat bonds, insurance and reinsurance companies jointly account for less than ten percent
of the current demand in the market. In order to be able to develop
explanations for this phenomenon, a deeper insight into the underlying
decision-making process is needed. Accordingly, our main research goal
in this paper is to identify major determinants of the cat bond investment decision of insurers. For this purpose, we conducted a comprehensive survey among senior executives in the European insurance industry.
Evaluating the corresponding data set by means of EFA and logistic
regression methodology, we are able to show that the firm’s expertise
and experience with regard to cat bond investments, their perceived fit
with its ALM philosophy, and the prevailing regulatory regime exert a
significant influence on an insurer’s propensity to invest. In contrast
to that, the perception of the asset class’s risk-return profile, diversification benefits, administrative complexity, as well as the availability of
data and information on cat bond transactions seem to be of lesser relevance. Similarly, we do not find evidence for an impact of firm size,
accounting treatment, or solvency capital requirements. These statistical results are complemented by further qualitative survey answers and
additional information from structured interviews with the investment
166
managers of four large dedicated cat bond funds.
Our findings should be highly relevant to cat bond issuers and policymakers alike. Since, in general, insurance companies represent a central
source of institutional investor demand in the capital markets, the reduction of existing investment barriers with regard to cat bonds could
generate a substantial growth impulse for this asset class. It appears
that, theoretically, issues relating to the first two determinants (lack of
expertise/experience, perceived fit with the strategic ALM goals) could
be simply overcome by properly educating prospective market participants and disseminating more information about the merits of adding
cat bond exposure to the balance sheet of a typical insurance company.
Particularly life insurers should be able to exploit the virtues of the instrument, since it may serve as a diversification tool for both their asset
and liability risks. Impediments arising due to regulatory constraints
such as the rigid legal investment guidelines set by FINMA, however,
seem a lot more difficult to address. In this respect, it might be helpful
to initiate an intensive dialogue with the regulators, highlighting that,
due to risk sharing and performance aspects, it may be economically
reasonable for insurance companies to allocate a certain fraction of their
asset portfolios to cat bonds.
Finally, future research could be aimed at overcoming some of the
limitations of our study. The most important aspect in this regard relates to sample size. Indeed, it would be helpful to verify our results
based on a much broader survey, for example, including U.S. insurance
companies or even adopting a global scope. In addition, the qualitative
information that was gathered through open survey questions and structured interviews raised completely new aspects for which measurement
variables had not been incorporated in our original questionnaire design.
Hence, an examination of the statistical significance of these potential
determinants is still outstanding.
6 Appendix
6
6.1
Appendix
Aspects Encouraging Cat Bond Investments
Risk-Return Profile and Diversification Benefits
- Attractive returns.
(Germany; Reinsurer; Director)
- Decent expected return proposition.
(Finland; Primary Insurer; Portfolio Manager)
- Attractive returns.
(Switzerland; Reinsurer; Head of Nonlife Risk Transformation)
- Potentially attractive spreads.
(Switzerland; Primary Insurer)
- Cat bond returns typically have a low fundamental correlation with other asset classes.
- Diversification.
Market Expansion/Complement the Traditional Insurance
Business
- For visibility in this market place – it can be better to
“write” cat bond business than to use the conventional
market.
(UK; Reinsurer; Managing Director)
- Market expansion.
(Italy; Primary Insurer; Managing Director)
- Certain risks are not available in form of traditional
reinsurance.
- Due to the evolution of the business it could be interesting to invest in this asset class.
(Italy; Primary Insurer; Director)
167
168
- Market making.
(Switzerland; Reinsurer; Head of Nonlife Risk Transformation)
- Cat Bonds are the most liquid asset class inside the nonlife risk category.
Expertise and Experience with Regard to the Asset Class
- To test the market.
(Sweden; Reinsurer; Group CFO)
- Obtaining knowledge of the cat bond market to use these
instruments in the future.
(Netherlands; Primary Insurer; Senior Risk Manager)
6.2
Aspects Opposing Cat Bond Investments
Fit with Strategic Asset and Liability Management Goals
- At the moment, they do not exactly fit in our asset and
liability management.
(Italy; Primary Insurer)
- The decision not to invest in cat bonds is based on a
total balance sheet view and the regional risk profile of
our company.
(Switzerland; Primary Insurer; Director)
- We are very conservative in our investment approach.
(Greece; Primary Insurer)
- Do not fit our ALM considerations.
(Austria; Primary Insurer; Market Risk Manager)
- Main focus in matching liabilities.
(Finland; Primary Insurer; Chairman of the Board)
- Strategic decision. Ultimately, we do not want to buy
risks that we already insure.
(Switzerland; Primary Insurer; CIO)
6.2 Aspects Opposing Cat Bond Investments
169
- We do not treat them as investments, since they are
correlated with our key cat risks.
- We acquire our cat risk through reinsurance and feel no
need to buy it through assets.
(Portugal; Primary Insurer; CFO)
- As a reinsurer we are already exposed to natural disasters risk. Investing in cat bonds would create a dependency between our insurance results and investment
results.
(Belgium; Reinsurer; Member of the Executive Board)
- We have not undertaken a particular effort due to regulatory constraints.
(Switzerland; Primary Insurer; Head of Investments)
- Not allowed to invest according to FINMA rules guiding “Gebundene Vermögen” (tied assets), therefore no
particular efforts undertaken.
(Switzerland; Primary Insurer; Head of Asset Management)
- All our investments need to qualify for tied assets. Considering the local asset management knowledge/team, these
investments will not qualify as “tied assets” and hence
we cannot invest.
(Switzerland; Primary Insurer; Risk Manager)
- See FINMA regulation covering “Gebundene Vermögen”
(tied assets).
(Switzerland; Primary Insurer; Head of Investments)
- We maintain a very low risk profile, which ensures that
we meet the Swiss regulatory tied assets requirement.
(Switzerland; Primary Insurer; Controller)
170
- Under Solvency II, cat risk is heavily penalized in terms
of capital requirements.
(Portugal; Primary Insurer; CFO)
Expertise and Experience with Regard to the Asset Class
- Missing know-how.
- We are more confident in traditional investment asset
classes.
- Opacity: we have difficulties to properly value cat bonds.
- The risk exposures and the risk accumulation are difficult to assess and monitor.
(Switzerland; Primary Insurer; CIO)
Other Reasons
- Although the risk-return profile is very interesting, it can
be difficult to explain to our stakeholders that we could
have made a loss due to a peril that our stakeholders
don’t expect to be our risk. For example, a Japanese
earthquake is not a peril that our stakeholders expect to
cause a loss for an insurer only active in Europe.
(Netherlands; Primary Insurer; Senior Risk Manager)
- Not very liquid
(Italy; Primary Insurer; Managing Director)
6.3
Further Comments
- We have only invested in a cat bond fund that has no
specific restrictions.
(Italy; Primary Insurer)
6.3 Further Comments
- Our ILS investments are not covered by our ordinary asset management activities but are managed by our dedicated ILS department, which is part of the reinsurance
division.
- The entity in our group that holds cat bonds is a Bermudian affiliate.
171
172
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Curriculum Vitae
177
Curriculum Vitae
Personal Information
Name:
Katja Müller
Date of Birth:
27th of August 1986
Place of Birth:
Berlin, Germany
Nationality:
German
Education
01/2011 − present
University of St. Gallen (HSG), St. Gallen, Switzerland
Doctoral Studies in Management
08/2008 − 05/2009
Florida Institute of Technology, Melbourne, USA
Master of Science in Applied Mathematics
10/2005 − 11/2010
University of Ulm, Ulm, Germany
Diplom-Wirtschaftsmathematikerin
09/1991 − 06/2000
Grosse Schule, Wolfenbüttel, Germany
Abitur (A-Levels)
Work Experience
01/2011 − present
Institute of Insurance Economics
University of St. Gallen, Switzerland
Project Manager and Research Associate
10/2009 − 07/2010
Teaching Assistant
08/2008 − 05/2009
Florida Institute of Technology, Melbourne, USA
Teaching Assistant
10/2007 − 07/2008
Teaching Assistant
02/2006 − 03/2006
Siemens AG, Braunschweig, Germany
Intern, Rail Automation

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