DYNAMIC ECONOMETRIC ANALYSIS OF
INSURANCE MARKETS WITH IMPERFECT
INFORMATION
ISBN 978 90 361 00946
Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul
This book is no.
442 of the Tinbergen Institute Research Series, established through
cooperation between Thela Thesis and the Tinbergen Institute.
already appeared in the series can be found in the back.
A list of books which
VRIJE UNIVERSITEIT
Dynamic Econometric Analysis of Insurance
Markets with Imperfect Information
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad Doctor aan
de Vrije Universiteit Amsterdam,
op gezag van de rector magnicus
prof.dr. L.M. Bouter,
in het openbaar te verdedigen
ten overstaan van de promotiecommissie
van de faculteit der Economische Wetenschappen en Bedrijfskunde
op dinsdag 13 januari 2009 om 13.45 uur
in de aula van de universiteit,
De Boelelaan 1105
door
Tibor Zavadil
geboren te ’a©a, Slowakije
promotor:
prof.dr. J.H. Abbring
venujem Jaapovi, Drahu²ke a rodi£om
Acknowledgments
First of all I would like to thank Jaap for accepting me for this PhD project. He showed
me the magic of science and was not only an excellent supervisor, but also my good friend.
Undoubtedly, without him this thesis would never see the light of day. Moreover, we had
so much fun during my PhD that I can honestly say that it is a pity that my PhD is over.
Finally, without Jaap I would have never nished my thesis on time. Last day before the
deadline we were working together whole night long (in Jaap's hotel room in Milan) to
accomplish the thesis.
Obviously, I have to thank also my girlfriend Drahu²ka who was always supporting
me, mainly in dicult situations, which I could not handle myself. Since we met she has
been a bright side of my life.
Then I need to thank my parents for their love and generosity; they were always
motivating me to work hard on myself. I would like to thank also my grandparents for
their curious questions about the progress of my PhD.
Further, I have to thank my closest colleagues: Ronald for being always available to
help me with technical issues, and Marcel for taking care of the printing of my thesis while
I was on holidays in south-east Asia. Then I want to thank also Sander an insurance
professional for his precious help with the data and explanation how it works in the
business.
I gratefully acknowledge nancial support by the Netherlands Organisation for Scientic Research (NWO) through a MaGW Free Competition grant (400-03-257).
Finally, I thank all my friends from Tinbergen Institute, Vrije Universiteit and Sushi
Me for making my stay in Amsterdam pleasant and funny.
I have never had so many
parties in my life before :-).
Peace and love,
Tibor
vii
Contents
Acknowledgments
vii
1 Introduction and Summary
1
1.1
Adverse Selection versus Moral Hazard
. . . . . . . . . . . . . . . . . . . .
3
1.2
Ex Ante and Ex Post Moral Hazard . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Asymmetric Information in Car Insurance
9
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Car Insurance in the Netherlands
. . . . . . . . . . . . . . . . . . . . . . .
12
2.3
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.1
Sample Selection
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3.2
Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.4
2.5
Asymmetric Information and Occurrence of Claims
9
. . . . . . . . . . . . .
18
2.4.1
Pair of Probits
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
2.4.2
Bivariate Probit . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4.3
χ2
. . . . . . . . . . . . . . . . . . . . . . . .
20
2.4.4
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.4.4.1
Actuarial Study . . . . . . . . . . . . . . . . . . . . . . . .
22
2.4.4.2
Experience Rating
26
2.4.4.3
Results for Young Drivers
. . . . . . . . . . . . . . . . . .
28
2.4.4.4
Results for Senior Drivers
. . . . . . . . . . . . . . . . . .
31
2.4.4.5
Negative Claim-Coverage Correlation . . . . . . . . . . . .
36
Asymmetric Information and Incurred Damages . . . . . . . . . . . . . . .
37
2.5.1
Classical Regression . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.5.2
Nonparametric Tests
39
Test of Independence
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
ix
CONTENTS
2.5.2.1
2.6
2.7
Implementation and Results . . . . . . . . . . . . . . . . .
40
Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.6.1
Liability Premium and Expected Damage
. . . . . . . . . . . . . .
43
2.6.2
Test for Asymmetric Information Based on Claim Frequency . . . .
45
2.6.3
Test for Asymmetric Information Based on Claim Severity
. . . . .
47
2.6.3.1
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.6.3.2
First Test . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.6.3.3
Second Test . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Appendix to Chapter 2
2.A
55
Technical Details for Tests from Subsection 2.6.3 . . . . . . . . . . . . . . .
55
2.A.1
Asymptotic Properties of the Test Statistic
T2
. . . . . . . . . . . .
55
2.A.2
Asymptotic Properties of the Test Statistic
T4
. . . . . . . . . . . .
56
3 State Dependence
59
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
State Dependence and Heterogeneity in Renewal . . . . . . . . . . . . . . .
60
3.3
Identiability
61
3.3.1
3.4
Occurrence Dependence and Duration Dependence
3.3.1.1
Full Information
3.3.1.2
Censored Data
. . . . . . . . .
62
. . . . . . . . . . . . . . . . . . . . . . .
63
. . . . . . . . . . . . . . . . . . . . . . . .
63
3.3.2
Occurrence Dependence and Lagged Duration Dependence
. . . . .
65
3.3.3
Occurrence Dependence and Nonstationarity . . . . . . . . . . . . .
65
Nonparametric Tests
3.4.1
x
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Simple Rank Test
67
. . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.4.1.1
The Case without Censoring . . . . . . . . . . . . . . . . .
68
3.4.1.2
The Case with Censoring
69
. . . . . . . . . . . . . . . . . .
CONTENTS
3.4.1.3
3.4.2
3.5
Sample Sizes
. . . . . . . . . . . . . . . . . . . . . . . . .
A Transformed Rank Test
71
. . . . . . . . . . . . . . . . . . . . . . .
75
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
4 Moral Hazard in Dynamic Insurance Data
79
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
4.2
Institutional Background and Data
. . . . . . . . . . . . . . . . . . . . . .
84
4.2.1
Experience Rating in Dutch Car Insurance . . . . . . . . . . . . . .
84
4.2.2
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
Model of Claim Rates and Sizes . . . . . . . . . . . . . . . . . . . . . . . .
90
4.3.1
Primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.3.2
Optimal Risk, Claims and Savings . . . . . . . . . . . . . . . . . . .
95
4.3.3
Dynamic Incentives from Experience Rating
. . . . . . . . . . . . .
98
. . . . . . . . . . . . . . . . . . . .
98
4.3
4.4
Measure of Incentives
4.3.3.2
Theoretical Characterization of Incentives
4.3.3.3
Numerical Characterization of Incentives . . . . . . . . . . 102
Empirical Analysis
. . . . . . . . . 100
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.1
Econometric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4.2
Structural Test on the Full Sample of Claim Times
4.4.3
Tests for State Dependence in Claim Times and Sizes . . . . . . . . 116
4.4.4
4.5
4.3.3.1
. . . . . . . . . 113
4.4.3.1
Theoretical Implications for the Claims Process . . . . . . 116
4.4.3.2
Distribution of First Claim Time
4.4.3.3
Marginal Distributions of First and Second Claim Times . 122
4.4.3.4
Joint Distribution of First and Second Claim Durations . . 125
4.4.3.5
Claim Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . . 121
Claim Withdrawals . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
xi
CONTENTS
Appendices to Chapter 4
135
4.A
Proofs of Results in Section 4.3
. . . . . . . . . . . . . . . . . . . . . . . . 135
4.B
Computation of Proposition 5's Function
4.C
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.D
Main (Corrected) Sample without Young Drivers
4.E
Cleaned Data
4.F
Sample Corrected Based on Initial Bonus-Malus Class . . . . . . . . . . . . 166
4.G
Main (Corrected) Sample Including Withdrawn Claims
Q
. . . . . . . . . . . . . . . . . 136
. . . . . . . . . . . . . . 142
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5 Conclusion
. . . . . . . . . . . 178
191
5.1
Integration of the Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.2
Dynamic Contract Choice
. . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.3
Observed Contract Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . 195
Summary in Dutch
204
References
212
xii
1
Introduction and Summary
Risk-averse agents benet from insurance against income shocks. Imperfect, asymmetric
information may lead to two problems in providing such insurance, moral hazard and
adverse selection. Moral hazard arises when agents change their behavior in favor of more
risky actions once they are insured. Adverse selection (on risk) arises when agents who are
inherently more risk-prone (bad risks) select into buying more insurance. Either way,
competitive markets may fail to provide ecient levels of insurance, which is a main reason
for regulation of insurance markets. Furthermore, moral hazard and adverse selection have
substantially dierent implications for optimal contract design. The empirical analysis of
the eects of asymmetric information on insurance markets is therefore of major interest.
1
It is a core topic in the recent empirical literature on the economics of contracts.
This PhD project analyzes asymmetric information in car insurance markets, which
represent one of the most important sectors in non-life insurance.
2
We use data from a
major Dutch car insurer, providing us with detailed information on insurees (age, sex,
address of residence), their cars (brand, model, price, engine volume, power, etc.), con-
1 See Chiappori and Salanié (2003) for an overview.
2 In the Netherlands, car insurance is the second most important sector of the nonlife insurance. With
the total gross premium income of 4.5 billion euro in 2006, car insurance covered almost 20% of the
income of the whole nonlife insurance market. The income from car insurance represents almost 1% of
the total GDP. Source: Statistics Netherlands (www.cbs.nl).
1
CHAPTER 1.
INTRODUCTION AND SUMMARY
tracts (coverage, premium, deductible, starting date, renewal date, etc.) and claims (type,
damage, impact). As argued by Chiappori and Salanié (1997), insurance data are ideal
for the empirical analysis of contract theory. Especially in car insurance, contracts are
highly standardized and can be exhaustively described by a small set of variables. A large
company typically covers hundreds of thousands of clients, which provides enough variation even with rare events. Finally, the empirical counterpart of the client's performance
is the occurrence of an accident and its cost, which are again precisely recorded in the
company's les. Thus, insurance data allow to test most predictions of contract theory in
a detailed way, using standard econometric tools.
One popular approach to testing for the presence of asymmetric information is based
on a general theoretical conclusion that, under asymmetric information,contracts with
more comprehensive coverage are chosen by agents with higher expected accident costs.
This conclusion holds under both adverse selection and moral hazard.
Under adverse
selection, high-risk agents, who expect to incur more losses, choose to buy more coverage
(Rothschild and Stiglitz, 1976). Under moral hazard, agents who buy (for whatever reason) more insurance, become more risky because the extensive coverage reduces incentives
for cautious behavior (Shavell, 1979, Holmström, 1979).
Chiappori and Salanié (2000) point out a direct implication of these predictions, which
is that, under asymmetric information, a positive correlation between coverage and frequency of accidents should be observed on observationally identical agents. They argue
3
that this prediction is robust to a variety of generalizations.
Using both parametric and
nonparametric tests, they do not nd evidence of asymmetric information in French car
insurance market among young drivers.
They suspect, however, that such information
asymmetry may arise in the course of time due to asymmetric learning about risk.
We adopt this approach in the rst part of the thesis and extend it in various ways.
First we will test for asymmetric information also among senior drivers, controlling for
3 However, as argued by de Meza and Webb (2001), this prediction might not hold in the presence of
both selection on risk preferences and moral hazard. See Section 2.7 for more discussion.
2
1.1.
ADVERSE SELECTION VERSUS MORAL HAZARD
their experience rating. Second, we will examine not only claim occurrences but also claim
sizes, since both are relevant for the (expected) claim costs. Finally, we will also explore
data on premia, which will facilitate applications of fully nonparametric methods.
The conditional-correlation approach is quite easy to implement, requiring only crosssectional data on contracts and claims. However, it cannot identify the type of information
asymmetry involved (if any).
Such distinction requires more advanced methods, which
are discussed in the next section.
1.1 Adverse Selection versus Moral Hazard
Chiappori (2001) discusses some ways how to distinguish adverse selection from moral
hazard. One of them is to use
natural experiments.
Assume that a given population faces
an incentive structure that is suddenly modied for some exogenous reason, such as a
policy reform. If this change in incentives coincides with a change in risk, this is evidence
4
of moral hazard.
Another possibility is to use
quasi natural experiments
in which identical agents face,
for exogenous reasons, dierent incentive schemes. If they also have dierent risks, this
is evidence of moral hazard. When using this approach, it is important to check that the
dierences in schemes are purely exogenous and do not reect some hidden characteristics
of the agents.
5
Next possibility is to do a
social experiment,
to contracts with dierent coverage.
in which agents are randomly assigned
If agents with better coverage claim more costs,
this is evidence of moral hazard. Such an experiment was done, for example, in health
insurance by Manning, Newhouse, Duan, Keeler, and Leibowitz (1987), who estimated
how cost sharing, i.e.
the portion of the bill the patient pays, aects the demand for
4 Natural experiments were exploited, among others, by Dionne and Vanasse (1997), Chiappori, Durand, and Geoard (1998) and Dionne, Maurice, Pinquet, and Vanasse (2005).
5 See, for example, Holly, Gardiol, Domenighetti, and Bisig (1998) and Cardon and Hendel (2001),
who estimate structural models of health insurance.
3
CHAPTER 1.
INTRODUCTION AND SUMMARY
medical services.
Last possibility is to explore
approaches can be chosen here.
dynamic aspects
of the contractual relationship.
Two
One assumes that existing contracts are optimal and
compares the observed features of these contracts to the theoretical predictions about
the form of optimal contracts, which are dierent under adverse selection and moral hazard.
6
The second approach does not rely on this optimality assumption. Instead, it takes
existing contracts as given and contrasts behavior, implied by the theory under adverse
selection and moral hazard, to observed behavior.
The idea is that particular features
of existing contracts (whether optimal or not) have dierent theoretical implications for
observed behavior under adverse selection and moral hazard. Thus, the two can be distinguished by a careful analysis of observed behavior.
This approach was introduced by Abbring, Chiappori, Heckman, and Pinquet (2003),
who observed that there is a close relation between the empirical analysis of moral hazard in a market with experience-rated insurance contracts and the classical problem of
distinguishing state dependence and heterogeneity in labor economics (Heckman, 1981).
Abbring, Chiappori, and Pinquet (2003) formalize this idea using dynamic economic theory. They show that in the French experience-rated car-insurance system each claim at
fault (i.e., that triggers a premium increase, or malus) increases incentives to avoid
further claims, and therefore reduces claim intensities under moral hazard. The resulting
negative occurrence dependence (Heckman and Borjas, 1980) of claims due to moral
hazard is possibly counteracted by the eects of unobserved heterogeneity: Agents who
incur claims are more likely to be bad drivers and to incur more claims in the future
anyhow. Abbring et al. (2003) extend the work of Heckman and Borjas to show that it
is possible to detect true occurrence dependence due to moral hazard in the presence of
such dynamic selection eects and possible non-stationarity of accident rates over time.
The second part of the thesis continues and broadens this line of work.
6 See Dionne and Doherty (1994) for an early example.
4
First, we
1.2.
EX ANTE AND EX POST MORAL HAZARD
develop a fully structured dynamic micro-econometric model to study moral hazard in
Dutch car insurance. This allows us to exploit the rich variation in incentives, induced by
the Dutch experience rating scheme, in our analysis of moral hazard. We also increase the
power of the tests by extending the analysis to longer panels of insurance data. Finally,
we distinguish two forms of moral hazard ex ante and ex post, by modeling both claim
occurrences and claim sizes. This distinction is important for the implications for market
outcomes, optimal contracts and public policy.
1.2 Ex Ante and Ex Post Moral Hazard
A central problem in the empirical analysis of insurance data is that insurance companies
can typically only provide data on claims that are actually led with the company, and
not on the occurrence of the relevant insured events directly.
Even in absence of false
claims, the two may not coincide if reporting an insured loss is to some extent at the
insuree's discretion. In such a context, it is useful to distinguish between an
hazard
eect on the occurrence of insured losses and an
ex ante moral
ex post moral hazard
eect on
the propensity to claim once a loss has occurred.
Abbring et al. (2003) focus on the occurrence of claims, and therefore on the combined
eects of ex ante and ex post moral hazard. From an insurance company's perspective,
this combined eect on claims may be all that matters. However, from an academic and
public-policy perspective the distinction between ex ante and ex post moral hazard is of
considerable interest.
We address this issue by extending Abbring et al.'s analysis of moral hazard by including not just the occurrences, but also the sizes of claims. The Dutch experience-rating
system only punishes the former, not the latter. In particular, given that a claim is led,
the claim amount does not aect future premia. Therefore, a Dutch insuree will be more
reluctant to report small losses than to report large losses. After all, the costs of reporting
(in terms of increased future premia, etc.) are the same in both cases, but the benets of
5
CHAPTER 1.
INTRODUCTION AND SUMMARY
reporting small losses are lower. Thus, instead of only analyzing whether the individual
claim intensity changes with the past driving history, we also analyze whether the claim
sizes change with the number of past claims. The latter is evidence of ex post moral hazard to the extent that, given the occurrence of an accident, insured losses are not aected
by ex ante moral hazard.
Complementary information on ex ante and ex post moral hazard can be found in our
Dutch car insurance data where agents have the option to withdraw a claim within six
months of the relevant accident, and thus avoid any experience-rating repercussions. This
option reduces incentives to underreport accidents when they occur, and provides direct
information on ex post moral hazard. In the extreme case that there are no costs to ling
and withdrawing a claim neither direct administrative costs nor indirect informational
costs and insured losses take time (beyond the initial ling period) to be assessed, initial
claims will be directly informative on ex ante moral hazard and all ex post moral hazard
will manifest itself as claim withdrawals.
1.3 Thesis Overview
This thesis is organized as follows.
In Chapter 2 we will test for the presence of asymmetric information in the Dutch
car insurance using the conditional-correlation approach. Under asymmetric information,
more comprehensive coverage is associated with higher risk, conditional on the information
available to the insurer.
We explore this prediction by analyzing whether agents with
better coverage have either higher frequency of claims or cause more severe accidents.
We use also data on premium, which allows us to develop novel nonparametric methods.
Controlling for agents' experience rating, we do not nd any evidence of asymmetric
information in this market. This is a common result from the empirical literature using
the conditional-correlation approach. This chapter is based on Zavadil (2008).
In Chapter 3 we will review and extend results on the identiability of, and nonpara-
6
1.3.
THESIS OVERVIEW
metric tests for, state dependence and heterogeneity in renewal models.
The renewal
models studied are analogous to linear panel data models with xed eects and lagged
endogenous regressors. This chapter focuses on the specic problems that arise with panel
duration data. Most importantly, it explores the implications of the fact that renewal data
can typically only be collected over a nite period of time. It shows that such censoring
invalidates existing identication results and, in particular if the renewal events are rare,
reduces the power of nonparametric tests. This chapter is based on Abbring and Zavadil
(2008) and develops some econometric theory used in the next chapter.
Chapter 4 constitutes the core of this thesis. It empirically analyzes moral hazard in
car insurance using a dynamic theory of an insuree's dynamic risk (ex ante moral hazard)
and claim (ex post moral hazard) choices and Dutch longitudinal micro data. We use the
theory to characterize the heterogeneous dynamic changes in incentives to avoid claims
that are generated by the Dutch experience-rating scheme, and their eects on claim times
and sizes under moral hazard. We develop tests that exploit these structural implications
of moral hazard and experience rating.
evidence of moral hazard.
Unlike much of the earlier literature, we nd
This chapter appeared as Abbring, Chiappori, and Zavadil
(2008).
Finally, Chapter 5 summarizes the main results from the previous chapters and discusses new ideas for future work. In particular, we argue that a natural next step is to
include dynamic contract choices in the analysis.
We provide some evidence that our
Dutch insurance data contain sucient variation in contract choices over time to support
such an analysis.
Throughout the whole thesis we will use the following connotation:
agent and
she
he
will refer to an
will refer to an insurance company. This connotation is based on linguistics;
in most European languages an agent is masculine and a company is feminine.
7
2
Asymmetric Information in Car Insurance
2.1 Introduction
Analysis of asymmetric information in insurance markets has become a core topic in the
1
recent empirical literature on the economics of contracts.
After the seminal work on
moral hazard and adverse selection by Arrow (1963), Pauly (1968, 1974) and Rothschild
and Stiglitz (1976), who showed that asymmetric information in competitive insurance
markets may lead to inecient outcomes and market failure, economic theorists devoted
much eort to development of adverse selection and moral hazard models.
In the last
two decades of the twentieth century, contract theory developed at a rapid pace, but
empirical applications lagged behind. At the turn of the millennium, this gap was lled
with numerous empirical papers analyzing asymmetric information in various insurance
markets; see Chiappori and Salanié (2003) for an excellent overview.
In the automobile insurance market, the initial empirical studies by Dahlby (1983,
1992) and Puelz and Snow (1994) suggested the existence of adverse selection in car
1 To mention just a few recent works: Israel (2004), Ceccarini and Pereira (2004), Cohen (2005), Dionne
et al. (2005), Dionne, Dahchour, and Michaud (2006), Chiappori, Jullien, Salanié, and Salanié (2006),
Pinquet, Dionne, Vanasse, and Maurice (2007) and Abbring, Chiappori, and Zavadil (2008) analyze
asymmetric information in car insurance, and Cardon and Hendel (2001), Hendel and Lizzeri (2003),
Fang, Keane, and Silverman (2006) study health and life insurance data.
9
CHAPTER 2.
insurance.
ASYMMETRIC INFORMATION IN CAR INSURANCE
These ndings were later challenged by subsequent research.
Particularly,
Chiappori and Salanié (2000) and Dionne, Gouriéroux, and Vanasse (2001) questioned
the results of Puelz and Snow's (1994) analysis, claiming that they used too constrained
functional forms relying on very few variables, and did not control for agent's seniority
and driving experience.
Chiappori and Salanié (2000) adopt an alternative approach based on a theoretical
conclusion that higher risks are associated with more comprehensive coverage. This result comes from the seminal works of Rothschild and Stiglitz (1976) and Wilson (1977)
who predict under adverse selection that high risk individuals choose higher insurance coverage and have more accidents (within risk classes). Under moral hazard, Shavell (1979)
and Holmström (1979) predict that those with higher insurance coverage have weaker
incentives for safe driving and should have more accidents.
Consequently, asymmetric
information leads to a positive correlation between coverage and frequency of accidents
(conditionally on all observables). This prediction is quite general; it does not require any
assumptions on preferences, neither on the rm's pricing policy.
Given that it is valid
under both adverse selection (bad risks buy more insurance) and moral hazard (comprehensive coverage decreases incentives to drive carefully), tests based on this prediction
2
cannot distinguish between the two.
Any test based on the correlation between choice of coverage and occurrence of claim
must control for all variables observed by the insurer because these are used to price
individual risk. Omitting any relevant characteristic observed by both parties can lead
to spurious informational asymmetry. Chiappori and Salanié (2000) point out that it is
quite problematic to control for the past driving record which is obviously endogenous.
They circumvent this problem by focusing on subpopulation of young drivers who have
no driving history yet. They do not nd any evidence of asymmetric information using
French car insurance data.
This chapter links directly to their work and extends it in
2 See Section 1.1 for a discussion how to disentangle moral hazard from adverse selection.
10
2.1.
INTRODUCTION
various ways.
First we test for asymmetric information also among senior drivers, controlling for their
driving experience observed by the insurer.
We argue that conditioning on experience
rating is innocuous and, in fact, indispensable because this way we control for eventual
(symmetric) learning of the insurer about agent's risk.
Second, we extend the analysis by including sizes of incurred damages.
Chiappori
(2001) predicts that, under asymmetric information, contracts with more comprehensive
coverage are chosen by agents with higher
expected accident costs.
factors: (1) the probability of claim and (2) its severity.
These depend on two
While Chiappori and Salanié
(2000) focus only on the rst factor by modeling claim frequencies, we take into account
also the second factor by exploring the data on observed losses. This is highly relevant in
the case when the distribution of incurred losses depends on agent's characteristics and,
especially, on the type of his contract.
Finally, our data provide also information on the actual premium, which represents the
insurer's estimation of the expected claim costs. We provide some empirical evidence that
the premium is a good predictor for both the occurrence and the sizes of claims. Under
the assumption that the premium is a sucient statistic for all risk factors observed by
the insurer in predicting the occurrence and the sizes of claims, we can model the accident
probability and the sizes of incurred losses using only the premium. Conditioning on one
variable gives us a space to explore novel fully nonparametric methods. In the end, we do
not nd any evidence of asymmetric information in Dutch car insurance data.
This chapter is organized as follows.
The next section describes the car insurance
system in the Netherlands. The third section presents the data. The fourth section tests
for asymmetric information in claim frequencies. The fth section tests for asymmetric
information in severity of claims. The sixth section uses premium to test for asymmetric
information. The last section concludes. Appendix provides some technical details for the
tests used in the Section 2.5.
11
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
2.2 Car Insurance in the Netherlands
Dutch law stipulates that all cars must have a
liability insurance
(LI) that covers damage
inicted to other drivers and their cars. Every car insurance company oers this coverage
along with other two noncompulsory products: a
limited comprehensive insurance
(Mini-
CASCO) which covers damage caused by nature, re, vandalism or thefts; and a
comprehensive insurance
full
(CASCO) which covers all risks, including damage at fault on
insured car.
The insurance premium is calculated in two steps. First, the insurer calculates a
premium
base
which depends only on the observed characteristics of the agent and his car.
Then the insurer takes into account also the agent's claim history (if observed) based on
which she determines a special discount or a surcharge on the base premium.
discuss this kind of
experience rating
We will
later in this section. Let us rst focus on the base
premium.
The base premium depends on various characteristics and, of course, on the type
of coverage.
Each coverage has a specic vehicle-dependent factor which is the most
important parameter used in the calculation of the premium. For LI it is the weight, for
Mini-CASCO the actual value and for CASCO the value at new. Beside that, the premium
is calculated using the following characteristics: agent's age and address (region), use of
car (private/business) and expected kilometrage (number of yearly driven kilometers).
CASCO premium depends also on the level of deductible.
3
Agents opting for higher
deductible get a discount on the premium. Sometimes higher deductible is compulsory in
which case there is no discount on the premium. In the period covered by our data (1995
4
2000), the standard level of deductible was 300 ,
currently it is 136 euro.
5
3 Mini-CASCO contracts have only standard level of deductible which is equal for all agents. There is
no deductible for the liability insurance.
4 (orijn) is a symbol for the Dutch guilder, the currency used in the Netherlands from 1279 to 2002
when it was replaced by the euro. 1 euro = 2.20371 Dutch guilders. Source: European Central Bank
(www.ecb.int).
5 At present, this is the standard level of deductible applied by most insurance companies in the
12
2.2.
CAR INSURANCE IN THE NETHERLANDS
Now, the experience rating is implemented by a so-called
bonus-malus (BM) system.
In this system, each agent is assigned to a certain BM class, which determines a height
of discount (resp.
surcharge) applied to his premium.
history yet, pay the full base premium.
with a
bonus
6
New agents, who have no claim
After each claim-free year, agents are awarded
which gives them a certain discount on their premium. On the other hand,
each claim at fault comes with a
malus
which causes a surcharge on the premium.
In this project we will work with the data from a major Dutch insurer which uses the
7
BM scheme given in Table 2.1.
There are 20 BM classes where the worst class is BM
class 1 and the best class is BM class 20. Every new insuree begins in BM class 2 where he
pays the full base premium (100%). After each year without a claim at fault, he advances
in BM scheme by one class up which awards him with an extra discount on the premium.
The maximum discount of 75% is provided in top BM classes 14 to 20. After each claim
at fault, the agent drops in BM scheme by 4 to 6 classes and usually pays higher premium.
The agent's BM class is updated at the beginning of each contract year and depends
on the BM class and the number of claims in the previous contract year. For instance,
an agent in BM class 12 pays 35% of his base premium.
If he has no accident during
the contract year, he will advance into BM class 13 where he will pay 30% of the base
premium. However, if he causes an accident, he will drop into BM class 7 where he will
pay 55% of the base premium.
If he causes 2 accidents, he will drop further into BM
class 3 and will pay 90% of the base premium. Finally, every agent causing three or more
claims in a year will be degraded into BM class 1 which implies a surcharge of 20% on
Nethelrands. Source: Dutch Association of Insurers (www.verzekeraars.nl).
6 Agents who switch from one insurer to the other can ask the old insurer for a statement which states
their claim history, usually in a form of a number of claim-free years they had. Based on this statement
they can apply their right for a premium discount by the new insurer.
7 This scheme is similar to the one proposed by de Wit et al. (1982) who made an extensive actuarial
study of the motor rating structure in the whole Netherlands.
The authors proposed a BM scheme
consisting of 14 classes, with a maximum discount of 70% in the top class 14 and a surcharge of 20% in
the bottom class 1. This scheme was broadly introduced in the Netherlands on January 1, 1982. In the
course of time, extra bonus classes were added, oering better protection against premium increase to
good customers. In our case, the highest BM class 20 gives an agent some kind of malus-deductible in
a way that his premium does not increase after one claim at fault.
13
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
the base premium.
2.3 Data
Our data describe contract and claim histories of a major Dutch car insurer in the period
from 1 January 1995 to 31 December 2000. They provide rich information about agents
(sex, age, address), their cars (brand, model, production year, price, weight, power, etc.)
and contracts (coverage, bonus-malus class, level of deductible, premium, renewal date).
If a claim occurs, we observe its type (accident, theft, windscreen, etc.), damage (material
damage, bodily injury) and whether the agent was guilty or not. The data are longitudinal, describing contract dynamics, like changes in coverage, insured subjects or agents'
addresses, and contract renewals or terminations. Given that this chapter applies static
econometric methods, we will use only cross-sectional data. The dynamic aspect of the
data will be explored in the Chapter 4.
2.3.1
Sample Selection
The raw data contains 163,194 unique contracts. There is, however, no information on
claims in 1995, therefore we excluded this year from the data. From the remaining 142,175
contracts we deleted 1,376 contracts that are not covered by the BM system
contracts that do not have LI.
9
8
and 563
As it will be explained later in this chapter, our approach
requires that all agents have the basic (liability) insurance.
Further we selected the contracts observed for at least one full contract year which is
the period between two contract renewal dates (or the period between the starting date
of the contract and its rst renewal date). By focusing only on fully observed contract
years we avoid problems with attrition.
In the data, there are 111,480 such contracts
8 These are the contracts covering companies' eets of cars. Such contracts have no individual BM
coecients but general eet discounts which are adjusted every year based on the eets' claim histories.
9 Some policyholders insured their car for liability cover at a dierent insurer. For example, Belgian
drivers must insure the liability cover at a Belgian insurer, but can choose an arbitrary insurer for the
comprehensive cover.
14
2.3.
DATA
Table 2.1: Bonus-Malus Scheme
Present Premium Future BM class after a contract year with
BM class paid no claim 1 claim 2 claims 3 or more claims
20
25%
20
14
8
1
19
25%
20
13
7
1
18
25%
19
12
7
1
17
25%
18
11
6
1
16
25%
17
10
6
1
15
25%
16
9
5
1
14
25%
15
8
4
1
13
30%
14
7
3
1
12
35%
13
7
3
1
11
37.5%
12
6
2
1
10
40%
11
6
2
1
9
45%
10
5
1
1
8
50%
9
4
1
1
7
55%
8
3
1
1
6
60%
7
2
1
1
5
70%
6
1
1
1
4
80%
5
1
1
1
3
90%
4
1
1
1
2
100%
3
1
1
1
1
120%
2
1
1
1
15
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
from which the vast majority (97.5%) starts or renews in 1996. We will select only these
contracts which is convenient for our analysis in the Section 2.6 where we condition on
the premium which needs to be calculated for all agents in the same way. By focusing on
the sample of contracts starting in one particular year we avoid problems with eventual
premium increase due to ination.
From the remaining 108,741 contracts we have to
delete 135 ones with erroneous or missing values of weight which is the key risk factor for
LI.
We are left with 108,606 contracts from which 24,101 cover business cars. We noticed
that for these cars the agents' characteristics, like sex and age, are missing because business
cars are usually used by many dierent drivers and therefore their premium cannot be
based on individual characteristics. Furthermore, we discovered that most business cars
are new and have full comprehensive coverage, so there is a lack of variety in this group.
For all these reasons we decided to discard business cars from our sample and concentrate
only on private users. This will allow us to use agents' characteristics in our model and,
more importantly, to distinguish between young and senior drivers.
Our nal sample consists of 84,505 contracts from which 34,251 have full (CASCO)
coverage and 50,254 only basic (LI) coverage.
2.3.2
Claims
A typical issue with using insurance data is that they refer to claims, not accidents.
Whether an accident, once it has occurred, becomes a claim (i.e.
is declared to the
insurance company) is left to the individual's discretion. Obviously, accidents that are not
covered will not be claimed. The estimated (conditional) claim-coverage correlation, based
on all observed claims, will be signicantly positive even in absence of any asymmetric
information fully insured agents claim more damages simply because they have better
coverage.
Therefore, if we want to compare claim occurrences between two types of
agents, ones with and the others without full insurance, we have to take into account only
16
2.3.
DATA
accidents which are covered by the contracts of all agents. Such accidents must involve
third-party damage which is covered by the liability insurance that is obligatory for all
agents. Moreover, by focusing on third-party claims we can avoid (at least partially)
post moral hazard
ex
eect, because accidents involving multiple parties are more likely to
be claimed to the insurer in any case.
Another issue is
insurance fraud
which arises when agents manufacture false claims
with the intent to fraudulently obtain payment from the insurer. The information asymmetry of insurance fraud is usually resolved by claim verication and monitoring.
10
Hard
insurance fraud is a special case of ex post moral hazard in which agents claim fake damages. It is more likely to be prevalent among agents with full coverage who can obtain
payment directly from their insurer by pretending to have incurred damage on their own
car. Staging credibly a fake accident with multiple cars is very dicult if not impossible. Therefore, by focusing on third-party claims we escape (substantially) problems of
eventual insurance fraud.
Lastly, we will focus only on claims at fault because these are directly informative on
agents' risk. Accidents where an agent is not guilty are covered by oender's insurance,
and therefore do not need to be claimed to the insurer.
From now on in this chapter, by a
claim
we will always mean a claim at fault with
a third-party damage. In our sample, we observe 80,790 contracts without a claim and
3,715 contracts with a claim. From those, 3,583 contracts have one claim, 124 contracts
two claims and 8 contracts three claims.
10 Many empirical studies, for example Cummins and Tennyson (1996) or Abrahamse and Carroll (1999),
found an evidence of fraud in automobile insurance markets.
Tennyson and Salsas-Forn (2002) claim
that the vast majority of suspicious claims involve potential buildup (exaggerated loss amounts) rather
than outright fraud (illegitimate claims). Insurance experts share the same experience. This means that
insurance fraud is more likely to distort our analysis on claimed amounts rather than on claim occurrences.
17
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
2.4 Asymmetric Information and Occurrence of Claims
As explained in the introduction, asymmetric information leads to a positive correlation
between coverage and claim costs, conditional on all information available to the insurer.
The claim costs depend on both the probability of an accident and the distribution of
incurred damages. In this section we will explore the st feature by testing for the conditional independence between the choice of better coverage and the occurrence of claim(s).
We will adopt the methods introduced by Chiappori and Salanié (2000), conditioning on
various sets of (relevant) variables observed by the insurer.
2.4.1
Pair of Probits
First we develop a simple parametric test based on two independent probits, one for the
choice of coverage and the other for the occurrence of claim.
Let
y
z
i = 1, . . . , n
denote agents. We dene two 0-1 endogenous variables:
choice of coverage y = 1
occurrence of claim z = 1
:
if agent
i
:
i
i
if agent
bought a full coverage and
i
had a claim at fault, otherwise
These denitions require some remarks. First,
zi
zi
equals zero. Second, the variable
y
if not.
zi = 0.
equals one only if an agent
least one claim at fault with a third party damage. If
blamed guilty,
yi = 0
i
i
had at
had no such claim or he was not
does not distinguish between agents
with only LI and those having also Mini-CASCO. Such distinction is not necessary here
given that Mini-CASCO insurance does not cover damages caused by trac accidents,
which are the only relevant ones for the denition of the variable
z.
Last remark concerns
the level of deductible which can dier among agents with CASCO insurance.
Ideally,
these contracts should be treated separately and not bundled together as we do here.
However, in our model, we concentrate only on liability claims which are not subject to
any deductible. Moreover, in the sample there are only 4% of agents with a non-standard
level of deductible, so there is no need to worry about this issue.
18
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
Now we can set up the two probit models. Let
of an agent
i.
i
and
be the set of observed risk factors
Then
yi = I(Xi β + i > 0)
where
Xi
ηi
and
zi = I(Xi γ + ηi > 0),
are some risk factors that are unobserved by the insurer, but possibly
observed by the agent. We allow these factors to be dependent on the covariates
i = Xi β ∗ + ∗i
assume that such dependence is linear, i.e.
and
ηi∗
are standard normal errors independent of
yi = I(Xi β̃ + ∗i > 0)
where
β̃ = β + β ∗
and
γ̃ = γ + γ ∗
and
Xi .
and
ηi = Xi γ ∗ + ηi∗ ,
Xi ,
but
where
∗i
Then we can write
zi = I(Xi γ̃ + ηi∗ > 0),
can be estimated by standard methods.
We estimate both probits independently by maximum likelihood method and compute
the generalized residuals
ˆ , where ˜∗ (β̃) and η̃ ∗ (β̃) are generalized
ˆ and η̃ˆ∗ = η̃ ∗ (β̃)
˜ˆ∗i = ˜∗i (β̃)
i
i
i
i
errors, dened by Gourieroux, Monfort, Renault, and Trognon (1987). For instance,
˜∗i (β̃)
is given by
˜∗i (β̃) ≡ E[∗i |yi , Xi ] = yi
where
φ
and
Φ
φ(Xi β̃)
φ(Xi β̃)
− (1 − yi )
,
Φ(Xi β̃)
Φ(−Xi β̃)
denote the density and the cumulative distribution function (cdf ) of a
standard normal distribution
N (0, 1).
Gourieroux et al. (1987, Section 3.7) dene a test statistic
P
n ˆ∗ ˆ∗
˜i η̃i
i=1 W =P n
i=1
˜ˆ∗i η̃ˆi∗
2
2 ,
which is, under the null of conditional independence
tributed as a
χ2 (1).
cov(∗i , ηi∗ ) = 0,
asymptotically dis-
This provides us with a test of the symmetric information assumption.
19
CHAPTER 2.
2.4.2
ASYMMETRIC INFORMATION IN CAR INSURANCE
Bivariate Probit
Estimating the two probits independently is appropriate under conditional independence,
but it is inecient under the alternative. Therefore we also estimate a bivariate probit
in which
∗i
coecient
and
%.
ηi∗
are jointly normal with zero mean, unit variance and a correlation
We will estimate this coecient together with its standard error which will
allow us to test the null of
2.4.3
χ2
% = 0.
Test of Independence
The two parametric procedures presented above rely on the functional forms, which are
quite restrictive since the latent models are linear and the errors are normal.
If the
underlying data generating process is driven by more complicated nonlinear functions of
X , our results could be biased in unpredictable ways.
To remedy this issue, Chiappori and
Salanié (2000) adopt a fully nonparametric procedure based on a
They create
M
χ2 test for independence.
groups of agents with similar characteristics observed by the insurer and,
for each group, they do the
χ2
test for independence.
In this way they obtain
M
test
statistics each of which is, under the null of independence, asymptotically distributed as
a
χ2 (1).
There are many ways how to use these data for a test of conditional independence.
One of them is a Kolmogorov-Smirnov test which compares the empirical cdf, calculated
from all obtained
M
test statistics, with its theoretical counterpart, the cdf of a
χ2 (1).
Under conditional independence, both functions should be asymptotically identical. As
the Kolmogorov-Smirnov test is known to have limited power, the authors provide two
other tests. The rst one is based on a sum of all
M
test statistics. Under conditional
independence this sum should be asymptotically distributed as
χ2 (M ).
The last approach
is based on the number of rejections in all cells. We reject the null of independence in a
cell if the test statistic exceeds 3.74, the 5% critical value of the
20
χ2 (1).
Under the null,
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
such rejection appears with a probability 0.05.
Thus, under conditional independence,
the total number of rejections should be distributed as a binomial
2.4.4
B(M, 0.05).
Implementation
Before we start testing we have to decide which variables to use in the model. Optimally,
we should condition on all information available to the insurer. In the data, there are more
than 20 descriptive variables about policyholders and their cars. We can easily include all
of them into
X.
Estimation of the probits will be still feasible and we can be sure that we
are not omitting any information which is known to the insurer. However, there are three
caveats in this procedure. (1) There are around 5% of agents for whom we do not observe
all variables. Consequently, if we want to use all characteristics in the model, we have to
exclude these agents from the sample. (2) Some variables, like for example weight and
engine volume, are highly correlated. The underlying multicollinearity does not reduce
the predictive power or reliability of the model as a whole, but it aects the calculation
of individual predictors.
Since we are not interested in the individual estimates of the
coecients, we do not need to worry so much about this issue here. (3) The most serious
problem with using all information is that we cannot condition on so many variables in
the non-parametric approach, because some groups of similar agents will be very small
or even empty. Here we face a trade-o between better conditioning (and thus a bigger
number of groups with similar agents) and not too small populations in each group (to
be able to apply asymptotic results).
To help us with this issue we asked the insurer
which characteristics are the most important in the determination of the agents' risk. We
were told that, in the period covered by the data, the following factors were used in the
calculation of the liability premium for private cars:
of policyholder.
weight, region, kilometrage
and
age
We will refer to these variables as premium risk factors.
Another possibility is to use directly the liability premium, which is also observed in
the data. It is natural to assume that the insurer's objective is to estimate the agents' risk
21
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
in the best possible way, using all available characteristics. Hence the premium should
truly reect the underlying third-party risk. Therefore, it could be sucient to condition
only on the premium. We will discuss and adopt this approach in Section 2.6.
The premium risk factors and the premium itself represent the information used by
the insurer to evaluate the agent's risk. We will verify whether this information is sufcient by testing the independence between the choice of coverage and the occurrence
of claims, conditioning only on these premium risk factors, resp. only on the premium.
If we reject the null of independence, we can conclude that the risk factors used by the
insurer are insucient to price the underlying risk. Then it is important to verify whether
this information asymmetry would still retain if we used extra risk factors that are still
observed by the insurer, but not used in the pricing. Therefore we will search for other
characteristics of agents, which have also strong predictive power for the claim occurrence.
We will lean on the
actuarial study,
provided to us by the insurer, which was made by
an independent consulting company, using exactly the same data as we do in this thesis.
This study suggests many improvements in the pricing policy, so the insurer's pricing of
the risk is not optimal in a sense that it does not explore all available information. We
will discuss here only the most important suggestions, concerning mainly a replacement
of some currently used premium risk factors by other factors which explain better the underlying risk. We will verify the relevance of these suggestions by estimating the bivariate
probit with the risk factors proposed by the actuarial study. If these risk factors improve
the explicative power of the bivariate probit (compared to the premium risk factors), we
will include them into our analysis. Our aim is to nd a small set of variables with the
strongest explicative power. As a benchmark, we will use our bivariate probit.
2.4.4.1 Actuarial Study
First we estimated the bivariate probit using only the premium risk factors. We found out
that only the coecients of the weight and the dummies for two regions were signicant
22
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
in both probits. This can originate from the fact that the underlying risk does not depend
proportionally on the premium risk factors.
The insurer prices the liability insurance
using a specic structure in which the relevant variables enter in a non-proportional way.
In what follows we will discuss in details all key risk factors used (in the period of our
data) by the insurer and possible improvements as proposed by the given actuarial study.
Region.
The insurer used four-level region code in her tari structure to discriminate
between urban and rural areas. The rst region code relates to the capital city of Amsterdam, the second code to other major cities in the Netherlands, the third code to small
towns, and the last code to countryside. The actuarial study suggests that such classication is not necessary and that it is sucient to distinguish only between big cities and
the rest of the country. In other words, they suggest to bundle rst region together with
the second, and the third region together with the fourth. Our analysis came to the same
conclusion. We made four dummies for each region. We cannot include all four dummies
into the model because then the matrix
dummy, say for region 1 (resp.
(resp.
region 4) is insignicant.
X
would be singular. Each time we exclude one
region 3), the coecient of the corresponding region 2
Therefore we create only one dummy, denoting both
regions 1 and 2, which relates to the use of a car in a
city.
Then its estimated coecient
is signicant in both probits; it is positive in the claim-occurrence probit and negative in
the coverage-choice probit.
Kilometrage.
Another factor used by the insurer in the tari structure was kilometrage.
Each agent entering the insurance was asked to estimate the number of driven kilometers
per year and choose one of the following three levels of kilometrage: below 12,000 km per
year, maximum 20,000 km per year and unlimited number of kilometers. This measure
turned out to be very unreliable because many agents underreported their actual level
of kilometrage, which is seldom checked by the insurer, to get extra discount on their
23
CHAPTER 2.
premium.
11
ASYMMETRIC INFORMATION IN CAR INSURANCE
Not surprisingly, we observe in the data that only 4.7% of the policyholders
have unlimited kilometrage. Furthermore, from the estimated bivariate probit it seems
that there is no signicant dierence between the estimated coecients of dummies for
the lowest and the highest kilometrage level; one of the two is insignicant when used in
the combination with a dummy for the middle kilometrage level.
The actuarial study suggests to use a
fuel type
of car instead of kilometrage.
As
a practical rating factor, the fuel type has an advantage over kilometrage in that it is
objective and veriable from the public vehicle licensing database.
12
Moreover, there is
an evidence in the data that diesel or gas fueled cars have approximately 32% higher
expected third party claims cost than petrol cars (all other factors being equal). This is
common in European motor markets and most probably results from the fact that the fuel
type is a proxy for kilometrage, with diesel- and gas-powered vehicles being more heavily
used. Therefore we decided to distinguish benzine powered cars from the rest. Estimated
coecient of the corresponding dummy is signicant in both probits; it is negative in the
claim-occurrence probit and positive in the coverage-choice probit.
Age of policyholder.
The last characteristic used by the insurer in the rating struc-
ture is the age of policyholder. The insurer discriminates only young drivers by giving
an extra surcharge to all policyholders aged below 28 years.
The actuarial study con-
rms that young policyholders have very high risk, compared to the policyholders aged
between 28 and 39 who represent the lowest risk.
Then the risk starts growing with
11 This happened mainly when the contract was underwritten via an insurer's intermediary who reported
lower kilometrage in order to get a good price for his clients. Kilometrage can be checked only by a claim
expert when surveying a specic claim. If the claim expert gets a proof that the actual kilometrage is
higher than was reported to the insurance company, the cover can be lapsed. However, many agents often
make up a good story why their actual kilometrage is so high. Since the insurer has no means to verify
the credibility of the actual kilometrage, she decided to drop it from her tari structure and use the fuel
type instead. This change took eect in 2002, i.e. two years after the period covered by our data.
12 We know about two databases providing technical data and price information about Dutch cars.
One is the RDW database (www.rdw.nl), freely accessible to public, which contains basic technical data
about all vehicles registered in the Netherlands. The other one RDC database (www.rdc.nl) is more
complex and accessible to car insurance companies against a payment. It contains all technical and price
information about the vehicles registered in the whole Benelux.
24
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
the age again and becomes signicantly higher for the policyholders aged between 50 to
59 probably because of the fact that some of these policyholders have high risk teenage
children starting to drive on their parents' insurance cover.
policyholders aged over 75 have the highest third party risk.
conrms these results.
The study concludes that
Our preliminary analysis
Estimated coecient for the age is individually insignicant in
the claim-occurrence probit when we use whole sample, apparently because there is no
proportional relation between the age of policyholders and the underlying risk. However,
when we estimate the claim-occurrence probit using only subsample of young drivers, the
estimated coecient becomes signicantly negative which could be explained by learning
eect. When we use a subsample of experienced drivers (aged 28 years or more), the estimated coecient for age is positive, though not very signicant. It becomes signicant
when we select a subsample of more senior drivers, aged above 40 years. Thus it seems
that the age of policyholders is an important risk factor.
Age of car.
The actuarial study suggests to use also this factor in rating, since there is
a strong statistical evidence that the underlying third party risk is higher for older cars.
Our preliminary analysis conrms it; estimated coecient of the age of car is signicantly
negative in the coverage-choice probit and signicantly positive in the claim-occurrence
probit. The age of car is evidently a very strong determinant in agent's decision whether to
buy a full insurance or not. Since the premium of the full insurance is based on the value
of car at new, but the car itself is insured against the maximum loss equal to its actual
value (depreciated by time), the full insurance is more advantageous for new cars than
for old cars. Indeed, we observe in the data that mainly new cars have CASCO coverage.
Agents cancel this coverage when their car gets older and usually switch to Mini CASCO.
Most old cars have only the liability cover.
13
For all these reasons we decided to use the
age of car in our tests.
13 Chapter 5 provides more details about the age distribution of cars among dierent types of coverage.
25
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
The actuarial study also suggests to use motor volume as a rating factor, since cars
with large engine have signicantly higher third party risk costs than cars with small
engine. Indeed, in our bivariate probit, the motor volume alone is a signicant predictor,
however, in a combination with the weight of car it is not so signicant any more. This
is probably because of a high correlation between both factors (around 70%).
We get
similar results when using engine power; it is also highly correlated with the previous two
14
variables. We will therefore use only the weight of car in our model.
To recapitulate, based on our bivariate probit, we found out that the following variables are the most relevant risk factors for the liability insurance:
policyholder, age of car
and the indicators for the
use in city
We will refer to these variables as actuarial risk factors.
weight of car, age of
and the fuel type
benzine.
In the bivariate probit, all
estimated coecients of these variables are signicant. Adding extra variables into the
model, like for example sex of policyholder, value of car or type of its body, does not signicantly improve the prediction power of the model; corresponding estimated coecients
are not signicant.
We therefore believe that it is sucient to condition only on these
characteristics.
2.4.4.2 Experience Rating
So far, we have discussed only exogenous characteristics that do not depend on agent's
claim history. We know, however, that the insurer also observes past driving records, which
are highly informative on probabilities of claim and, as such, are used for pricing. Omitting
experience rating from the tests can generate spurious information asymmetry, because
the corresponding information is treated as being private, whereas it is in fact common
to both parties. Indeed, Puelz and Snow (1994) found evidence of adverse selection in car
insurance market when they neglected experience rating. Such omission can generate a
bias that tends to overestimate the level of asymmetric information.
14 In some countries, like for example Slovakia, the key risk factor for the liability insurance is the motor
volume. In the Netherlands, as suggested by de Wit et al. (1982), it is the weight of car.
26
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
This remark clearly suggests that the tests should control for driving experience. Chiappori and Salanié (2000) point out that the introduction of such a variable is problematic
because of its obvious endogeneity. They circumvent this problem by focusing on a subpopulation of beginning drivers for whom no driving history is observed yet. They nd
no evidence of asymmetric information among young drivers, but suspect that such informational asymmetry can arise in the course of insurance relationship due to
learning.
asymmetric
This entails that drivers learn faster about their risk than insurers because they
accumulate information also on near misses and small accidents which they do not report
to the insurer.
Hence, it is interesting to test for presence of asymmetric information
especially among experienced drivers.
Such analysis, however, requires conditioning on
the observed claim history. We will assume that all claim history, observed by the insurer
(and naturally also by the agent) is suciently expressed by the experience rating, namely
the agent's BM class. In the rest of this section we will justify the introduction of this
variable in our model.
If we want to include experience rating in our model, we cannot avoid a discussion
about dynamic aspects of the contractual relationship, namely
learning.15
The learn-
ing can be either symmetric, when both parties learn equally about the agent's risk, or
asymmetric, when one party (usually the agent) learns faster than the other party (the
insurer).
Under the null of
symmetric information,
i.e.
neither adverse selection nor moral
hazard, both the agent and the insurer share the same information about the agent's
risk. If there is no learning, then the experience rating reects just some random shocks
and does not provide extra information about the agent's risk. If there is some learning,
then this learning must be symmetric, since both parties share the same information. In
this case it is important to condition on the experience rating, which reects the general
15 The basic reference on learning is Harris and Holmstrom (1982) who studied a case of symmetric
learning in labor market. Their model was further applied to life insurance by Hendel and Lizzeri (2003).
Learning in car insurance markets was recently studied by Dionne et al. (2006) and Cohen (2005, 2008).
27
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
knowledge (common to both parties) about the agent's risk. Omitting experience rating
can create spurious informational asymmetry.
Under the alternative of
or moral hazard (or both).
asymmetric information,
there is either adverse selection
Under pure moral hazard (without learning), conditioning
on experience rating is not necessary, but does not harm either; it leaves variation in
CASCO coverage that, under moral hazard, should be related to risk. Under pure adverse
selection, the agent either has private information about his risk from the start, or acquires
it through asymmetric learning over the course of time. Either way, the experience rating
will not fully reect the agent's risk. Conditioning on the experience rating may reduce
the magnitude of the information asymmetry, but will not cancel it completely out. In any
case, it is useful to condition on the experience rating because this is directly informative
on the level of the insurer's knowledge about the agent's risk.
From the above reection we can conclude that it is never harmful to condition on
the experience rating. The main reason for conditioning on the experience rating is to
control for the insurer's learning about the agent's risk. Dionne et al. (2006) also include
the BM coecient into their model, claiming that it provides additional information on
the riskiness of policyholders. Their results suggest that the agents in high BM classes
(who receive high bonus discount on their premium) tend to both buy more insurance
and have less accidents.
In what follows we will present the results of the tests separately for young and senior
drivers. It is not necessary to condition on the experience rating for young agents since
they have very little driving experience.
We expect this will be important mainly for
senior drivers.
2.4.4.3 Results for Young Drivers
First we will turn attention to young drivers. As already mentioned, the insurer charges
all policyholders younger than 28 years an extra fee because there is a strong empirical
28
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
evidence that this group represents a higher risk. Therefore we will focus on a subsample
of agents below the age of 28 years. Driving experience of such agents is certainly less
than 10 years given that in the Netherlands, the legal age for obtaining a driving licence
for a car is 18 years. Despite this, we observe a big variety among the BM classes. Some
agents are even in the BM class 20 which they would normally reach after having driven
for 18 years without a claim. This comes from the fact that in the period covered by our
data, the insurer gave new drivers a possibility to inherit a favorable BM coecient from
their parents.
16
Our subsample of young drivers consists of 4,319 individuals which represents 5% of
the whole sample. More than a third of these agents are at the margin age of 27 years.
Another 28% are 26 years old.
Just less than a quarter of agents are younger than 25
years.
First part of Table 2.2 gives the distribution of young agents according to their coverage
and occurrence of claim. From the rst sight we can see that the agents with full coverage
have proportionally less accidents (3.8%) that the agents with only basic insurance (5.5%).
This is somehow surprising. Consequently, the
is negative (-0.029), but insignicant: the
χ2
unconditional
claim-coverage correlation
test of independence does not reject the null
17
at a conventional 5% level.
When we estimate the two independent probits using all available exogenous characteristics,
18
we get a value of the
bivariate probit estimates
%
at
W -statistic
−0.058
equal to
0.842,
with a
with a standard error of
p-value
0.076.
of
0.359.
The
We are far from
rejecting the null.
Conditioning only on the premium risk factors gives
0.031,
and
%̂ = −0.098
with a standard error of
0.049.
W = 4.667
with a
p-value
of
Surprisingly, both tests reject the
16 Nowadays, it is still possible to get the same BM-discount on the second car in a household, but only
under strict conditions, preventing young drivers from starting with a high discount.
17 The rejection of the null is stronger when we exclude 27 year old agents. Then the
test is
p-value
of the
0.239.
χ2
18 There are 228 agents with missing values for some variables, thus the probits are estimated using
only 4,091 observations with full information on all variables.
29
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
null at a 5% level. However, when we exclude the oldest drivers from the sample (27 year
old agents), the null is not rejected any more. Then
and
%̂ = −0.064
with a standard error of
W = 1.216
and
%̂ = −0.021
p-value
of
0.270,
0.062.
Conditioning only on the actuarial risk factors gives
0.562,
with a
with a standard error of
0.070.
W = 0.336
with a
p-value
of
Both models reject the null even
more strongly than when we conditioned on all exogenous characteristics.
In the nonparametric approach we rst group agents based on the premium risk factors.
Beside 4 levels for the region and 3 levels for the kilometrage, we make 2 levels for the
weight: light cars (below 1,000 kg) and heavy cars (above 1,000 kg). In this way we got
24 groups. 3 groups had, however, very few individuals, so we add them to a bigger group
with similar characteristics. We ended up with 21 groups where the smallest one has 11
individuals and the biggest one 1,882 individuals. All three proposed tests are far from
rejecting the null of independence. The Kolmogorov-Smirnov test statistic is
p-value
of
0.938.
All
χ2
test statistics summed up to
the 5% critical value of the
so the
p-value
of
χ2 (21).
B(21, 0.05)
13.468
0.115 with a
which is much below
32.671,
Finally, there was no rejection of the null at any cell,
is 1.
Grouping agents with regard to the actuarial risk factors gives the same result. As
before, we make 2 levels for the weight of car.
For the age of car, we distinguish 3
levels: new (0 - 4 years), as good as new (5 - 9 years) and old (10 years and more). By
conditioning also on the remaining two 0-1 variables (the indicators for a use in city and
a benzine fueled car), we created 23 groups. The smallest one has 12 individuals and the
biggest one 1,070 individuals. Again, none of the three proposed tests rejects the null of
independence. The value of the Kolmogorov-Smirnov test statistic is
of
0.550.
All
χ2
tests statistics summed up to
critical value of the
χ2 (23).
9.859
0.174 with a p-value
which is much below
35.172,
the 5%
Furthermore, there is no rejection of the null in any cell.
All tests gave the same result: conditional (and also unconditional) correlation is not
signicant. This means that there is no asymmetric information between young drivers
30
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
and the insurer. This result is consistent with Chiappori and Salanié (2000), who did not
nd any evidence of asymmetric information among young drivers either.
It seems that beginners have no informational advantage over the insurance company.
Young drivers have very little driving experience to determine whether they will turn out
to be good or bad drivers. Therefore they have no extra (private) information on their
risk when they choose an insurance coverage for the rst time. New policyholders either
randomize their contract choice or distribute across the menu of contracts based on their
preferences, which are uncorrelated to risk. In the course of time, however, agents can
gain some information advantage due to asymmetric learning. Therefore, we will repeat
all tests on the subsample of senior drivers.
2.4.4.4 Results for Senior Drivers
Here we will focus on senior drivers, i.e. on all agents aged 28 years or more. There is no
guarantee that all these agents are experienced drivers because some of them could have
started driving in later age. However, in the data we observe that more than a half of
these agents have been insured at our insurance company for at least 10 years. Thus we
can be sure that majority of agents in our subsample are experienced drivers.
Our subsample of senior drivers consists of 80,186 individuals. Around 28% of them
are aged below 40 years. Less than 5% of agents are above 75 years old. Second part of the
Table 2.2 gives the distribution of senior agents according to their coverage and occurrence
of claim.
We can see again that the agents with full coverage have proportionally less
accidents (3.9%) that the agents with only basic insurance (4.7%). The
rejects the null of (unconditional) independence.
coverage correlation is, however, quite small:
The estimated
χ2
test strongly
unconditional
claim-
−0.017.
The correlation does not change too much when we condition on all exogenous variables.
19
The two independent probits give
W = 12.440
with a
p-value
of
0.000.
The
19 By conditioning on all variables we loose 4,050 observations which have some variables with missing
values. All probits are therefore estimated using only subsample of 76,136 agents for whom all variables
31
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
Table 2.2: Distribution of Agents According to Their Coverage and Occurrence of Claim
YOUNG DRIVERS
Claim
Coverage yes no Total
basic
full
Total
Test of independence:
189
3,271
3,460
(5.5%)
(94.5%)
(100%)
33
826
859
(3.8%)
(96.2%)
(100%)
222
4,097
4,319
(5.1%)
(94.9%)
(100%)
χ2 (1) = 3.707
p-value
with
= 0.054
SENIOR DRIVERS
Claim
Coverage yes no Total
basic
full
Total
Test of independence:
32
2,178
44,616
46,794
(4.7%)
(95.3%)
(100%)
1,315
32,077
33,392
(3.9%)
(96.1%)
(100%)
3,493
76,693
80,186
(4.4%)
(95.6%)
(100%)
χ2 (1) = 24.003
with
p-value
= 0.000
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
bivariate probit estimates
%̂ = −0.046
0.013,
with a standard error of
so the correlation
is in both cases signicant. However, as soon as we add the BM class,
rejected any more:
error of
W = 0.572
with a
p-value
of
0.450
and
20
%̂ = −0.010
the null is not
with a standard
0.014.
When we condition only on the premium risk factors, we get again signicant correlation:
0.010.
a
W = 36.367
with a
p-value
of
0.000
and
%̂ = −0.060
with a standard error of
The correlation stays signicant even after adding the BM class:
p-value of 0.005 and %̂ = −0.029 with a standard error of 0.010.
estimate the probits using
with a
p-value
of
0.112
only
and
with a standard error of
Conditioning on the actuarial risk factors gives
and
%̂ = −0.048 with a standard error of 0.013,
W = 15.613
%̂ = −0.011
with a standard error of
W = 2.530
0.010.
with a
p-value
of
0.000
so the estimated correlation is signicant.
It becomes insignicant when we add the BM class:
and
Interestingly, when we
the BM class, the correlation is insignicant:
%̂ = −0.016
with
This result is surprising
21
and may come from the fact that our model is not well specied.
W = 8.068
W = 0.781
with a
p-value
of
0.377
0.013.
From all these results we can conclude that omitting the experience rating (i.e. the BM
class) from the model creates a spurious correlation between the coverage choice and the
claim occurrence. This result is consistent with Dionne et al. (2006), who also discovered
that the BM class mask a correlation between coverage and accidents.
Further we noticed that the results are very similar when we condition on all observed
variables as when we condition only on the actuarial risk factors, which we thoroughly
selected in the previous section.
This conrms our earlier claim that it is enough to
condition only on these factors.
What is surprising is the fact that by conditioning only on the premium risk factors (i.e.
are observed.
20 Given that BM class is an ordinal variable, we create dummies for each class (except for the rst one)
and add them into
X.
21 Chiappori (2001) points out that any misspecication can lead to a spurious correlation. Parametric
approaches, in particular, are highly vulnerable to this type of aws, especially when they rely upon some
simple, linear form.
33
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
the variables actually used by the insurer in the premium rating), we found a signicantly
negative correlation, even after adding the BM class into the model. It is quite improbable
that the insurer would use such weak variables (with low predictive power) in the premium
rating. Much better explanation is that our parametric model is not exible enough to
capture the non-proportional structure used by the insurer in rating of the underlying risk.
If this is true then our nonparametric approach should overcome this problem. Another
possibility is to use directly the premium calculated by the insurer, which we will do later.
In the non-parametric approach we leaned again on our actuarial study. We distinguish
two groups of senior agents by age: younger (28 - 39 years), having lower risk, and older
(40 and more), having higher risk.
22
As in the previous section, we distinguish 2 levels of
the weight: light cars (below 1,000 kg) and heavy cars (above 1,000 kg).
First, we condition on the premium risk factors, i.e. on the weight (2 levels), the region
(4 levels), the kilometrage (3 levels) and the age of policyholder (2 levels). In this way
we get 48 groups of similar agents. The smallest group has 13 individuals and the biggest
one 16,982 individuals. The value of the Kolmogorov-Smirnov test statistic is
a corresponding
a
p-value
of the
p-value
χ2 (48)
23
rejected in 6 cells
of
0.002.
All 48
χ2
test statistics sum up to
distribution equal to
which gives a
0.001.
84.400
0.266
with
which gives
Finally, the null of independence is
p-value of the B(48, 0.05) equal to 0.032.
We see that all
three tests reject the null of independence. This happened also earlier when we ignored
experience rating. Conditioning on all 20 BM classes is not appropriate, because some
groups would be very small. Therefore we will distinguish only two levels: low BM classes
(1 to 10) and high BM classes (11 to 20). In this way we get 96 groups. Three groups are
too small, having less than 10 individuals, so we merge two of them, which are similar,
together; and we attach the remaining small group to another similar group which is
bigger. We end up with 94 groups; the smallest one has 10 individuals, the biggest one
22 We do not make a special group for the agents older than 75, who have the highest risk, because
there are very few (less than 5% of ) such old agents in the data.
23 All cells where the null was rejected are big, having more than 550 individuals.
asymptotic results of the
34
χ2
tests of independence are reliable.
Therefore the
2.4.
ASYMMETRIC INFORMATION AND OCCURRENCE OF CLAIMS
14,018 individuals. Now, none of the tests rejects the null anymore. The KolmogorovSmirnov test statistic is
0.102
with a
p-value
105.760 which gives a p-value of 0.191.
p-value
of
B(94, 0.05)
equal to
0.697.
of
0.255.
All 94
χ2
test statistics sum up to
Finally, there are only 4 rejections
24
which gives a
We see now that the null is not rejected any more.
The rejection of the null we got earlier in the parametric approach was very probably
caused by a misspecication.
Conditioning on the actuarial risk factors gives similar results. We create 48 groups
based on the weight (2 levels), the use in a city (2 levels), the benzine fueled car (2 levels),
the age of policyholder (2 levels) and the age of car (3 levels). The smallest group has 13
individuals and the biggest one 10,649 individuals. The Kolmogorov-Smirnov test does
not reject the null; the test statistic is
the
with a
p-value
of
0.729.
p-value
of
As in the previous case, the null is rejected in 6 cells which gives a
p-value
of
B(48, 0.05)
equal to
0.032.
test statistics sum up to
68.675
But the other two
which gives a
tests reject the null. All 48
0.027.
χ2
0.095
This time, however, one rejection appears in the smallest
group (with 13 individuals), so we cannot rely on the asymptotic properties of the tests.
When we exclude this group from the analysis, we do not reject the null any more at a
conventional 5% level.
25
Anyway, this issue disappears when we condition also on the BM
class. Again, by distinguishing low and high BM classes, as in the previous case, we get
96 groups. Two similar groups are very small, so we merge them together. The merged
group has still only 7 individuals, but there is no rejection of the null in this group. The
biggest group has 9,006 individuals. Now, none of the tests reject the null any more. The
value of the Kolmogorov-Smirnov test statistic is
of 95
χ2
test statistics values
rejections in the cells
26
100.115
which gives a
0.107
which gives a
p-value
of
with a
p-value
B(95, 0.05)
of
p-value
0.340.
equal to
of
0.221.
The sum
Finally, there are 8
0.103.
Most of the nonparametric tests reject the null when we condition only on exogenous
24 Again, all rejections are in big cells. The smallest cell with a rejection has 397 individuals.
25 Without this group, the sum of the remaining 47 test statistics is 62.716 which gives a p-value of the
χ2 (47)
equal to
0.062.
With 5 rejections only, the
p-value
of the
B(47, 0.05)
is
0.085.
26 All cells with a rejection are big enough. The smallest cell has 146 individuals.
35
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
variables. Once we add the BM class, no test rejects the null any more. These results
are consistent with the ones we obtained in the parametric approach, except for the one
rejection which we got when we used only premium risk factors. As discussed earlier, this
rejection is very likely caused by a misspecication.
To conclude, we did not nd any evidence of the asymmetric information between
agents and the insurer, even among senior drivers.
2.4.4.5 Negative Claim-Coverage Correlation
It is interesting that the estimated claim-coverage correlation is signicantly
negative
when
we do not condition on the BM class. The estimation results from our probits suggest
that the BM class is positively correlated with coverage choice and negatively correlated
with claim probability. This result is supported also by the study of Dionne et al. (2006).
On top of that, the observed claim frequencies from the Table 2.2 also suggest that agents
with full coverage claim
less
than agents with only basic coverage. The theory predicts
opposite result under adverse selection or moral hazard.
One explanation for such relationship could be
advantageous selection.
de Meza and
Webb (2001) postulate that individuals have private information about both their risk
type and their risk aversion.
The advantageous selection appears if more risk-averse
agents have lower risk and buy more insurance. The authors also relate the advantageous
selection to exaggerated optimism and mistaken reluctance to purchase insurance. Those
who are reluctant to purchase insurance are also disinclined to take precautions.
In a context of the car insurance we have another explanation which is based on the
experience rating. Agents who do not claim for many years receive a huge discount on
their premium. Then the full insurance, which is otherwise quite expensive, becomes more
aordable. On the other hand, agents with many claims pay very high premium which
can discourage them from buying the full insurance or can lead to its cancelation, if they
already have one.
36
2.5.
ASYMMETRIC INFORMATION AND INCURRED DAMAGES
In any case, the obtained results suggest that the steep experience rating used by our
insurer is quite eective in ghting against eventual threats of asymmetric information.
First it battles adverse selection by making the full insurance attractive mainly for good
drivers who are oered a considerable discount on the premium. Second, it may reduce
moral hazard eects by giving the insurees proper incentives to drive more carefully. It
would be interesting to better quantify both eects, which is left for future research.
2.5 Asymmetric Information and Incurred Damages
As discussed in the introduction, asymmetric information leads to a positive correlation
between the coverage and the expected claim costs which depend on (1) the probability
of claim occurrence and (2) the distribution of incurred losses in the case a claim has
already occurred. In the previous section we developed the rst aspect by focusing on the
occurrence
of claims. Now we will explore the second aspect by examining the
severity
of
claims. In particular, we would like to gure out whether there is a relation between the
choice of coverage and the distribution of incurred losses.
In case of adverse selection, agents with higher expected losses buy more insurance.
Under moral hazard, agents with full coverage cause more (serious) accidents because of
decreased incentives for safe driving. One could therefore expect that fully covered agents
have not only higher frequency of claims but also claim higher amounts. We will test for
this prediction by comparing the claim sizes of fully covered and basically covered agents.
We expect the former to incur larger losses than the latter.
In this section we will focus only on the agents who had at least one claim.
are 3,715 such agents in our sample.
There
2,367 of them have only basic insurance, while
the remaining 1,348 agents have full insurance. Obviously, each claim involves a certain
(positive) third-party damage. For each contract, we calculate a total sum of all observed
damages and denote it by
L.
These amounts vary a lot, from the lowest total damage
of 50 to the highest total damage of 1,250,000 . An average total damage is 6,205 37
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
for all contracts (with a standard error of 39,677). For fully covered contracts it is 6,714
(with a standard error of 45,748) and for basically covered contracts 5,915 (with a
standard error of 35,769). This already suggests that the agents with full coverage cause
more damages than the agents with only basic coverage. The estimated standard errors
are, however, so big, that the dierence in means is not signicant.
In what follows, we will do more formal statistical tests which take into account also
the observed characteristics of the agents and their cars. Our null hypothesis will be that
the distribution of incurred damages,
agents (y
= 1)
F,
conditional on
as well as basically insured agents (y
1) = F (L|L > 0, X, y = 0).
X,
= 0),
is the same for fully insured
i.e.
H0 : F (L|L > 0, X, y =
We will test the null against a one-sided alternative that
fully-covered agents cause more damage than basically-covered agents, i.e.
0, X, y = 1) < F (L|L > 0, X, y = 0).
dominance
H1 : F (L|L >
In other words, we will test for the
stochastic
of the damages caused by the agents with full insurance. As earlier, we will
use two approaches, one parametric and the other nonparametric.
2.5.1
Classical Regression
First approach relies on a simple regression estimate of the coecient for the indicator
of better coverage,
y,
conditioning on all (important) characteristics. We specify a fully
parametric model in the following way:
Li = Xi β + αyi + ui ,
where
agent
ui
i
is some zero mean error and
Li
is the total third-party damage caused by an
during the whole contract year under consideration. We would like to stress here
again that we focus only on realized damages, so we ignore agents who have no claims;
therefore
i,
Li > 0.
As before,
Xi
denotes all (relevant) observed characteristics of an agent
including a dummy for his BM class.
38
2.5.
ASYMMETRIC INFORMATION AND INCURRED DAMAGES
Under the null we expect
α=0
parameters of the model by OLS
27
and under the alternative
α > 0.
and test for the signicance of
We can estimate all
α̂ > 0.
As earlier we
will use three groups of characteristics: (1) all characteristics, (2) premium risk factors
and (3) actuarial risk factors.
28
By using all characteristics,
standard error of
2, 023.38.
a standard error of
standard error of
α
equal to
Premium risk factors give similar result:
1, 425.49.
1, 849.01.
we get the estimate of
Finally, actuarial risk factors produce
All estimates of
α
842.62
with a huge
α̂ = 1, 377.13
α̂ = 815.81
with
with a
are insignicant, so we cannot reject the
null that the incurred damages do not depend on the type of coverage.
2.5.2
Nonparametric Tests
Since any parametric approach involves a risk of misspecication, we will develop also
a fully nonparametric method.
As in the previous section, we will make
M
groups of
agents with similar characteristics, and within each group we will test for the null of
independence, conditionally on
X.
First test we will do is the Kolmogorov-Smirnov test which compares empirical distributions of the total incurred damages,
with full coverage (y
= 1)
F̂ (L|L > 0, X, y),
for both types of agents, ones
and the others with only basic coverage (y
rejects the null in favor of the alternative if
F̂ (L|L > 0, X, y = 1)
= 0).
The test
lies signicantly below
F̂ (L|L > 0, X, y = 0).
As an alternative, we will also do the Wilcoxon rank-sum test which compares the
sums of ranks of incurred damages in the whole sample between the agents with full
insurance and the agents with only basic insurance. The test rejects the null in favor of
the alternative if the sum of ranks for the fully covered agents is signicantly bigger than
27 It is quite possible that, under the alternative,
yi
is positively correlated with the error. This kind of
endogeneity is not a problem here, because we are not interested in any causality. We just want to test
whether agents with full coverage cause more damage
relative
to agents with only basic coverage.
28 228 contracts have missing values for some characteristics, therefore the model is estimated using
only 3,487 contracts.
39
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
the sum of ranks for the basically covered agents.
While executing both tests independently, we will register in each cell whether the null
was rejected at a 5% level or not. Then, for each test, we will count the total number
of rejections. Given that, under the null, a rejection appears with a probability of 5%,
the total number of rejections (for each test separately) should be distributed as binomial
B(M, 0.05).
We will globally reject the null in favor of the alternative if the number of
rejections is much bigger than
M/20,
its expected value under the null.
We will do one more test which is based on simple comparison of average damages
between both groups of agents. Under the null, there is a 50% probability in each cell that
the average damage of one group is bigger than the average damage of the other group.
Under the alternative, it is more probable that the average damage of the fully covered
agents is bigger than the average damage of the basically covered agents. For each group
we will register whether the latter is true, and then we will count the total number of such
cases. Under the null, this number should be distributed as binomial
reject the null if this number is much bigger than
M/2,
B(M, 0.5).
We will
which is its expected value under
the null.
2.5.2.1 Implementation and Results
Before we start implementing the nonparametric method, we have to choose the characteristics, based on which we will group the (similar) agents. Here we have to be especially
careful with a choice of the appropriate characteristics because now our subsample of contracts with a claim is much smaller than the whole sample, which we used in the previous
section. Again, we are facing a trade-o between better conditioning (thus bigger number
of cells) and not too small cells (to be able to apply asymptotic properties of the tests).
As earlier, we make two levels for the weight: light cars (below 1,000 kg) and heavy
cars (above 1,000 kg). Then we make three levels for the age: young drivers (below 28
years), experienced drivers (28 to 49 years) and older drivers (50 and more).
40
First we
2.5.
ASYMMETRIC INFORMATION AND INCURRED DAMAGES
grouped the agents based on the premium risk factors, i.e. beside the age and the weight
we also conditioned on the region (4 levels) and the kilometrage (3 levels). We observed,
however, that many cells had only couple of individuals which was inconvenient. Then
we tried to condition on all actuarial risk factors. Again we got many small cells which
prevented us from doing the tests. Therefore we decided to do a compromise between the
premium risk factors and the actuarial risk factors. We replaced two premium factors by
the actuarial ones, namely the four-level region code by the two-level indicator of the use
in city and the three-level kilometrage by the two-level indicator for the benzine-fueled
cars. Now the situation is much better, however, there are still couple of very small cells,
all of them concerning young drivers. We noticed that there are only 222 young drivers in
our subsample. Therefore we decided to discriminate between young drivers only based
on the weight of their car which is, naturally, the most important determinant of incurred
damages (heavier cars cause more damage than light cars). In this way we get 2 groups
for young drivers and 16 groups for senior drivers, so in total 18 groups. The smallest
group has 19 individuals and the biggest group 648 individuals.
We observe that only
in 7 groups, the average damage of the fully insured agents is bigger than the average
damage of the basically insured agents. Furthermore, both tests reject the null only in
29
one (dierent) cell
which gives a
p-value
of
B(18, 0.05)
equal to
0.603.
We are very far
from rejecting the null.
When we condition also on the BM class, distinguishing, as earlier, low (1 - 10) and
high (11 - 20) BM classes, we get 36 groups. We merge 2 small groups, having less than 10
individuals, with other similar groups. In this way we get 34 groups, the smallest one with
19 individuals and the biggest one with 487 individuals. During the testing procedure, we
have to exclude one more group (with 26 individuals) because it includes only agents with
basic insurance. In the remaining 33 groups we nd only 12 groups where the average
damage of the fully insured agents is bigger than the average damage of the basically
29 The Kolmogorov-Smirnov test rejects the null in a cell consisting of 44 agents and the Wilcoxon
rank-sum test rejects the null in a cell with 28 agents.
41
CHAPTER 2.
insured agents.
ASYMMETRIC INFORMATION IN CAR INSURANCE
Furthermore, the Kolmogorov-Smirnov test does not reject the null in
any cell, while the Wilcoxon rank-sum test does it only in one cell (with 115 individuals).
All these results suggest that there is no signicant dierence in the incurred damages
between the agents with full insurance and the agents with only basic insurance.
2.6 Premium
In the previous two sections we tested for the asymmetric information on agent's risk. One
of the practical issues we had to solve was a proper selection of explanatory variables which
we used for conditioning. We had to do so mainly in the nonparametric approach where
conditioning on all characteristics was not possible due to the curse of dimensionality.
Therefore we decided rst to use the premium risk factors which are the characteristics
used by the insurer in premium pricing.
These characteristics performed well in the
nonparametric part, where the null of independence was not rejected when we controlled
for the experience rating.
However, we encountered some problems in the parametric
approach, where the estimated conditional correlation was signicantly negative even
after adding the BM class into
the parametric model.
X.
Such result might be caused by a misspecication of
Therefore we undertook the actuarial study, in which we chose
a group of variables with high explanatory power.
By conditioning on these actuarial
risk factors and the BM class, we obtained an insignicant estimate of the claim-coverage
correlation.
Now we would like to take a dierent approach by using the premium instead of all
(important) risk factors. Insurers usually use more complicated pricing structure, in which
the premium risk factors enter in a nonproportional way. Such structure is periodically
reviewed and updated, so that the calculated premium reects the underlying risk in the
best possible way. Moreover, the premium also represents the insurer's knowledge about
the agent's risk. The latter determines not only the expected claim costs, but also the
agent's decision whether to buy a full insurance or not. This suggests that, when testing
42
2.6.
PREMIUM
for asymmetric information, it could be sucient to condition only on the premium that
incorporates all the relevant insurer's information about the agent's risk.
The possibility to condition only on one variable the premium accords certain advantages. The rst one is that parametric models can be specied in much more exible
way with one variable than with multiple variables, which may involve some mutual interactions or cross-eects. The second advantage is that conditioning on one variable gives
a huge space for the application of nonparametric methods because of the low dimensionality. In this section we will prot from both these advantages. First we will repeat the
Section 2.4's tests for asymmetric information in claim occurrences, conditioning only on
the premium. Then we will develop a fully nonparametric method to test for asymmetric
information in incurred damages.
Before we start with the tests, let us rst clarify our idea in more details with some
empirical support from the data.
2.6.1
Liability Premium and Expected Damage
If we want to condition on a premium we have to use the premium which is observed
for all agents and calculated in the same way for the agents with full insurance as well
as for the agents with only basic insurance. Such premium is naturally the premium for
the liability cover, which is obligatory for all drivers.
Moreover, the liability premium
is directly informative on the third-party risk that is highly relevant for the third-party
claims, on which we focus in this chapter.
In the data, we observe for each agent the yearly
using only the exogenous premium risk factors. The
base premium
which is calculated
actually paid premium
depends on
the agent's BM class as given in the Table 2.1. Our previous results revealed that omitting
experience rating from the model causes a spurious informational asymmetry. Therefore
in our further analysis we will focus only on the actually paid liability premium, which
includes also the BM discount, resp. surcharge. From now on in this chapter we will call
43
CHAPTER 2.
it simply
premium
ASYMMETRIC INFORMATION IN CAR INSURANCE
and denote it by
q.
A natural conjecture is that the premium reects expected underlying risk costs plus
some insurer's overheads, i.e.
q(X) = h(E[L|X]),
where
E[L|X]
is the expected damage caused by an agent with characteristics
X,
and
h
is some strictly increasing function whose graph lies above the diagonal (since the insurer
has to cover loading costs and make some prot). We can verify the validity of the above
formula empirically by smoothing a scatter plot of all observed damages (including also
zero damages) against all paid premia. In Figure 2.1, we display the scatter plot of the
actually paid liability premia against the observed third-party damages, smoothed by
Lowess method with bandwidth 0.8.
We can see that the smoothed line, representing
average incurred damage, is indeed strictly increasing with the premium and lies slightly
below the diagonal, which is what we expected.
The above formula allows us to write
E[L|X] = h−1 (q(X)) = E[L|q(X)],
premium is a sucient statistic for the expected damage.
thus the
The latter can be further
expressed as a product of the probability of a claim and the expected size of
incurred
loss
once a claim has occurred:
E[L|q(X)] = Pr[L > 0|q(X)] · E[L|L > 0, q(X)].
Further we assume that the premium is a sucient statistic for both the probability
of a claim and the expected size of incurred losses. This means that conditioning on the
premium is the same as conditioning on all risk factors, i.e.
and
E[L|L > 0, q(X)] = E[L|L > 0, X].
Pr[L > 0|q(X)] = Pr[L > 0|X]
This assumption is quite strong and might not be
valid in some special cases when the eects of the premium on the probability of a claim
and the expected size of incurred loses work in opposite directions. Imagine, for example,
two drivers with the same expected claim costs (i.e. paying the same premium), but one
44
2.6.
PREMIUM
has a light car and high probability of claim, and the other one has a heavy car and low
probability of claim. The rst agent causes many small accidents while the second one
few severe accidents. In product, their expected damage is the same, but the eect of the
premium on the probability of claim and the expected size of incurred loss is reversed.
Anyhow, we believe that our assumption is in general true. We have a strong empirical
evidence that the premium is a good predictor for the probability of claim; its coecient
in the claim-occurrence probit is signicantly positive, suggesting that agents paying high
premium cause indeed more accidents.
On the other hand, agents with high premium
incur also larger losses. When we regress the size of incurred losses on the premium, its
coecient is positive, though its signicance is only at the 10% level.
In what follows we will use our assumption to test for asymmetric information in claim
occurrences and claim severities, by conditioning only on the premium. The possibility
to condition only on one variable considerably simplies our previous tests, where we had
to battle with the curse of dimensionality (in the nonparametric part) and worry about
the misspecication (in the parametric part). Furthermore, due to low dimensionality we
will be able to develop new fully nonparametric methods.
2.6.2
Test for Asymmetric Information Based on Claim Frequency
As argued earlier, the premium is highly informative on the agent's risk, at least from
the insurer's perspective. The risk in turn inuences the decision whether to buy a full
coverage or not, and naturally determines the probability of an accident. Therefore, when
testing for the asymmetric information between the choice of coverage and the occurrence
of claim, it should be sucient to condition only on the premium.
This dramatically
simplies the implementation of the tests from Section 2.4. Let us briey summarize the
output of these tests when using only the premium.
Estimation of the two independent probits with the premium as the only covariate,
gives
W = 2.374
with a
p-value
of
0.123.
The bivariate probit estimates
%̂ = −0.016
45
with
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
Figure 2.1: Damage by Premium for All Agents
Note: Observed damages above 4,000 are not displayed in the graph, but were taken into account when
smoothing.
46
2.6.
a standard error of
0.010.
PREMIUM
In both cases, the estimated correlation is insignicant. This
result again conrms our intuition that the rejection of the null, we got in the previous
section when using all premium risk factors together with the BM class, was caused by a
misspecication. The premium we use here is a function of exactly the same factors, but
it depends on these factors apparently in a nonlinear way (otherwise we would get similar
results whether using the premium or the premium risk factors).
Using the nonparametric approach we group all agents into 20 cells, based on their
premium. The rst cell groups the agents who pay the lowest premium, below 100 a
year. The second cell groups agents with the premium between 100 and 199 , and so on
up to the group 19 where the agents pay a premium between 1,800 and 1,899 . The last
cell groups the agents whose premium is above 1,900 guilders. The smallest group has
74 individuals and the biggest one 25,156 individuals. None of the tests reject the null
of independence. The Kolmogorov-Smirnov test statistic is
All 20 test statistics sum up to
equal to
0.751.
15.443
which gives a
0.129
p-value
with a
of the
p-value
χ2 (20)
of
0.842.
distribution
Finally, there is only one rejection of the null in the group 18 (with 123
individuals), which gives the
p-value
of the
B(20, 0.05)
distribution equal to
0.642.
We found no evidence of asymmetric information when we conditioned only on the
premium. It should be perhaps emphasized again how important it is to control for the
experience rating. When we repeated all these tests using only the base premium, which
does not include the BM discount (resp. surcharge), the null of independence was strongly
rejected.
2.6.3
Test for Asymmetric Information Based on Claim Severity
In the Section 2.5 we tested whether fully insured agents cause more damage than basically
insured agents and we found no evidence of it. Our suspicion, however, remains, mainly
after we juxtaposed incurred damages with paid premia.
Figure 2.2 displays a scatter
plot of the incurred damages against the premia, smoothed again by Lowess method with
47
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
bandwidth 0.8. At rst glance we can see that the line representing smoothed damage
caused by fully insured agents lies
above
the line representing smoothed damage caused
by basically covered agents, at least at the tails.
In what follows we will test whether
this dierence is signicant. As discussed earlier, we will condition only on the premium,
which allows us to apply a new fully nonparametric method, introduced by Koul and
Schick (1997).
2.6.3.1 Model
Let
E[L|L > 0, q] = µ(q)
pays the premium
q.
denotes the expected size of incurred damage for an agent who
We want to test whether this depends on the agent's type of coverage.
Therefore we denote by
µy (q) = E[L|L > 0, q, y]
an agent with coverage
y.
Recall that
coverage. Our null hypothesis is that
µ
y =0
the expected size of incurred damage for
y =1
refers to basic coverage and
is independent of
y,
i.e.
H0 : µ0 = µ1 .
to full
Under the
alternative we suppose that the agents with full coverage cause more damage than the
agents with only basic coverage, i.e.
for all values of
q
H1 : µ1 > µ0 ,
where
with strict inequality for at least one
q.
µ1 > µ 0
means
µ1 (q) ≥ µ0 (q)
Note that these hypotheses are,
with the next paragraph's independence assumptions, equivalent to the ones we dened
in Section 2.5.
Our observations consist of bivariate data
is the premium of an agent
i
with coverage
y
(qy,i , Ly,i ), i = 0, . . . , ny , y = 0, 1
who caused a damage of size
qy,i
where
Ly,i > 0.
We
assume that the following relations are satised:
Ly,i = µy (qy,i ) + εy,i ,
where the errors
f.
i = 1, . . . , ny ,
y = 0, 1,
εy,i are all mutually independent and identically distributed with a density
The covariates
qy,i
are also all mutually independent and have a common density
g.
In
30
addition, we assume that the covariates are independent of the errors.
30 This assumption seems to be innocuous despite the fact that the incurred damages are always positive.
48
2.6.
PREMIUM
Figure 2.2: Incurred Damage by Premium for Fully and Basically Insured Agents
Note: Observed damages above 60,000 are not displayed in the graph, but were taken into account
when smoothing.
49
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
Koul and Schick (1997) propose two tests for a common design, when
two tests for a distinct design, when
n0 6= n1 .
n0 = n1 ,
and
Given that, in our case, the sizes of the two
subsamples are dierent, we will focus only on the tests suitable for the distinct design.
The Appendix 2.A provides some technical details.
2.6.3.2 First Test
The rst test requires a kernel regression estimate of
µ,
based on the pooled sample:
Pny
i=1 Ly,i wa (qy,i − q)
y=0
,
P1 Pny
i=1 wa (qy,i − q)
y=0
P1
µ̂(q) =
where
wa
31
is some kernel function
with a bandwidth
a.
q ∈ R,
Throughout this chapter we will
32
use the Epanechnikov kernel which has favorable theoretical properties.
Set
ε̂y,i = Ly,i − µ̂(qy,i ),
so that
ε̂y,i
mimics
εy,i
i = 1, . . . , ny ,
y = 0, 1,
under the null hypothesis. Koul and Schick (1997) propose the
following test which rejects the null hypothesis for large values of the test statistic
r
T2 =
where
ψ
n0 n1
n0 + n1
Pn1
ψ(ε̂1,i )
−
n1
i=1
is a nondecreasing measurable function.
Pn0
ψ(ε̂0,i )
n0
i=1
,
If the error density
zero mean, the authors suggest to use the specication
ψ(ε) = ε
f
is normal with
which makes the test
locally asymptotically most powerful.
The support of the errors could be limited for negative values if the expected size of incurred damage
E(L|L > 0, q)
the covariates.
gets close to zero for some premium
q.
Then the errors would not be independent of
Such pattern is, however, not observed in the data.
From Figure 2.2 we can see that
the incurred damages are in average far from zero for all premia. The situation would be dierent if we
modeled just the expected damage
E(L|q) which gets indeed close to zero for small values of the premium;
see Figure 2.1.
31 More precisely,
which is positive
wa (x) = a1 w xa , x ∈ R, a > 0, where w
on (−1, 1) and vanishes o (−1, 1).
is a symmetric Lipschitz continuous density
32 Epanechnikov kernel is optimal in the sense of minimization of the asymptotic mean integrated
squared error (Jones and Wand, 1995, Section 2.7).
50
2.6.
The test statistic
T2 basically compares ψ -averages of the errors in the two subsamples.
Under the null, the errors in both subsamples should be equal, therefore
to zero.
statistic
Under the alternative
T2
PREMIUM
µ1 > µ0 ,
the errors
ε1
T2 should be close
ε0 ,
should be bigger than
so the
should be positive.
The authors prove that under the null and some mild additional assumptions on
f, g, µ, ψ
and
a
(see Appendix 2.A.1 for details),
mean and a variance
τ2
T2
is asymptotically normal with zero
which can be consistently estimated by
ny
1 X
X
1
ψ 2 (ε̂y,i ) −
τ̂2 =
n0 + n1 y=0 i=1
We estimated the function
µ(q)
!2
ny
1 X
X
1
ψ(ε̂y,i ) .
n0 + n1 y=0 i=1
by the Nadaraya-Watson estimator and looked for a
suitable bandwidth by the cross-validation method. We found an optimal value for the
bandwidth at 1,297. We used the suggested specication of
which produced
T2 = 26, 790.3
the approximated
p-value
of
ψ
for normal errors,
with an estimated standard error of
0.250,
√
ψ(ε) = sign(ε) ε.
The rst one is concave for negative
ε
ψ(ε) = sign(ε)ε2
ε<0
so it makes big errors smaller. The rst specication produced
with an estimated standard error of
The second specication gave
the approximated
p-value
of
3.889 · 1010
and
and convex for positive
so it makes big errors even bigger. The second one is convex for
ε > 0,
This gives
so we do not reject the null.
As a robustness check we also tried two other specications: (1)
(2)
39, 662.7.
ψ(ε) = ε,
and the approximated
ε,
and concave for
T2 = 2.377 · 1010
p-value
of
0.271.
T2 = 39.988 with an estimated standard error of 72.992 and
0.292.
In any case we cannot reject the null.
51
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
2.6.3.3 Second Test
The second test proposed by Koul and Schick (1997) rejects the null hypothesis for large
values of the test statistic
r
T4 =
where
ρ
n1
n0 X
n0 n1
1 X
ρ(L1,j − L0,i )wa (q1,j − q0,i ),
n0 + n1 n0 n1 i=1 j=1
is a measurable odd function.
suggest to use the specication
If the error density
ρ(x) = x
f
is normal, the authors
which makes the test locally asymptotically
most powerful.
Note that this test does not require estimation of the common mean function
µ.
It
just averages weighted dierences between the damages incurred by the fully insured
agents (L1 ) and the basically insured agents (L0 ) who pay similar premium
null, these damages should be roughly equal, so
alternative
L1 > L0 ,
T4
q.
Under the
should be close to zero. Under the
the test statistic should be signicantly positive.
The authors prove that under the null and some mild additional assumptions on
f, g, µ, ρ
and
a
(see Appendix 2.A.2 for details),
mean and a variance
τ4
τ̂4 =
T4
is asymptotically normal with zero
which can be consistently estimated by
1
2
!
n1
n0
X
X
1
1
2
2
Ūj,· − Ū·,· +
Ū·,i − Ū·,·
,
n1 j=1
n0 i=1
where
Uj,i = ρ(Y1,j − Y0,i )wa (X1,j − X0,i ),
Ūj,·
n0
1 X
Uj,i
=
n0 i=1
n0 X
n1
1 X
Uj,i ,
Ū·,· =
n0 n1 i=1 j=1
n1
1 X
and Ū·,i =
Uj,i .
n1 j=1
Since we do not know an optimal value for the bandwidth
a, we will try a wide range of
suitable values, say from 1 to 2,000. This will allow us to see how the estimated
52
p-value of
2.6.
PREMIUM
the test is sensitive to dierent values of bandwidth. Furthermore, to be able to determine
robustly a rejection level of the test, we will need to know what is the minimum of all
estimated
p-values.
First we tried the suggested specication
normal.
We observe that the estimated
ρ(x) = x
p-value
which is optimal if the errors are
of the test is quite unstable for small
values of bandwidth. This happens probably because observations with unique values of
the premium are not taken into account if bandwidth is too small. Therefore the size of
bandwidth should be reasonably big. Indeed, the estimated
p-value stabilizes around 0.25
for bandwidths bigger than 20. It reaches its minimum of 0.220 at the bandwidth of size
300. Based on this result, we cannot reject the null.
As a robustness check, we tried two dierent specications for the function
ρ(x) = sign(x)x2
ψ
and
√
ρ(x) = sign(x) x.
in the previous test.
ρ,
namely
We used the same specications for the function
As earlier, estimated
p-values
bandwidth, but get stable for bandwidths above 20.
are unstable for small values of
The minimum
p-value
is reached
again for a bandwidth around 300. It is 0.124 for the rst specication and 0.493 for the
second one. In any case we cannot reject the null.
Finally, because the observed damages have a huge variance (see the beginning of
the Section 2.5), as a robustness check, we repeated both tests,
transformed damages,
log(L).
T2
and
T4 ,
using log-
This should make the errors more homoscedastic. The null
was again not rejected.
Our new nonparametric approach delivers the same result as the tests from the Section
2.5. The null, that the expected size of incurred damage does not depend on the type of
coverage, is not rejected. Looking back to the Figure 2.2 we remark that the two damage
lines visibly deviate only at the right tale where very few data are observed. Most of the
observations (above 90%) have the premium below 1,000 , where the two lines almost
coincide.
53
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
2.7 Conclusion
We did not nd any evidence of asymmetric information between agents and the insurer.
When controlling for all relevant characteristics, the choice of coverage has no inuence
on the occurrence of claims, neither on the size of incurred damages.
These results should be, however, interpreted with care. de Meza and Webb (2001)
pointed out that the conditional-correlation approach may fail to detect asymmetric information in the presence of both selection on
risk preferences
and moral hazard. Notably, if
more risk averse drivers tend to buy more insurance and drive more cautiously, the correlation between the coverage and the occurrence of claims can be even negative. Moreover,
our analysis uses only third-party claims. If asymmetric information is particularly strong
for accidents involving only one car, we cannot detect it.
In Chapter 4 we will explore dynamic features of the data, using all observed claims at
fault together with their sizes and timing. We will develop dynamic econometric methods
to test for the presence of moral hazard. Since dynamic methods exploit more information from the data, they are in general more powerful in detecting traces of asymmetric
information.
54
2.A.
TECHNICAL DETAILS FOR TESTS FROM SUBSECTION 2.6.3
APPENDIX TO CHAPTER 2
2.A Technical Details for Tests from Subsection 2.6.3
In the subsection 2.6.3, we used two tests, introduced by Koul and Schick (1997), based on
the statistics
T2
under the null
and
T4 .
The authors derive asymptotic properties of these test statistics
H0 : µ0 = µ1 = µ.
Using the same notation as in the subsection 2.6.3, we
assume that
• µ
is a measurable function,
• q, ε
and
ε0
are independent random variables with respective densities
is a measurable function such that
• ψ
is a nondecreasing measurable function such that
• ρ
is an odd measurable function such that
such a way that
2.A.1
n0
and
n1
and
f,
and
0 < Ev 2 (q) < ∞,
• v
For the asymptotics, we let
v≥0
g, f
0 < var(ψ(ε)) < ∞,
and
0 < Eρ2 (ε − ε0 ) < ∞.
tend to innity and let
a
depend on
n0
and
n1
in
a → 0.
Asymptotic Properties of the Test Statistic
T2
Koul and Schick (1997, Theorem 2.1): Suppose additionally that
1.
f
has zero mean and nite variance,
2.
g
is bounded and bounded away from 0 on a closed interval
3.
µ
is continuous,
4.
ψ
is Lipschitz-continuous, and
5.
(n0 + n1 )a2 → ∞.
I,
and vanishes o
55
I,
CHAPTER 2.
ASYMMETRIC INFORMATION IN CAR INSURANCE
Then
r
T2 =
n1
n0
1 X
1 X
v(q1,i )ψ(ε̂1,i ) −
v(q0,i )ψ(ε̂0,i )
n1 i=1
n0 i=1
n0 n1
n0 + n1
is asymptotically normal with zero mean and variance
!
τ2 = var(v(q)ψ(ε))
which can be
consistently estimated by
ny
1 X
X
1
τ̂2 =
v 2 (qy,i )ψ 2 (ε̂y,i ) −
n0 + n1 y=0 i=1
!2
ny
1 X
X
1
v(qy,i )ψ(ε̂y,i ) .
n0 + n1 y=0 i=1
The authors study local asymptotic power of the test and recommend to choose
if
g
is unknown and
ψ(ε) = ε
if
f
exact distribution of the premia,
v≡1
is normal with zero mean. Since we do not know the
g,
we took
v ≡ 1.
What concerns the function
ψ,
we
tried the specication recommended for normal errors and two other specications as a
robustness check.
2.A.2
Asymptotic Properties of the Test Statistic
T4
Let
Z
r(t) = Eρ(t − ε) =
ρ(t − x)f (x)dx, t ∈ R,
Z
2
R(t) = Eρ (t + ε − ε0 ) = ρ2 (t + x1 − x2 )f (x1 )f (x2 )dx1 dx2 ,
1
(v(q) + v(q + s))r(t + µ(q) − µ(q + s))g(q + s),
2
hs (q, t) =
Koul and Schick (1997, Theorem 2.4): Suppose that
Z
sup
n0 a → ∞
and
for some
η>0
n1 a → ∞,
2.1 and
Z Z
lim
s→0
56
q, s, t ∈ R.
(v(q) + v(q + s))2 R(µ(q) − µ(q + s))g(q + s)g(q)dq < ∞
|s|<η
t ∈ R,
|hs (q, t) − h0 (q, t)|2 f (t)g(q)dqdt = 0.
2.2 2.A.
TECHNICAL DETAILS FOR TESTS FROM SUBSECTION 2.6.3
Then
r
T4 =
n1
n0 X
1
n0 n1
1 X
(v(q1,j + v(q0,i ))ρ(L1,j − L0,i )wa (q1,j − q0,i )
n0 + n1 n0 n1 i=1 j=1 2
is asymptotically normal with zero mean and variance
τ4 = Ev 2 (q)g 2 (q)r2 (ε)
which can
be consistently estimated by
1
τ̂4 =
2
!
n1
n0
2
2
1 X
1 X
Ūj,· − Ū·,· +
Ū·,i − Ū·,·
,
n1 j=1
n0 i=1
where
1
(v(q1,j ) + v(q0,i ))ρ(Y1,j − Y0,i )wa (X1,j − X0,i ),
2
n0
n1
1 X
1 X
=
Uj,i ,
Ū·,i =
Uj,i
and
n0 i=1
n1 j=1
Uj,i =
Ūj,·
Ū·,·
n0 X
n1
1 X
Uj,i .
=
n0 n1 i=1 j=1
The authors claim (in Remark 2.5) that if
r
and
R
v
and
g
are bounded,
v
is continuous, and
are Lipschitz-continuous, then (2.1) and (2.2) are implied by
Z
lim
s→0
|µ(q + s) − µ(q)|2 g(q)dq = 0.
This condition is satised, for example, when
µ
is continuous.
The authors also study local asymptotic power of the test and recommend to choose
v ≡ 1
if
g
is unknown and
ρ(ε) = ε
if
f
be nondecreasing and Lipschitz-continuous.
the function
ρ,
is a normal density.
As before, we took
Otherwise,
v ≡ 1
ρ
should
and, concerning
we tried the specication recommended for normal errors and two other
specications as a robustness check.
57
3
State Dependence
3.1 Introduction
Distinguishing state dependence from heterogeneity in renewal data is of major substantial
interest in economics, but hard (Heckman, 1991). Standard techniques for linear panel
data with xed eects, which exploit within-subject variation and dynamic instruments,
do not readily apply to renewal, or panel duration, data. Problems arise for two reasons.
First, renewal models are inherently nonlinear. Second, only a selection of renewal events
can usually be observed in a nite observational period. This chapter studies identiability
of and testing for state dependence in renewal models with censored data.
They are many examples of substantial economic problems that can be reduced to the
analysis of state dependence in renewal data. For example, Abbring, Chiappori, Heckman,
and Pinquet (2003) relate state dependence of claim intensities in car insurance to moral
hazard. They argue that, under the existing experience-rating schemes, incentives change
with each insurance claim.
Consequently, under moral hazard and given unobserved
determinants, claim intensities depend on the occurrence of past claims.
This chapter's analysis builds on the pioneering work on state dependence in eventhistory data by Bates and Neyman (1952), Heckman (1981), and Heckman and Borjas
59
CHAPTER 3.
STATE DEPENDENCE
(1980). We rst explore the literature on the identication of event-history models with
state dependence and heterogeneity initiated by Elbers and Ridder (1982) and Heckman
and Singer (1984) and reviewed by Van den Berg (2001). We point out that censoring may
substantially invalidate existing results for the identication of panel duration models and
provide some constructive identication results for this censored case.
Subsequently, we consider testing for state dependence with panel duration data. We
focus on a set of (paired) rank tests that were inspired by Holt and Prentice's (1974)
and Chamberlain's (1985) partial likelihood methods for paired duration data and further
developed and applied by Abbring, Chiappori, and Pinquet (2003). These tests compare
two subsequent durations for each subject, in the subsample for which two such durations
are observed. They naturally handle the fact that this subsample may only be a strict
selection of the full sample if there is censoring. Our main contribution is to illustrate
that the tests have little power in the case in which this selection is very strong. This, for
example, happens with data on rare events, such as insurance events.
The remainder of the chapter proceeds as follows. Section 3.2 presents a framework
for the analysis of state dependence and heterogeneity in renewal data. Section 3.3 reviews and develops identication results for a range of models in this framework. Section
3.4 explores the power of some nonparametric tests for state dependence.
Section 3.5
concludes.
3.2 State Dependence and Heterogeneity in Renewal
Consider a sequence of similar, renewal, events in continuous time
t ∈ R+ .
Examples
include claims incurred on an insurance contract in contract economics, nominal price
changes in macroeconomics, a consumer's purchases of a given good or service in marketing, and transactions of a particular stock in nance.
1
Suppose that the renewal events
1 For example, Abbring, Chiappori, and Pinquet (2003), Dionne, Dahchour, and Michaud (2006) and
Abbring et al. (2008) analyze car insurance claims.
Campbell and Eden (2007) analyze grocers' price
changes. Chintagunta and Dong (2006) review the application of duration analysis in marketing; Jain
60
3.3.
occur at times
t.
up to time
0 = T0 < T1 < · · · ,
and let
IDENTIFIABILITY
N (t) ≡ max{k ∈ N|Tk ≤ t}
Denote durations between renewals by
all unobserved heterogeneity is captured by a vector
count such events
∆Tk ≡ Tk − Tk−1 .
λ.
Suppose that
We suppress observed covariates
2
throughout.
If the renewal intensity
• t conditional
on
θ (t|H(t), λ)
(H(t), λ),
• H(t) ≡ {N (u); 0 ≤ u < t}
• λ
conditional on
H(t),
at time
t
depends on
we say that there is
conditional on
λ,
we say that there is
time dependence,
we say that there is
or
nonstationarity ;
state dependence ;
unobserved heterogeneity.
Following Heckman and Borjas (1980), state dependence can be further classied in
rence dependence
and
(dependence on
lagged duration dependence
N (t−)), duration dependence
(dependence on
(dependence on
occur-
TN (t−) ),
∆T1 , . . . , ∆TN (t−) ).
For example, the dynamic economic models of insurance claim times under moral
hazard and experience rating in Abbring, Chiappori, and Pinquet (2003) and Abbring
et al. (2008) predict nonstationarity and occurrence dependence, but no (lagged) duration
dependence, of claim intensities. They face the empirical challenge of distinguishing these
eects, which are of substantial interest, from those of unobserved heterogeneity in risk.
3.3 Identiability
Separating duration dependence and heterogeneity in data on a single renewal duration,
single spell data, is notoriously hard (Lancaster, 1979, Heckman and Borjas, 1980).
Elbers and Ridder (1982), Heckman and Singer (1984), Ridder (1990), Kortram, Lenstra,
Ridder, and van Rooij (1995) show that strong, separability and other, assumptions are
and Vilcassim (1991) provide one key example of the empirical analysis of repeated consumer's purchases.
Engle and Russell (1998) discuss methods for the analysis of transaction times in nance, and apply these
to the analysis of IBM stock transactions.
2 Where of interest, we discuss the way they could enter the analysis.
61
CHAPTER 3.
STATE DEPENDENCE
needed. In particular, data on external covariates are needed, and strong assumptions on
the variation in these covariates and the way they enter the duration model.
Data on sequences of renewal times for each subject, multiple spell or panel duration data, facilitate the identication of duration dependence under fewer assumptions.
Such data also allow for the analysis of state dependence across spells, that is occurrence
dependence and lagged duration dependence (Heckman and Borjas, 1980, Honoré, 1993).
In this section, we will review and develop some identication results for panel duration
data. A novel aspect of our analysis is that we explicitly deal with the common problem
that renewal events are selectively observed because data are only collected for a nite
amount of time. This censoring problem is shown to invalidate existing panel identication
results, and greatly reduces the identifying power of panel duration data.
Section 3.4
subsequently explores the eects of censoring on common tests for state dependence.
3.3.1
Occurrence Dependence and Duration Dependence
A close analogy with static linear panel data analysis arises in the special case in which
there is occurrence and duration dependence, but no lagged duration dependence nor
nonstationarity. Consider a specication for this special case in which
OD
θ (t|H(t), λ) = ξN (t−)+1 (t − TN (t−) )λ,
∆T1 , . . . , ∆Tk , . . .
λ
is a nonnegative
Here, the baseline hazard
ξk : R+ → (0, ∞)
are mutually independent conditional on
random variable with some distribution
G.
reects duration dependence. It may dier between spells
Rt
0
ξk (u)du
for all nite
t.3
k
λ,
and
and has an integral
We exclude defects by assuming that
Ξk (t) ≡
limt→∞ Ξ(t) = ∞.4
3 This can be extended to allow for a nite support of ∆T .
k
4 For results on identiability of mixture duration models with defects, see Abbring (2002, 2007).
62
3.3.
IDENTIFIABILITY
3.3.1.1 Full Information
First, consider the case in which always at least two renewal events, and therefore
T2 ,
are observed. Honoré (1993) shows that
Proposition 1
Ξ1 , Ξ2 , G
(Honoré, 1993, Theorem 1)
T1
and
are identied in this case.
. The functions Ξ , Ξ
1
2
and G in the model
specication (OD) are uniquely determined from the distribution of (T1 , T2 ).
Some intuition for this result follows from the analogy with linear panel data that arises
if we rewrite the model as a panel transformation model:
log Ξ1 (∆T1 ) = − log λ + log E1
3.1 and
log Ξ2 (∆T2 ) = − log λ + log E2 ,
with
E1 and E2 unit exponential variables that are mutually independent and independent
λ.
Because Honoré's result does not require data and assumptions on covariates, it can
of
be interpreted as a result conditional on covariates in the case in which data on covariates
are available. In particular, this implies that identiability extends to the case in which
duration dependence and heterogeneity may vary in arbitrary ways with the observed
covariates.
3.3.1.2 Censored Data
Now suppose that renewal events are only observed up to and including some random time
C ⊥⊥{Tk ; k ∈ N}.
Let
Dk ≡ I(C ≥ Tk )
be an indicator of complete observation of
this notation, we have data on the distribution of
{min{Tk , C}, Dk ; k ∈ N}.
the independence assumption, this identies the distribution of
C̄); k ∈ N}, with C̄
of
(T1 , T2 )
the upper bound of the support of
is identied.
5
C.
If
Tk .
In
Because of
{Tk · I(Tk ≤ C̄), I(Tk ≤
C̄ = ∞, then the distribution
Consequently, in that case Proposition 1 continues to apply.
6
5 In an extension in which T may have nite support, it is sucient that Pr(T > C̄) = 0.
2
2
6 The literature has focused on the practical inference problems that arise in this case. In particular,
even with independent censoring, the second duration
∆T2
is censored at a random time
C̄ − ∆T1
63
that is
CHAPTER 3.
STATE DEPENDENCE
If, on the other hand,
C̄ < ∞,
the (selected) subpopulation
then the distribution of
{T2 ≤ C̄}.
(T1 , T2 )
is only identied on
This case commonly arises in empirical work,
where panels are nitely lived, and complicates identication. Honoré (1993)'s proof of
Proposition 1 does not readily extend to this case.
To partially resolve this identication problem, suppose that duration-dependency patterns between spells are identical, and that occurrence dependence simply proportionally
shifts the hazard rate:
∗
OD
θ (t|H(t), λ) = β N (t−) ξ(t − TN (t−) )λ,
where
β >0
reects occurrence dependence there is none if and only if
ξ : R+ → (0, ∞) reects duration dependence,
t.
with integral
Intuitively, in this model we can tell whether
direction of the asymmetry in the distribution of
Proposition 2
Ξ(t) ≡
β < 1, β = 1,
or
Rt
0
β = 1
and
ξ(u)du for all nite
β>1
by checking the
(∆T1 , ∆T2 ) on its domain of observation.
.
(Identication of Occurrence Dependence from Censored Renewal Data)
The sign of β − 1 in the model specication (OD∗ ) is uniquely determined from the distribution of (T1 , T2 ) on [0, C̄]2 , C̄ > 0.
Proof.
The proof is constructive. Let
and note that
of
(T1 , T2 )
on
Z
Z ≡ {(t1 , t2 ) ∈ R2+ : Ξ(t1 ) < Ξ(t2 )
is identied from the distribution of
[0, C̄]2
∆T1
on
[0, C̄].
t1 + t2 ≤ C̄}
Next, the distribution
gives
Pr (∆T1 ≤ t1 , ∆T2 ≤ t2 )
= βξ(t1 )ξ(t2 )L00 [Ξ(t1 ) + βΞ(t2 )]
dt1 dt2
for almost all
and
and
Pr (∆T1 ≤ t2 , ∆T2 ≤ t1 )
= βξ(t1 )ξ(t2 )L00 [Ξ(t2 ) + βΞ(t1 )]
dt1 dt2
R∞
(t1 , t2 ) ∈ Z . Here L(s) ≡ 0 exp(−sv)dG(v) is the Laplace
transform of
typically not independent of it. Consequently, even under the assumption of independent censoring, the
second duration cannot be analyzed in isolation from the rst duration (Visser, 1996).
64
3.3.
G.
Because
L00
IDENTIFIABILITY
is strictly monotonic, this identies the sign of
Ξ(t1 ) + βΞ(t2 ) − Ξ(t2 ) − βΞ(t1 ) = (β − 1) [Ξ(t2 ) − Ξ(t1 )]
for almost all
sign of
3.3.2
(t1 , t2 ) ∈ Z .
Because
Ξ(t2 ) − Ξ(t1 ) > 0
for all
(t1 , t2 ) ∈ Z ,
this equals the
β − 1.
Occurrence Dependence and Lagged Duration Dependence
The specication in (OD) can be extended with lagged duration dependence:
LD
θ (t|H(t), λ) = µN (t−)+1 (∆T1 , . . . , ∆TN (t−) )ξN (t−)+1 (t − TN (t−) )λ,
where
µ1 = 1.
The function
∆T1 , . . . , ∆Tk−1 ,
for given
µk
captures the dependence of
N (t−) = k − 1, TN (t−) ,
and
θ (t|H(t), λ) on past durations
t, k = 2, 3, . . ..
Honoré (1993) presents identication results for a two-spell version of (LD) and com-
7
plete data.
His analysis allows
λ to vary across spells, but requires proportional variation
with external observed covariates, and does not directly carry over to (LD). However, it
does strongly suggest that identication of (LD) requires richer external variation in the
renewal durations than that required for identication of the basic model in (OD), even
without censoring.
3.3.3
Occurrence Dependence and Nonstationarity
In many applications, the renewal process takes place in a nonstationary environment.
One example is the contracting environment of the car insurance claims process analyzed
by Abbring, Chiappori, and Pinquet (2003) and Abbring et al. (2008). Insurance premia
are often updated annually, at the time of contract renewal. With forward looking agents
who suer from moral hazard, this leads to (contract) time eects in the claims process.
7 Abbring and Van den Berg (2003) present results for a related extension of the model.
65
CHAPTER 3.
STATE DEPENDENCE
Time eects can easily be mistaken for state dependence of substantial interest. Therefore, controlling for time eects is important if they cannot be excluded
a priori.
Consider
the following specication of the renewal intensity with occurrence dependence and time
eects (Abbring, Chiappori, and Pinquet, 2003):
NS
θ (t|H(t), λ) = β N (t−) ψ(t)λ.
Here,
ψ : R+ → (0, ∞)
captures time eects. It has an integral
Ψ(t) ≡
Rt
0
ψ(u)du
for all
t ∈ R+ .
Nonstationarity breaks the previous subsection's analogy to the linear static panel
data model. To see this, again rewrite the model as a transformation model,
log Ψ(T1 ) = − log λ + log E1
3.2 and
log [Ψ(T2 ) − Ψ(T1 )] = − log β − log λ + log E2 ,
with
of
λ,
E1 and E2 unit exponential variables that are mutually independent and independent
and
E2
independent of
T1 .
Thus, in terms of appropriately transformed times, this
nonstationary model with occurrence dependence is a dynamic panel data model, with
the usual one-factor structure on the errors, but with endogenous variables that cannot
be separated.
The following partial identication result for this model parallels Proposition 2.
Proposition 3
. The sign of
(Abbring, Chiappori, and Pinquet, 2003, Proposition 2)
β − 1 in the model specication (NS) is uniquely determined from the distribution of
(N (C̄), T1 , T2 ) on [0, C̄]2 , C̄ > 0.
In addition, Abbring, Chiappori, and Pinquet conjecture that the parameter
identied without further assumptions. They also show that
β
is known, under the additional assumption that
L
E[λ] < ∞.
and
Ψ
β is point-
are identied once
The latter assumption has
been common in the analysis of the mixed proportional hazard model since the early work
66
3.4.
NONPARAMETRIC TESTS
of Elbers and Ridder (1982), but is not innocuous (Ridder, 1990).
3.4 Nonparametric Tests
Tests for state dependence in renewal data can be distinguished by the way they control
for unobserved heterogeneity.
One approach follows the intuition from linear panel data analysis, and uses withinsubject variation, variation between spells for each given subject. In fact, it is clear from
Section 3.3 that, in some special cases, renewal models with heterogeneity can be written
as linear panel data models in log durations. In these cases, standard methods for linear
panel data with xed eects can be applied to control for heterogeneity, provided that
there is no censoring. By and large, this is the regression approach to the analysis of state
dependence forwarded by Heckman and Borjas (1980, Section II.b). More generally, linear
panel data methods cannot be applied, but Holt and Prentice's (1974) and Chamberlain's
(1985) methods for paired duration data can be.
Another approach follows the seminal work of Bates and Neyman (1952) and Heckman
and Borjas (1980, Section II.a) and exploits that, in a stationarity environment without
state dependence, the number of claims in a given data period is a sucient statistic
for the unobserved heterogeneity in the claim intensities.
Consequently, any signs of
nonstationarity or state dependence of these intensities in subsamples with a given number
of events in a period cannot be explained by heterogeneity and are direct evidence of time
and state dependence.
Note that a test that directly exploits this last result is in fact an omnibus test against
time and state dependence.
This is true for many tests for state dependence (see e.g.
Heckman and Borjas, 1980). Abbring, Chiappori, and Pinquet (2003) use such omnibus
tests, but also develop more advanced tests that allow for nonstationarity under the null,
and that are designed to have power against particular types of state dependence.
This section investigates the power of a particularly simple rank test for state depen-
67
CHAPTER 3.
STATE DEPENDENCE
dence that they used. Throughout, we will maintain Section 3.3.3's model with occurrence
dependence and time eects,
NS
θ (t|H(t), λ) = β N (t−) ψ(t)λ.
Section 3.4.1 studies a version of the test that directly compares the durations of each
subject's rst and second spells, and has power against both time and state dependence.
It is only a test specically against state dependence if stationarity (ψ
= 1)
is assumed.
An extension based on appropriately transformed durations, briey studied in Section
3.4.2, allows for nonstationarity (general
Ψ)
under the null, and tests specically against
state dependence.
3.4.1
A Simple Rank Test
Throughout this section, we assume stationarity (ψ
= 1);
it is implicitly understood that
this section's tests do not allow for nonstationarity under the null. Moreover, we will derive
all results conditional on
λ
that is, we will take
λ
to be a xed nuisance parameter.
Because the distributions of this section's statistics with heterogeneous
from mixing over their distributions for given
λ,
λ
follow directly
power and other results for the case of
general heterogeneity follow easily, and are not explicitly discussed.
3.4.1.1 The Case without Censoring
For given
λ,
parameter
λβ k−1 , k ∈ N.
the durations
∆Tk
are independently and exponentially distributed with
Consequently,
π(λ, β) ≡ Pr(T1 ≥ T2 − T1 |λ) =
Note that
π(λ, β) does not depend on λ, and simply write π(β).
dependence,
68
β
β+1
π(1) = 1/2.
Under the alternative,
π(λ, β) < 1/2
Under the null of no state
if
β<1
and
π(β) > 1/2
if
3.4.
NONPARAMETRIC TESTS
β > 1.
Now, suppose that we have a sample of
n renewal histories and that at least two renewal
times are observed for each subject. Thus, we have a sample
and reject the null that
β=1
if the empirical analog of
((T1,1 , T2,1 ), . . . , (T1,n , T2,n )),
π(β),
n
1X
π̂n ≡
I(T1,i ≥ T2,i − T1,i ),
n i=1
is far enough away from
1/2.
we would reject the null if
For example, if we test against the alternative that
π̂n
is suciently small.
β < 1,
This is simply a binomial test with
standard power and size properties. Note that these do not depend on
λ,
so that they
hold for general heterogeneity.
3.4.1.2 The Case with Censoring
Unfortunately, the previous section's standard test is not feasible in the typical case that
we can only observe renewal events for a nite period of time. For expositional convenience, suppose that we observe all renewal events up to a xed time,
C̄ = 1.
C̄ ,
and normalize
Then, we have a sample
(T1,1 , . . . , TN1 (1),1 ; N1 (1)), . . . , (T1,n , . . . , TNn (1),n ; Nn (1)) .
Consider the feasible statistics
π̂C,n ≡ P∞
k=2
π̂C2 ,n ≡
where
k
Mk,n ≡
Pn
i=1
n
X
1
Mk,n
I(T1,i ≥ T2,i − T1,i , Ni (1) ≥ 2)
n
1 X
I(T1,i ≥ T2,i − T1,i , Ni (1) = 2),
M2,n i=1
I (Ni (1) = k)
is the number of observations in the sample for which
renewal events are observed. We again reject the null if
away from
and
i=1
π̂C,n
or
π̂C2 ,n
are suciently far
1/2.
69
CHAPTER 3.
STATE DEPENDENCE
For given subsample sizes
P∞
k=2
Mk,n
and
M2,n , π̂C,n
and
π̂C2 ,n
are again standard
binomial tests, but now with underlying Bernouilli probabilities
πC (λ, β) ≡ Pr(T1 ≥ T2 − T1 |λ, N (1) ≥ 2) = π(β) · A(λ, β)
and
πC2 (λ, β) ≡ Pr(T1 ≥ T2 − T1 |λ, N (1) = 2) = π(β) · B(λ, β),
where
1
1 − β + (β + 1)e−λ − 2e− 2 λ(β+1)
A(λ, β) ≡
1 − β + βe−λ − e−λβ
and
1
2
β + 1 2β + 1 + e−λ(β −1) − 2(β + 1)e− 2 λ(β−1)
·
.
B(λ, β) ≡
2β + 1
β + e−λ(β 2 −1) − (β + 1)e−λ(β−1)
Both
A(λ, 1) = 1
and
B(λ, 1) = 1,
so that
πC (λ, 1) = πC2 (λ, 1) = 1/2.
For given
subsample sizes, this gives the standard one-sided or two-sided rejection regions for a
binomial test of a fair Bernouilli trial. Moreover, the rejection probabilities are continuous
functions of respectively
|πC (λ, β) − 1/2|
and
πC (λ, β)
πC2 (λ, β).
and
|πC2 (λ, β) − 1/2|
Thus, the tests are more powerful if
are larger, for the relevant alternatives. Therefore,
for characterizing the power of the tests for given subsample sizes, it suces to characterize
|πC (λ, β) − 1/2|
and
First, consider
β > 1,
|πC2 (λ, β) − 1/2|,
πC (λ, β).
we have that
as functions of
Although
A(λ, β) > 1
if
|πC (λ, β) − 1/2| < |π(λ, β) − 1/2|
β
πC (λ, β) < 1/2
β <1
for all
and
and the nuisance parameter
if
β < 1
A(λ, β) < 1
β 6= 1,
so that
if
π̂C,n
and
β > 1.
πC (λ, β) > 1/2
limλ→∞ A(λ, β) = 1,
Next, consider
70
limλ→0 πC (λ, β) = 1/2 for all β .
and the dierences between
πC2 (λ, β).
Again,
π(λ, β)
is less powerful than
π̂n ,
λ, π̂C,n
has
For large
and
if
This implies that
even if they would be based on (sub-)samples of the same size. For very small
very little power, because
λ.
λ,
πC (λ, β)
on the other hand,
vanish.
πC2 (λ, β) < 1/2 if β < 1 and πC2 (λ, β) > 1/2 if β > 1.
3.4.
However, for
β < 1,
we now have that
λ.
for large values of
it has low power if
λ
However, if
λ
B(λ, β) > 1
In both cases,
π̂C2 ,n
and
π̂C2 ,n
π̂C2 ,n = 1
may be more powerful than
π̂n .
is more powerful than
β < 1,
β <1
β 6= 1.
and
A(λ, β) < B(λ, β)
then
Consequently,
π̂C2 ,n
dependence. If
λ
if
but
B(λ, β) < 1
for small
λ is small.
π̂C,n
and
so that
π̂C,n
π̂n ,
λ,
but
Again,
if both are based on
unique λ̃(β)
such that
|πC2 (λ, β) − 1/2| > |π(λ, β) − 1/2|,
β > 1, limλ→∞ πC2 (λ, β) = 1,
π̂C2 ,n .
β
2β+1
so that
< π(β).
For all
λ, A(λ, β) > B(λ, β)
|πC (λ, β) − 1/2| < |πC2 (λ, β) − 1/2|
is more powerful than
of the same size (note though that
tests are equally poor if
π̂n
there exist a
limλ→∞ πC2 (λ, β) =
β > 1,
if
then
Moreover, if
Finally, we can compare the power of
if
β 6= 1
λ > λ̃(β),
(see Figure 3.1). If
almost surely. If
B(λ, β) < 1
has lower power than
subsamples of the same size. In fact, for each
B(λ̃(β), β) = 1
i.e.
λ,
limλ→0 πC2 (λ, β) = 1/2.
is very small:
is large,
π̂C2 ,n
for small values of
β > 1,
The opposite holds for
B(λ, β) > 1 for large λ.
NONPARAMETRIC TESTS
π̂C,n
if
if both are based on subsamples
is typically based on a
larger
subsample). Both
is close to zero, and perform equally well with extreme state
λ is very large, π̂C,n
is almost as good as, and
π̂C2 ,n
outperforms,
π̂n , again
for equal (sub-)sample sizes.
Table 3.1 summarizes these ndings. In general, we can conclude that the tests for
censored data have limited power if renewal events are rare, i.e. if
λ
is small, for given
subsample sizes. In such cases, the only way to increase the tests' power is to increase the
number of observations. In the next section, we characterize the sample sizes needed for
the tests to have reasonable power at various values of
λ
and
β.
3.4.1.3 Sample Sizes
In this section, we will focus on the statistic
asymptotically normal with mean
1/2
π̂C2 ,n .
and variance
Under the null that
[4nP2 (λ, 1)]−1 ,
β = 1, π̂C2 ,n
where
2
P2 (λ, β) ≡ Pr(N (1) = 2|λ) =
βe−λ − (β + 1)e−λβ + e−λβ
.
(β + 1)(β − 1)2
71
is
CHAPTER 3.
STATE DEPENDENCE
Figure 3.1: Graph of Function
B(λ, β)
and
λ̃(β)
Ž
ΛHΒL
BHΛ,ΒL 1
0
0
5Λ
Β1
2
β 6= 1, λ̃(β) is the unique solution λ of B(λ, β) = 1. The graph of λ̃ is represented
an intersection of the function B with the horizontal plane at 1.
Note: For each positive
in the above gure as
72
10
3.4.
NONPARAMETRIC TESTS
Table 3.1: Comparison of
General properties of
Parameters
π -functions
π , πC
β<1
and
πC2
β=1
β>1
λ < λ̃(β)
π < πC2 < πC <
1
2
π = πC2 = πC =
1
2
π > πC2 > πC >
1
2
λ > λ̃(β)
πC 2 < π < π C <
1
2
π = πC2 = πC =
1
2
πC 2 > π > π C >
1
2
Note: The function
λ̃
is dened in Section 3.4.1 and plotted in Figure 3.1.
Limiting properties of
Limit
λ→0
πC
and
πC 2
0<λ<∞
λ→∞
λ
(e 2 −1)2
eλ (λ−1)+1
β→0
πC = πC2 =
1
2
0<β<1
πC = πC2 =
1
2
See table
1<β<∞
πC = πC2 =
1
2
above
β→∞
πC = πC2 =
πC =
π
are that
limβ→0 π(β) = 0
β
β+1
πC =
π C = πC 2 = 1
does not exist
Note: Limiting properties of
πC = πC2 = 0
> πC2 =
β
β+1
β
2β+1
< πC2 = 1
πC = πC2 = 1
and
limβ→∞ π(β) = 1.
73
CHAPTER 3.
Under the null,
M2,n
STATE DEPENDENCE
n−1 M2,n
is a consistent estimator of
P2 (λ, 1).
In practice, we would take
α
for the normal distribution
as given and compute a critical region for a given size
1/2
with mean
and variance
(4M2,n )−1 .
In this section, we characterize, for dierent
λ and β , the minimal sample size nmin (λ, β)
needed to ensure that the null is rejected at the expected value
M2,n
relevant subsample size
equals its expected value
πC2
of
nP2 (λ, β).
π̂C2 ,n , given that the
Note that this roughly
corresponds to the minimum sample size needed to reject the null half of the times at the
given values of
eect of
β
λ
and
λ and β .
β
Also note that the results from this computation reect both the
on the test's power for a given subsample size, and the eects of
λ
and
on that subsample size.
For concreteness, we focus on a one-sided test of
β = 1 against β < 1.
In this case, we
will reject the null if
π̂C2 ,n ≤
where
for
u(α)
M2,n
is the
and
α-quantile
πC2 ,n (λ, β)
for
3.3 u(α)
1
+ p
,
2 2 M2,n
of the standard normal distribution. Substituting
π̂C2 ,n ,
and solving for the sample size
n
nP2 (λ, β)
for which (3.3) holds
with equality, gives
u(α)2
nmin (λ, β) =
P2 (λ, β) [1 − 2πC2 (λ, β)]2
That is,
πC2 ,n (λ, β)
is the critical value for rejecting the null
have a subsample with
nmin P2 (λ, β)
one-sided tests against
β>1
Note that
β=1
β < 1.
in favor of
β<1
if we
A similar computation for a
leads to the same expression for
nmin (λ, β)
for
β > 1.
nmin (λ, β) is large if the denominator in the right-hand side of (3.4) is small,
which happens if
πC2 (λ, β)
observations, for
3.4 P2 (λ, β)
is close to
is close to 1) or if
λ
1/2,
is small (that is, if
λ
and/or
β
is small).
It is also large if
which happens if there is little state dependence (that is, if
is small. On the other hand,
nmin (λ, β)
will be small if
λ
and
β
β
are
large.
Table 3.2 plots
74
nmin (λ, β)
for various values of
λ
and
β,
for
α = 5%.
Note that we
3.4.
NONPARAMETRIC TESTS
need millions of observations if the renewal events are rare, in particular if
λ
is smaller
than 0.2.
3.4.2
A Transformed Rank Test
Abbring, Chiappori, and Pinquet (2003) developed a variant of Section 3.4.1's test
that allows for general nonstationarity (Ψ) under the null. It is eectively
π̂C2 ,n
π̂C2 ,n
applied to
an appropriate empirical transformation of the observed renewal times. Because the testt
involves this empirical transformation, the analysis of its distributional properties cannot
be derived by mixing over its properties in the case of a homogeneous
explicitly deal with mixing over the distribution
First, suppose we know
Ψ.
G
λ
of
instead of
and
λ
T1
T1
and
T2
Ψ(T2 ) − Ψ(T1 )
and
βλ,
T2 .
and
Note that
is increasing on the
Ψ(T1 )
and
Ψ(T2 )
λ, Ψ(T1 )
are again independent exponential random variables with parameters
respectively. We can directly apply the earlier analysis for the stationary case,
Ψ.
Then, we can construct
Ψ(T1,i )
π̂C2 ,n
Ψ,
so that
π̂C2 ,n (Ψ)
is not feasible.
Chiappori, and Pinquet show that we can estimate
of
H1 (t) ≡ Pr(T1 ≤ t|N (1) = 1)
gests substituting
Ĥ1,n
Ψ(T2,i )
and therefore
to arbitrary, but still known, nonstationarity.
In general, we do not know
Ĥ1,n
and
n
1 X
I (Ψ(T1,i ) ≥ Ψ(T2,i ) − Ψ(T1,i ), Ni (T ) = 2) .
M2,n i=1
This is a generalization of
ance
Ψ
So, we can work with the transformed times
π̂C2 ,n (Ψ) ≡
null that
in this section.
without loss of information. This is convenient, as, for given
provided that we know
analog
Therefore, we
Then, we can deal with possible nonstationarity by working
in integrated-hazard time instead of calendar time.
supports of
λ.
for
β = 1, π̂n (Ĥ1,n )
Ψ
and using
Ψ
consistently by the empirical
under the null that
π̂C2 ,n (Ĥ1,n )
However, Abbring,
β = 1.
as our test statistic.
is asymptotically normal with expectation
[4n Pr(N (1) = 2)]−1 + [6n Pr(N (1) = 1)]−1
1/2
This sugUnder the
and vari-
(Abbring, Chiappori, and Pinquet, 2003,
75
CHAPTER 3.
STATE DEPENDENCE
Table 3.2: Sample size needed to reject the null that
β = 1,
for various values of
λ
and
β
β
λ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1
25,184,288
16,011,558
0.2
1,647,802
1,052,549
925,740
955,190
1,112,960
1,466,716
2,263,569
0.3
340,706
218,653
193,280
200,503
234,958
311,519
483,841
0.4
112,826
72,749
64,632
67,410
79,448
105,976
165,656
0.5
48,361
31,330
27,976
29,336
34,774
46,669
73,420
0.6
24,403
15,884
14,256
15,030
17,919
24,196
38,311
0.7
13,780
9,012
8,130
8,618
10,334
14,040
22,374
0.8
8,450
5,552
5,034
5,366
6,472
8,847
14,190
0.9
5,517
3,643
3,320
3,558
4,316
5,937
9,584
1
3,785
2,511
2,301
2,479
3,025
4,187
6,803
1.2
1.3
1.4
1.5
66,330,810 237,500,978 197,255,624 45,567,046 18,850,249
9,931,289
5,985,657
14,012,010 14,380,528 16,660,656 21,824,318 33,467,702
β
λ
0.1
0.8
0.9
1.1
0.2
4,516,366
16,284,972
13,729,815
3,197,037
1,333,547
708,634
430,904
0.3
971,876
3,529,081
3,020,317
708,906
298,145
159,787
98,020
0.4
334,993
1,225,023
1,064,261
251,785
106,766
57,706
35,709
0.5
149,475
550,478
485,463
115,764
49,491
26,975
16,837
0.6
78,525
291,238
260,721
62,665
27,009
14,845
9,346
0.7
46,172
172,462
156,723
37,967
16,497
9,143
5,805
0.8
29,482
110,906
102,307
24,980
10,942
6,115
3,915
0.9
20,049
75,958
71,127
17,504
7,729
4,355
2,812
1
14,329
54,673
51,969
12,889
5,737
3,259
2,122
Note: This table gives the sample size
value
πC2
M2,n
equals its expected value
76
of
π̂C2 ,n (λ, β),
nmin (λ, β)
needed to reject the null that
β =1
at the expected
5% one-sided test based on π̂C2 ,n , given that the relevant subsample size
nP2 (λ, β), for various values of the parameters λ and β .
using a
3.5.
Proposition 7). The variance can be estimated consistently as
Substitution of
fact that
Ĥ1,n
Ĥ1,n
for
Ψ
some of the occurrence dependence if
distribution
G
of
λ,
1/(4M2,n ) + 1/(6M1,n ).
comes at the price of lower power.
is only a consistent estimator of
the population analog
CONCLUSION
β 6= 1.
πC2 (H1 (β, G); β, G)
strictly increases near
Ψ
This is due to the
under the null, and generally captures
Abbring, Chiappori, and Pinquet show that
of the statistic, as a function of
β=1
if
G
β
for given
is nondegenerate with at least two
positive points of support. Here, we add that the presence of nontrivial heterogeneity in
λ
is crucial for this result: If
G
is degenerate, then
πC2 (H1 (β, G); β, G) = 1/2
for all
β.
Simulation results not reported here conrm that the test only has power if there is
substantial heterogeneity in
λ.
3.5 Conclusion
Typically, renewal data can only be collected over a nite period of time. This chapter
shows that this seriously hampers the analysis of state dependence in the presence of
general heterogeneity, and possibly, nonstationarity. In particular, existing identication
results for panel duration models do not apply to renewal data that are censored this way.
And, nonparametric tests for state dependence loose their power if the renewal events are
rare.
77
4
Moral Hazard in Dynamic Insurance Data
4.1 Introduction
Four decades of theoretical research on asymmetric information have rmly established
its importance for insurance relations and competitive insurance markets. The practical
relevance of this research, and of the results on the eciency of insurance markets and
the design of optimal contracts that it has produced, depends critically on the empirical
relevance of asymmetric information.
A substantial and fast growing literature is now
assessing this relevance for a variety of markets, using microeconometric methods and
micro data on contracts and insurees. For some markets, notably car insurance, evidence
is surprisingly mixed and muted, often pointing to a lack of asymmetric information problems. Much of the literature, however, uses static theory and cross-sectional data, which
limits both its versatility in dealing with truly dynamic aspects of insurance markets,
such as experience rating, and variation in the data that can be turned into robust empirical results. The empirical distinction between moral hazard and selection eects using
static methods has turned out to be particularly hard; as argued by Abbring, Chiappori,
Heckman, and Pinquet (2003), this is the standard econometric problem of distinguishing
causal and selection eects.
79
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
In this chapter we instead analyze moral hazard in car insurance using panel data on
contracts and claims provided by a Dutch insurance company. The analysis exploits some
remarkable properties of the Dutch experience-rating system. Specically, we theoretically
analyze the Dutch scheme as a repeated contract between an insuree and an insurer,
in which each period's interaction involves memory of the relationship's past history.
Using control theory, we study the endogenous changes this structure induces in the
incentives agents face at each point in time. Under moral hazard, these changes, in turn,
generate specic patterns in the time prole of claim occurrences and sizes that we fully
characterize; these patterns are specic in the sense that they would not appear under
the null of no moral hazard. Finally, we develop structural econometric tests based on
this theory and apply them to the Dutch micro data. The tests are exibly parametric
and nonparametric, and valid in the presence of unobserved heterogeneity of a general
type. In contrast to much of the earlier literature, we nd evidence of moral hazard in
car insurance.
We also discuss the empirical distinction between ex ante and ex post moral hazard. Ex
ante moral hazard entails that agents respond to changes in incentives by changing the risk
of losses. Ex post moral hazard concerns the eects of incentives on claiming actual losses.
The distinction between ex ante and ex post moral hazard is important because of their
dierent welfare consequences (e.g. Chiappori, 2001). Because an insurer's administrative
data typically only contain data on claims, and not on losses, distinguishing between
ex ante and ex post moral hazard requires additional structural assumptions.
Under a
reasonable set of such assumptions, we nd that at least some of the detected moral
hazard is due to ex post moral hazard.
Our theoretical model species agent's optimal dynamic savings, loss prevention effort, and claim choices under the experience-rating (bonus-malus) scheme in Dutch car
insurance. It produces predictions on the joint behavior of the claim occurrence, claim
size, and experience-rating processes, for given individual risk and other characteristics.
80
4.1.
INTRODUCTION
Ex ante moral hazard is captured by the endogenous loss prevention eort; ex post moral
hazard by the endogenous claim choice. Endogenous savings allow for self-insurance.
The model provides a characterization of the dynamic heterogeneous incentives to
avoid claims inherent to the Dutch experience-rating scheme, and their behavioral consequences under moral hazard. In particular, incentives are dened as the loss in expected
discounted utility that would be incurred if a claim would be led. We show how incentives
vary with the current bonus-malus state and contract time, and jump with each claim
because of its foreseeable eect on the future bonus-malus state. We present an algorithm
for numerically characterizing these eects and provide a quantitative analysis of incentives. We restrict attention to computations under the null of no moral hazard. Because
claim rates are constant under the null, these computations are relatively straightforward.
Our tests for moral hazard build on these theoretical computations. We rst focus on
the timing of claims. Under the null that there is no moral hazard, claim rates do not
vary with incentives; under moral hazard, on the other hand, claim rates are lower when
incentives are stronger.
Our main test exploits the full model structure.
It is a score
(Lagrange multiplier) test for the dependence of claim rates on incentives in a version of
the structural model that allows for exible heterogeneity in risk. Because it only requires
the computation of incentives under the null, it is easy to implement using our algorithm.
We nd strong evidence that claim rates decrease with incentives, and reject the null of
no moral hazard at all conventional levels.
In addition to this structural parametric test for moral hazard, we also present and
apply a range of nonparametric tests for state-dependence and contract-time eects on
claim rates, controlling for risk heterogeneity. Our theory implies that any such eects
must be due to moral hazard and, in this way, identies the substantial problem of testing
for moral hazard with the classical statistical problem of distinguishing state dependence
and heterogeneity.
This is a hard problem, but one that has been studied at length
in statistics and econometrics (Bates and Neyman, 1952, Heckman and Borjas, 1980,
81
CHAPTER 4.
Heckman, 1981).
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Our tests rely on this literature's key insight that, without contract-
time and state dependence, the number of claims in a given period is a sucient statistic
for the unobserved heterogeneity in the conditional distribution of claim times in that
period. Consequently, any signs of time or state dependence in subsamples with a given
number of claims in a period are evidence of moral hazard.
Moreover, an implication
of the Dutch experience-rating system is that incentives may jump up or down at the
time a claim is led, depending on the current bonus-malus state. Therefore, we not only
test for state dependence, but also for appropriate changes in its sign across bonus-malus
states. Even though the nonparametric tests have relatively little power with the type of
rare events found in insurance data (see Chapter 3), they corroborate the results from the
structural test.
Our theory also attributes a moral-hazard interpretation to state-dependence and
contract-time eects on the sizes of claims.
Under the assumption that ex ante moral
hazard only aects the occurrence, but not the size, of insured losses, the latter are informative on ex post moral hazard. We complement this analysis of ex post moral hazard
with data on claim withdrawals. Agents in our data set can withdraw a claim within six
months and avoid malus. Under some assumptions, which we spell out in detail, claim
withdrawals are observed manifestations of ex post moral hazard.
This chapter contributes to a rich literature on asymmetric information in insurance
markets.
The seminal work on moral hazard and adverse selection by Arrow (1963),
Pauly (1974, 1968), and Rothschild and Stiglitz (1976) showed that competitive insurance
markets may be inecient if information is asymmetric.
A vast theoretical literature
followed up on their key insights. Increasingly, attention has shifted from the development
of theory to the empirical analysis of its relevance (see, e.g., Chiappori, 2001, Chiappori
and Salanié, 2003, for reviews).
1
Chiappori (2001) forwarded the idea to exploit the
1 Car-insurance data were studied, among others, by Dionne and Vanasse (1992), Puelz and Snow
(1994), Dionne and Doherty (1994), Chiappori and Salanié (1997), Dionne, Gouriéroux, and Vanasse
(1999), Richaudeau (1999), Chiappori and Salanié (2000), Dionne et al. (2001), Abbring, Chiappori, and
Pinquet (2003), Cohen (2005), Dionne et al. (2006), Chiappori et al. (2006), and Pinquet et al. (2007).
82
4.1.
INTRODUCTION
rich variation that can be derived from dynamic theory and found in longitudinal data;
Abbring, Chiappori, Heckman, and Pinquet (2003) suggested that we base a test for
moral hazard on the dynamic variation in individual risk with the idiosyncratic variation
in incentives due to experience rating.
The empirical papers most closely related this chapter are Abbring, Chiappori, and
Pinquet (2003), Dionne et al. (2006) and Pinquet et al. (2007).
from and extends these works in several ways.
Our analysis diers
First, we precisely model the forward-
looking behavior of an agent in the actual institutional environment characterizing the
insurance market studied. We use this model to dene and compute dynamic incentives
and construct a structural test that exploits these computations in detail. Secondly, we
explicitly distinguish ex ante and ex post moral hazard, which requires a formal analysis
of the claim ling behavior. Finally, we model both claim occurrences and claim sizes.
Together, this allows us to confront a novel and precise set of dynamic implications for
claim occurrences and sizes under moral hazard to longitudinal data.
The remainder of this chapter is organized as follows. Section 4.2 briey discusses the
Dutch car-insurance market, with specic attention for the experience-rating scheme used.
It also introduces the data. Section 4.3 develops the theory. We use the theory to analyze
the dynamic incentives inherent to experience rating, and to derive the implications of
moral hazard for claim rate and size dynamics. Section 4.4 develops an econometric framework for testing the eects of moral hazard from data on claim rates and sizes and presents
the empirical results. Section 4.5 concludes. Appendices 4.A and 4.B provide proofs and
computational details for Section 4.3. Appendix 4.C gives additional information on the
data. Appendices 4.D 4.G provide robustness checks.
Health and life insurance data were analyzed by, for example, Holly et al. (1998), Chiappori et al. (1998),
Cardon and Hendel (2001), Hendel and Lizzeri (2003) and Fang et al. (2006). Finkelstein and Poterba
(2002) studied annuities.
83
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
4.2 Institutional Background and Data
4.2.1
Experience Rating in Dutch Car Insurance
In 2006, the 16.3 million inhabitants of the Netherlands were driving 7.2 million private
cars.
2
Because liability insurance is mandatory in the Netherlands, this comes with a
substantial demand for car insurance. In the same year, 74 insurance companies served this
demand.
3
Even though these companies are supervised by the Dutch nancial authorities,
they are to great extent free to set their premia and contractual conditions. In doing so,
the Dutch insurance companies, united in the Dutch Association of Insurers, have to great
extent coordinated their experience-rating systems in car insurance.
Before 1982, car insurers employed a limited experience-rating scheme. This scheme
was commonly considered to be inadequate to price observed risk. In the early 1980s, six
of the market's leading rms proposed a much ner
a large actuarial study (de Wit et al., 1982).
4
bonus-malus (BM) system,
based on
Early 1982, this system was introduced
in Dutch car insurance in a coordinated way. After some early market turbulence, the
insurers by and large settled on similar bonus-malus schemes.
In this chapter we use the same data as in the Chapter 2.
The data come from
one of the six companies that were leading the introduction of the bonus-malus system
in Dutch car insurance.
During the data period, January 1, 1995December 31, 2000,
5
this company used the bonus-malus scheme given in Table 4.1 . The premium discount
depends on the insuree's current
contract renewal date.
bonus-malus class,
which is determined at each annual
Twenty bonus-malus classes are distinguished, from 1 (highest
2 Source: Statistics Netherlands (www.cbs.nl).
3 Source: Dutch Association of Insurers (www.verzekeraars.nl).
4 Information on the development of the BM system in Dutch car insurance is scattered throughout
the professional literature. de Wit et al. (1982) provides information on the actuarial research underlying
the bonus-malus system, and some very early history. Assurantiemagazine (2004) provides more recent
historical reection.
5 This is the same BM scheme as the one given earlier, in the Table 2.1. We repeat it here for the
convenience and with a special notation introduced later in this chapter.
84
4.2.
INSTITUTIONAL BACKGROUND AND DATA
premium) to 20 (lowest premium).
Every new insuree starts in class 2 and pays the
corresponding premium. We will refer to this premium as the
base premium.
After each
claim-free year, an insuree advances one class, up to class 20. Each claim at fault sets
an insuree back into a lower class.
the base premium.
The worst class is 1, and implies a surcharge to
This scheme is representative for the bonus-malus schemes used in
6
the Netherlands in this period.
Consequently, throughout this chapter we assume that
the drivers in our data set cannot escape Table 4.1's bonus-malus system by switching
insurers.
The empirical analysis in this chapter exploits that the incentives to avoid a claim jump
with each claim led, and vary with contract time and across bonus-malus classes. To gain
some rst insight in the dierences in the cost of a claim to an insuree across dierent
bonus-malus classes and dierent numbers of claims, we have computed the change in
the premium at the next renewal date with each claim in a contract year, for dierent
bonus-malus classes. Table 4.2 gives the percentage premium change after a claim-free
contract year, and the subsequent marginal percentage changes in the premium after each
claim in the contract year. For example, after a claim-free year in class 8, an insuree will
be upgraded to class 9 and pay 45% instead of 50% of the base premium. This amounts
to a 10% reduction in the premium.
If he les one claim in the contract year, he will
instead be downgraded to class 4 and pay 80% of the base premium. This amounts to
a
(80 − 45)/45 = 78%
increase relative to the premium that would be paid without the
claim. A second claim would take him down further to class 1, and a premium equal to
120% of the base (a 50% increase relative to having one claim). A third claim would have
6 The scheme is similar to the one originally proposed by de Wit et al. (1982), extended with multiple
(maximum-bonus) levels that oer good customers some protection against premium increases. Evidence
on the development of the bonus-malus system is sketchy see Footnote 4 but strongly suggests that
the sector actively coordinated on similar bonus-malus schemes in the course of the 1980s, before the start
of our data period. Moreover, further major innovations to car insurance pricing were only introduced
recently, after the end of our data period. We also compared Table 4.1's scheme to schemes
currently
oered by Dutch insurers and found only minor dierences. The maximum discount on premium ranges
from 70% to 80% and the maximum surcharge is in the range of 15% to 30%. Some insurance companies
oer also collective insurance with more advantageous bonus-malus schemes.
85
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
no further eect on the premium.
Clearly, unlike the French scheme studied by Abbring, Chiappori, and Pinquet (2003),
this scheme is not proportional. The premium increases after a rst claim are largest for
those in the intermediate bonus-malus classes, and smallest for those in the top and bottom
classes.
The marginal premium increases after a second or third claim, however, are
increasing nearly monotonically with the bonus-malus class, from 0% in the lowest classes
to 100140% in class 20.
7
This all suggests that incentives to avoid a claim jump down
after a rst claim for insurees in low classes and jump up after a rst claim for insurees in
high classes. In Section 4.3, we formally dene incentives in a dynamic theoretical setting
and provide some numerical computations to formalize this intuition.
Finally, note that insurees are contractually obliged to claim all their insured losses as
soon as possible. However, the contract leaves them the option to withdraw their claims
within six months from the loss date. Withdrawn claims do not count as at-fault claims
in determining the insuree's bonus-malus class and therefore do not aect the premium.
Therefore, throughout most of this chapter we treat withdrawn claims like unclaimed
losses. That is, we ignore them, together with losses that were not claimed in the rst
place. Section 4.4.4 discusses the fact that withdrawals are in fact observed manifestations
of ex post moral hazard.
4.2.2
Data
Our data provide the contract and claim histories of personal car insurance clients of a
major Dutch insurer from January 1, 1995 to December 31, 2000. The raw data consist
of 1,730,559 records.
Each record registers a change in a particular contract (renewal,
change of car, etcetera), or a claim.
on
drivers
The data include 75 variables, with information
(sex, age, occupation, postcode),
cars
(brand, model, production year, price,
7 In the lowest classes, therefore, the bonus-malus scheme itself does not give incentives to avoid a
second or third claim. However, the insurance company reserves the right to cancel contracts with three
or more claims at fault in a year. Because claims at fault are fairly rare, this is unlikely to aect insurees'
decisions a lot. Therefore, we ignore contract cancelations in our theoretical and empirical analysis.
86
4.2.
INSTITUTIONAL BACKGROUND AND DATA
Table 4.1: Bonus-Malus Scheme
Present Premium Future BM class (B(K, N)) after a contract year with
BM class paid no claim 1 claim 2 claims 3 or more claims
(K) (q = A(K)) (N = 0) (N = 1) (N = 2)
(N ≥ 3)
20
25%
20
14
8
1
19
25%
20
13
7
1
18
25%
19
12
7
1
17
25%
18
11
6
1
16
25%
17
10
6
1
15
25%
16
9
5
1
14
25%
15
8
4
1
13
30%
14
7
3
1
12
35%
13
7
3
1
11
37.5%
12
6
2
1
10
40%
11
6
2
1
9
45%
10
5
1
1
8
50%
9
4
1
1
7
55%
8
3
1
1
6
60%
7
2
1
1
5
70%
6
1
1
1
4
80%
5
1
1
1
3
90%
4
1
1
1
2
100%
3
1
1
1
1
120%
2
1
1
1
Note: The notation in parentheses is taken from Section 4.3's model.
87
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.2: Percentage Premium Change after a Claim-Free Contract Year and Marginal
Percentage Changes in the Premium after each Claim, by Bonus-Malus Class
Present Premium change Increase in premium after
BM class if no claim 1st claim 2nd claim 3rd claim
(K)
( N = 0)
(N = 1) (N = 2) (N = 3)
20
0%
0%
100%
140%
19
0%
20%
83%
118%
18
0%
40%
57%
118%
17
0%
50%
60%
100%
16
0%
60%
50%
100%
15
0%
80%
56%
71%
14
0%
100%
60%
50%
13
-17%
120%
64%
33%
12
-14%
83%
64%
33%
11
-7%
71%
67%
20%
10
-6%
60%
67%
20%
9
-11%
75%
71%
0%
8
-10%
78%
50%
0%
7
-9%
80%
33%
0%
6
-8%
82%
20%
0%
5
-14%
100%
0%
0%
4
-13%
71%
0%
0%
3
-11%
50%
0%
0%
2
-10%
33%
0%
0%
1
-17%
20%
0%
0%
Note: The notation in parentheses and below is taken from Section 4.3's model.
The second column
reports
New premium after claim-free year
− Old
premium
Old premium
for each bonus-malus class
K.
=
A [B(K, 0)] − A(K)
A(K)
The third, fourth and fth columns report
A [B(K, N )] − A [B(K, N − 1)]
A [B(K, N − 1)]
N = 1, 2, 3, for all
K with N claims.
for respectively
a year in class
88
bonus-malus classes
K.
Here,
A [B(K, N )]
is the new premium after
4.2.
weight, power, etc.),
renewal date), and
contracts
claims
INSTITUTIONAL BACKGROUND AND DATA
(coverage, bonus-malus class, level of deductible, premium,
(type of claim, damage, etc.).
The raw data contains 163,194 unique contracts. Because they do not contain information on claims in 1995, we excluded this year from the data.
8
contracts that are not covered by the bonus-malus system.
We also excluded the
This leaves 140,799 unique
contracts with a total of 101,074 claims. Of these claims, 34,491 are claims at fault that
may lead to a malus.
9
However, in 2,463 of these cases, insurees have avoided a malus by
10
withdrawing their claim.
Throughout most of the chapter, we treat withdrawn claims
as unclaimed losses, and simply exclude them from the analysis. Section 4.4.4 specically
studies the withdrawal data to learn about moral hazard. Appendix 4.C shows that the
empirical results presented in the main text are robust to alternative ways of dealing with
withdrawals.
We restrict our analysis to the claim histories from the contracts' rst renewal (or
start) date in the sample onwards. In the data, there are 124,021 contracts with observed
renewal date. Of these contracts, 6,787 were interrupted for some period of time. In these
cases, we only use the contract history from its rst observed renewal date to its rst
interruption.
For each contract, we registered the claim history, with information on the times and
sizes of claims at fault that were not withdrawn. We examined the bonus-malus transitions between all observed contract years, corrected some inconsistencies (see Appendix
4.C for details), and registered the initial bonus-malus class (i.e., the bonus-malus class
established at the rst renewal date). Along the way, we discovered that the data on the
bonus-malus class after the 2000 renewal are not reliable.
11
Therefore, we excluded con-
8 These are the contracts covering companies' eets of cars. Such contracts have no individual BM
coecients, but general eet discounts. These discounts are adjusted every year based on the eets' claim
histories.
9 The data also include so called nil claims, which are mostly pro forma claims of amounts below
the deductible.
These may correspond to an at fault event, but typically do not aect the agent's
bonus-malus status. Therefore, we treat all nil claims as claims not-at-fault.
10 We use both direct and indirect information to identify withdrawals. See Appendix 4.C for details.
11 40,104 out of 68,515 bonus-malus transitions in 2000 were incorrect.
89
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
tracts that started in 2000, and the history of ongoing contracts after their 2000 renewal
date.
Our full nal sample consists of 123,169 unique contracts with 23,396 claims at fault.
Table 4.3 shows that many of these are observed for the maximum period of 4 years.
We illustrate some of the data's key features using only data on the rst fully observed
contract years in the sample. Table 4.4 gives the number of contracts in this subsample by
bonus-malus class and by number of claims at fault led in the contract year. Contracts
with one claim and contracts with two or more claims will be important, for dierent
reasons, to our empirical analysis.
There are a lot of contracts with one claim in our
subsample but, because claims are rare, there are only 278 contracts with at least two
claims.
Figure 4.1 plots the distribution of contracts in our subsample across bonus-malus
classes.
Classes 1 3 have less than 1% of the contracts each; more than 26% of the
contracts are in the highest class 20. The majority of contracts (over 57%) is in the high
bonus-malus classes 14 20, where the premium is just 25% of the base premium.
Figure 4.1 also plots the shares of contracts in our subsample with at least one and at
least two claims at fault, by bonus-malus class. These shares drop substantially with the
bonus-malus class. It may be tempting to relate this variation in the number of claims
over bonus-malus classes to our discussion of incentives. However, the overall pattern can
be well explained by heterogeneity in risk, with high-risk individuals sorted into the lower
bonus-malus classes.
4.3 Model of Claim Rates and Sizes
This section characterizes the dynamic incentives to avoid car insurance claims that are
inherent to the Dutch bonus-malus scheme.
We do so by analyzing a model of a sin-
gle agent's risk prevention and claim behavior that combines features of Mossin's (1968)
static model of insurance and Merton's (1971) continuous-time analysis of optimal con-
90
4.3.
MODEL OF CLAIM RATES AND SIZES
Table 4.3: Contract Exposure Durations in the Sample
Number of years
Y
1
Number of contracts observed
exactly Y years between Y − 1 and Y years
Total
8,097
11,775
19,872
4,709
9,616
14,325
3
6,262
7,387
13,649
4
68,820
6,503
75,323
87,888
35,281
2
Total
123,169
Table 4.4: Number of Contracts Observed for At Least One Full Contract Year, by BonusMalus Class and Number of Claims in the First Contract Year
BM
Number of contracts with
class no claim 1 claim 2 claims 3 claims 4 claims
1
562
118
24
4
2
749
94
11
1
1
Total
709
855
3
962
81
10
1,053
4
1,311
100
9
1,420
5
1,876
112
13
1
2,002
6
2,514
160
14
2
2,690
7
3,363
207
16
8
4,232
273
16
9
4,889
249
15
10
6,490
293
11
11
6,063
279
12
12
6,004
285
16
6,305
13
5,879
266
11
6,156
14
6,669
311
13
6,993
15
6,165
301
6
6,472
16
6,377
297
13
17
5,671
249
7
5,927
18
4,367
204
10
4,581
3,586
4,521
2
1
2
3,855
214
5
20
27,652
1,373
29
2
105,650
5,466
261
15
6,795
6,356
1
19
Total
5,155
6,688
4,074
29,056
2
111,394
Note: Nil and withdrawn claims were excluded from the sample.
91
CHAPTER 4.
Figure 4.1:
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Distribution of Contracts Observed for At Least One Full Contract Year
Across Bonus-Malus Classes; and Shares of Those Contracts with At Least One and At
Least Two Claims at Fault in the First Contract Year, by Bonus-Malus Class
30%
25%
All contracts
Contracts with at least 1 claim
Contracts with at least 2 claims
Share
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10
11
BM class
92
12
13
14
15
16
17
18
19
20
4.3.
sumption.
MODEL OF CLAIM RATES AND SIZES
Our model is related to Briys (1986), but focuses on experience rating and
its moral-hazard eects. It is an extension of Abbring, Chiappori, and Pinquet's (2003)
model with heterogeneous losses and endogenous claiming, carefully adapted to the Dutch
institutional environment. Also, unlike Abbring et al.'s analysis of experience rating in
French car insurance, we make the nonstationarity arising from annual premium revision
explicit. This is important for our empirical analysis because in the Dutch bonus-malus
system, unlike in the French one, both the number of past claims and their distribution
across contract years matter for the current bonus-malus status.
4.3.1
Primitives
We consider the behavior and outcomes of an agent
i
in continuous time
τ
with innite
horizon. Time is measured in contract years and has its origin at the moment the agent
entered the insurance market.
The wealth of agent
time
0,
agent
i
i
at time
τ
is denoted by
Wi (τ )
is endowed with some initial wealth
τ + dτ , agent i receives a return ρWi (τ )dτ
any other income, such as labor income.
Wi (0) > 0.
Then, between
τ
and
ci (τ )dτ .
We ignore
with some probability
pi (τ )dτ .13
on his wealth and consumes
12
The agent causes an accident between
If so, he incurs some monetary loss.
and accumulates as follows. At
τ
and
τ + dτ
Denote the
j -th
We assume that
Lij
(Li1 , . . . , Li(j−1) ),
from some time-invariant distribution
loss incurred by agent
i
by
Lij .
is drawn independently of the agent's insurance history, including
by an insurance contract involving a xed deductible
Fi .14
Di
The losses
and a premium
Lij
are covered
qi (τ )dτ
that is
paid continuously. The deductible is applied on a claim-by-claim basis, i.e. if a claim for
12 For the purpose of our analysis, this is equivalent to assuming that any such income is perfectly
foreseen by the agent (Merton, 1971, Section 7).
13 Accidents that are not caused by the agent are fully covered and have no impact on future premia.
Such accidents can be and are disregarded in our analysis. From now on, by accident or claim we always
mean accident or claim
at fault.
14 This assumption is violated if agents can inuence
Fi
ex ante by choosing to drive more or less
carefully. Then, data on claim sizes do not distinguish between ex ante and ex post moral hazard, but
are still informative on the overall presence of moral hazard.
93
CHAPTER 4.
Lij
a loss
is led, the insurer pays
The premium
Table 4.1.
i's
MORAL HAZARD IN DYNAMIC INSURANCE DATA
qi (τ )
Lij − Di
to the agent.
is determined by agent
Thus, we can write
i's
qi (τ ) = Ai (Ki (τ )),
bonus-malus class into his ow premium.
Ki (τ )
bonus-malus class
where
Ai
according to
is a mapping from agent
Because the base premium to which the
discounts in Table 4.1 are applied depends on agent
i's
characteristics, the mapping
Ai
15
will be heterogeneous across agents.
Agent
i
is endowed with an initial bonus-malus class
Ki (0).
The bonus-malus class is
updated at the beginning of each contract year, the renewal date, according to the rule in
Table 4.1. Thus,
date
is a right-continuous process, with discrete steps at each renewal
τ ∈ N depending on the past contract year's bonus-malus class and number of claims.
Denote by
time
Ki (τ )
τ.
Ni (τ )
That is,
the number of claims in the ongoing contract year up to and including
Ni (τ )
is a claim-counting process that is set to zero at the beginning of
each contract year. Then, at each renewal date
τ ∈ N,
4.1 Ki (τ ) = B(Ki (τ −), Ni (τ −)),
where
Ki (τ −)
and
Ni (τ −)
are agent
past contract year, respectively, and
i's
B
bonus-malus class and number of claims in the
represents Table 4.1's bonus-malus updating rule.
Note that this rule is common to all agents.
Recall that it moves agents who survive
a contract year without claims to a higher bonus-malus class, corresponding to a lower
premium, and all other agents to a lower class, with a higher premium.
Insurance claims led by agents are potentially aected by ex ante and ex post moral
hazard (Chiappori, 2001). Ex ante moral hazard arises if an agent can aect the probability of an accident. We model this by allowing, at each time
the intensity
cost
pi (τ )
Γi (pi (τ )).
τ,
the agent to choose
of having an accident from some bounded interval
We assume that
Γi
is twice dierentiable on
(pi , pi ),
[pi , pi ],
with
at a utility
Γ0i < 0, Γ00i > 0.
15 Here, we abstract from time-varying characteristics other than K . There is not much harm in treating
i
e.g. age as a time-invariant characteristic, as our empirical analysis will focus on events in only one or a
few contract years.
94
4.3.
MODEL OF CLAIM RATES AND SIZES
In words, reducing accident rates is costly and returns to prevention are decreasing. For
deniteness, we also assume that
Γ0i (pi +) = −∞
and
Γ0i (pi −) = 0.
In addition, we allow
for ex post moral hazard by allowing the agent to hide a loss he has actually incurred
from the insurer. For clarity of exposition, we assume that claiming and hiding losses are
costless, but that the agent cannot claim losses that have not actually been incurred.
The agent's instantaneous utility from consuming
tensity
pi (τ ) at time τ
is
ui (ci (τ ))−Γi (pi (τ )).
ci (τ )
16
and driving with accident in-
We assume that
ui is strictly increasing and
concave. The agent chooses consumption, prevention and claiming plans that maximize
17
total expected discounted utility
Z
∞
e
E
−ρτ
[ui (ci (τ )) − Γi (pi (τ ))] dτ ,
0
subject to the intertemporal budget constraint
limτ →∞ e−ρτ W (τ ) = 0 and given the wealth
and premium dynamics described above.
At each time
τ,
the agent observes his wealth, bonus-malus class and claim histories.
As we have implicitly assumed that any labor and other income is perfectly foreseen by
the agent, he only has to form expectations on future accidents and their implications.
4.3.2
Optimal Risk, Claims and Savings
For notational convenience, we now drop the index
i.
It should be clear, however, that
all results are valid at the individual level, irrespective of the distribution of preferences
and technologies across agents. In particular, the results hold for any type of unobserved
heterogeneity in these primitives of the model.
Because our model is Markovian and, apart from annual contract renewal, timehomogeneous, the optimal consumption, prevention and claim decisions at time
τ
only
16 Section 4.3.3.1 discusses a simple extension of the model in which hiding losses is costly. Such an
extension is needed to formalize variation in the degree of ex post moral hazard in general, and the
extreme case that agents report all losses (above the deductible) and do not suer from ex post moral
hazard in particular.
17 For simplicity, we assume that subjective discount rates equal the interest rate.
95
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
depend on the past history through the agent's current wealth
K(τ ),
the number of claims at fault
N (τ ),
and the time
W (τ ),
t ≡ τ − [τ ]
bonus-malus class
past in the ongoing
contract year.
Let
V (t, W, K, N )
denote the agent's optimal expected discounted utility at time
the contract year if his wealth equals
N
W,
he is in bonus-malus class
K,
t
in
and has claimed
losses in the ongoing contract year. This value function satises the Bellman equation
V (t, W, K, N ) =
n
max u(c)dt − Γ(p)dt + e−ρdt ×
c,p,X
h
(1 − pdt)V (t + dt, (1 + ρdt)W − cdt − A(K)dt, K, N )
Z
+ pdt
V (t + dt, (1 + ρdt)W − min{l, D} − cdt − A(K)dt, K, N + 1)dF (l)
X
Z
io
V (t + dt, (1 + ρdt)W − l − cdt − A(K)dt, K, N )dF (l) ,
+ pdt
Xc
4.2 with
4.3 V (1, W, K, N ) ≡ lim V (t, W, K, N ) = V (0, W, B(K, N ), 0).
t↑1
Equation (4.2) can be interpreted as follows.
Between
t
and
t + dt
the agent derives
ows of utility from his consumption and disutility from his prevention eort. The value
V (t, W, K, N )
equals the net value of these utility ows, at the optimal consumption
and prevention levels, plus the expected optimal discounted utility at time
probability
1 − pdt
no accident occurs.
t + dt.
With
Then, the agent's wealth is increased with the
interest ow minus consumption and the premium, and the number of claims at fault,
stays unchanged. If the agent causes an accident, with probability
pdt,
N,
he will incur an
additional wealth loss. The size of this wealth loss is subject to ex post moral hazard.
If the damage
L
caused by the accident lies in the optimal choice of the
claims for insurance compensation and only looses the minimum of
D.
Then, the number of claims at fault,
96
N,
increases by 1. If
L
L
claim set X ,
he
and the deductible
lies in the complement
4.3.
Xc
MODEL OF CLAIM RATES AND SIZES
of the optimal claim set, however, he does not claim and pays the full loss
the number of claims at fault,
N,
L.
Then,
stays unchanged.
Equation (4.3) reects the eects of annual premium renewal. It requires that the value
in class
with
0
K
with
N
claims just before a renewal time equals the value in class
B(K, N )
claims just after renewal.
Bellman equation (4.2) can be rewritten in a more familiar form by rearranging and
taking limits
dt ↓ 0,
n
ρV (S) = max u(c) − Γ(p)
c,p,X
hZ
+p
V (t, W − min{l, D}, K, N + 1)dF (l)
X
Z
i
+
V (t, W − l, K, N )dF (l) − V (S)
4.4 Xc
o
+ VW (S) [ρW − c − A(K)] + Vt (S) ,
where
W.
Vt
and
VW
are the partial derivatives of
V
with respect to, respectively,
t
and
The left-hand side of (4.4) is the ow (or perpetuity) value attached by the agent
to state
S ≡ (t, W, K, N ).
It equals the (optimal) instantaneous ow of utility from his
consumption net of the disutility from his prevention eort plus three expected value
(capital) gains terms, (i) the expected value gain because of an accident, (ii) the value
gain due to net accumulation of wealth, and (iii) the appreciation of the value over time.
Standard arguments guarantee that (4.4), with (4.3), has a unique solution
that an optimal consumption-prevention-claim plan exists.
that the value function
V
is strictly increasing in wealth
weakly increasing in the bonus-malus class
claims at fault
N
K
V,
and
In Appendix 4.A, we prove
W
(Lemma 1) and that it is
and weakly decreasing in the number of
(Lemma 2).
One direct implication is that the agent follows a threshold rule for claiming.
Proposition 4. The optimal claim set in state S is given by X (S) ≡ (x (S), ∞), for
∗
∗
some claim threshold x∗ (S) ≥ D.
97
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Thus, if the agent incurs a loss
K
L
and number of claims at fault
at time
N
t
then, for given wealth
W,
bonus-malus class
right before t, he claims if and only if
L > x∗ (S).
The
threshold is implicitly dened as the loss at which he is indierent between claiming and
not claiming:
4.5 V (t, W − D, K, N + 1) = V (t, W − x∗ (S), K, N ).
This assumes an internal solution and, in particular, ignores the trivial, and empirically
irrelevant, case in which
X = ∅.
Optimality of the two remaining choices, consumption and prevention, requires that
the corresponding rst-order conditions are satised,
u0 (c∗ (S)) = VW (S)
0
Z
∗
4.6
4.7
and
x∗ (S)
V (t, W − l, K, N )dF (l)
−Γ (p (S)) = V (S) −
Z
0
∞
−
V (t, W − D, K, N + 1)dF (l),
x∗ (S)
where
p∗ (S) and c∗ (S) are, respectively, the optimal accident and consumption intensities
in state
S ≡ (t, W, K, N ).
The rst equation is the standard Euler condition, which
balances the marginal utilities from current and future consumption. The second condition
requires equality of the marginal cost of prevention and the marginal cost of an accident.
4.3.3
Dynamic Incentives from Experience Rating
4.3.3.1 Measure of Incentives
First, consider ex ante moral hazard. The rst-order condition (4.7) embodies two distinct
aspects of ex ante moral hazard, the agent's ability to reduce risk and the incentives he
is given to do so. If the marginal cost
−Γ0
of reducing risk quickly increases from
0
to
∞,
changes in incentives have little eect on risk and moral hazard is limited. In the limiting
case in which
98
Γ(p) = 0
if
p ≥ p0
and
Γ(p) = ∞
if
p < p0 ,
for some
p0 > 0,
the agent will
4.3.
choose an accident rate
p0
MODEL OF CLAIM RATES AND SIZES
irrespective of incentives to avoid claims. We will refer to this
limiting case as the case of no (ex ante) moral hazard.
The right-hand side of (4.7) is the expected discounted utility cost of a claim. This
is a measure of the incentives to avoid an accident, for a given prevention technology
In this section, we characterize the variation in these incentives with, in particular,
and
t.18
Γ.
K, N
In the next section, we use this characterization to test for moral hazard.
We focus on the dynamic incentives inherent to the bonus-malus scheme and set the
deductible
D
to
0.
This simplies the presentation and does not greatly interfere with our
objective of learning about
changes
in incentives across states. Section 4.3.3.2 formalizes
this point in the context of a particular model specication.
We will also restrict attention to incentives in the case without moral hazard. This
will be sucient for computing a score test for moral hazard and for interpreting local
behavior of econometric tests near the null of no moral hazard. Without moral hazard,
the optimal accident rate
p∗ (S)
equals a xed number
p0 > 0
in all states
S
and all losses
are claimed, so the right-hand side of (4.7) simplies to
4.8 V (t, W, K, N ) − V (t, W, K, N + 1).
We will characterize incentives in the case of no moral hazard by characterizing this
dierence in utility values as a function of the state
(t, W, K, N ).
Before we move to these computations, note that (4.8) is also a measure of incentives
to avoid a claim given that an accident has occurred. Linearizing (4.5) as a function of
the threshold around the deductible
D=0
gives
4.9 x∗ (t, W, K, N )VW (t, W, K, N ) ≈ V (t, W, K, N ) − V (t, W, K, N + 1).
18 Abbring, Chiappori, and Pinquet (2003) obtain unambiguous theoretical results on the change in
incentives after each claim in French car insurance. These results rely on the proportional nature of the
French bonus-malus system, and do not carry over to the Dutch system. Moreover, Abbring et al. do not
model the nonstationarity arising from annual contract renewal and, therefore, do not provide results on
contract-time eects.
99
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
The right-hand side of this equation is again the expected discounted utility cost of a
claim in (4.8).
The left-hand side is the marginal cost, in expected discounted utility
units and at a time a claim decision needs to be taken, of increasing the threshold just
above the deductible
D = 0.
Note that this cost, unlike the cost
is not a free parameter of the model.
Γ
of loss prevention,
This is a direct consequence of our assumption
that claiming and hiding losses is costless, and implies that the model does not have
a parameter that indexes the degree of ex post moral hazard.
extend the model with such a parameter.
capital loss of
to
γL,
for some parameter
γx∗ (t, W, K, N )VW (t, W, K, N ).
in which
γ → ∞,
It is straightforward to
For example, if hiding a loss
γ ≥ 1,
L
leads to a
then the left-hand side of (4.9) generalizes
Then, the null of no moral hazard is the limiting case
and hiding losses is prohibitively expensive. Throughout this chapter,
it is implicitly understood that the null of no (ex post) moral hazard can be generated
this way. For expositional convenience, we will not make this explicit in the notation.
4.3.3.2 Theoretical Characterization of Incentives
We compute the value function and the incentives for the constant absolute risk aversion
(CARA) class of utility functions, which is given by
u(c) =
with
α>0
1 − e−αc
,
α
the coecient of absolute risk aversion,
arises as a limiting case if we let
α ↓ 0.
−u00 (c)/u0 (c).
Linear utility,
u(c) = c,
The CARA class brings analytical and compu-
tational simplications that we believe outweigh, for the purpose of this chapter at least,
its disadvantages (see e.g. Caballero, 1990, for some discussion).
Merton's (1971) results that, with CARA utility, the value and utility functions have
the same functional forms and consumption is linear in wealth, provide intuition for
Proposition 5. In the case of no moral hazard with accident rate p , D = 0, and CARA
100
0
4.3.
MODEL OF CLAIM RATES AND SIZES
utility,
c∗ (S) = ρ [W − Q(t, K, N )]
and
V (S) =
1 − e−αρ[W −Q(t,K,N )]
,
αρ
with S ≡ (t, W, K, N ) and Q the unique solution to the system of dierential equations
ρQ(t, K, N ) = π(K) + p0
4.10
4.11
eαρ[Q(t,K,N +1)−Q(t,K,N )] − 1
+ Qt (t, K, N )
αρ
Q(1, K, N ) = Q(0, B(K, N ), 0).
Here, Qt (t, K, N ) is the partial derivative of Q(t, K, N ) with respect to t.
Proposition 5 is proved in Appendix 4.A.
function
V
It provides a characterization of the value
that can be used to compute incentives under the null of no moral hazard.
To gain some insight in Proposition 5's characterization of optimal consumption and
the value function, rst note that equation (4.10) reduces to
ρQ(t, K, N ) = π(K) + p0 [Q(t, K, N + 1) − Q(t, K, N )] + Qt (t, K, N )
if we let
α ↓ 0.
Q(t, K, N )
Thus, in the limiting case of linear utility that is, a risk-neutral agent
reduces to the expected discounted ow of future premia. The agent simply
consumes the ow value
V (S) = W − Q(t, K, N ).
ρ[W − Q(t, K, N )]
of his
net
wealth, which produces a value
The expected discounted utility cost of a claim in state
S
is
given by
V (S) − V (t, W, K, N + 1) = Q(t, K, N + 1) − Q(t, K, N ).
Conveniently, incentives are independent of the level of wealth in this case.
With a risk-averse agent that is, for xed
α>0
the right-hand side of equation
(4.10) involves an additional term
p0
eαρ[Q(t,K,N +1)−Q(t,K,N )] − 1
− [Q(t, K, N + 1) − Q(t, K, N )] ,
αρ
101
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
which is strictly positive for all
(t, K, N ).
As a consequence,
Q(t, K, N )
strictly exceeds
the expected discounted ow of premia, and optimal consumption is lower than with
S are now given by
eαρQ(t,K,N +1) − eαρQ(t,K,N ) , so that a wealth-
linear utility. This reects precautionary savings. Incentives in state
V (S) − V (t, W, K, N + 1) = (αρ)−1 e−αρW
invariant measure of incentives is given by
∆V (t, K, N + 1) ≡
V (S) − V (t, W, K, N + 1)
eαρQ(t,K,N +1) − eαρQ(t,K,N )
.
=
e−αρW
αρ
Note that this measure again reduces to
Q(t, K, N + 1) − Q(t, K, N )
if we let
α ↓ 0.
Before we move to a numerical characterization of incentives, briey consider the case
of a general but state-invariant deductible
state
S
D.
In this case, with linear utility, incentives in
Q(t, K, N + 1) − Q(t, K, N ) in the expected discounted
are the sum of the increase
premium ow and the deductible
D.
Because the deductible is not state dependent,
changes in incentives across states are not aected.
Consequently, tests that focus on
changes in incentives across states within agents are robust to an extension to general
deductibles (see Section 4.4.2).
4.3.3.3 Numerical Characterization of Incentives
In the remainder of this section, we will numerically characterize incentives by computing
∆V (t, K, N + 1)
for various values of
the underlying function
Q
(t, K, N ), α,
and
p0 .
is presented in Appendix 4.B.
consistent with a 4% annual interest and discount rate.
we take
p0 = 0.053,
An algorithm for computing
which corresponds to a
contract year. This equals the share
94.8%
We set
in particular,
π(K)
to be
In our baseline computations
probability of having no claim in the
105,650
of contracts without claims in our subsample
111,394
of single contract years (see Table 4.4). We measure the premium
the base premium. That is,
ρ = ln(1.04)
π(K)
in multiples of
is set equal to the premium reported in Table 4.1 and,
π(2) = 1.
Figure 4.2 plots the (wealth-invariant measures of the) present discounted utility costs
102
4.3.
of a rst (∆V
MODEL OF CLAIM RATES AND SIZES
(1, K, 1)), a second (∆V (1, K, 2)) and a third (∆V (1, K, 3)) claim just before
contract renewal, as a function of the bonus-malus class
to the linear-utility case
α=0
K.
The bold graphs correspond
and give the expected discounted premium cost of a claim
in multiples of the base premium. The other graphs correspond to
in that order and with the graphs corresponding to
a consumption level equal to
to coecients of
relative
20
α = 0.1
closest to the bold graph. At
α = 0, 0.1, . . . , 0.5
times the base premium,
risk aversion equal to
α = 0.1, 0.2, . . . , 0.5,
0, 2, . . . , 10,
correspond
respectively. This is roughly
the range considered, with some empirical support, by Caballero (1990).
Incentives near the null of no moral hazard are considerable. In the linear case, total
wealth drops by more than the annual base premium. Recall that the base premium is
four times the premium in class 20 paid by most insurees in our sample. The cases with
risk aversion are very similar.
Incentives also vary a lot between bonus-malus classes.
The incentives to avoid a rst claim are small in the lowest classes, where the premium
paid is already high. They then increase substantially, and again fall to a lower level in
the highest classes. Robustly across the values of
α,
these incentives are larger than the
incentives to avoid a second or a third claim in low classes
K.
K,
and smaller in high classes
Thus, for agents in high bonus-malus classes, the Dutch bonus-malus system has
implications that are similar to those of the French proportional experience-rating scheme
studied by Abbring, Chiappori, and Pinquet (2003): The rst and also the second claim
in a contract year lead to jump up in incentives, and therefore jump down in claim rates
under moral hazard. However, the Dutch system allows us to contrast this implication
with the eects of low bonus-malus classes, where incentives jump
Figure 4.3 plots the change
∆V (1, K, N + 1) − ∆V (1, K, N )
down after a rst claim.
for
N =1
(resp.
N = 2)
in incentives to avoid a claim when a rst (resp. second) claim is led just before contract
renewal, again for dierent degrees of risk aversion.
after a rst claim jump down for low
claim do not change for low
K
K
This graph shows that incentives
and up for high
K.
The incentives after a second
(they are already equal to zero), but jump down for middle
103
CHAPTER 4.
K
and up for high
point,
t = 1,
MORAL HAZARD IN DYNAMIC INSURANCE DATA
K.
These eects, and those in Figure 4.2, are computed at a specic
in time, but are robust to considering alternative times.
To illustrate this, Figure 4.3 also plots the changes over the course of a contract
year in incentives to avoid, respectively, a rst, a second and a third claim.
∆V (t, K, N ) − ∆V (0, K, N )
these graphs of
is for all
N = 1, 2, 3
Because
close to linear as a function of time
t,
∆V (1, K, N )−∆V (0, K, N ) summarize well the time patterns in incentives,
and the variation in these time patterns between bonus malus classes.
The changes in
incentives over time are small relative to the jumps in incentives when a claim is led
just before contract renewal. The
and
∆V (1, K, N ) − ∆V (0, K, N )
N = 2.
dierences between ∆V (1, K, N + 1) − ∆V (0, K, N + 1)
are even smaller for
Because these dierences give the change in
the contract year, this implies that the graphs of
N =1
and
2,
N = 1
and almost the same for
∆V (t, K, N + 1) − ∆V (t, K, N )
over
∆V (1, K, N +1)−∆V (1, K, N ) for both,
indeed characterize well the jump in incentives after a rst and a second
claim at all times across the contract year.
Even if the time-variation in incentives is small relative to the jumps in incentives at
the times of a claim, it may still aect some of our empirical procedures that focus on
the latter. After all, the time-variation in incentives aects all contracts, but only some
contracts experience jumps in incentives.
We will return to this in Section 4.4 in the
specic context of an econometric model.
Finally, we explore the robustness of these results to changes in the accident intensity
under the null,
p0 .
Figure 4.4 again plots the jumps in incentives after a rst and a second
claim and changes in incentives over the course of a contract year for dierent levels of
risk aversion, but for
p0 = 0.
The graphs are qualitatively similar to those in Figure 4.3
for the average-risk case. Time eects are smaller in the zero-risk case because agents do
not expect any accident during the contract year; time preference is the only source of
nonstationarity in this extreme case.
Figure 4.5 plots the same graphs for
104
p0 = 0.232,
which is the average risk level consis-
4.4.
EMPIRICAL ANALYSIS
tent with the share of contracts without a claim in the worst bonus-malus class,
K = 1.
At this risk level, incentives at the time of a rst claim only increase in the highest bonusmalus classes, and decrease at low and intermediate bonus-malus classes. The jumps in
incentives at the time of a second claim have similar features as before.
Time eects
are now substantial. Because agents are very likely to experience an accident during the
contract year anyhow, incentives do not jump much early in the year even if they would
jump a lot close to renewal.
In sum, the qualitative conclusions for our baseline case with average risk continue to
hold as long as
p0
is average or low, but change for very large
p0 .
Nevertheless, we can
robustly conclude that incentives at the time of a rst claim drop in all low classes, and
increase in very high classes. The results on jumps in incentives at the time of a second
claim are more robust to changes in accident intensity; these incentives do not change in
low classes, drop in middle classes and increase in high classes.
4.4 Empirical Analysis
The empirical analysis uses the data set introduced in Section 4.2.2. We formalize this
using Section 4.3's notation, with an appropriate change of the time scale's origin.
The full sample consists of
i ∈ {1, . . . , n},
we let time
τ
n
contracts. Time is measured in years. For each contract
have its origin at the start of the contract's rst contract
year included in the sample. Then,
Contract
Let
i
is the initial bonus-malus class in the
is observed until some random attrition time
Ni (τ )
until time
Ci
is
N̄i (Ci ),
τ.
sample.
Ci .
count the number of claims at fault on contract
year up to and including time
i
Ki (0)
i
in the ongoing contract
Note that the total number of claims incurred on contract
with
N̄i (τ ) ≡ Ni (τ ) +
[τ ]
X
Ni (u−).
u=1
105
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.2: Incentives to Avoid First, Second and Third Claim; at an Average Risk Level
5
∆ V(1,K,1)
∆ V(1,K,2)
∆ V(1,K,3)
4.5
4
3.5
∆V
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
Note: This gure plots
hazard, for
π(K)
p0 = 0.053
∆V (1, K, N )
for
N = 1, 2, 3
as functions of
K
for the CARA case without moral
and dierent values of the coecient of absolute risk aversion
is measured in multiples of the base premium, as in Table 4.1.
the linear-utility case
α = 0
respectively.
106
The premium
The bold graphs correspond to
and give the expected discounted premium cost of a claim in terms of
α = 0.1, 0.2, . . . , 0.5, in that order and with the
α = 0.1 closest to the bold graph. At a consumption level equal to 20 times the
α = 0, 0.1, . . . , 0.5 correspond to coecients of relative risk aversion equal to 0, 2, . . . , 10,
the base premium.
The other graphs correspond to
graphs corresponding to
base premium,
α.
4.4.
EMPIRICAL ANALYSIS
Figure 4.3: Change in Incentives to Avoid a Claim after a First and a Second Claim, and
Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course of a
Contract Year; at an Average Risk Level
4
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
2
∆ V
1
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
∆V (1, K, N + 1) − ∆V (1, K, N ) for N = 1, 2, and ∆V (1, K, N ) − ∆V (0, K, N )
for N = 1, 2, 3 as functions of K for the CARA case without moral hazard, for p0 = 0.053 and dierent
values of the coecient of absolute risk aversion α. The premium π(K) is measured in multiples of the
base premium, as in Table 4.1. The bold graphs correspond to the linear-utility case α = 0 and give the
Note: This gure plots
expected discounted premium cost of a claim in terms of the base premium. The other graphs correspond
to
α = 0.1, 0.2, . . . , 0.5,
graph.
At a consumption level equal to
coecients of
relative
α = 0.1 closest to the bold
α = 0, 0.1, . . . , 0.5 correspond to
in that order and with the graphs corresponding to
20
risk aversion equal to
times the base premium,
0, 2, . . . , 10,
respectively.
107
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.4: Change in Incentives to Avoid a Claim after a First and a Second Claim, and
Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course of a
Contract Year; at a Zero Risk Level
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
2
∆ V
1
0
−1
−2
−3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
∆V (1, K, N + 1) − ∆V (1, K, N ) for N = 1, 2, and ∆V (1, K, N ) − ∆V (0, K, N )
for N = 1, 2, 3 as functions of K for the CARA case without moral hazard, for p0 = 0 and dierent
values of the coecient of absolute risk aversion α. The premium π(K) is measured in multiples of the
base premium, as in Table 4.1. The bold graphs correspond to the linear-utility case α = 0 and give the
Note: This gure plots
expected discounted premium cost of a claim in terms of the base premium. The other graphs correspond
to
α = 0.1, 0.2, . . . , 0.5,
graph.
At a consumption level equal to
coecients of
108
relative
α = 0.1 closest to the bold
α = 0, 0.1, . . . , 0.5 correspond to
in that order and with the graphs corresponding to
20
risk aversion equal to
times the base premium,
0, 2, . . . , 10,
respectively.
4.4.
EMPIRICAL ANALYSIS
Figure 4.5: Change in Incentives to Avoid a Claim after a First and a Second Claim, and
Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course of a
Contract Year; at a High Risk Level
3
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
2
1
2
∆ V
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
∆V (1, K, N + 1) − ∆V (1, K, N ) for N = 1, 2, and ∆V (1, K, N ) − ∆V (0, K, N )
for N = 1, 2, 3 as functions of K for the CARA case without moral hazard, for p0 = 0.232 and dierent
values of the coecient of absolute risk aversion α. The premium π(K) is measured in multiples of the
base premium, as in Table 4.1. The bold graphs correspond to the linear-utility case α = 0 and give the
Note: This gure plots
expected discounted premium cost of a claim in terms of the base premium. The other graphs correspond
to
α = 0.1, 0.2, . . . , 0.5,
graph.
At a consumption level equal to
coecients of
relative
α = 0.1 closest to the bold
α = 0, 0.1, . . . , 0.5 correspond to
in that order and with the graphs corresponding to
20
risk aversion equal to
times the base premium,
0, 2, . . . , 10,
respectively.
109
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Denote the time and size of the
Then, for each contract
and its claim history
i,
j th
we observe
Hi [0, Ci )
claim on contract
Ci ,
up to time
i
Tij
with
and
Lij ,
respectively.
the contract's initial bonus-malus state
Ci ,
Ki (0),
where
Hi [0, τ ) ≡ ({Ni (u); 0 ≤ u < τ }; Li1 , . . . , LiN̄i (τ ) ).
Note that the bonus-malus history up to time
Ci , {Ki (τ ); 0 ≤ τ < Ci },
within contract years, and can be constructed from
The full unbalanced sample is
Ki (0)
and
{Ni (u); 0 ≤ u < Ci }.
{Ci , Ki (0), Hi [0, Ci ); i = 1, . . . , n}.
We assume that it is
a random sample from the distribution of its population counterpart
The claim history
H ≡ H[0, ∞),
does not vary
{C, K(0), H[0, C)}.
and its relation to the bonus-malus class
K(0)
initially
occupied by the agent, are the focus of our empirical analysis.
4.4.1
Econometric Model
At the core of our econometric model is the intensity
time
θl
of claims of size
l ∈ R+
or up at
τ , conditional on the claim history H[0, τ ) up to time τ , the initial bonus-malus class
K(0),
and a nonnegative individual-specic eect
λ.
We specify the following model:
θl (τ |λ, H[0, τ ), K(0)) = ϑ (t|λ, N (τ −), K(τ )) · F (max{l, x∗ (t, λ, N (τ −), K(τ ))}|λ) ,
4.12
where
t ≡ τ −[τ ] is time elapsed in the contract year and ϑ (t|λ, N (τ −), K(τ )) is the rate at
which losses are incurred at time
τ
by an agent with characteristics
K(τ ) and has claimed N (τ −) times in the year up to time τ .
class
λ who has been in class
Recall that the bonus malus
K(τ ) is fully determined by the initial class K(0) and the claim history H[0, τ ).
The
second factor,
F (max{l, x∗ (t, λ, N (τ −), K(τ ))}|λ), is the conditional probability that the
loss is of size
L≥l
and that it is claimed, i.e.
L ≥ x∗ .
This specication incorporates
Section 4.3's assumption that losses are drawn from an exogenous and time-invariant
distribution
110
F (·|λ) = 1 − F (·|λ) that may dier between agents.
It also reects the result
4.4.
EMPIRICAL ANALYSIS
that agents follow a threshold rule for claiming (Proposition 4).
Without further loss of generality, and to facilitate a discussion of theory's implications
for (4.12), we write
4.13 ϑ (t|λ, N (τ −), K(τ )) = λ · ψ(t) · β (t|λ, N (τ −), K(τ )) ,
with
and
ψ
β
a continuous and positive function representing external contract time eects,
an almost surely bounded and positive function. We frequently use the notation
Ψ(t) ≡
Rt
0
ψ(u)du
conditional on
and normalize
Ψ(1) = 1.
We assume that
λ
has distribution
K(0) = K .
Together with equation (4.12), this fully species the distribution of
assume independent censoring, that is
C ⊥⊥ H|K(0).
H|K(0).
θl (τ |H[0, τ ), K(0))
We
This is a standard assumption
in event-history analysis (e.g. Andersen, Borgan, Gill, and Keiding, 1993).
that
GK
It ensures
can be identied with the claim rate among surviving contracts,
θl (τ |H[0, τ ), K(0), C > τ ).
We are now ready to dene the tests' hypotheses within the context of the econometric
model. First, consider the simplication of (4.12) that is implied by the absence of moral
hazard. In the empirical analysis, we will refer to this case as the
null of no moral hazard.
Prediction 1. The claims process under the null of no moral hazard. Without
moral hazard, β ≡ 1 and x∗ ≡ 0, so that
θl (τ |λ, H[0, τ ), K(0)) = λψ(t)F (l|λ).
Given λ, claim rates and sizes do not depend on the past number of claims N (τ −) or the
bonus-malus class K(τ ); they only depend on contract time through the function ψ . That
is, there is no state dependence in the claims process.
Taken literally, Section 4.3's theory implies that
λF (l|λ)
is time-invariant, with
λ = p0 .
Thus,
ψ
ψ ≡ 1,
so that
θl (τ |λ, H[0, τ ), K(0)) =
captures contract-time eects that are
111
CHAPTER 4.
external
MORAL HAZARD IN DYNAMIC INSURANCE DATA
to the model, that is, that are independent of the claim history and the bonus-
malus class. We entertain the possibility of such eects because, if they are there for some
reason, they are likely to confound our analysis of state dependence.
both tests that assume
ψ≡1
19
We will present
(stationarity) and tests that allow for nonparametric
ψ.
The proportional specication of (4.13) will then capture the rst-order eects of any
external contract-time eects. Note that in addition we explicitly allow, through
x∗ ,
for contract-time eects that arise
internally
β
and
because of the fact that contracts are
renewed at discrete times. These internal time eects will in general enter the claim rate
nonproportionally.
Under the alternative of moral hazard, Prediction 1 generally fails.
In that case,
θl (τ |λ, H[0, τ ), K(0)) depends negatively on incentives, which in turn vary with t, λ, and,
in particular,
N (τ −)
and
K(τ ).
In Subsection 4.4.2, we impose the full structure of
Section 4.3's theory on the econometric model, including
ψ ≡ 1.
We present and apply a
score test that can be interpreted as a Lagrange multiplier test for moral hazard in the
structural model. In Section 4.4.3, we use more general tests for state dependence in claim
times and sizes. There, we only rely on qualitative predictions of the eect of incentives
on claim rates and sizes, without directly using incentive computations.
Before we present these tests, we briey reect on the possibility that they pick up
alternative sources of state dependence in the claims process, such as learning, fear, or
cautionary responses to accidents, that are unrelated to nancial incentives and moral
hazard. In Section 4.3, we have assumed these away by specifying the prevention tech-
19 The theoretical and econometric models only recognize contract time and do not explicitly consider
the eects of calendar time (or duration since last event for that matter). In our sample, dierent contracts
have dierent renewal dates, so that contract time and calendar time do not coincide. If renewal dates
are evenly distributed over calendar time, seasonal calendar-time eects are not likely to matter much
to the empirical analysis.
However, in our sample we observe that the share of contracts starting in
January is 13.3% which is more than twice as much as at the end of the calendar year (6.1% contracts
start in November and 6.2% in December). This variation could be explained by the fact that it is more
advantageous to buy a new car at the beginning of a calendar year because the ageing of a car in years
depreciates its value much faster than the ageing in months. On the other hand, the shares of contracts
starting in other (middle) months of the year are almost equal; they range from 7.1% (August) to 9.4%
(April).
112
4.4.
nology, as represented by the cost function
Γ,
EMPIRICAL ANALYSIS
to be independent of the accident history.
There are two reasons not be overly concerned about this. First, many of these alternative
sources of state dependence are expected to work in one direction, unlike the nancial incentives in the Dutch bonus-malus system. Learning from accidents, for example, is likely
to reduce the accident rate in all states, irrespective of nancial incentives. Therefore, it
is unlikely that, for example, learning exactly replicates the implications of moral hazard.
Second, learning eects are likely to be small for older drivers. We have conrmed the robustness of the empirical conclusions that follow by repeating our analysis on a subsample
of insurees of 28 years old and up; see Appendix 4.D for details.
4.4.2
Structural Test on the Full Sample of Claim Times
We rst focus on the timing of claim and ignore information on claim sizes. Section 4.3
proves that ex ante and ex post moral hazard work in the same direction (see also Section
4.4.3). Thus, we can view tests based on claim times as overall tests for moral hazard.
Assume that there are no external time eects,
ψ ≡ 1,
so that all nonstationarity
arises from behavioral responses to variation in incentives over time. In addition, suppose
that there are
R
risk-types
λ1 , . . . , λ R
of agents. Consider the following auxiliary model
of claim rates,
4.14 θ(τ |λ, H[0, τ ), K(0)) = λ · exp [−β∆V (t, K(τ ), N (τ −) + 1|λ)]
with
Pr(λ = λr |K(0) = K) = ξr (K), r = 1, . . . , R,
and
1, . . . , 20, and ∆V (·|λ) equal to ∆V (·) evaluated at p0 = λ.
K
have the same supports
{λ1 , . . . , λR }
across
K,
β=0
moral hazard, we expect to nd evidence that
r=1 ξr (K)
= 1,
The distributions of
for
K =
λ|K(0) =
but with dierent probability masses
at each support point because of sorting into classes
Under the null of no moral hazard,
PR
K.
and claim rates are time-invariant. Under
β > 0.
We will now argue that a score test
113
CHAPTER 4.
for
β=0
MORAL HAZARD IN DYNAMIC INSURANCE DATA
in (4.14) can be interpreted as a structural test for moral hazard.
The auxiliary model's specication corresponds exactly to theory under the null; in
that case
θ(τ |λ, H[0, τ ), K(0)) = λ = θ0 (τ |λ, H[0, τ ), K(0)).
It can be seen as an ap-
proximation to the theoretical model under local alternatives to the null and a specic
functional form of
Γ.
with cost function
Γλ (p) = (p/β̃) [ln(p/λ) − 1] + λ/β̃ ,
Suppose that an agent with characteristics
so that
λ
chooses
p
from
Γ0λ (p) = β̃ −1 ln(p/λ).
(0, λ],
Sub-
stituting in the rst-order condition (4.7) and assuming that there is no ex post moral
hazard gives
p∗ (S) = λ · exp −β̃ [V (t, W, K, N |λ) − V (t, W, K, N + 1|λ)]
h
i
≈ λ · exp −β̃e−αρW ∆V (t, K, N + 1|λ) ,
4.15
where the approximation in the second line holds near the null of no moral hazard. Thus,
the auxiliary model (4.14) is a good approximation to the optimal claiming hazard near
the null, that is, for small
β̃ ,
with
β = β̃e−αρW .
Note that
β = β̃
is homogeneous in
the population, as in the auxiliary model, in the limiting case of a risk-neutral agent
(α
= 0).
In this case, the derivative of
p∗ (S)
with respect to
β
at
β =0
exactly equals
the corresponding derivative of the auxiliary model's claim rate in (4.14). Consequently,
a score test for
β=0
in the auxiliary model exactly equals a Lagrange multiplier test for
moral hazard in the structural model.
The score test for moral hazard has so far been narrowly developed for the case without
ex post moral hazard, a specic functional form of the cost function
Γ,
a zero deductible,
and linear utility. However, the intuition for a test based on the auxiliary model (4.14) does
not rest on this example's specic assumptions, and we expect such a test to have power
against moral hazard more generally.
deductible
D,
For example, with a general but state-invariant
the approximation in (4.15) becomes
h
i
p∗ (S) ≈ λ̃ · exp −β̃∆V (t, K, N + 1|λ) ,
114
4.4.
with
λ̃ = λ exp −β̃D .
Clearly, a score test for
β=0
EMPIRICAL ANALYSIS
in the auxiliary model continues
to be a test for moral hazard in this extension.
We estimate both restricted (β
= 0)
and unrestricted versions of the auxiliary model
with parametric maximum likelihood, using the full unbalanced sample and computing
∆V
using the linear specication (α
= 0).
We compute the likelihood using a discrete
(daily) approximation, building on
θ(τ |λ, H[0, τ ), K(0))
1
− − N (τ −) = 1 λ, N (τ −) , K(τ ) ≈
,
Pr N τ +
365
365
τ ∈
k
;k
365
∈ Z+
.
Each likelihood computation for the unrestricted model, and the
computation of the score test statistic, embed the algorithm in Appendix 4.B to compute
∆V (·|λr )
(that is,
∆V (·)
at
p0 = λr ), r = 1, . . . , R,
at daily times.
In addition to the
score test, we also compute Wald and likelihood-ratio statistics to test for
the alternative that
β 6= 0.
β=0
against
Because the latter two tests involve estimates of the auxiliary
model under the alternative of moral hazard, where it only approximates the structural
model, their interpretation as structural tests is less clear cut.
However, because the
approximation holds near the null, we expect them to have good power against, at least,
local moral-hazard alternatives.
We estimated the unrestricted model with various numbers of support points for the
distribution of
R = 4
and
λ, R = 2, 3, 4, 5,
R = 5,
and obtained stable estimates of
β.
Moreover, between
the maximum log likelihood only increased by 5.83 points, even
though 21 parameters were added.
20
Table 4.5 gives the estimates of
β
and the
λs
in the
unrestricted model, with their estimated standard errors, for the specication of the model
with 3, 4 and 5 support points. It also presents the score, Wald and likelihood-ratio test
statistics for the hypothesis that there is no moral hazard:
β = 0.
The estimate of
β
is
signicantly positive. All three tests reject the null of no moral hazard at all conventional
20 Computation time also became an issue: Estimating the model with ve support points took almost
a week on a standard PC.
115
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
levels.
Figure 4.6 plots the probability masses
points for each class
K.
ξr (K) of the unrestricted model with 3 support
For expositional convenience, the estimates of
are in ascending order, i.e.
λ̂1 < λ̂2 < λ̂3 .
λs
in Table 4.5
With this in mind, it is easy to see that the
probability masses are slowly moving from the highest risk (λ̂3 ) in bonus-malus class 1
to the lowest risk (λ̂1 ) in bonus-malus class 20. This pattern is consistent with dynamic
sorting of agents across bonus-malus classes.
4.4.3
Tests for State Dependence in Claim Times and Sizes
The previous section presents a tightly structured test for state dependence. It is tightly
structured in the sense that it concentrates on local alternatives in which all state dependence is channeled through the dynamic incentives computed using Section 4.3's theory.
In this section, we explore the application of more universal, nonparametric tests for state
dependence from the literature. The interpretation and, in a few cases, construction of
these tests rely on the theory's
qualitative
predictions on the claims process for given
λ
under moral hazard. We rst develop and present these predictions.
4.4.3.1 Theoretical Implications for the Claims Process
The theoretical analysis of Section 4.3 can now be applied to predict the properties of
the claims process for given
λ
under local moral-hazard alternatives. First note that the
theory implies that incentives to avoid claims vary between initial bonus-malus classes
K.
However, in data the resulting moral-hazard eects on claims are confounded with
sorting of agents with dierent characteristics
λ
into dierent classes
K.
The problem of
empirically separating these selection eects from the causal eects of incentives is the
standard problem of causal inference from cross-sectional data. This is a notoriously hard
problem that we avoid here. Instead, we exploit that there is idiosyncratic variation in
incentives over time.
116
4.4.
EMPIRICAL ANALYSIS
Table 4.5: Maximum-Likelihood Estimation of the Auxiliary Model (4.14) with Three,
Four, and Five Support Points
Three Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
0.4229
0.0428
0.0427
0.0050
0.1770
0.0254
0.3514
0.0234
Tests of
β=0
•
LM test: 26.14,
p-value
= 0.00
•
LR test: 89.74,
p-value
= 0.00
•
Wald test: 97.54,
p-value
= 0.00
Four Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
0.3810
0.0337
0.0000
0.0000
0.0552
0.0074
0.2211
0.0125
0.8715
0.1260
Tests of
β=0
•
LM test: 72.73,
p-value
= 0.00
•
LR test: 94.96,
p-value
= 0.00
•
Wald test: 127.90,
p-value
= 0.00
Five Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
λ5
0.4017
0.0390
0.0135
0.0118
0.0531
0.0117
0.1889
0.0164
0.2629
0.0171
0.8896
0.1219
Tests of
β=0
•
LM test: 76.56,
•
LR test: 106.39,
•
Wald test: 106.09,
p-value
p-value
= 0.00
= 0.00
p-value
= 0.00
Note: The left side of each panel presents maximum-likelihood estimates of the relevant parameters in
the unrestricted auxiliary model (4.14). The right side of each panel presents Lagrange multiplier (LM),
likelihood-ratio (LR) and Wald tests for moral hazard.
117
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.6: Estimated Probability Masses
ξr (K) of the Auxiliary Model (4.14) with Three
Mass Points
BM class 1
BM class 2
BM class 4
1
1
1
0.5
0.5
0.5
0.5
0
λ1
λ2
λ3
0
BM class 5
λ1
λ2
λ3
0
BM class 6
λ1
λ2
λ3
0
BM class 7
1
1
1
0.5
0.5
0.5
0.5
0
0
0
λ1
λ2
λ3
λ1
λ2
λ3
BM class 10
λ1
λ2
λ3
0
BM class 11
1
1
1
1
0.5
0.5
0.5
λ1
λ2
λ3
0
BM class 13
λ1
λ2
λ3
0
BM class 14
λ1
λ2
λ3
0
BM class 15
1
1
1
1
0.5
0.5
0.5
λ1
λ2
λ3
0
BM class 17
λ1
λ2
λ3
0
BM class 18
λ1
λ2
λ3
0
BM class 19
1
1
1
0.5
0.5
0.5
0.5
0
0
0
λ2
λ3
λ1
λ2
λ3
λ1
λ2
λ3
λ1
λ2
λ3
λ1
λ2
λ3
λ1
λ2
λ3
BM class 20
1
λ1
λ3
BM class 16
0.5
0
λ2
BM class 12
0.5
0
λ1
BM class 8
1
BM class 9
118
BM class 3
1
0
λ1
λ2
λ3
4.4.
EMPIRICAL ANALYSIS
Prediction 2. Dependence of claims on N (τ −), by class K(τ ) under moral hazard. Conditional on λ, loss rates jump down (β(t|λ, 0, K) > β(t|λ, 1, K) > β(t|λ, 2, K))
and claim sizes increase (x∗ (t, λ, 0, K) < x∗ (t, λ, 1, K) < x∗ (t, λ, 2, K)) at the times
of the rst and the second claims in high classes K . In contrast, in low classes K
loss rates jump up (β(t|λ, 0, K) < β(t|λ, 1, K) ≤ β(t|λ, 2, K)) and claim sizes decrease
(x∗ (t, λ, 0, K) > x∗ (t, λ, 1, K) ≥ x∗ (t, λ, 2, K)) after the rst and the second claims. There
is no change in loss rates and claim sizes after the second claim in classes K ≤ 5. Because the state-dependence eects on loss rates and claim probabilities work in the same
direction, the results for the loss rates carry over to claim rates.
Next, for expositional convenience, suppose that there are no external time eects,
ψ ≡ 1.
Then, we have
Prediction 3. Dependence of claims on time t, by class K(τ ) under moral hazard. Conditional on λ, loss rates of an agent with 0 claims, resp. 1 claim (or, more
particularly, β(t|λ, 0, K), resp. β(t|λ, 1, K)) weakly decrease with t in most classes K , but
may increase in the highest classes. Loss rates of an agent with 2 claims (β(t|λ, 2, K))
are time-invariant in classes K ≤ 9 and strictly decrease with t in classes K > 9. The
opposite results hold for claim thresholds x∗ , so that the eects on loss rates carry over to
claim rates. All these time eects are small compared to the jumps at the time of a claim
(Prediction 2), except for very high loss rates.
If there are external contract-time eects, that is if
ψ is nontrivial, then Prediction 3 holds
relative to these external eects.
Predictions 1-3 are all conditional on
ual contract. Because
λ;
they are predictions at the level of an individ-
λ is not observed, tests based on contrasting the predicted behavior
under the null (Prediction 1) with the predicted behavior under the moral-hazard alternative (Predictions 2 and 3) are not feasible. The econometric challenge is to develop tests
that use these predictions without requiring data on
λ.
119
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Our tests exploit the dynamics of claims implied by Predictions 2 and 3. Rather than
studying cross-sectional variation in incentives, and trying to separate these from selection eects, we exploit variation in incentives over time. The problem of separating the
corresponding dynamic moral hazard eects from dynamic selection is the classic problem
of distinguishing state dependence and heterogeneity. Like the problem of distinguishing
causal eects and selection eects in a static setting, this is a hard problem. However, it is
a richer problem that has been well-studied in the statistics and econometrics literature.
A key result from this literature implies that, under the null, the total number of claims
in the contract year is a sucient statistic for the unobserved heterogeneity in the loss
intensities. We use this result to control for unobserved heterogeneity in the loss rates.
We build on Abbring, Chiappori, and Pinquet's (2003) adaptations and extensions of the
tests developed in the seminal work by Bates and Neyman (1952), Heckman and Borjas
(1980) and Heckman (1981).
We rst study time eects in claim rates. Prediction 1 implies that, under the null
and after controlling for heterogeneity, time eects should be identical between classes
K.
Moreover, there should be no time eects at all under the theory's assumption of
stationarity (ψ
≡ 1).
On the other hand, both Predictions 2 and 3 imply that there will
be time eects under moral hazard.
Time eects in claim rates are likely to be small and tests for moral hazard based
on observed time eects are not likely to be very powerful. More importantly, they may
be confounded by external time eects (ψ ).
Therefore, we quickly move to comparing
(distributions of ) rst and second claim times and sizes. Here, Prediction 2 takes center
stage. Because the jumps in incentives at the time of a claim are much larger than the
time-variation in incentives, Prediction 2's structural occurrence dependence (Heckman
and Borjas, 1980) eects dominate Prediction 3's time eects. Therefore, we can test for
moral hazard by testing the implications of Prediction 2 for the relation between rst
and second claim times and sizes, across classes
120
K
and controlling for heterogeneity and,
4.4.
EMPIRICAL ANALYSIS
possibly, external time eects.
For the state-dependence tests, we use the balanced subsample consisting of the rst
fully observed contract years, presented in the Table 4.4.
We will only use data on
contracts with one claim and contracts with (exactly or at least) two claims in the contract
year. We will use the same notation as before, i.e.
Ki (0)
will denote the initial bonus-
malus class (which is the bonus-malus class in the rst observed contract year);
Lij
will refer to the time and size of the
j th
Tij
and
claim (in the rst contract year).
4.4.3.2 Distribution of First Claim Time
Consider the distribution of the rst claim time
T1
in the subpopulation with exactly one
claim in the contract year and in one of the bonus-malus classes in
K,
H1 (t|K) = Pr(T1 ≤ t|N (1−) = 1, K(0) ∈ K),
and its empirical counterpart
Ĥ1,n (t|K) =
where
n
n
X
M1,K,n
i=1
I(Ti1 ≤ t, Ni (1−) = 1, Ki (0) ∈ K),
is the total number of contracts in the sample and
k, Ki (0) ∈ K)
exactly
1
k
Mk,K,n ≡
Pn
i=1
I(Ni (1−) =
is the number of contracts in the sample of contracts in a class in
K
with
claims.
Under the null of no moral hazard,
H1 (·|K) = Ψ(·)
(Prediction 1 and Abbring, Chi-
appori, and Pinquet, 2003). Under the moral hazard alternative,
depend on the choice of
K
and dier from
Ψ(·).
H1 (·|K)
will typically
This variation is caused by both changes
in incentives at the time of a claim (Prediction 2) and changes in incentives over time
(Prediction 3). We tested the null that
H1 (·|K)
is equal for all
K ∈ {1, 2, . . . , 20}
using
the Kruskal-Wallis test and do not reject the null at conventional levels (see Table 4.6).
Figure 4.7 plots
Ĥ1,n (t|K(0) ∈ K)
for low BM classes
K = {1, . . . , 10}
and high BM
121
CHAPTER 4.
classes
MORAL HAZARD IN DYNAMIC INSURANCE DATA
K = {11, . . . , 20}.
The dierence between these two empirical distributions is not
p-values
of Wilcoxon rank-sum and Kolmogorov-Smirnov tests (given in
signicant: The
rst lines of the Table 4.6) are above conventional levels.
Suppose now that
ψ ≡ 1.
Then, under the null of no moral hazard,
form distribution. In the Figure 4.7, both empirical distributions of
H1 should be a uni-
H1
(for low and high
BM classes) lie below the diagonal which suggests that agents le claims later in the year
in all bonus-malus classes. This is consistent with the theory's Prediction 2 under moral
hazard for low bonus-malus classes, but violates this prediction for high classes. Moreover,
this apparent anomaly is signicant since the
uniformity of
classes it is
H1
0.002
is
0.015
p-value
of the Kolmogorov-Smirnov test for
for high classes. For low classes, the
p-value
is
0.083;
and for all
(see Table 4.7).
These results should, however, be interpreted with considerable care, because Predictions 2 and 3 correspond to only small eects of
K
on
H1
directions. Therefore, even small external time eects in
and, moreover, work in opposite
ψ
can explain the anomaly and,
together with moral hazard, generate the pattern observed in Figure 4.7. To see this, note
that the estimate of
H1
for high bonus-malus classes lies above that for low classes. Thus,
consistently with Prediction 2, agents in high classes claim
earlier
in the year
relative to
agents in low classes.
By comparing across bonus-malus classes, we have controlled for external time eects.
Another way to control for such eects is to compare rst and second claim times.
4.4.3.3 Marginal Distributions of First and Second Claim Times
Consider the distribution of the second claim time
T2
in the subpopulation with exactly
two claims in the contract year and in one of the bonus-malus classes in
H2 (t|K0 ) = Pr(T2 ≤ t|N (1−) = 2, K(0) ∈ K0 ),
122
K0 ,
4.4.
Table 4.6: Nonparametric Tests Based on Comparison of
EMPIRICAL ANALYSIS
H1
and
H2
for Dierent Bonus-
Malus Classes
Kruskal - Wallis test
H1 (K)
H2 (K)
equal for all
equal for all
p-value
K
K
0.592
0.271
Wilcoxon test
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.006
Kolmogorov - Smirnov test
p-value
0.629
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.676
0.004
Note: This table is computed using a subsample of rst fully-observed contract years from Table 4.4's
sample. The values in
bold imply rejection of the null of no moral hazard at a 5% level.
Low classes are
BM classes 1 10 and high classes are BM classes 11 20.
Table 4.7: Kolmogorov-Smirnov Test Comparing
2
with H1 for Dierent Bonus-Malus Classes
H1
with the Uniform Distribution and
H2
Kolmogorov - Smirnov test
H1 (all K )
H1 (low K )
H1 (high K )
H2 (all K )
H2 (low K )
H2 (high K )
H2 (low K )
H2 (high K )
∼
∼
∼
∼
∼
∼
∼
∼
U nif orm
U nif orm
U nif orm
H12 (all K )
H12 (low K )
H12 (high K )
H12 (high K )
H12 (low K )
p-value
0.002
0.015
0.083
0.524
0.065
0.344
0.149
0.541
Note: This table is computed using a subsample of rst fully-observed contract years from Table 4.4's
sample. The values in
bold imply rejection of the null of no moral hazard at a 5% level.
Low classes are
BM classes 1 10 and high classes are BM classes 11 20.
123
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.7: Comparison of
Ĥ1
with the Uniform Distribution for Low and High Bonus-
Malus Classes
1
Uniform CDF
Empirical H1 for low BM 1 − 10
0.9
Empirical H1 for high BM 11 − 20
0.8
0.7
H1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
124
0.6
0.7
0.8
0.9
1
4.4.
EMPIRICAL ANALYSIS
and its empirical counterpart,
1
n
X
M2,K0 ,n
i=1
0
Ĥ2,n (t|K ) =
I(Ti2 ≤ t, Ni (1−) = 2, Ki (0) ∈ K0 ).
Abbring, Chiappori, and Pinquet's (2003) analysis implies that, under the null of no
moral hazard,
H2 (t|K0 ) = H1 (t|K)2 , for all ψ
and
K, K0 .
They also show that this equality
breaks down under moral hazard, and is likely to do so in one direction. The immediate
implication of this result is that under no moral hazard,
the choice of
K0 .
A Kruskal-Wallis test for the null that
{1, 2, . . . , 20} gives a p-value of 0.271.
low (1 10) and high (11 20).
H2 (t|K0 )
H1 (·|K)
will not depend on
is equal for all
K ∈
The result changes if we group the BM classes into
Then, both Wilcoxon and Kolmogorov-Smirnov tests
reject the null at conventional levels; see Table 4.6.
Another test of moral hazard compares
choices of
K
{1, . . . , 10}
and
K0 .
some evidence that
H2 > H12
and
Ĥ1,n (·|K), Ĥ1,n (·|K)2
Figure 4.8 plots
(low BM classes) and for
Ĥ2,n (·|K0 )
and
K = K0 = {11, . . . , 20}
in low classes, and that
Ĥ1,n (·|K)2
for appropriate
Ĥ2,n (·|K0 )
for
K = K0 =
(high BM classes). We nd
H2 < H12
in high classes.
From
Abbring, Chiappori, and Pinquet's (2003) analysis and Prediction 2, we may expect the
21
opposite rankings under moral hazard.
for
H2 = H12
with dierent choices of
K
However, none of the Kolmogorov-Smirnov tests
and
K0
rejects the null; see Table 4.7. This is
consistent with our ndings from Chapter 3 that nonparametric state-dependence tests,
unlike Section 4.4.2's structural test, have little power with data on rare events.
4.4.3.4 Joint Distribution of First and Second Claim Durations
So far, we have only compared marginal distributions of rst and second claim times.
Intuitively, much can be gained by comparing rst and second claim times within contracts, that is, by studying the
joint
distribution of rst and second claim times. Thus,
21 Abbring, Chiappori, and Pinquet (2003) focus on local behavior near the null. In Chapter 3 we
showed that the global implications are less clear-cut. This may also explain some of this result.
125
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.8: Comparison of
Ĥ1
with the Uniform Distribution and of
and High Bonus-Malus Classes, with
Ĥ1
and
Ĥ2
Ĥ12
with
Ĥ2
Estimated on the Same Classes
Low BM classes 1 − 10
1
0.9
Uniform CDF
Empirical H1
0.8
Empirical H21
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
t
High BM classes 11 − 20
1
0.9
Uniform CDF
Empirical H1
0.8
Empirical H21
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
126
0.6
for Low
4.4.
T1
we compare the time of the rst claim
claim
T2 − T1
EMPIRICAL ANALYSIS
and the time between the rst and the second
in the subpopulation with exactly two claims in the contract year.
Assuming stationarity (ψ
≡ 1)
and under the null of no moral hazard, we have that
Pr(T1 ≥ T2 − T1 |N (1−) = 2, K(0) ∈ K) =
for all
K.
1
2
Under moral hazard, on the other hand, we would expect this probability to
be larger than
smaller than
1/2
1/2
in low classes, where incentives jump down after the rst claim, and
in high classes, where incentives jump up. Note that here we again use
that these jumps in incentives dominate the changes in incentives over time.
Thus, under stationarity (ψ
of contracts in classes in
K
≡ 1),
a test for moral hazard can be based on the share
with two claims for which the time to the rst claim is larger
than the time between the rst and the second,
π̂n (K) =
1
n
X
M2,K,n
i=1
I(Ti1 ≥ Ti2 − Ti1 , Ni (1−) = 2, Ki (0) ∈ K).
Under the null of no moral hazard,
variance
Pk,K
1/(4nK P2,K ), where nK
π̂n (K)
is asymptotically normal with mean
is the total number of contracts in all classes from
is more generally the probability that a contract in a class in
contract year. The variance of
1/2
π̂n (K)
K
has
can be consistently estimated by
k
and
K, and
claims in the
1/(4M2,K,n ).
Another test for moral hazard under stationarity can be based on
[
ln
βn (K) =
1
n
X
M2,K,n
i=1
ln
Ti1
Ti2 − Ti1
I(Ni (1−) = 2, Ki (0) ∈ K)
which is asymptotically normal under the null of no moral hazard, with expectation
variance
π 2 /(3nK P2,K ).
The variance can be consistently estimated by
The rst two columns of Table 4.8 give
standard errors for various choices of
K.
π̂n (K)
and
[
ln
βn (K)
0 and
π 2 /(3M2,K,n ).
with their estimated
The two statistics' values, and their variation
127
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
with classes, are consistent with moral hazard. However, the null of no moral hazard is
not rejected at a 5% level, because the small numbers of observations imply low precision.
This is consistent with our results from Chapter 3 that these tests have limited power
with data on rare events.
Precision can be increased by pooling classes at both ends of the bonus-malus scheme,
but reversing the comparison for the high classes. For example, we can use
π̂n (KL , KH ) =
1
M2,KL ∪KH ,n
n
X
[I(Ti1 ≥ Ti2 − Ti1 , Ni (1−) = 2, Ki (0) ∈ KL ) +
i=1
I(Ti1 ≤ Ti2 − Ti1 , Ni (1−) = 2, Ki (0) ∈ KH )],
with
KL
and
KH
disjoint sets of low and high bonus-malus classes, respectively. Under
moral hazard, we would expect this share to be larger than
1/2.
Therefore, we can use
one-sided test.
The rst two columns of Table 4.9 give the values of
[
ln
βn (KL , KH )
of
[
ln
βn (K).
π̂n (KL , KH ) and a similar variant
We expect the latter to be positive under moral hazard. The
results are again consistent with moral hazard, now with some rejections of the null at a
5% level in very high and very low BM classes.
Abbring, Chiappori, and Pinquet (2003) develop a variant
π̂n (K) that allows for general external time eects ψ .
the transformed durations
classes in
K.
As before,
K
H1 (T1 |K0 )
and
K0
and
π̂n∗ (K|K0 )
of the statistic
Adapted to our setting, it compares
H1 (T2 |K0 ) − H1 (T1 |K0 )
in the subsample with
can be wisely chosen to maximize power.
Proposition 7 in Abbring, Chiappori, and Pinquet implies that, under the null of
no moral hazard,
ance
π̂n∗ (K|K0 )
is asymptotically normal with expectation
1/(4nK P2,K ) + 1/(6nK0 P1,K0 ),
1/(6M1,K0 ,n ).
which can be consistently estimated by
1/2
1/(4M2,K,n ) +
The last two columns of Tables 4.8 and 4.9 plot the values of
π̂n∗
estimated standard errors for dierent bonus-malus classes. First we estimated
all bonus-malus classes (taking
128
and vari-
with the
H1
using
K0 = {1, . . . , 20}) and then using only the tested (current)
4.4.
bonus-malus classes (taking
K0 = K).
The values of
EMPIRICAL ANALYSIS
π̂n∗ (K|K0 ) statistic and their variation
with classes, are again consistent with moral hazard. However, the null of no moral hazard
is not rejected at a 5% level, because the
πn∗
test has lower power than the
πn
test (see
Chapter 3).
4.4.3.5 Claim Sizes
As discussed in Section 4.4.3.1, the jumps in incentives at the time of a claim dominate the
variation in incentives over time. Therefore, in comparing claim sizes within a contract
year, we can focus on Prediction 2's occurrence-dependence eects, and ignore Prediction
3's time eects.
This facilitates a test for ex post moral hazard based on a comparison of the sizes of
agents' rst and second claims in a contract year, even though these occur at dierent
times.
Under ex post moral hazard the size
stochastically larger than the size
L1
L2
of a second claim in a contract year is
of a rst claim in high classes where incentives jump
up after the rst claim. On the other hand, rst claim sizes are stochastically larger than
second claim sizes in low classes.
Under the null of no moral hazard, rst and second claim sizes share the same distribution
F (·|λ).
Table 4.10 reports
p-values
of Wilcoxon and sign tests for this hypothesis
against one-sided and two-sided alternatives, using subsamples of contracts with two or
more claims in dierent bonus-malus classes. They suggest that
L1
and
L2
are not identi-
cally distributed. In particular, the second claim is stochastically larger in the subpopulation in higher classes. This is consistent with ex post moral hazard: Agents in high classes
K
increase their claiming thresholds
x∗
after experiencing a jump up in their incentives
at the time of their rst claim.
129
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.8: Tests Based on Comparison of First and Second Claim Durations, for Dierent
Bonus-Malus Classes
BM
classes
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
62.5% (10.2%)
0.392 (0.370)
54.2% (10.2%)
54.2% (10.9%)
1 2
65.7% (8.5%)
0.570 (0.307)
60.0% (8.5%)
60.0% (8.9%)
1 3
62.2% (7.5%)
0.384 (0.270)
57.8% (7.5%)
57.8% (7.8%)
1 4
55.6% (6.8%)
0.211 (0.247)
51.9% (6.8%)
53.7% (7.1%)
1 5
55.2% (6.1%)
0.180 (0.222)
52.2% (6.1%)
55.2% (6.4%)
1 6
54.3% (5.6%)
0.046 (0.202)
50.6% (5.6%)
54.3% (5.8%)
1 7
53.6% (5.1%)
0.086 (0.184)
50.5% (5.1%)
53.6% (5.3%)
1 8
53.1% (4.7%)
0.085 (0.171)
50.4% (4.7%)
53.1% (4.9%)
1 9
52.3% (4.4%)
0.083 (0.160)
49.2% (4.5%)
50.8% (4.6%)
1 10
53.2% (4.2%)
0.106 (0.154)
50.4% (4.3%)
51.1% (4.4%)
All
52.9% (3.1%)
0.107 (0.112)
50.6% (3.1%)
50.6% (3.1%)
11 20
52.5% (4.5%)
0.109 (0.164)
50.8% (4.6%)
50.8% (4.6%)
12 20
51.8% (4.8%)
0.035 (0.173)
50.0% (4.8%)
50.0% (4.8%)
13 20
51.1% (5.2%)
0.070 (0.187)
50.0% (5.2%)
50.0% (5.2%)
14 20
53.0% (5.5%)
0.132 (0.199)
51.8% (5.5%)
51.8% (5.5%)
15 20
51.4% (6.0%)
0.101 (0.217)
50.0% (6.0%)
50.0% (6.0%)
16 20
48.4% (6.3%)
-0.040 (0.227)
46.9% (6.3%)
46.9% (6.3%)
17 20
45.1% (7.0%)
-0.188 (0.254)
43.1% (7.0%)
43.1% (7.1%)
18 20
47.7% (7.5%)
-0.105 (0.273)
45.5% (7.6%)
47.7% (7.6%)
19 20
44.1% (8.6%)
-0.057 (0.311)
41.2% (8.6%)
44.1% (8.6%)
20
41.4% (9.3%)
-0.112 (0.337)
41.4% (9.3%)
44.8% (9.3%)
1
Note: This table is computed using a subsample of rst fully-observed contract years from Table 4.4's
sample. The values in
bold imply rejection of the null of no moral hazard at a 5% level (two-sided test).
In the computation of
π̂n∗ ,
we rst used all bonus-malus classes to estimate
(current) bonus-malus classes listed in the rst column.
130
H1 ,
and then only the tested
4.4.
EMPIRICAL ANALYSIS
Table 4.9: Tests Based on Comparison of First and Second Claim Durations that Pool
Low and High Bonus-Malus Classes
BM classes
low high
1
Test statistics (std. error)
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
60.4% (6.9%)
0.239 (0.249)
56.6% (6.9%)
56.6% (6.9%)
(6.3%)
0.363 (0.227)
59.4% (6.3%)
59.4% (6.3%)
(6.0%)
0.317 (0.218)
59.4% (6.0%)
59.4% (6.1%)
0.311 (0.204)
57.0% (5.7%)
58.2% (5.7%)
(0.196)
58.1% (5.4%)
58.1% (5.5%)
0.243 (0.204)
58.2% (5.7%)
58.2% (5.7%)
π̂n
20
62.5%
60.9%
59.3%
59.5%
1 2
20
1 2
19 20
1 2
18 20
1 2
17 20
1 3
19 20
1 3
18 20
57.3% (5.3%)
0.246 (0.192)
56.2% (5.3%)
57.3% (5.4%)
1 4
17 20
55.2% (4.9%)
0.200 (0.177)
54.3% (4.9%)
54.3% (4.9%)
1 5
16 20
53.4% (4.4%)
0.112 (0.158)
52.7% (4.4%)
52.7% (4.4%)
1 6
15 20
51.7% (4.1%)
-0.022 (0.148)
50.3% (4.1%)
50.3% (4.1%)
1 7
14 20
50.6% (3.7%)
-0.014 (0.135)
49.4% (3.8%)
49.4% (3.8%)
1 8
13 20
51.2% (3.5%)
0.015 (0.126)
50.2% (3.5%)
50.2% (3.5%)
1 9
12 20
50.4% (3.2%)
0.028 (0.118)
49.6% (3.3%)
49.6% (3.3%)
1 10
11 20
50.6% (3.1%)
0.006 (0.112)
49.8% (3.1%)
49.8% (3.1%)
58.2% (5.6%)
(5.4%)
(5.6%)
0.343
Note: This table is computed using a subsample of rst fully-observed contract years from Table 4.4's
sample. The values in
bold imply rejection of the null of no moral hazard at a 5% level (one-sided test).
In the computation of
π̂n∗ ,
we rst used all bonus-malus classes to estimate
H1 ,
and then only the tested
(current) bonus-malus classes listed in the rst column.
131
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.10: Comparison of First and Second Claim Sizes for Various Bonus-Malus Classes
BM # Wilcoxon test Sign test L ∼ L against
classes obs.
p
L L L ≺ L L 6∼ L
1
-value
1
2
1
2
2
1
1
29
0.336
0.068
0.969
0.136
1 2
41
0.791
0.378
0.734
0.755
1 3
51
0.633
0.500
0.610
1.000
1 4
60
0.802
0.449
0.651
0.897
1 5
74
0.942
0.546
0.546
1.000
1 6
90
0.755
0.701
0.376
0.752
1 7
106
0.458
0.809
0.248
0.497
1 8
122
0.219
0.913
0.120
0.239
1 9
139
0.173
0.913
0.117
0.235
1 10
151
0.240
0.028
0.043
0.035
0.029
0.038
0.873
0.164
0.329
0.090
0.986
0.010
0.010
0.007
0.012
0.011
0.009
0.003
0.027
0.019
0.021
0.014
0.025
0.023
0.019
0.007
0.052
0.104
0.879
0.203
0.405
0.763
0.360
0.720
All
278
11 20
127
12 20
113
13 20
97
14 20
86
15 20
73
16 20
67
17 20
53
18 20
46
0.127
0.973
19 20
36
0.388
20
31
0.493
0.061
0.055
0.993
0.994
0.996
0.993
0.994
0.995
0.998
2
0.053
Note: This table is computed using a subsample of contracts with at least two claims in the rst fullyobserved contract year from the Table 4.4's sample. The values in
moral hazard at a 5% level.
132
bold imply rejection of the null of no
4.5.
4.4.4
CONCLUSION
Claim Withdrawals
So far, we have ignored withdrawn claims. We will now argue that withdrawals are directly
informative on moral hazard, and present some evidence.
Suppose that it takes time for loss amounts to be assessed, so that agents have to le
a claim before the loss amount is fully known. Furthermore, suppose that there are no
costs administrative or informational of ling and withdrawing claims. Then, agents
will report all losses to the insurer to secure an option on compensation, and typically
withdraw those claims for losses that fall below the threshold. Our data on claims and
withdrawals are thus directly informative on ex post moral hazard (withdrawals), and
ex ante moral hazard (initial claims). If we relax our assumptions, some ex post moral
hazard will end up reducing initial claims. In any case, the mere fact that some claims
are withdrawn in the sample points to the evidence of ex post moral hazard.
Under the null of no ex-post moral hazard, agents will claim all accidents to the insurer
and withdraw only those which damage falls below the level of deductible. The agent's
decision whether to withdraw a claim or not will therefore depend only on the size of a
damage and not on the bonus-malus class. Consequently, the shares of withdrawn claims
should be roughly the same among all BM classes.
Figure 4.9 plots the shares of withdrawn claims for each bonus-malus class. We observe
that the shares are small for low and high bonus-malus classes and big for the bonus-malus
classes in between. This is consistent with the incentives to avoid a rst (∆V
and a second (∆V
(1, K, 2))
(1, K, 1))
claim that we presented in the Figure 4.2.
4.5 Conclusion
Putting novel theoretical insights into the dynamic incentives implied by experience rating
to empirical use, we nd evidence of moral hazard in Dutch car insurance. The earlier
literature often fails to nd such evidence.
133
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.9: Share of Withdrawn Claims per Bonus-Malus Class
8%
7%
Withdrawals
6%
Share
5%
4%
3%
2%
1%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
BM class
Note: This graph only includes withdrawals that are directly observed. See Appendix 4.C.
134
20
4.A.
PROOFS OF RESULTS IN SECTION 4.3
APPENDICES TO CHAPTER 4
4.A Proofs of Results in Section 4.3
Lemma 1. The value function V
Proof.
Consider a state
is strictly increasing in wealth W .
(t, W, K, N )
and denote the (stochastic) optimal consumption-
prevention-claim plan following this state by
with
W0 > W,
(c∗ , p∗ , X ∗ ).
Then, in state
the agent can attain an expected discounted utility equal to
by following the same plan
and instantaneous utility
(c∗ , p∗ , X ∗ ).
u
Because consuming
is strictly increasing,
Lemma 2. The value function V
(t, W 0 , K, N )
V (t, W, K, N )
c∗ +ρ(W 0 −W ) > c∗
is feasible
V (t, W 0 , K, N ) > V (t, W, K, N ).
is weakly increasing in the bonus-malus class K and
weakly decreasing in the number of claims at fault N .
Proof.
Consider a state
(t, W, K, N )
and denote the (stochastic) optimal consumption-
prevention-claim plan following this state by
with
K0 ≥ K
and
N0 ≤ N,
(c∗ , p∗ , X ∗ ).
Then, in state
(t, W, K 0 , N 0 )
the agent can attain an expected discounted utility equal to
V (t, W, K, N ) by following the same plan (c∗ , p∗ , X ∗ ).
In this case future insurance premia
are weakly smaller, in the sense of stochastic dominance, than under optimal behavior
in state
in
N,
(t, W, K, N ),
because
B(K, N )
premia are weakly decreasing in
future claims. Therefore, choosing
is weakly increasing in
K,
K
and weakly decreasing
and the agent faces the same distribution of
(c∗ , p∗ , X ∗ ) in state (t, W, K, N ) is feasible and, indeed,
V (t, W, K 0 , N 0 ) ≥ V (t, W, K, N ).
Proof of Proposition 5.
First, note that the proposition's specications of the consump-
tion rule and value function satisfy the Euler equation:
u0 (c∗ (S)) = e−αρ[W −Q(t,K,N )] = VW (S)
135
CHAPTER 4.
Second, note that
MORAL HAZARD IN DYNAMIC INSURANCE DATA
ρV (S) = u(c∗ (S)),
so that Bellman equation (4.4) is satised if
h
i
0 = p0 V (t, W, K, N + 1) − V (S) + VW (S) [ρW − c∗ (S) − A(K)] + Vt (S).
Because
−αρ[W −Q(t,K,N )]
V (t, W, K, N + 1) − V (S) = e
1 − eαρ[Q(t,K,N +1)−Q(t,K,N )]
αρ
VW (S) [ρW − c∗ (S) − A(K)] = e−αρ[W −Q(t,K,N )] [ρQ(t, K, N ) − A(K)] ,
,
and
Vt (S) = −e−αρ[W −Q(t,K,N )] Qt (t, K, N ),
this is guaranteed by equation (4.10). Third, the Bellman equation's premium renewal
conditions (4.3) are satised by equation (4.11):
1 − e−αρ[W −Q(1,K,N )]
αρ
−αρ[W −Q(0,B(K,N ),0)]
1−e
=
= V (0, W, B(K, N ), 0).
αρ
V (1, W, K, N ) =
Finally, using standard methods it can be proved that there exists a unique solution
Q to
the system (4.10)(4.11).
4.B Computation of Proposition 5's Function Q
Let
A
and
B
be given by Table 4.1 and attach some values to the parameters
ρ, α ,
and
p0 .
In the limiting case
α ↓ 0, the corresponding initial-value problem (4.10)(4.11) has an
explicit analytical solution
136
Q.
In particular, the initial values
Q(0, ·, 0)
can be computed
4.B.
COMPUTATION OF PROPOSITION 5'S FUNCTION
Q
directly using
Here,
I
is the




 Q(0, 1, 0)

.

.
.


Q(0, 20, 0)
 π(1) 

 . 
 1 − e−ρ

=
.
(I − e−ρ T )−1 
 . .

ρ



π(20)
20 × 20
identity matrix and
trix among bonus-malus classes implied by
T
p0
4.16
is the annual transition probability maand
B.
The solution
Q
then satises the
recursive system
π(K)
π(K)
−ρ(1−t)
Q(t, K, N ) =
Q(0, 1, 0) −
+e
ρ
ρ
for
N ≥ 3,
Q(t, K, 2) = Q(t, K, 3) + e−(p0 +ρ)(1−t) [Q(0, B(K, 2), 0) − Q(0, 1, 0)],
Q(t, K, 1) = Q(t, K, 2) + e−(p0 +ρ)(1−t) {Q(0, B(K, 1), 0) − Q(0, B(K, 2), 0)
+p0 (1 − t)[Q(0, B(K, 2), 0) − Q(0, 1, 0)]},
and
Q(t, K, 0) = Q(t, K, 1) + e−(p0 +ρ)(1−t) {Q(0, B(K, 0), 0) − Q(0, B(K, 1), 0)
+p0 (1 − t)[Q(0, B(K, 1), 0) − Q(0, B(K, 2), 0)]
1
+ p20 (1 − t)2 [Q(0, B(K, 2), 0) − Q(0, 1, 0)]}.
2
In the general case, the function
Q
can be computed iteratively using
Algorithm 1. Give starting values to Q(0, K, 0), K = 1, . . . , 20, and repeat
1. set Q∗ (0, K, 0) = Q(0, K, 0), K = 1, . . . , 20;
2. for K = 1, . . . , 20,
(a) for N ≥ 3, set
π(K)
π(K)
−ρ(1−t)
Q(t, K, N ) =
+e
Q(0, 1, 0) −
;
ρ
ρ
137
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
(b) for N = 2, 1, 0, set Q(·, K, N ) to the numerical solution of the corresponding
single-equation initial-value problem in (4.10)(4.11);
until maxK |Q(0, K, 0) − Q∗ (0, K, 0)| ≤ ε, for some small ε > 0.
The values
Q(0, ·, 0)
for the linear-utility case, those that satisfy (4.16), can be used
as starting values in Algorithm 1. Note that in the linear-utility case itself, this produces
Q
in one iteration; in cases with
to compute
Q
α > 0,
for multiple values of
α,
more iterations are typically needed. If we have
we can use the linear-utility values of
starting values for the computations with the lowest value of
α,
the resulting
starting values for the computations with the second-lowest value of
α,
Q(0, ·, 0)
as
Q(0, ·, 0)
as
etcetera.
4.C Data
Recall from Section 4.2.2 that the data provide contract and claim histories of personal
car insurance clients of a major Dutch insurer from January 1, 1995 to December 31,
2000.
All data, except information on claim withdrawals, came in a single le with
1,730,559 records for 163,194 unique contracts. A second le provided information on the
withdrawal of claims by agents after they were led. Recall from Section 4.2 that agents
had the option to avoid a malus after ling a claim by timely withdrawing it.
We excluded information on the year 1995, because it lacked information on claims.
From the remaining 142,175 contracts, we deleted 1,376 contracts that were not subject
22
to the bonus-malus system.
We also deleted 16,778 contracts with unobserved renewal
date. Most of these contracts started in 1995 and did not renew in 1996. Many were also
short-term contracts covering only a couple of weeks or months.
This left a sample of
124,021 contracts.
We matched the second le's withdrawal information to the main data set based on
contract and claim identiers, but this matching was not complete. This is important,
22 We will use this raw sample, consisting of 140,799 contracts, in the last chapter to give some
motivation for the future research.
138
4.C.
DATA
because consistency of the claim and BM information is crucial to this chapter's empirical
analysis. The remainder of this section discusses the ways we enforced such consistency
by correcting the claim withdrawal and BM information, and checked for our empirical
work's robustness to these corrections.
First, few BM transitions in 2000 were recorded correctly. Therefore, we truncated all
contract histories that were renewed in 2000 at the 2000 renewal date. This cut another
219 contracts that were rst observed in 2000 from the sample, leaving 123,802 unique
contracts.
Of these 123,802 contracts, 103,930 are observed for more than one year.
For each
such contract we observe the sequence of BM classes in consecutive contract years, with
the number of claims at fault that were not withdrawn in each contract year. For 14,206
contracts, we observed one or more deviations from Table 4.1's BM updating rule.
Many of these deviations can be explained by unobserved withdrawals, which may
exist because observed withdrawals could not be perfectly matched to the main data le.
For example, some contracts were awarded a bonus after a contract year with a claim.
This is only consistent with the BM system if the claim was withdrawn. Therefore, we
decided to treat those claims as (unobserved) withdrawals.
Consequently, we excluded
them from the sample. All in all, we found 1,355 unobserved withdrawals in the sample
constructed so far.
Even after excluding unobserved withdrawals, the sample still contained incorrect BM
transitions. We corrected these anomalies by constructing the most appropriate BM class
for the rst contract year that is, the class that minimized changes to the raw data and deriving the BM classes in all consecutive contract years from this initial BM class
and claims using the BM updating rule.
Of the 14,325 contracts observed for two years only, 1,269 have an incorrect BM
transition.
In these 1,269 cases, we simply set the BM class in the second year to be
consistent with the BM class and the number of claims observed in the rst year.
139
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
For most contracts with inconsistencies observed for more than 2 years, a BM sequence
based on the rst year's BM class delivered the best t to the observed BM sequence. A
single inconsistency in the middle of a BM sequence, sandwiched between consistent BM
classes, also often occurred.
Then, we simply corrected this single BM class using the
previous year's BM class and claim information.
In some cases with more than two years of data, the rst year's BM class was inconsistent with the BM classes in all later years. Then, we forced the rst year's BM class to
be consistent with the rst BM class later in the sequence that was consistent with later
BM classes and claims. Because the BM updating rule in Table 4.1 is not a one-to-one
mapping, there were often more consistent choices of a rst BM class. For example, an
agent who was downgraded to BM class 1 in the second contract year after having one
claim in the rst, could have been in any of the BM classes 1 5 in the rst year. In
these cases, we chose the highest consistent BM class. In a very few cases we were not
able to correct the rst year's BM class this way. For example, no choice of a rst year's
BM class is consistent with a claim in the rst year and a BM class 15 or higher in the
second year. We deleted 12 such contracts, observed for 4 years, from the sample.
Finally, we deleted 621 contracts that were observed for more than 2 years and had
only inconsistent BM transitions. This leaves a nal sample with 123,169 unique contracts
and 23,396 claims at fault that were not withdrawn. All empirical results reported in this
chapter are based on this sample.
We checked the robustness of these results with respect to the ways we have selected the
sample and corrected the BM classes and claim withdrawal information by recomputing
all results on dierent samples employing dierent ways of correcting for inconsistencies.
First, we used an alternative sample that included only observations with consistent raw
data on claims and BM classes. No corrections were applied to these data. The results
are presented in Appendix 4.E. Second, we used a sample that was alternatively corrected
for inconsistent BM information by deriving all BM sequences from the initial BM classes
140
4.C.
DATA
observed, using the BM updating rule; see Appendix 4.F for the results. Third, we used
the main, corrected sample, but included all withdrawn claims as claim-at-fault events;
see Appendix 4.G for the outcome.
We found that all results reported in this chapter are robust.
141
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
4.D Main (Corrected) Sample without Young Drivers
Tables and gures numbered 4.D.1 and up report the results of redoing all analyses on a
subsample of insurees of 28 years old and up of the main sample (see Section 4.4.1). Like
in the main analysis, all claims observed to be withdrawn are excluded.
142
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Table 4.D.3: Contract Exposure Durations in the Sample
Number of years
Y
1
Number of contracts observed
exactly Y years between Y − 1 and Y years
Total
8,074
11,389
19,463
2
5,107
9,329
14,436
3
6,576
7,111
13,687
4
66,654
6,144
72,798
86,411
33,973
Total
120,384
Table 4.D.4: Number of Contracts Observed for At Least One Full Contract Year, by
Bonus-Malus Class and Number of Claims in the First Contract Year
BM
Number of contracts with
class no claim 1 claim 2 claims 3 claims 4 claims
1
Total
1
519
112
23
4
2
704
84
11
1
659
3
902
75
10
987
4
1,266
96
8
1,370
5
1,682
101
9
2
1,794
6
2,330
146
13
2
2,491
7
3,142
192
14
3,348
8
3,857
252
15
4,124
800
9
4,584
235
14
10
6,018
274
11
2
11
5,763
262
10
12
5,895
287
16
6,198
13
5,807
258
10
6,075
14
6,683
311
13
7,007
15
6,241
298
7
6,546
16
6,429
297
13
17
5,692
250
7
5,949
18
4,380
202
10
4,592
2
3,868
213
5
20
27,651
1,372
28
2
103,413
5,317
247
16
6,304
6,037
1
19
Total
4,835
1
6,740
4,086
29,053
2
108,995
143
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.D.1: Distribution of Contracts Observed for At Least One Full Contract Year
Across Bonus-Malus Classes; and Shares of Those Contracts with At Least One and At
Least Two Claims at Fault in the First Contract Year, by Bonus-Malus Class
30%
25%
All contracts
Contracts with at least 1 claim
Contracts with at least 2 claims
Share
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10
11
BM class
144
12
13
14
15
16
17
18
19
20
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Figure 4.D.2: Incentives to Avoid First, Second and Third Claim; at an Average Risk
Level
5
∆ V(1,K,1)
∆ V(1,K,2)
∆ V(1,K,3)
4.5
4
3.5
∆V
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
145
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.D.3: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at an Average Risk Level
4
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
∆2 V
1
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
Figure 4.D.4: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a Zero Risk Level
unchanged
146
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Figure 4.D.5: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a High Risk Level
3
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
2
1
∆2 V
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
147
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.D.5: ML Estimation of the Auxiliary Model (4.14) with 3 and 4 Support Points
Three Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
0.4193
0.0390
0.0393
0.0032
0.2023
0.0197
0.4172
0.0322
Tests of
β=0
•
LM test: 120.11,
p-value
= 0.00
•
LR test: 103.66,
p-value
= 0.00
•
Wald test: 115.75,
p-value
= 0.00
Four Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
Table 4.D.6:
0.3795
0.0342
0.0000
0.0000
0.0551
0.0080
0.2240
0.0124
0.8919
0.1273
Tests of
β=0
•
LM test: 57.25,
•
LR test: 100.75,
•
Wald test: 123.05,
Nonparametric Tests Based on Comparison of
H1
Bonus-Malus Classes
Kruskal - Wallis test
H1 (K)
H2 (K)
equal for all
equal for all
K
K
Wilcoxon test
0.736
0.285
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.005
Kolmogorov - Smirnov test
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
148
p-value
0.370
0.433
0.002
p-value
and
p-value
= 0.00
= 0.00
p-value
H2
= 0.00
for Dierent
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Figure 4.D.7: Comparison of
Ĥ1
with the Uniform Distribution for Low and High Bonus-
Malus Classes
1
Uniform CDF
Empirical H1 for low BM 1 − 10
0.9
Empirical H1 for high BM 11 − 20
0.8
0.7
H1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Table 4.D.7: Kolmogorov-Smirnov Test Comparing
2
with H1 for Dierent Bonus-Malus Classes
H1
with the Uniform Distribution and
H2
Kolmogorov - Smirnov test
H1 (all K )
H1 (low K )
H1 (high K )
H2 (all K )
H2 (low K )
H2 (high K )
H2 (low K )
H2 (high K )
∼
∼
∼
∼
∼
∼
∼
∼
U nif orm
U nif orm
U nif orm
H12 (all K )
H12 (low K )
H12 (high K )
H12 (high K )
H12 (low K )
p-value
0.003
0.018
0.052
0.649
0.055
0.250
0.107
0.515
149
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.D.8: Comparison of
Ĥ1
with the Uniform Distribution and of
Low and High Bonus-Malus Classes, with
Ĥ1
and
Ĥ2
Ĥ12
with
Estimated on the Same Classes
Low BM classes 1 − 10
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
t
High BM classes 11 − 20
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
150
0.6
Ĥ2
for
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Table 4.D.8: Tests Based on Comparison of First and Second Claim Durations, for Different Bonus-Malus Classes
BM
classes
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
60.9% (10.4%)
0.408 (0.378)
56.5% (10.4%)
56.5% (11.1%)
1 2
64.7% (8.6%)
0.586 (0.311)
61.8% (8.6%)
61.8% (9.1%)
1 3
61.4% (7.5%)
0.392 (0.273)
59.1% (7.6%)
59.1% (7.9%)
1 4
53.8% (6.9%)
0.196 (0.252)
51.9% (7.0%)
51.9% (7.3%)
1 5
54.1% (6.4%)
0.177 (0.232)
52.5% (6.4%)
54.1% (6.7%)
1 6
52.7% (5.8%)
0.033 (0.211)
51.4% (5.8%)
51.4% (6.0%)
1 7
51.1% (5.3%)
0.061 (0.193)
50.0% (5.4%)
50.0% (5.5%)
1 8
52.4% (4.9%)
0.090 (0.179)
51.5% (5.0%)
53.4% (5.1%)
1 9
52.1% (4.6%)
0.092 (0.168)
50.4% (4.7%)
49.6% (4.8%)
1 10
53.1% (4.4%)
0.117 (0.160)
51.6% (4.5%)
51.6% (4.5%)
1
All
53.4% (3.2%)
0.137 (0.115)
51.4% (3.2%)
51.4% (3.2%)
11 20
53.8% (4.6%)
0.158 (0.166)
51.3% (4.6%)
51.3% (4.6%)
12 20
52.3% (4.8%)
0.075 (0.174)
49.5% (4.8%)
49.5% (4.8%)
13 20
51.6% (5.2%)
0.117 (0.188)
49.5% (5.2%)
49.5% (5.2%)
14 20
51.8% (5.5%)
0.148 (0.199)
50.6% (5.5%)
50.6% (5.5%)
15 20
51.4% (6.0%)
0.124 (0.217)
50.0% (6.0%)
50.0% (6.0%)
16 20
47.6% (6.3%)
-0.042 (0.229)
46.0% (6.3%)
46.0% (6.4%)
17 20
44.0% (7.1%)
-0.192 (0.257)
42.0% (7.1%)
42.0% (7.1%)
18 20
46.5% (7.6%)
-0.109 (0.277)
44.2% (7.6%)
46.5% (7.7%)
19 20
42.4% (8.7%)
-0.060 (0.316)
39.4% (8.7%)
42.4% (8.8%)
20
39.3% (9.4%)
-0.118 (0.343)
39.3% (9.5%)
42.9% (9.5%)
151
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.D.9: Tests Based on Comparison of First and Second Claim Durations that Pool
Low and High Bonus-Malus Classes
BM classes
low high
1
20
1 2
20
1 2
19 20
1 2
18 20
1 2
17 20
1 3
19 20
1 3
18 20
1 4
1 5
Test statistics (std. error)
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
60.8% (7.0%)
0.249 (0.254)
58.8% (7.0%)
58.8% (7.1%)
(6.4%)
0.375 (0.230)
(6.1%)
0.327 (0.222)
58.4% (5.7%)
0.320 (0.207)
π̂n
62.9%
61.2%
59.5%
59.7%
(5.5%)
(5.7%)
0.352
(0.198)
0.250 (0.207)
61.3%
61.2%
59.5%
59.7%
(6.4%)
(6.1%)
58.4% (5.7%)
(5.5%)
(5.7%)
(6.4%)
(6.2%)
(5.8%)
(5.5%)
(5.8%)
57.5% (5.4%)
0.252 (0.194)
17 20
54.9% (5.0%)
0.194 (0.180)
54.9% (5.0%)
54.9% (5.0%)
16 20
53.2% (4.5%)
0.108 (0.163)
53.2% (4.5%)
53.2% (4.6%)
1 6
15 20
50.7% (4.2%)
-0.043 (0.151)
50.7% (4.2%)
50.7% (4.2%)
1 7
14 20
49.7% (3.8%)
-0.040 (0.139)
49.7% (3.9%)
49.7% (3.9%)
1 8
13 20
50.5% (3.6%)
-0.008 (0.130)
51.0% (3.6%)
51.0% (3.6%)
1 9
12 20
50.0% (3.3%)
0.012 (0.121)
50.4% (3.4%)
50.4% (3.4%)
1 10
11 20
49.8% (3.2%)
-0.015 (0.115)
50.2% (3.2%)
50.2% (3.2%)
152
57.5% (5.4%)
61.3%
61.2%
59.7%
59.5%
59.7%
57.5% (5.4%)
4.D.
MAIN (CORRECTED) SAMPLE WITHOUT YOUNG DRIVERS
Table 4.D.10:
Comparison of First and Second Claim Sizes for Various Bonus-Malus
Classes
BM # Wilcoxon test Sign test L ∼ L against
classes obs.
p
L L L ≺ L L 6∼ L
1
-value
1
2
1
2
2
1
1
28
0.466
0.092
0.956
0.185
1 2
40
0.628
0.437
0.682
0.875
1 3
50
0.490
0.556
0.556
1.000
1 4
58
0.667
0.448
0.653
0.896
1 5
69
0.888
0.595
0.500
1.000
1 6
84
0.656
0.707
0.372
0.744
1 7
98
0.570
0.760
0.307
0.614
1 8
113
0.371
0.827
0.226
0.452
1 9
129
0.344
0.811
0.241
0.481
1 10
141
0.446
0.750
0.307
0.614
0.018
0.007
0.009
0.016
0.020
0.017
0.005
0.035
0.037
0.015
0.018
0.032
0.040
0.034
0.009
0.068
0.135
0.845
0.250
0.500
0.708
0.428
0.856
All
265
0.075
0.987
11 20
124
0.099
0.996
12 20
112
0.092
0.995
13 20
96
0.078
0.991
14 20
86
0.064
0.989
15 20
73
0.100
0.991
16 20
66
0.056
0.998
17 20
52
0.131
0.982
18 20
45
0.185
0.964
19 20
35
0.534
20
30
0.688
2
0.070
153
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
4.E Cleaned Data
Tables and gures numbered 4.E.1 and up report the results of redoing all analyses with
a sample that includes only observations with consistent raw information on claims and
bonus-malus classes (see Appendix 4.C). No corrections are applied to these data. Like
in the main analysis, all claims observed to be withdrawn are excluded.
154
4.E.
CLEANED DATA
Table 4.E.3: Contract Exposure Durations in the Sample
Number of years
Y
1
Number of contracts observed
exactly Y years between Y − 1 and Y years
Total
8,097
11,775
19,872
2
4,296
8,760
13,056
3
5,487
6,560
12,047
4
59,183
5,438
64,621
77,063
32,533
Total
109,596
Table 4.E.4: Number of Contracts Observed for At Least One Full Contract Year, by
Bonus-Malus Class and Number of Claims in the First Contract Year
BM
Number of contracts with
class no claim 1 claim 2 claims 3 claims 4 claims Total
1
363
96
19
1
479
2
570
78
11
1
660
3
788
66
9
863
4
1,059
88
8
1,155
5
1,483
94
9
1
1,587
6
2,019
126
10
2
2,157
7
2,754
172
16
2,942
8
3,570
222
13
3,805
9
4,122
204
14
10
5,574
264
10
2
11
5,197
225
8
12
5,235
255
15
5,505
13
4,777
221
10
5,008
14
5,565
244
9
5,818
15
5,360
262
4
5,626
16
5,600
261
11
5,872
17
4,997
223
5
5,225
18
3,842
177
8
4,027
2
19
3,606
183
4
20
26,424
1,224
26
2
92,905
4,685
219
11
Total
4,342
1
5,849
5,432
3,793
27,676
1
97,821
155
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.E.1: Distribution of Contracts Observed for At Least One Full Contract Year
Across Bonus-Malus Classes; and Shares of Those Contracts with At Least One and At
Least Two Claims at Fault in the First Contract Year, by Bonus-Malus Class
30%
25%
All contracts
Contracts with at least 1 claim
Contracts with at least 2 claims
Share
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10
11
BM class
156
12
13
14
15
16
17
18
19
20
4.E.
CLEANED DATA
Figure 4.E.2: Incentives to Avoid First, Second and Third Claim; at an Average Risk
Level
5
∆ V(1,K,1)
∆ V(1,K,2)
∆ V(1,K,3)
4.5
4
3.5
∆V
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
157
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.E.3: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at an Average Risk Level
4
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
∆2 V
1
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
Figure 4.E.4: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a Zero Risk Level
unchanged
158
4.E.
CLEANED DATA
Figure 4.E.5: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a High Risk Level
2
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
1
∆2 V
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
159
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.E.5: ML Estimation of the Auxiliary Model (4.14) with 3 and 4 Support Points
Three Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
0.4203
0.0528
0.0268
0.0051
0.1869
0.0375
0.3023
0.0156
Tests of
β=0
•
LM test: 5.70,
•
LR test: 33.08,
•
Wald test: 63.47,
p-value
= 0.02
p-value
= 0.00
p-value
= 0.00
Four Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
0.3688
0.0417
0.0000
0.0000
0.0843
0.0091
0.1992
0.0147
0.5643
0.0881
Tests of
β=0
•
LM test: 16.13,
p-value
= 0.00
•
LR test: 50.10,
p-value
= 0.00
•
Wald test: 78.23,
Table 4.E.6: Nonparametric Tests Based on Comparison of
Kruskal - Wallis test
equal for all
equal for all
K
K
Wilcoxon test
160
p-value
0.406
0.463
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.036
Kolmogorov - Smirnov test
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
= 0.00
H1 and H2 for Dierent Bonus-
Malus Classes
H1 (K)
H2 (K)
p-value
0.336
0.433
0.054
4.E.
Figure 4.E.7: Comparison of
Ĥ1
CLEANED DATA
with the Uniform Distribution for Low and High Bonus-
Malus Classes
1
Uniform CDF
Empirical H1 for low BM 1 − 10
0.9
Empirical H1 for high BM 11 − 20
0.8
0.7
H1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Table 4.E.7: Kolmogorov-Smirnov Test Comparing
2
with H1 for Dierent Bonus-Malus Classes
H1
with the Uniform Distribution and
H2
Kolmogorov - Smirnov test
H1 (all K )
H1 (low K )
H1 (high K )
H2 (all K )
H2 (low K )
H2 (high K )
H2 (low K )
H2 (high K )
∼
∼
∼
∼
∼
∼
∼
∼
U nif orm
U nif orm
U nif orm
H12 (all K )
H12 (low K )
H12 (high K )
H12 (high K )
H12 (low K )
p-value
0.012
0.219
0.082
0.542
0.109
0.660
0.215
0.746
161
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.E.8: Comparison of
Ĥ1
with the Uniform Distribution and of
Low and High Bonus-Malus Classes, with
Ĥ1
and
Ĥ2
Ĥ12
with
Estimated on the Same Classes
Low BM classes 1 − 10
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
t
High BM classes 11 − 20
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
162
0.6
Ĥ2
for
4.E.
CLEANED DATA
Table 4.E.8: Tests Based on Comparison of First and Second Claim Durations, for Different Bonus-Malus Classes
BM
classes
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
57.9% (11.5%)
0.356 (0.416)
47.4% (11.5%)
47.4% (12.2%)
1 2
63.3% (9.1%)
0.577 (0.331)
56.7% (9.1%)
56.7% (9.6%)
1 3
59.0% (8.0%)
0.361 (0.290)
53.8% (8.0%)
53.8% (8.4%)
1 4
53.2% (7.3%)
0.177 (0.265)
48.9% (7.3%)
48.9% (7.6%)
1 5
53.6% (6.7%)
0.141 (0.242)
50.0% (6.7%)
51.8% (7.0%)
1 6
50.0% (6.2%)
-0.042 (0.223)
47.0% (6.2%)
47.0% (6.4%)
1 7
50.0% (5.5%)
0.023 (0.200)
47.6% (5.6%)
48.8% (5.7%)
1 8
50.5% (5.1%)
0.082 (0.186)
48.4% (5.2%)
47.4% (5.3%)
1 9
50.5% (4.8%)
0.089 (0.174)
47.7% (4.8%)
46.8% (4.9%)
1 10
51.3% (4.6%)
0.113 (0.166)
48.7% (4.6%)
49.6% (4.7%)
1
All
51.6% (3.4%)
0.094 (0.123)
49.3% (3.4%)
49.3% (3.4%)
11 20
52.0% (5.0%)
0.072 (0.181)
50.0% (5.0%)
49.0% (5.1%)
12 20
52.2% (5.2%)
0.045 (0.189)
50.0% (5.2%)
50.0% (5.3%)
13 20
50.6% (5.7%)
0.088 (0.207)
49.4% (5.7%)
49.4% (5.8%)
14 20
52.2% (6.1%)
0.159 (0.222)
50.7% (6.1%)
50.7% (6.2%)
15 20
51.7% (6.6%)
0.111 (0.238)
50.0% (6.6%)
50.0% (6.6%)
16 20
50.0% (6.8%)
0.006 (0.247)
48.1% (6.8%)
48.1% (6.9%)
17 20
46.5% (7.6%)
-0.165 (0.277)
44.2% (7.6%)
44.2% (7.7%)
18 20
47.4% (8.1%)
-0.135 (0.294)
44.7% (8.1%)
47.4% (8.2%)
19 20
43.3% (9.1%)
-0.106 (0.331)
40.0% (9.1%)
43.3% (9.2%)
20
42.3% (9.8%)
-0.143 (0.356)
42.3% (9.8%)
42.3% (9.9%)
163
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.E.9: Tests Based on Comparison of First and Second Claim Durations that Pool
Low and High Bonus-Malus Classes
BM classes
low high
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
1
20
57.8% (7.5%)
0.233 (0.270)
53.3% (7.5%)
55.6% (7.5%)
1 2
20
60.7% (6.7%)
0.375 (0.242)
57.1% (6.7%)
58.9% (6.8%)
1 2
19 20
60.0% (6.5%)
0.341 (0.234)
58.3% (6.5%)
60.0% (6.5%)
1 2
18 20
57.4% (6.1%)
0.330 (0.220)
55.9% (6.1%)
57.4% (6.1%)
1 2
17 20
57.5% (5.9%)
0.334 (0.212)
56.2% (5.9%)
56.2% (5.9%)
1 3
19 20
58.0% (6.0%)
0.250 (0.218)
56.5% (6.0%)
58.0% (6.1%)
1 3
18 20
55.8% (5.7%)
0.250 (0.207)
54.5% (5.7%)
55.8% (5.8%)
1 4
17 20
53.3% (5.3%)
0.171 (0.191)
52.2% (5.3%)
52.2% (5.3%)
1 5
16 20
51.8% (4.8%)
0.069 (0.173)
50.9% (4.8%)
50.9% (4.8%)
1 6
15 20
49.2% (4.5%)
-0.074 (0.163)
48.4% (4.5%)
48.4% (4.6%)
1 7
14 20
49.0% (4.1%)
-0.059 (0.149)
48.3% (4.1%)
48.3% (4.2%)
1 8
13 20
50.0% (3.8%)
0.005 (0.138)
49.4% (3.9%)
49.4% (3.9%)
1 9
12 20
49.3% (3.5%)
0.027 (0.128)
48.8% (3.6%)
48.3% (3.6%)
1 10
11 20
49.8% (3.4%)
0.029 (0.123)
49.3% (3.4%)
49.3% (3.4%)
164
4.E.
Table 4.E.10:
CLEANED DATA
Comparison of First and Second Claim Sizes for Various Bonus-Malus
Classes
BM # Wilcoxon test Sign test L ∼ L against
classes obs.
p
L L L ≺ L L 6∼ L
1
-value
1
2
1
2
2
1
1
20
0.296
0.058
0.979
0.115
1 2
32
0.736
0.430
0.702
0.860
1 3
41
0.781
0.500
0.622
1.000
1 4
49
0.846
0.500
0.612
1.000
1 5
59
0.916
0.603
0.500
1.000
1 6
71
0.705
0.762
0.318
0.635
1 7
87
0.412
0.858
0.196
0.391
1 8
100
0.203
0.933
0.097
0.193
1 9
116
0.153
0.943
0.082
0.163
1 10
127
0.195
0.021
0.022
0.024
0.011
0.015
0.013
0.922
0.107
0.214
0.087
0.992
All
231
11 20
104
12 20
94
13 20
79
14 20
69
15 20
60
16 20
56
17 20
45
18 20
40
0.088
0.992
19 20
32
0.246
0.945
20
28
0.495
0.828
0.051
0.998
0.998
0.999
0.998
0.999
0.999
1.000
0.003
0.004
0.001
0.003
0.002
0.001
0.001
0.018
0.019
0.006
0.008
0.003
0.007
0.004
0.003
0.002
0.036
0.038
0.286
0.572
0.108
2
0.215
165
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
4.F Sample Corrected Based on Initial Bonus-Malus
Class
Tables and gures numbered 4.F.1 and up report the results of redoing all analyses on a
sample constructed using an alternatively correction for inconsistent bonus-malus classes
(see Appendix 4.C).
This alternative correction takes the initial bonus-malus class ob-
served as given and constructs all further bonus-malus classes from observed claims and
withdrawals data.
excluded.
166
Like in the main analysis, all claims observed to be withdrawn are
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Table 4.F.3: Contract Exposure Durations in the Sample
Number of years
Y
1
Number of contracts observed
exactly Y years between Y − 1 and Y years
Total
8,097
11,775
19,872
2
4,709
9,616
14,325
3
6,428
7,572
14,000
4
69,059
6,546
75,605
88,293
35,509
Total
123,802
Table 4.F.4: Number of Contracts Observed for At Least One Full Contract Year, by
Bonus-Malus Class and Number of Claims in the First Contract Year
BM
Number of contracts with
class no claim 1 claim 2 claims 3 claims 4 claims
1
Total
1
475
230
45
7
2
745
126
16
1
758
3
948
115
11
1,074
4
1,295
149
13
1,457
5
1,796
162
16
1
1,975
6
2,431
261
23
2
2,717
7
3,347
306
18
3,671
8
4,227
326
22
4,575
888
9
4,842
314
15
10
6,473
351
12
2
11
6,039
330
10
12
6,003
349
17
6,369
13
5,504
287
10
5,801
14
6,452
588
14
7,054
15
6,220
297
6
6,523
16
6,432
300
13
17
5,753
253
7
6,013
18
4,426
205
10
4,641
2
3,922
210
4
20
27,832
1,375
29
2
105,162
6,534
311
18
6,837
6,381
1
19
Total
5,173
1
6,746
4,136
29,238
2
112,027
167
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.F.1: Distribution of Contracts Observed for At Least One Full Contract Year
Across Bonus-Malus Classes; and Shares of Those Contracts with At Least One and At
Least Two Claims at Fault in the First Contract Year, by Bonus-Malus Class
30%
25%
All contracts
Contracts with at least 1 claim
Contracts with at least 2 claims
Share
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10
11
BM class
168
12
13
14
15
16
17
18
19
20
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Figure 4.F.2: Incentives to Avoid First, Second and Third Claim; at an Average Risk
Level
5
∆ V(1,K,1)
∆ V(1,K,2)
∆ V(1,K,3)
4.5
4
3.5
∆V
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
169
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.F.3: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at an Average Risk Level
4
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
∆2 V
1
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
Figure 4.F.4: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a Zero Risk Level
unchanged
170
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Figure 4.F.5: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a High Risk Level
1.5
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
1
0.5
0
∆2 V
−0.5
−1
−1.5
−2
−2.5
−3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
171
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.F.5: ML Estimation of the Auxiliary Model (4.14) with 3 and 4 Support Points
Three Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
0.3201
0.0435
0.0512
0.0037
0.1987
0.0240
0.3319
0.0136
Tests of
β=0
•
LM test: 5.47,
•
LR test: 36.92,
•
Wald test: 54.26,
p-value
= 0.02
p-value
= 0.00
p-value
= 0.00
Four Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
0.3356
0.0339
0.0000
0.0000
0.0644
0.0066
0.2070
0.0117
0.3376
0.0123
Tests of
β=0
•
LM test: 21.79,
p-value
= 0.00
•
LR test: 11.74,
p-value
= 0.00
•
Wald test: 98.28,
Table 4.F.6: Nonparametric Tests Based on Comparison of
Kruskal - Wallis test
equal for all
equal for all
K
K
Wilcoxon test
172
p-value
0.619
0.414
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.014
Kolmogorov - Smirnov test
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
= 0.00
H1 and H2 for Dierent Bonus-
Malus Classes
H1 (K)
H2 (K)
p-value
0.287
0.530
0.017
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Figure 4.F.7: Comparison of
Ĥ1
with the Uniform Distribution for Low and High Bonus-
Malus Classes
1
Uniform CDF
Empirical H1 for low BM 1 − 10
0.9
Empirical H1 for high BM 11 − 20
0.8
0.7
H1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Table 4.F.7: Kolmogorov-Smirnov Test Comparing
H2 with H12 for Dierent Bonus-Malus Classes
H1
with the Uniform Distribution and
Kolmogorov - Smirnov test
H1 (all K )
H1 (low K )
H1 (high K )
H2 (all K )
H2 (low K )
H2 (high K )
H2 (low K )
H2 (high K )
∼
∼
∼
∼
∼
∼
∼
∼
U nif orm
U nif orm
U nif orm
H12 (all K )
H12 (low K )
H12 (high K )
H12 (high K )
H12 (low K )
p-value
0.014
0.015
0.867
0.601
0.152
0.489
0.133
0.380
173
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.F.8: Comparison of
Ĥ1
with the Uniform Distribution and of
Low and High Bonus-Malus Classes, with
Ĥ1
and
Ĥ2
Ĥ12
with
Estimated on the Same Classes
Low BM classes 1 − 10
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
t
High BM classes 11 − 20
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
174
0.6
Ĥ2
for
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Table 4.F.8: Tests Based on Comparison of First and Second Claim Durations, for Different Bonus-Malus Classes
BM
classes
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
1
62.2% (7.5%)
0.488 (0.270)
55.6% (7.5%)
64.4% (7.9%)
1 2
62.3% (6.4%)
0.418 (0.232)
57.4% (6.4%)
59.0% (6.8%)
1 3
61.1% (5.9%)
0.372 (0.214)
56.9% (5.9%)
59.7% (6.2%)
1 4
57.6% (5.4%)
0.326 (0.197)
54.1% (5.4%)
55.3% (5.7%)
1 5
56.4% (5.0%)
0.263 (0.180)
53.5% (5.0%)
56.4% (5.2%)
1 6
53.2% (4.5%)
0.082 (0.163)
50.0% (4.5%)
52.4% (4.7%)
1 7
52.8% (4.2%)
0.099 (0.152)
50.0% (4.2%)
53.5% (4.3%)
1 8
52.4% (3.9%)
0.107 (0.142)
50.0% (3.9%)
53.7% (4.0%)
1 9
52.0% (3.7%)
0.104 (0.136)
49.2% (3.8%)
54.2% (3.8%)
1 10
52.9% (3.6%)
0.123 (0.131)
50.3% (3.7%)
53.4% (3.7%)
All
52.4% (2.8%)
0.108 (0.103)
50.5% (2.9%)
50.5% (2.9%)
11 20
51.7% (4.6%)
0.084 (0.166)
50.8% (4.6%)
50.0% (4.6%)
12 20
51.8% (4.8%)
0.040 (0.173)
50.9% (4.8%)
50.0% (4.8%)
13 20
50.5% (5.2%)
0.046 (0.188)
50.5% (5.2%)
49.5% (5.2%)
14 20
51.8% (5.5%)
0.098 (0.199)
51.8% (5.5%)
50.6% (5.5%)
15 20
50.7% (6.0%)
0.091 (0.218)
49.3% (6.0%)
49.3% (6.1%)
16 20
47.6% (6.3%)
-0.053 (0.229)
46.0% (6.3%)
46.0% (6.4%)
17 20
44.0% (7.1%)
-0.207 (0.257)
42.0% (7.1%)
42.0% (7.1%)
18 20
46.5% (7.6%)
-0.126 (0.277)
44.2% (7.6%)
46.5% (7.7%)
19 20
42.4% (8.7%)
-0.082 (0.316)
39.4% (8.7%)
42.4% (8.8%)
20
41.4% (9.3%)
-0.112 (0.337)
41.4% (9.3%)
44.8% (9.3%)
175
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.F.9: Tests Based on Comparison of First and Second Claim Durations that Pool
Low and High Bonus-Malus Classes
BM classes
low high
Test statistics (std. error)
π̂n
60.8%
61.1%
60.6%
58.7%
59.5%
60.0%
58.3%
1
20
1 2
20
1 2
19 20
1 2
18 20
1 2
17 20
1 3
19 20
1 3
18 20
1 4
17 20
1 5
16 20
54.9% (3.9%)
1 6
15 20
51.8% (3.6%)
1 7
14 20
1 8
13 20
1 9
1 10
176
(5.8%)
(5.3%)
(5.2%)
(4.9%)
(4.7%)
(4.9%)
(4.7%)
57.0% (4.3%)
[
ln
βn
π̂n∗ (all)
0.341 (0.211)
56.8% (5.8%)
(0.191)
57.8% (5.3%)
0.300 (0.187)
58.5% (5.2%)
(0.178)
56.7% (4.9%)
(0.172)
57.7% (4.8%)
0.319
0.297
0.323
0.28
0.282
0.281 (0.177)
(0.169)
58.1%
(4.9%)
56.5% (4.7%)
π̂n∗ (current)
56.8% (5.9%)
60.0%
59.6%
58.7%
59.0%
58.3%
(5.4%)
(5.2%)
(5.0%)
57.7% (4.8%)
(5.0%)
(4.7%)
(0.156)
55.6% (4.3%)
56.3% (4.4%)
0.183 (0.142)
53.7% (3.9%)
53.7% (4.0%)
0.020 (0.131)
50.3% (3.6%)
50.3% (3.7%)
51.1% (3.3%)
0.026 (0.121)
49.3% (3.4%)
49.3% (3.4%)
51.4% (3.1%)
0.052 (0.113)
49.8% (3.2%)
49.8% (3.2%)
12 20
50.5% (2.9%)
0.049 (0.107)
49.1% (3.0%)
49.5% (3.0%)
11 20
51.1% (2.8%)
0.043 (0.103)
49.8% (2.9%)
49.8% (2.9%)
4.F.
SAMPLE CORRECTED BASED ON INITIAL BONUS-MALUS
CLASS
Table 4.F.10:
Comparison of First and Second Claim Sizes for Various Bonus-Malus
Classes
BM # Wilcoxon test Sign test L ∼ L against
classes obs.
p
L L L ≺ L L 6∼ L
1
-value
1
2
1
2
2
1
1
53
0.780
0.392
0.708
0.784
1 2
70
0.706
0.640
0.452
0.905
1 3
81
0.596
0.672
0.412
0.824
1 4
94
0.612
0.697
0.379
0.757
1 5
111
0.897
0.776
0.285
0.569
1 6
136
0.246
0.928
0.099
0.198
1 7
154
0.201
0.937
0.085
0.171
1 8
176
0.104
0.979
1 9
193
0.086
0.978
1 10
206
0.148
0.019
0.030
0.030
0.017
0.031
0.020
0.049
0.959
All
331
11 20
125
12 20
113
13 20
96
14 20
86
15 20
72
16 20
66
17 20
52
18 20
45
0.068
0.982
19 20
35
0.238
0.912
20
31
0.493
0.763
0.060
0.998
0.994
0.998
0.995
0.997
0.997
0.999
0.991
0.030
0.030
0.003
0.010
0.004
0.009
0.006
0.006
0.002
0.018
0.036
2
0.059
0.061
0.054
0.109
0.006
0.020
0.008
0.018
0.013
0.013
0.004
0.036
0.155
0.311
0.360
0.720
0.072
177
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
4.G Main (Corrected) Sample Including Withdrawn Claims
Tables and gures numbered 4.G.1 and up report the results of redoing all analyses using
the contract and bonus-malus information from the main (corrected) sample, but including
all withdrawn claims as claim-at-fault events (see Appendix 4.C).
178
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Table 4.G.3: Contract Exposure Durations in the Sample
Number of years
Y
1
Number of contracts observed
exactly Y years between Y − 1 and Y years
Total
8,097
11,775
19,872
2
4,709
9,616
14,325
3
6,261
7,385
13,646
4
68,808
6,501
75,309
87,875
35,277
Total
123,152
Table 4.G.4: Number of Contracts Observed for At Least One Full Contract Year, by
Bonus-Malus Class and Number of Claims in the First Contract Year
BM
Number of contracts with
class no claim 1 claim 2 claims 3 claims 4 claims
1
Total
1
562
119
24
4
2
746
95
11
1
710
3
957
83
11
4
1,308
100
9
5
1,876
112
13
1
2,002
6
2,509
162
14
2
2,687
7
3,360
209
17
3,586
8
4,232
272
16
4,520
853
1,051
1,417
9
4,882
252
16
10
6,486
298
11
2
11
6,056
282
12
12
6,002
287
16
6,305
13
5,874
269
11
6,154
14
6,663
313
13
6,989
15
6,161
302
9
6,472
16
6,375
298
14
17
5,669
252
7
5,928
18
4,366
207
11
4,584
2
3,854
214
7
20
27,651
1,374
29
2
105,589
5,500
271
15
6,796
6,352
1
19
Total
5,152
1
6,688
4,075
29,056
2
111,377
179
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.G.1: Distribution of Contracts Observed for At Least One Full Contract Year
Across Bonus-Malus Classes; and Shares of Those Contracts with At Least One and At
Least Two Claims at Fault in the First Contract Year, by Bonus-Malus Class
30%
25%
All contracts
Contracts with at least 1 claim
Contracts with at least 2 claims
Share
20%
15%
10%
5%
0%
1
2
3
4
5
6
7
8
9
10
11
BM class
180
12
13
14
15
16
17
18
19
20
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Figure 4.G.2: Incentives to Avoid First, Second and Third Claim; at an Average Risk
Level
5
∆ V(1,K,1)
∆ V(1,K,2)
∆ V(1,K,3)
4.5
4
3.5
∆V
3
2.5
2
1.5
1
0.5
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
181
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.G.3: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at an Average Risk Level
4
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
3
2
∆2 V
1
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
Figure 4.G.4: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a Zero Risk Level
unchanged
182
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Figure 4.G.5: Change in Incentives to Avoid a Claim after a First and a Second Claim,
and Changes in Incentives to Avoid a First, a Second and a Third Claim over the Course
of a Contract Year; at a High Risk Level
3
∆ V(1,K,2) − ∆ V(1,K,1)
∆ V(1,K,3) − ∆ V(1,K,2)
∆ V(1,K,1) − ∆ V(0,K,1)
∆ V(1,K,2) − ∆ V(0,K,2)
∆ V(1,K,3) − ∆ V(0,K,3)
2
1
∆2 V
0
−1
−2
−3
−4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
K
183
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.G.5: ML Estimation of the Auxiliary Model (4.14) with 3 and 4 Support Points
Three Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
0.4710
0.0353
0.0435
0.0037
0.2623
0.0110
0.3560
0.0126
Tests of
β=0
•
LM test: 9.22,
•
LR test: 73.61,
•
Wald test: 178.15,
p-value
p-value
= 0.00
= 0.00
p-value
= 0.00
Four Support Points
Parameter Estimate Std. Error
β
λ1
λ2
λ3
λ4
Table 4.G.6:
0.4692
0.0330
0.0396
0.0036
0.2481
0.0150
0.2672
0.0105
0.3611
0.0130
Tests of
β=0
•
LM test: 229.96,
•
LR test: 48.01,
•
Wald test: 201.78,
Nonparametric Tests Based on Comparison of
H1
Bonus-Malus Classes
Kruskal - Wallis test
H1 (K)
H2 (K)
equal for all
equal for all
K
K
Wilcoxon test
0.667
0.247
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
0.008
Kolmogorov - Smirnov test
p-value
H1 (low K ) ∼ H1 (high K )
H2 (low K ) ∼ H2 (high K )
184
p-value
0.504
0.528
0.003
p-value
p-value
and
= 0.00
= 0.00
p-value
H2
= 0.00
for Dierent
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Figure 4.G.7: Comparison of
Ĥ1
with the Uniform Distribution for Low and High Bonus-
Malus Classes
1
Uniform CDF
Empirical H1 for low BM 1 − 10
0.9
Empirical H1 for high BM 11 − 20
0.8
0.7
H1
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Table 4.G.7: Kolmogorov-Smirnov Test Comparing
H2 with H12 for Dierent Bonus-Malus Classes
H1 with the Uniform Distribution and
Kolmogorov - Smirnov test
H1 (all K )
H1 (low K )
H1 (high K )
H2 (all K )
H2 (low K )
H2 (high K )
H2 (low K )
H2 (high K )
∼
∼
∼
∼
∼
∼
∼
∼
U nif orm
U nif orm
U nif orm
H12 (all K )
H12 (low K )
H12 (high K )
H12 (high K )
H12 (low K )
p-value
0.002
0.011
0.052
0.545
0.059
0.389
0.127
0.658
185
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Figure 4.G.8: Comparison of
Ĥ1
with the Uniform Distribution and of
Low and High Bonus-Malus Classes, with
Ĥ1
and
Ĥ2
Ĥ12
with
Estimated on the Same Classes
Low BM classes 1 − 10
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
t
High BM classes 11 − 20
1
Uniform CDF
Empirical H1
0.9
Empirical H21
0.8
Empirical H2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
t
186
0.6
Ĥ2
for
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Table 4.G.8: Tests Based on Comparison of First and Second Claim Durations, for Different Bonus-Malus Classes
BM
classes
1
Test statistics (std. error)
π̂n
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
62.5% (10.2%)
0.392 (0.370)
54.2% (10.2%)
54.2% (10.9%)
1 2
65.7% (8.5%)
0.570 (0.307)
60.0% (8.5%)
60.0% (8.9%)
1 3
60.9% (7.4%)
0.346 (0.267)
56.5% (7.4%)
56.5% (7.7%)
1 4
54.5% (6.7%)
0.183 (0.245)
50.9% (6.8%)
52.7% (7.0%)
1 5
54.4% (6.1%)
0.157 (0.220)
51.5% (6.1%)
54.4% (6.3%)
1 6
53.7% (5.5%)
0.029 (0.200)
50.0% (5.5%)
53.7% (5.7%)
1 7
53.5% (5.0%)
0.080 (0.182)
50.5% (5.1%)
53.5% (5.2%)
1 8
53.0% (4.7%)
0.080 (0.169)
50.4% (4.7%)
53.0% (4.8%)
1 9
51.9% (4.4%)
0.064 (0.158)
48.9% (4.4%)
50.4% (4.5%)
1 10
52.8% (4.2%)
0.089 (0.152)
50.0% (4.2%)
50.7% (4.3%)
All
52.8% (3.0%)
0.090 (0.110)
50.2% (3.1%)
50.2% (3.1%)
11 20
52.7% (4.4%)
0.091 (0.160)
50.4% (4.4%)
51.2% (4.5%)
12 20
52.1% (4.6%)
0.020 (0.168)
49.6% (4.7%)
50.4% (4.7%)
13 20
51.5% (5.0%)
0.049 (0.180)
49.5% (5.0%)
50.5% (5.0%)
14 20
53.3% (5.3%)
0.104 (0.191)
51.1% (5.3%)
52.2% (5.3%)
15 20
51.9% (5.7%)
0.072 (0.207)
49.4% (5.7%)
50.6% (5.8%)
16 20
48.5% (6.1%)
-0.074 (0.220)
47.1% (6.1%)
47.1% (6.1%)
17 20
44.4% (6.8%)
-0.250 (0.247)
42.6% (6.8%)
42.6% (6.9%)
18 20
46.8% (7.3%)
-0.182 (0.265)
44.7% (7.3%)
46.8% (7.4%)
19 20
41.7% (8.3%)
-0.210 (0.302)
38.9% (8.4%)
41.7% (8.4%)
20
41.4% (9.3%)
-0.112 (0.337)
41.4% (9.3%)
44.8% (9.3%)
187
CHAPTER 4.
MORAL HAZARD IN DYNAMIC INSURANCE DATA
Table 4.G.9: Tests Based on Comparison of First and Second Claim Durations that Pool
Low and High Bonus-Malus Classes
BM classes
low high
1
20
1 2
20
1 2
19 20
1 2
18 20
1 2
17 20
1 3
19 20
1 3
18 20
1 4
Test statistics (std. error)
[
ln
βn
π̂n∗ (all)
π̂n∗ (current)
60.4% (6.9%)
0.239 (0.249)
56.6% (6.9%)
56.6% (6.9%)
(6.3%)
0.363 (0.227)
59.4% (6.3%)
59.4% (6.3%)
π̂n
62.5%
62.0%
59.6%
59.8%
(5.9%)
58.5% (5.5%)
(5.3%)
(0.215)
(0.200)
60.6%
(6.0%)
57.3% (5.5%)
60.6%
(6.0%)
58.5% (5.6%)
(0.192)
58.4% (5.3%)
58.4% (5.4%)
0.286 (0.200)
58.5% (5.5%)
58.5% (5.6%)
57.0% (5.2%)
0.263 (0.188)
55.9% (5.2%)
57.0% (5.3%)
17 20
55.0% (4.8%)
0.216 (0.174)
54.1% (4.8%)
54.1% (4.9%)
1 5
16 20
52.9% (4.3%)
0.116 (0.156)
52.2% (4.3%)
52.2% (4.4%)
1 6
15 20
50.9% (4.0%)
-0.020 (0.144)
50.3% (4.0%)
49.7% (4.0%)
1 7
14 20
50.3% (3.6%)
-0.008 (0.132)
49.7% (3.7%)
49.7% (3.7%)
1 8
13 20
50.9% (3.4%)
0.019 (0.123)
50.5% (3.4%)
50.0% (3.5%)
1 9
12 20
50.0% (3.2%)
0.025 (0.115)
49.6% (3.2%)
49.2% (3.2%)
1 10
11 20
50.2% (3.0%)
0.003 (0.110)
49.8% (3.1%)
49.8% (3.1%)
188
(5.5%)
0.387
0.348
0.376
4.G.
MAIN (CORRECTED) SAMPLE INCLUDING WITHDRAWN
CLAIMS
Table 4.G.10:
Comparison of First and Second Claim Sizes for Various Bonus-Malus
Classes
BM # Wilcoxon test Sign test L ∼ L against
classes obs.
p
L L L ≺ L L 6∼ L
1
-value
1
2
1
2
2
1
1
29
0.336
0.068
0.969
0.136
1 2
41
0.791
0.378
0.734
0.755
1 3
52
0.649
0.445
0.661
0.890
1 4
61
0.815
0.399
0.696
0.798
1 5
75
0.941
0.500
0.591
1.000
1 6
91
0.755
0.662
0.417
0.834
1 7
108
0.413
0.807
0.250
0.501
1 8
124
0.195
0.911
0.121
0.243
1 9
142
0.121
0.923
0.104
0.208
1 10
154
0.174
0.147
0.295
All
288
11 20
134
12 20
120
13 20
104
14 20
93
15 20
80
16 20
71
17 20
56
0.021
0.050
0.042
0.035
0.043
0.887
0.119
0.978
18 20
49
0.171
0.957
19 20
38
0.335
20
31
0.493
0.070
0.068
0.008
0.010
0.007
0.012
0.011
0.009
0.004
0.041
0.016
0.019
0.013
0.024
0.022
0.018
0.009
0.076
0.152
0.872
0.209
0.418
0.763
0.360
0.720
0.994
0.994
0.996
0.993
0.994
0.995
0.998
2
0.081
189
5
Conclusion
In this chapter we will summarize the key empirical results from the previous chapters
and integrate them into one consistent conclusion. Then, we will outline new ideas for
future research, providing some support from the data.
5.1 Integration of the Results
In Chapter 2, we tested for general asymmetric information using the conditional-correlation
approach. Our tests focused mainly on adverse selection and ex ante moral hazard, by
considering only the third-party claims, which are very likely to be reported to the insurer.
Using cross-sectional data, which covered one particular contract year, we did not nd
any evidence of asymmetric information. In Chapter 4, we tested for general moral hazard
using structural dynamic econometric methods and longitudinal data covering multiple
contract years. We found evidence of moral hazard and concluded that part of this moral
hazard is due to ex post moral hazard. Taken at face value, these results imply that there
is ex post moral hazard, but little or no ex ante moral hazard and no adverse selection in
our data.
However, the results of Chapter 2 should be interpreted with some care. First, the tests
191
CHAPTER 5.
CONCLUSION
in this chapter use only cross-sectional data on a single contract year and, therefore, may
not be very powerful. Second, as argued by de Meza and Webb (2001), the conditionalcorrelation method is fragile to certain types of multidimensional asymmetric information.
Notably, if there is asymmetric information on risk preferences and moral hazard, and more
risk averse drivers tend to buy more insurance and drive more cautiously, the conditional
correlation between insurance coverage and occurrence of claims can even be negative.
Third, the analysis in this chapter used only third-party claims. If asymmetric information
is particularly relevant to accidents involving only one car, we cannot detect it.
Chapter 4 provides a more advanced analysis, which explores dynamic features of the
data, such as the timing of claims, in relation to contract dynamics, notably experience
rating.
Our full sample covers data over a period of (up to) four contract years.
This
is important, because previous empirical studies using data on only one contract year
failed to detect asymmetric information, and particularly moral hazard.
take into account all claims at fault, i.e.
1
Further, we
also those involving only one car.
This is
important, because ignoring accidents involving only one car can obscure the presence of
moral hazard. Under moral hazard, fully covered agents, who started driving less carefully
because of decreased incentives, are both more likely to cause accidents with third-party
damages and accidents where only their car is damaged. Finally, our analysis is robust to
general selection on unobservables (including de Meza and Webb's (2001) advantageous
selection), since we fully control for the (unobserved) heterogeneity in agents' risks. This
is not the case in the previous analysis. Therefore we have good reasons to believe that the
results from this chapter are more robust than the results from Chapter 2. In particular,
it is quite possible that there exists also ex ante moral hazard, not just ex post moral
hazard.
1 Abbring et al. (2003) did not nd evidence of moral hazard when using dynamic methods and longitudinal data covering only one contract year. Two other studies (Israel, 2004, Dionne et al., 2005) applied
their tests on longer panels and found evidence of moral hazard.
192
5.2.
DYNAMIC CONTRACT CHOICE
5.2 Dynamic Contract Choice
Chapter 2 explores the cross-sectional relation between contract choice and risk in a single
contract year. Chapter 4 studies risk dynamics, but ignores contract choice. This way,
this PhD thesis so far does not exploit a potentially rich source of variation, dynamic
contract choice.
There are (at least) four reasons to extend the analysis by including
dynamic contract selection.
First, such an analysis is informative on the importance of adverse selection. In particular, one may argue that young, inexperienced drivers cannot assess their driving ability
any better than insurers can.
This seems to suggest that adverse selection is not rele-
vant for car-insurance markets. However, even if agents and insurers are symmetrically
informed initially, asymmetric information on risk may arise in the course of an insurance
relationship due to
asymmetric learning.2
This entails that drivers learn faster about their
risk than insurers, for example because they accumulate information on near-accidents and
small accidents that they do not report to the insurer. In turn, asymmetric learning may
aect dynamic contract selection. Thus, a dynamic form of adverse selection arises. It is
therefore of interest to investigate to what extent an analysis of claims and contract-choice
processes, based on appropriate dynamic economic models, is informative on asymmetric
learning.
Second, any such analysis should also account for
symmetric learning,
in relation
to experience rating. This will provide a more rigorous foundation for the conditionalcorrelation analysis in Chapter 2, which conditioned on the bonus-malus class to control
for symmetric learning.
Third, dynamic contract selection cannot be ignored in a dynamic study of moral
hazard if it is endogenous to the claims process under investigation. An extreme example
is the problem of nonignorable attrition that arises, in particular, if multiyear panels are
used. Such problems may arise even if contract choice is only aected by events that are
2 See Subsection 2.4.4.2 for more discussion.
193
CHAPTER 5.
CONCLUSION
symmetrically observed by agents and insurers. For example, agents and insurers can be
expected to symmetrically learn from accidents that are reported to the insurer. In turn,
this may aect contract choice, or even the decision to terminate a contract. A claim at
fault may also aect contract choice and termination through its eect on the premium. A
joint analysis of claims and contract-selection processes can shed a light on these dynamic
selection eects. This would take the analysis beyond the usual independent censoring
assumption in event-history analysis, which we have also maintained in Chapter 4.
Fourth, dynamic contract-choice data may enhance the analysis of asymmetric information by distinguishing learning eects from the nancial-incentive eects of claims
through the premium. In the previous chapter we have already addressed this issue in
the context of moral hazard. We exploited that learning is probably more important for
young drivers than senior ones, and investigated the robustness of our results on moral
hazard to possible learning eects; see Appendix 4.D. Additional information on learning
from dynamic contract choice would allow us to further enhance our inference on moral
hazard.
Dynamic contract selection in car insurance was recently studied by Dionne et al.
(2006), who jointly analyze dynamic contract choices and accidents under experience
rating.
They propose a causality test that separates moral hazard from learning and
adverse selection.
Using three years of French longitudinal survey data with dynamic
information on both claims and accidents, they estimate a dynamic panel data model
and nd some evidence of moral hazard among policyholders with signicant driving
experience (5-15 years).
We intend to build on Dionne et al.'s novel approach to the dynamic empirical analysis
of insurance markets by amending and/or extending Chapter 4's structural event-history
methods to include learning and dynamic contract choice. By using a more structured
and continuous-time approach, we can further clarify the dynamic relations between risk,
learning and contract choice in the data. It allows us to fully exploit the continuous-time
194
5.3.
OBSERVED CONTRACT DYNAMICS
variation in our data, which is relatively rich compared to the annual discrete-time panel
data used by Dionne et al., and develop more powerful tests.
As we will see in the next section, many dynamic contract changes may be triggered
by more or less external events, such as the purchase of a new car. Therefore, our analysis
should control for such external events, in order to enhance our understanding of contract
changes that are truly endogenous, such as those related to learning about risk.
Obviously, the empirical analysis of dynamic contract choices would be impossible if
we would not have data on such dynamics. Therefore, we nish this thesis with a brief
exploration of contract-choice dynamics in our data.
5.3 Observed Contract Dynamics
In this section, we explore the feasibility of the analysis of dynamic contract choice with
the Dutch data by providing some evidence that agents change their coverage in the course
of time in these data. We will distinguish, as in Chapter 2, only two levels of coverage:
basic (liability) coverage and full (comprehensive) coverage.
3
In the raw data
we observe 29,979 changes in coverage, of which 16,324 are from basic
coverage to full coverage (further called
coverage (further called
1-0 change ).
0-1 change ) and 13,655 from full coverage to basic
These correspond to 16,056 unique contracts with
a 0-1 change and 13,424 unique contracts with a 1-0 change. Obviously, some contracts
experience multiple changes in coverage across the whole observed period: 1 contract has
even 9 changes, 2 contracts 6 changes, 3 contracts 5 changes, 55 contracts 4 changes, 365
contracts 3 changes, 4,293 contracts 2 changes, 20,042 contracts 1 change and 116,038
contracts have no change in coverage across the whole observed period.
Table 5.1 gives the number of changes in coverage, which take place at the renewal date
and during the contract year. We can see that most 1-0 changes occur at the renewal date.
This can be explained by the fact that policy rules allow agents to cancel or reduce their
3 See Appendix 4.C, footnote 22 for more details.
195
CHAPTER 5.
CONCLUSION
insurance coverage only at the contract renewal date, except in the cases of emigration,
death of the insuree or change of the insured object. On the other hand, most of the 0-1
changes happen in the course of the contract year, which can be explained by the fact that
policy rules give no timing restrictions on extension of the coverage. These changes can
be explained by aging of an insured car (1-0 change), purchase of a new car (0-1 change),
or by
learning
an agent might want to change his coverage once he learns more about
his risk.
It is easy to determine how many changes in coverage correspond to a change of car,
since each car in the Netherlands has a unique license number, which we directly observe
in the data. Thus any change of license number in a contract corresponds to a change of
insured object. Table 5.2 gives the number of changes in coverage with regard to a change
4
of car.
From this table we can see that most of the 0-1 changes indeed correspond to
a change of car.
Agents buying a new car usually buy also extra (full) insurance.
the other hand, the 1-0 changes are mostly not related to a change of car.
On
They can
correspond to learning or aging of insured cars.
5
Table 5.3 gives the age distribution of cars with respect to their coverage.
From this
table we can see that most of the new cars have full (CASCO) insurance. Older cars have
usually restricted comprehensive coverage (Mini-CASCO) and very old cars only liability
coverage (LI). Clearly, some, but not all, of the 1-0 variation can indeed be explained by
aging of cars.
Our preliminary exploration of the data suggests that there is a lot of contract-choice
dynamics in the data.
Many changes in coverage are due to changes of car (purchase
of full insurance) or are to great extent explained by aging of cars (cancelation of full
insurance), but there is ample unexplained variation that may be related to learning and
other aspects of our future research.
4 We had to exclude 140 contracts with a missing license number of the insured car.
5 This table includes only cars, which are old from 0 to 20 years. In the sample, we observe also some
few older cars, the oldest one being 65 years old.
196
5.3.
OBSERVED CONTRACT DYNAMICS
Table 5.1: Number of Changes in Coverage at Renewal Date and During Contract Year
Change in
Number of changes
coverage at renewal date during contract year Total
0 1
314
16.010
16.324
1 0
11.166
2.489
13.655
11.480
18.499
29.979
Total
Table 5.2: Number of Changes in Coverage Related to Change of Car
Change
in
coverage
YES
Change of car
NO
at renewal date
during con. year
at renewal date
during con. year
0 1
260
15,432
53
526
1 0
131
1,749
11,032
725
Table 5.3: Age Distribution of Cars with Respect to Coverage
Age
of car
CASCO
Number of cars with
Mini-CASCO LI only
0
27,523 (98.2%)
271
1
34,240 (97.4%)
2
36,620 (95.5%)
3
36,286 (90.7%)
3,080
4
5
Total
# cars
(1.0%)
309
(1.1%)
28,038
546
(1.6%)
532
(1.5%)
35,153
1,289
(3.4%)
888
(2.3%)
38,347
(7.7%)
1,427
(3.6%)
40,028
35,225 (82.0%)
6,497 (15.1%)
2,431
(5.7%)
42,956
31,012 (71.0%)
10,739 (24.6%)
3,622
(8.3%)
43,698
6
25,899 (58.4%)
15,324 (34.5%)
5,043 (11.4%)
44,367
7
20,331 (45.4%)
19,493 (43.5%)
6,905 (15.4%)
44,814
8
15,008 (33.6%)
22,366 (50.1%)
9,110 (20.4%)
44,667
9
10,596 (23.2%)
24,159 (52.9%)
12,502 (27.4%)
45,642
10
6,994 (15.5%)
23,319 (51.6%)
16,132 (35.7%)
45,183
11
4,250 (10.3%)
19,231 (46.6%)
18,701 (45.3%)
41,288
12
2,455
(7.0%)
14,069 (40.4%)
18,882 (54.2%)
34,867
13
1,411
(5.2%)
9,363 (34.8%)
16,506 (61.3%)
26,931
14
707
(3.9%)
5,335 (29.8%)
12,028 (67.2%)
17,904
15
365
(3.4%)
2,789 (26.3%)
7,533 (71.0%)
10,606
16
237
(3.9%)
1,449 (23.8%)
4,428 (72.8%)
6,081
17
173
(4.6%)
860 (22.9%)
2,751 (73.1%)
3,761
18
126
(5.4%)
548 (23.3%)
1,695 (72.1%)
2,351
19
95
(6.0%)
390 (24.7%)
1,104 (69.8%)
1,581
20
66
(6.1%)
271 (25.0%)
753 (69.4%)
1,085
Note: Due to overlapping in coverage, the total number of cars is smaller than the sum of columns.
197
Dynamische Econometrische Analyse van
Verzekeringsmarkten met Imperfecte
Informatie Samenvatting
Asymmetrische Informatie in Verzekering
De meeste mensen houden niet van inkomensrisico's.
verkleinen door ze te delen met anderen.
Gelukkig kunnen ze deze risico's
In moderne economieën wordt dit soort risi-
codeling aangeboden in de vorm van verzekeringen.
Levensverzekeringen, bijvoorbeeld,
beschermen tegen het inkomensrisico dat kleeft aan onverwacht kort of lang leven. Voor
elk individu afzonderlijk is dit levensrisico aanzienlijk, maar het gemiddeld sterfteverloop
in een grote groep verzekerden is goed te voorspellen.
Een verzekeraar kan zijn lev-
ensverzekeringsklanten dus hun levensrisico laten delen zonder dat ze elkaar ooit hoeven
te ontmoeten.
In zo'n moderne, anonieme verzekeringsmarkt is er een goede kans dat verzekerden voor
de verzekering relevante informatie kunnen verbergen. Verzekeraars en verzekerden zijn
dan 'asymmetrisch geïnformeerd'. Asymmetrische informatie kan twee vormen aannemen.
Ten eerste is het denkbaar dat mogelijke klanten hun risico beter kunnen inschatten dan
de verzekeraar. Dit kan leiden tot 'negatieve selectie' (adverse
selection ) op risico:
klanten
die een relatief hoog risico lopen ten opzichte van andere klanten met dezelfde door de
verzekeraar waargenomen risicofactoren zullen zich relatief goed verzekeren. Ten tweede
heeft de verzekeraar vaak geen volledige controle over het risicogedrag van de klant. In
dat geval ontstaat er 'moreel gevaar' (moral
hazard ):
klanten met een betere dekking
gedragen zich risicovoller.
Verzekeraars passen hun verzekeringsaanbod aan zulke asymmetrische-informatieproblemen
aan. Autoverzekeringen, bijvoorbeeld, hebben doorgaans een eigen risico. Ze bieden dan
199
Summary in Dutch
geen volledige verzekering tegen autoschade. De automobilist blijft deels verantwoordelijk
voor de gevolgen van zijn rijgedrag; het moreel gevaar wordt beperkt ten koste van de risicodeling. De meeste verzekeraars bieden verder een menu aan contracten aan, vaak met
keuze uit verschillende eigen risico's. Dit menu kan worden ontworpen zodat klanten met
verschillende verborgen eigenschappen verschillende contracten kiezen. De keuze uit het
menu verraadt dan de verborgen eigenschappen van de klant. Tot op zekere hoogte staat
dit de verzekeraar toe om zijn verzekering aan te passen aan de verborgen eigenschappen
van de klant.
Als er geen asymmetrische informatie is, dan kunnen verzekeringsmarkten op een efciënte manier zorgen voor risicodeling. Asymmetrische informatie kan echter de marktwerking verstoren. Dit kan een reden zijn voor overheidsingrijpen in verzekeringsmarkten;
het verklaart de sterke interesse van economen voor het asymmetrische-informatieprobleem.
Het is dus belangrijk om vast te stellen of asymmetrische informatie in de praktijk echt een
rol speelt en, zo ja, in welke vorm. Onderzoek naar dit soort problemen aan de hand van
gegevens uit de verzekeringspraktijk is het laatste decennium goed van de grond gekomen;
zie Chiappori and Salanié (2003) voor een overzicht.
Nederlandse Autoverzekering en Gegevens
In dit PhD project gebruiken we rijke administratieve gegevens van een grote Nederlandse
autoverzekeraar. Onze longitudinale gegevens dekken een tijdsperiode van 5 jaar en bevatten volledige informatie over verzekerden (leeftijd, geslacht, adres), hun auto's (merk,
prijs, motorinhoud, kracht, etc.), polissen (dekking, premie, eigen risico, etc.) en claims
(type, schade, invloed).
De Nederlandse verzekeraar gebruikt de ervaringsrating die in Tabel 2.1 is aangegeven.
De premies worden jaarlijks herzien aan de hand van het claimgedrag. De premie in het
volgende polisjaar hangt af van de bonus-malus (BM) trede en het aantal claims-doorschuld (claims
200
at fault )
in het vorige polisjaar. In lage en gemiddelde BM treden levert
Samenvatting in het Nederlands
een jaar zonder claims een premieverlaging (bonus ) op. Aan de andere kant, elke claim
waaraan de verzekerde schuld heeft leidt bij de eerstvolgende premieherziening tot een
verhoging (malus ), behalve in de hoogste en de laagste BM treden.
Een belangrijk probleem met alle administratieve autoverzekeringsgegevens is dat
alleen claims en geen schades worden geregistreerd. Als verzekerden een zekere vrijheid
hebben om te kiezen of ze een schade melden aan de verzekeraar, dan zijn claims en
schades niet gelijk. In het algemeen zullen verzekerden kleine schades niet melden, omdat de even kleine vergoedingen geen premieverhoging waard zijn. Dit wordt wel
moreel gevaar
genoemd.
ex post
Natuurlijk is het optimaal om alleen schades te claimen die
groter zijn dan een bepaalde drempel, die hoger dan het eigen risico is.
Het eect van verzekering op het schaderisico zelf heet
ex ante moreel gevaar.
Het
onderscheid tussen ex ante en ex post moreel gevaar is van economisch belang. Onder ex
ante moreel gevaar wordt door verzekering het schaderisico zelf beïnvloed; ex post moreel
gevaar beïnvloedt alleen de verdeling van risico tussen de klant en de verzekeraar en niet
het risico zelf. De welvaartsgevolgen van beide vormen van moreel gevaar zijn dus radicaal
verschillend.
Onze autoverzekeringsgegevens zijn bijzonder interessant omdat ze directe informatie
over ex post moreel gevaar geven.
De Nederlandse verzekeraar biedt haar klanten de
mogelijkheid om een claim binnen 6 maanden terug te trekken en zo een premieverhoging
te vermijden. Dit genereert informatie over beide vormen van moreel gevaar. Veronderstel
dat schades binnen een bepaalde termijn gemeld moeten worden om voor vergoeding in
aanmerking te komen, dat het melden van schades kostenloos is en dat het exact vaststellen
van het schadebedrag langer duurt dan de meldingstermijn.
Dan worden alle schades
gemeld bij de verzekeraar om de optie van vergoeding open te houden en uit ex post
moreel gevaar zich volledig in het terugtrekken van claims als de schade uiteindelijk te
laag uitvalt. Zowel ex post als ex ante moreel gevaar kunnen dan dus direct bestudeerd
worden. In de praktijk zal het aantal teruggetrokken claims een onderschatting van het
201
Summary in Dutch
ex post moreel gevaar geven en een analyse van de gemelde schades een overschatting van
het ex ante moreel gevaar. Een eerste blik op de Nederlandse gegevens suggereert dat ex
post moreel gevaar belangrijk is; er wordt geregeld gebruik gemaakt van de mogelijkheid
om claims terug te trekken.
Toetsen voor Asymmetrische Informatie
Een eenvoudige toets op het belang van asymmetrische informatie kan worden gebaseerd
op de waargenomen relatie tussen claims en de door een verzekerde gekochte dekking. Als
beter gedekte verzekerden meer claimen, dan wijst dit op asymmetrische informatie. Neem
bijvoorbeeld autoverzekeringen. In het geval van negatieve selectie kiezen automobilisten
die, beter dan de verzekeraar, weten dat ze goed kunnen autorijden een verzekering met
een hoger eigen risico. In het geval van moreel gevaar kiezen automobilisten met een hoger
eigen risico ervoor om voorzichtiger te rijden. Omdat er geen reden is voor een structureel
verband tussen claims en de mate van verzekering als er geen asymmetrische informatie
is, is deze relatie informatief over de aanwezigheid van asymmetrische informatie.
Asymmetrische informatie leidt dus tot een positieve voorwaardelijke correlatie tussen
dekking en claims, wat een eenvoudig toets toestaat. Deze voorwaardelijke-correlatiebenadering
(conditional-correlation
approach )
is eerst toegepast door Chiappori and Salanié (2000),
die toetsten voor asymmetrische informatie in Franse autoverzekeringen. Om problemen
met de endogeniteit van de ervaringsrating te omzeilen, hebben zij zich geconcentreerd op
jonge bestuurders zonder claimsgeschiedenis. De auteurs hebben geen bewijs van asymmetrische informatie gevonden, maar vermoeden dat een asymmetrie kan ontstaan in de
loop van de verzekeringsrelatie door
asymmetrisch leren.
In dit geval weten jonge auto-
mobilisten initieel weliswaar niet meer over hun risico dan de verzekeraar, maar worden ze
snel wijzer van ervaringen, zoals bijna-ongelukken, die ze niet delen met de verzekeraar.
In het eerste deel van dit proefschrift adopteren wij de methode van Chiappori and
Salanié en breiden deze op verschillende wijzen uit. Ten eerste onderzoeken we ook de
202
Samenvatting in het Nederlands
asymmetrische informatie bij gevorderde bestuurders door te controleren voor hun ervaringsrating.
Daarnaast bekijken we niet alleen het voorkomen van claims maar ook
de grootte van claims.
Tot slot verkennen we ook de gegevens van premies, wat het
mogelijk maakt om nieuwe nonparametrische methoden toe te passen. Met gebruik van
alleen cross-sectionele gegevens over een jaar, vinden we geen sporen van asymmetrische
informatie.
Nadeel van deze voorwaardelijke-correlatiebenadering is dat noch de dynamische keuze
van verzekeringen noch het eect van de dynamische structuur van de verzekeringen op
het rij- en claimgedrag kan worden onderzocht.
In de praktijk worden deze, blijkens
het veelvuldige voorkomen van een bonus-malusstructuur, wel erg belangrijk gevonden.
Daar komt bij dat de toets geen onderscheid maakt tussen negatieve selectie en moreel
gevaar. Dit onderscheid is belangrijk, omdat beide vormen van asymmetrische informatie
verschillende implicaties voor optimale contracten en de werking van verzekeringsmarkten
hebben.
Een voor de hand liggende oplossing is om het claimgedrag van verzekerden dynamisch te analyseren. In het Nederlandse BM systeem wijzigen de prikkels om claims
te voorkomen met de huidige BM trede, de tijd in het lopende polisjaar en het aantal
claims dat al is ingediend. We meten deze wijzigingen in de prikkels en tonen aan dat
Nederlanders die aan moreel gevaar lijden met elke claim door schuld hun claimsintensiteit
veranderen.
Dit resultaat verbindt moreel gevaar in Nederlandse autoverzekeringen met zogenaamde 'toestandsafhankelijkheid' (state
dependence )
in het individuele claimproces: de
snelheid waarmee verzekerden claimen hangt af, via de prikkels die uitgaan van het BM
systeem, van het claimverleden. We kunnen dus leren over moreel gevaar door te meten
of er zulke toestandsafhankelijkheid in het claimproces is. Hierbij moeten we werkelijke,
individuele toestandsafhankelijkheid onderscheiden van de eecten van niet-waargenomen
heterogeniteit (Heckman, 1981).
203
Summary in Dutch
Deze benadering is eerst geïntroduceerd door Abbring, Chiappori, Heckman, and Pinquet (2003) en toegepast in Franse autoverzekering door Abbring, Chiappori, and Pinquet
(2003). In het tweede deel van dit proefschrift breiden we deze werklijn op verschillende
wijzen uit. Eerst ontwikkelen we een volledig structureel dynamisch micro-econometrisch
model om moreel gevaar in de Nederlandse motorrijtuigverzekering te analyseren.
We
gebruiken namelijk rijke variatie in de prikkels die uitgaan van het Nederlandse BM systeem. Vervolgens versterken wij de statistische kracht van de toetsen door verlenging van
de analyse naar langere panels van verzekeringsgegevens, wat ons toestaat de informatie
in onze gegevens volledig te benutten. Tot slot maken wij onderscheid tussen ex ante en
ex post moreel gevaar door zowel voorkomen als grootte van claims te modelleren.
Onze dynamische analyse geeft bewijs voor moreel gevaar in de Nederlandse autoverzekering.
Een deel ervan kan door ex post moreel gevaar verklaard worden.
Ons
resultaat is bijzonder in de empirische literatuur, die meestal geen bewijs voor asymmetrische informatie in autoverzekering vindt.
204
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212
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