Data analytics for insurance loss modeling, telematics pricing and claims reserving

195 Pages·2017·6.73 MB·English

FACULTEIT ECONOMIE EN
BEDRIJFSWETENSCHAPPEN
KU LEUVEN
Data analytics for insurance loss modeling,
telematics pricing and claims reserving
Proefschrift Voorgedragen tot
het Behalen van de Graad van
Doctor in de Toegepaste
Economische Wetenschappen
door
Roel VERBELEN
NUMMER 563 2017

Committee
Advisor:
Prof. Dr. Gerda Claeskens KU Leuven
Co-Advisor:
Prof. Dr. Katrien Antonio KU Leuven and University of Amsterdam
Chair:
Prof. Dr. Robert Boute KU Leuven and Vlerick Business School
Members:
Prof. Dr. Jan Beirlant KU Leuven and University of the Free State
Prof. Dr. Jan Dhaene KU Leuven
Prof. Dr. Edward W. Frees University of Wisconsin–Madison
Prof. Dr. Montserrat Guill´en University of Barcelona
Daar de proefschriften in de reeks van de Faculteit Economie en
Bedrijfswetenschappen het persoonlijk werk zijn van hun auteurs, zijn alleen
deze laatsten daarvoor verantwoordelijk.
i

Acknowledgments
“Pursuing a PhD? No thanks, not for me...”
ThatIneededquitesomeconvincingissomethingthatKatrienwillsurelyconfirm.
In fact, she already started about 6 years ago when she guided me to the best
possible educational path after my bachelor studies in mathematics at Ghent
University. I started wondering about a future career in actuarial science and
reached out to Katrien requesting help on the next step to take. I was blown
away by her well-founded and comprehensive reply – that she wrote while being
on vacation. She invited me for a personal meeting at her office in Leuven and
convinced me to switch to KU Leuven and start a master program in statistics.
Alreadybackthen, shesuggestedthepossibilityofafterwardsstartingaPhD.As
supervisor of my master thesis, she continued to advocate a PhD and together
with Gerda eventually persuaded me to embark on an interdisciplinary research
project, combining actuarial science and statistics. Not a moment has gone by
that I regretted this decision.
First and foremost, I would like to express my profound gratitude to Gerda
and Katrien. During all these years, I have had the privilege to work closely with
them and have them as mentors. Thank you for convincing me to take up this
challenge and supporting me along the way. You were always there for me when
I needed guidance, answered promptly to any of my questions and read article
drafts with the utmost attention to detail. I have been impressed and inspired
by your level of dedication to the university and to research and how you both
manage to combine it with your family life.
Further, I want to thank my doctoral committee members from KU Leuven
(Prof. Jan Beirlant and Prof. Jan Dhaene), my external jury members (Prof. Ed-
ward W. Frees and Prof. Montserrat Guill´en) and chairman Prof. Robert Boute.
Thank you for the thorough reading of my thesis and for all insightful comments
iii
iv Acknowledgements
during the doctoral seminars and preliminary defense. Your constructive sug-
gestions helped to improve the quality of this thesis. In particular, thank you
Prof. Montserrat Guill´en for travelling to Leuven to be present at my public de-
fense.
I would also like to thank my co-authors of several research papers. Thanks
to Prof. Andrei Badescu, Prof. Sheldon Lin and Lan Gong from the University
of Toronto to introduce me to the class of mixtures of Erlang distributions which
formedthestartingpointofmyresearchinthefieldoflossmodeling. I’mgrateful
for the collaboration with Tom Reynkens and Prof. Jan Beirlant which lead to a
related follow-up paper. Thank you Jonas Cr`evecoeur for the joint brainstorms
and your feedback on the reserving project. I also very much enjoyed working
with Maxime Clijsters and Roel Henckaerts on a research topic in the context of
insurance pricing. I am proud to have co-supervised your master theses which
(cid:12)
were both rewarded with the IA(cid:12)BE thesis prize and additionally with the Johan
de Witt thesis prize for Maxime.
Datahavebeenindispensableforthesuccessofmyprojectinactuarialstatis-
tics. In this regards, I could benefit from the professional network of Katrien in
the actuarial community and the academic network of Gerda in the statistical
community. Special thanks go out to Jonas Onkelinx for going to great lengths
to share the telematics data with us and for his valuable support in handling this
challenging data set. These telematics data created a strategic advantage and
were of crucial importance for our research on usage-based insurance pricing. I
also wish to thank Hans Laevens for permission to use the interesting mastitis
data in the multivariate mixture of Erlangs chapter.
I am very grateful to the government agency for Innovation by Science and
Technology(IWT),nowcalledFlandersInnovation&Entrepreneurship(VLAIO),
for financing this doctorate and to the Flemish Supercomputer Centre VSC for
computational support involving simulation studies and analyzing big data sets.
Thanks to KU Leuven for providing me with the facilities to carry out this
doctoral research. I have enjoyed being part of the stimulating, interactive and
international research environment that the Faculty of Economics and Business
hastooffer. SincethisFacultyhostsbotharesearchgroupinstatistics,aswellas
in actuarial science, this has been the perfect place to conduct my PhD research.
I am very thankful for having been surrounded by so many nice colleagues
from ORSTAT, AFI, LSTAT in Heverlee, down the hall... It has been a pleasure
to be part of the statistics group at ORSTAT as well as of the insurance group
at AFI. I have been fortunate to get close with many of you and you formed a
Acknowledgements v
big part in my life over the recent years. Thanks for all those social activities
we shared together: the barbecues at Thomas’s, the Turkish dinners at Deniz
and Mehmet’s, Peter’s wedding, the Romania trip for the Pircalabelu wedding,
the volleyball games, the soccer with the guys from MSI, the kayaking in the
Ardennes, organizing the EDC quiz, the nights out at Oude Markt... I have
enjoyed them all very much! In particular, I would like to thank Lore for having
been such an incredible office mate. Thanks for all those laughs as well as sincere
discussions on both life and research. I am lucky to have had an office mate who
grewintoaclosefriend. WhenLoreleftforBoston,Iwasluckyoncemoretohave
Daumantas join the office. Thank you for all those little jokes and creating such
a pleasant working atmosphere. Special thanks go out to Ines for all the advice,
support and practical help over the years and for compensating my forgetfulness
from time to time.
I want to thank Prof. Meelis K¨a¨arik for the enriching research stay at the
University of Tartu in the final months of my doctorate. My office mate Annika
has made me feel very welcome, ait¨ah!
Pursuing a PhD gave me the opportunity to present my work at international
conferences, to attend scientific workshops and to meet many academics from all
over the world. Above all, I was fortunate to have met Liivika in this way. Our
impressive series of foreign encounters has turned into more than I could ever
dreamof. Thankyouforallthewonderfuljourneysandexperiencestogether. As
a graduating PhD student yourself you understood as no one else how intensive
thesedoctoralyearshavebeenformeandyoumanagedtocalmmedownwhenever
I was stressed. I felt strengthened having you by my side and your support while
writing this dissertation was invaluable. I’m grateful and proud to have you as a
partnerinlifeandIamexcitedtodiscoverwhatliesaheadinourfuturetogether.
Last but definitely not least I want to thank my family. Words can do no
justicetowhatyoumeantome. Youhavemydeepestandsincerestgratitudefor
your encouragement and unconditional support in every step of my life.
Roel Verbelen Leuven, June 2017.

Table of Contents
Committee i
Acknowledgements iii
1 Introduction 1
1.1 Innovations in loss modeling . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Innovations in car insurance pricing through telematics technology 4
1.3 Innovations in claims reserving . . . . . . . . . . . . . . . . . . . . 5
2 Fittingmixtures of Erlangs to censored and truncated data using
the EM algorithm 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Mixtures of Erlangs with a common scale parameter . . . . . . . . 13
2.3 The EM algorithm for censored and truncated data . . . . . . . . . 14
2.3.1 Truncated mixture of Erlangs . . . . . . . . . . . . . . . . . 15
2.3.2 Construction of the complete data vector . . . . . . . . . . 16
2.3.3 Initial step . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.4 E-step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.5 M-step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Choice of the shape parameters . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Adjusting the shapes . . . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Reducing the number of Erlangs . . . . . . . . . . . . . . . 23
2.4.4 Comparetheresultingfitusingdifferentinitializingparam-
eters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5.1 Simulated censored and truncated bimodal data . . . . . . 25
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viii Table of Contents
2.5.2 Unemployment duration . . . . . . . . . . . . . . . . . . . . 29
2.5.3 Secura Re, Belgian insurance data . . . . . . . . . . . . . . 31
2.5.4 Simulated generalized Pareto data . . . . . . . . . . . . . . 35
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 Appendix A: Denseness . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 Appendix B: Partial derivative of Q . . . . . . . . . . . . . . . . . 39
3 Multivariate mixtures of Erlangs for density estimation under
censoring 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Multivariate Erlang mixtures with a common scale parameter . . . 46
3.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.1 Randomly censored and fixed truncated data . . . . . . . . 48
3.3.2 Construction of the complete data likelihood . . . . . . . . 50
3.3.3 The EM algorithm for censored and truncated data . . . . 51
3.4 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4.1 Initialization and first run of the EM algorithm . . . . . . . 54
3.4.2 Reduction of the shape vectors . . . . . . . . . . . . . . . . 57
3.4.3 Adjustment of the shape vectors . . . . . . . . . . . . . . . 59
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.2 Old faithful geyser data . . . . . . . . . . . . . . . . . . . . 65
3.5.3 Mastitis study . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Appendix: Partial derivative of Q . . . . . . . . . . . . . . . . . . . 72
4 Unraveling the predictive power of telematics data in car insur-
ance pricing 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Statistical background and related modeling literature . . . . . . . 79
4.3 Telematics insurance data . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.1 Data processing. . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Risk classification using policy and telematics information . 85
4.4 Model building and selection . . . . . . . . . . . . . . . . . . . . . 89
4.4.1 Generalized additive models . . . . . . . . . . . . . . . . . . 90
4.4.2 Compositional data . . . . . . . . . . . . . . . . . . . . . . 91
4.4.3 Model selection and assessment . . . . . . . . . . . . . . . . 97
4.5 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99