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UNIVERSITY OF PIRAEUS DEPARTMENT OF STATISTICS AND INSURANCE SCIENCE POSTGRADUATE PROGRAM IN ACTUARIAL SCIENCE AND RISK MANAGEMENT OPTIMAL CAPITAL ALLOCATION BETWEEN INSURANCE COMPANIES By Georgios P. Kasapidis MSc Dissertation Submitted to the University of Piraeus in partial fulfillment of the requirements for the degree of Master in Actuarial Science and Risk Management Piraeus, February 2014 This thesis was approved unanimously by the three-member Commission of Inquiry appointed by the GA Department of Statistics and Isurance Science of the University of Piraeus No. ........ meeting in accordance with the bylaws of Σ Ω the Postgraduate Program in Actuarial Science and Risk Management Ι Α Ρ Ι Committee membersΕ were: Π - ...................................... ( Supervisor ) Ο - ...................................... Ι Μ - ...................................... Η Τ Σ Ι The approvalΠ of the thesis by the Department of Statistics and Ε Insurance Science, University of Piraeus does not imply acceptance of Ν Α opinions of the author Π ii Σ Ω Ι Α Ρ Ι Ε Π Ο Ι Μ Η Τ Σ Ι Π Ε Ν Α Π iii ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΕΙΡΑΙΩΣ Σ Ω ΤΜΗΜΑ ΣΤΑΤΙΣΤΙΚΗΣ ΚΑΙ ΑΣΦΙΑΛΙΣΤΙΚΗΣ Α Ρ ΕΠΙΣΤΗΜΗΣ Ι Ε ΜΕΤΑΠΤΥΧΙΑΚΟ ΠΡΟΓΡΑΜΜΠΑ ΣΠΟΥΔΩΝ ΣΤΗΝ ΑΝΑΛΟΓΙΣΤΙΚΗ ΕΠΙΣΤΗΜΗ ΚΑΟΙ ΔΙΟΙΚΗΤΙΚΗ ΚΙΝΔΥΝΟΥ Ι Μ Η ΒΕΛΤΙΣΤΕΣ ΚΑΤΑΤΝΟΜΕΣ ΚΕΦΑΛΑΙΩΝ ΜΕΤΑΞΥ Σ ΑΣΦΑΛΙΣΤΙΚΩΝ ΕΤΑΙΡΙΩΝ Ι Π Ε Γεώργιος Π. Κασαπίδης Ν Α Διπλωματική εργασία Π Υποβλήθηκε στο Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης του Πανεπιστημίου Πειραιώς ως μέρος των απαιτήσεων του Μεταπτυχιακού Διπλώματος Ειδίκευσης στην Αναλογιστική Επιστήμη και Διοικητικής του Κινδύνου. Πειραιάς, Φεβρουάριος 2014. iv Η παρούσα ∆ιπλωµατική Εργασία εγκρίθηκε οµόφωνα από την Τριµελή Εξεταστική Επιτροπή που ορίσθηκε από τη ΓΣΕΣ του Τµήµατος Στατιστικής και Ασφαλιστικής Επιστήµης του Πανεπιστηµίου Πειραιώς στην υπ’ αριθµ. …….. συνεδρίασή του σύµφωνα µε τον Εσωτερικό Κανονισµό Λειτουργίας του Σ Ω Προγράµµατος Μεταπτυχιακών Σπουδών στην Αναλογιστική Επιστήμη και Διοικητική Ι του Κινδύνου Α Ρ Ι Ε Τα µέλη της ΕπιτροΠπής ήταν: - ………………………Ο……….. (Επιβλέπων) Ι Μ - ……………………………….. Η - ……………………………….. Τ Σ Ι Π Η έγκριση της ∆ιπλωµατική Εργασίας από το Τµήµα Στατιστικής και Ε Ν Ασφαλιστικής Επιστήµης του Πανεπιστηµίου Πειραιώς δεν υποδηλώνει αποδοχή των Α Π γνωµών του συγγραφέα. v Dedicated to the first mathematician that I met, my Godfather Σ Ω Ι Α Ρ Ι Ε Π Ο Ι Μ Η Τ Σ Ι Π Ε Ν Α Π vi Acknowledgments I would like to thank and express my appreciation to my supervisor teacher Dr. Efstathios Chadjiconstantinidis for his knowledgeable guidance and continuing support to fulfill this dissertation. Also I would like to express my special thanks to Dr. Martin Nigsch, my line manager in Methods and Processes team in Swiss Re, giving me the unique opportunity to be a Σ member of his team and further a member of Swiss Re. Without his valuable guidance I wouldn't Ω be able to finish on time my thesis. Further I would like to give my special thanks to Daniele Ι Calabresi and Robert Lynch, for their valuable help provided me, cΑontinually in order to write the reinsurance arrangement model in python programming languagΡe and also for their valuable suggestions. Ι Ε Π Ο Ι Μ Η Τ Σ Ι Π Ε Ν Α Π vii Σ Ω Ι Α Ρ Ι Ε Π Ο Ι Μ Η Τ Σ Ι Π Ε Ν Α Π viii Optimal capital allocation between insurance companies Keywords: risk measures, robust, coherent risk measures, value – at – risk, expected shortfall, truncated tail – value – at – risk, quota – share, stop – loss, premium principles, optimal Σ reinsurance, risk transfers, reinsurance arrangements, risk adjusted value, expected policyholder Ω deficit, single claims liabilities. Ι Α Ρ Ι Abstract Ε Π The level of capital to be held by an insurance compa ny is essential and should be defined, so that Ο with high probability the company can meet its obligations. Thus the determination of a target Ι level of capital under an appropriate risk measuΜre is essential. In this thesis we study the effect of risk transfer between insurance companies on this target level of required capital. This thesis is Η based on two papers, aiming to present optimal capital allocation – risk transfers methods, Τ between insurance companies, under certain capital requirements – thus risk measures. Further Σ based on a certain theory and on a Ιstochastic model of single claims development, we study the Π impact of risk transfer arrangements between insurance companies numerically and hopefully we Ε present interesting conclusions about optimal risk transfer in this model environment. Ν Α Π ix Σ Ω Ι Α Ρ Ι Ε Π Ο Ι Μ Η Τ Σ Ι Π Ε Ν Α Π x
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