Risk management and regulatory aspects of life insurance companies with a special focus on disability insurance

Transcription

1 Risk managemen and regulaory aspecs of life insurance companies wih a special focus on disabiliy insurance Disseraion zur Erlangung des akademischen Grades eines Dokors der Wirschafswissenschafen (Dr. rer. pol. an der Fakulä für Mahemaik und Wirschafswissenschafen der Universiä Ulm Vorgeleg von: Andreas Johannes Niemeyer Amierender Dekan: Prof. Dr. Dieer Rauenbach Guacher: Jun. Prof. Dr. Marcus C. Chrisiansen Prof. Dr. Hans-Joachim Zwiesler Prof. Dr. Daniel Bauer Tag der Promoion: 10. Juli 2014

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3 Acknowledgmens I would like o hank everyone who suppored me when wriing my hesis. Firs, I am deeply graeful o my supervisor Jun. Prof. Dr. Marcus C. Chrisiansen for his grea suppor. I learned so many hings from him and he helped me o pu his hesis ino he righ direcion. Furhermore, I would like o hank Prof. Dr. Hans-Joachim Zwiesler and Prof. Dr. Daniel Bauer for he grea discussions and for being reviewers. I also would like o hank Prof. Dr. An Chen and Prof. Dr. Marin Eling for heir helpful commens. Likewise, I am very graeful o all my colleagues from he Insiue of Insurance Science and he Research Training Group 1100 for heir suppor and heir companionship. The pleny discussions wih hem helped me o improve he hesis. Addiionally, I wan o hank Jessie Dorsz, David Prevo, and Tara Prevo for heir helpful commens o improve he linguisic syle of he hesis. Furhermore, I would like o hank my family for heir suppor. Especially, I am graeful for he endless suppor of Kaja Schilling. The discussions wih her and her feedback were very inspiring. Ulm, July 2014 Andreas Niemeyer iii

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5 Conens Overview of research papers Co-auhorship vii ix Research conex and summary of research papers 1 1 Field of research Moivaion and objecives Summary of research papers Research papers 1 Fundamenal definiion of he Solvency Capial Requiremen in Solvency II 21 1 Inroducion The regulaory framework Discussion of Recial The SCR for an arbirary poin in ime Comparison of he differen SCR definiions Convergence of SCR definiions Invariance propery wih applicaion o insurance groups Risk Margin Conclusion A Appendix On he forward rae concep in muli-sae life insurance 55 1 Inroducion Definiion of forward raes Some sufficien condiions for independence of diffusion processes A specific srucure of he difference of wo diffusion processes Disabiliy insurance Applicaion o oher models Conclusions A Appendix Modeling and forecasing duraion-dependen moraliy raes 87 1 Inroducion v

6 2 Descripion of he daa and esimaion of he empirical moraliy raes The Lee-Carer model and single effecs Exensions of he Lee-Carer model Proofs of he proposiions Forecasing of he moraliy raes Conclusions A Appendix Safey margins for sysemaic biomeric and financial risk in a semi-markov life insurance framework Inroducion Semi-Markov model, risk decomposiion, and surplus Safey margins for sysemaic biomeric and financial risk Two-sae Markov acive-dead model Markov model for disabiliy insurance wih ineres rae risk Semi-Markov invalid-dead model Conclusions A Appendix Zusammenfassung 151 Curriculum Viae 157 vi

7 Overview of research papers Research papers included in his disseraion 1. Chrisiansen, M. C. and Niemeyer, A. (2012. Fundamenal definiion of he Solvency Capial Requiremen in Solvency II. ASTIN Bullein, Available on CJO 2014 doi: /asb Chrisiansen, M. C. and Niemeyer, A. (2013. On he forward rae concep in mulisae life insurance. To appear in Finance and Sochasics. 3. Chrisiansen, M. C., Niemeyer, A. and Teigiszerová, L. (2014. Modeling and forecasing duraion-dependen moraliy raes. Submied o Compuaional Saisics & Daa Analysis (under review. 4. Niemeyer, A. (2014. Safey margins for sysemaic biomeric and financial risk in a semi-markov life insurance framework. Submied o Insurance: Mahemaics and Economics (under review. vii

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9 Co-auhorship Jun. Prof. Dr. Marcus C. Chrisiansen Marcus C. Chrisiansen is a professor a he Universiy of Ulm. He received a Diploma in mahemaics from he Oo von Guericke Universiy Magdeburg in 2003 and a docoral degree from he Universiy of Rosock in His research ineress are acuarial science and risk heory. Lucia Teigiszerová Lucia Teigiszerová is an underwrier a Hannover Re. She received a Bachelor in applied mahemaics from he Masaryk Universiy (Brno, Czech Republic in 2010 and a Maser in mahemaics and managemen from he Universiy of Ulm in ix

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11 Research conex and summary of research papers 1 Field of research This cumulaive hesis conribues o he field of risk managemen and regulaory aspecs of life insurance companies wih a special focus on disabiliy insurance. The insurance marke and, in paricular, he insurance regulaion in Europe has changed during he las decades, especially wih he inroducion of Solvency II (cf., e.g., Eling e al., 2007; Rees e al., Wih he inroducion of Solvency II a rend owards a more riskoriened calculaion mehod of he Solvency Capial Requiremen (SCR and a consideraion of marke values sared. The Solvency II proposal allows he calculaion of he SCR eiher by a sandard formula or a company-specific inernal model. The laer mehod yields a beer quanificaion of he company s risk (cf., e.g., Doff, 2008, bu needs a comprehensive valuaion of asses and liabiliies for which sochasic models are necessary. Solvency II has no been inroduced ye, bu will probably become effecive in Unil hen, many unsolved problems and deails need o be specified. Besides he SCR, in order o ensure solvency insurance companies charge conservaive premiums which include safey margins. Life insurance companies prefer he calculaion of premiums by using a deerminisic valuaion basis wih implici safey margins and do no add safey margins on op of he expeced claims as i is done in non-life insurance (cf., e.g., Chrisiansen, The ineracion beween SCR and safey margins is discussed in Kriele and Wolf (2007. However, in insurance pracice he safey margins are ofen jus ime-consan percenages of he bes esimae valuaion basis (cf., e.g., Kolser e al., 1998 and he resuling safey level is usually no specified. The German marke for disabiliy insurance has significanly grown during he las years due o reducion of public disabiliy insurance (cf., e.g., Maegebier, In imes of low ineres raes, insurance producs wih a high biomeric risk seem o be increasingly aracive o insurance companies, since he company s profis resul, amongs ohers, from keeping some par of he surplus which arises from premiums ha are on average oo high. While mos classical life insurance producs are modeled wihin a Markov framework, for disabiliy insurance a semi-markov framework is mosly used, since he ransiion raes from disabled o acive and dead depend on he ime since disablemen (cf., e.g., Haberman and Piacco, 1999; Piacco, Sochasic models for such ransiion raes are rarely discussed in he lieraure, while here exis relaively many publicaions on moraliy rae models (cf., e.g., Piacco e al.,

12 Research conex and summary of research papers This disseraion conribues o he discussion of inerpreing he Solvency II direcive. Furhermore, we invesigae he implicaions of pricing life insurance conracs wih forward raes. We also develop sochasic models for he moraliy raes of disabled individuals. Finally, we propose a mehod for he calculaion of safey margins for sysemaic biomeric and financial risk in life insurance conracs. 2 Moivaion and objecives The Solvency II framework is based on hree pillars where he firs pillar describes he capial requiremens, he second pillar he qualiaive requiremens, and he hird pillar he public disclosure rules (cf., e.g., Holzmüller, While here exis specified formulas for he calculaion of he SCR using he sandard formula approach (cf. CEIOPS, 2010, he formulas needed for an inernal model are only described in words (cf. European Parliamen and he Council, Many papers analyze he capial requiremens and invesigae differen aspecs of i. For he main definiion of he SCR we find differen mahemaical inerpreaions in lieraure (cf., e.g., Barrieu e al., 2012; Bauer e al., 2012; Möhr, 2011, bu i is no discussed in lieraure how hese formulas relae o each oher. For he comparabiliy of he resuling SCRs i is essenial ha hey are based on he same definiion. For insurance companies as well as for he insurance regulaor i is herefore imperaive o be aware of he properies and economic meanings of he differen SCR definiions. In addiion, he Solvency II concep is based on marke values of asses and liabiliies. Since for he liabiliies in many cases no marke values are available, he Solvency II direcive proposes he use of a cos-of-capial approach o obain marke-consisen values (cf., e.g., Salzmann and Wührich, The difference beween marke value and bes esimae reserve is called Risk Margin (RM and he cos-of-capial approach should be applied o he projeced SCRs (cf. CEIOPS, Consequenly, fuure SCRs are needed for he calculaion of he RM for which a generally valid definiion is missing in lieraure. The Solvency II balance shee is based on marke values and in Solvency II a RM should be used o calculae marke-consisen values for non-raded insurance liabiliies. However, here exis many oher approaches for he calculaion of marke-consisen values of liabiliies (cf., e.g., Sheldon and Smih, 2004; Wührich e al., One approach uses forward raes. They have been used a long ime in he ineres rae world and forward ineres raes have become a sandard in he financial mahemaics lieraure (cf., e.g., Bingham and Kiesel, The Heah-Jarrow-Moron mehodology provides a convenien way o model forward ineres raes. As a resul of is success, he idea of forward ineres raes has been ransferred o forward moraliy raes during he las decade (cf., e.g., Bauer e al., 2008; Cairns e al., 2006; Dahl, Wihin a life insurance framework mos auhors assume ha ineres raes and moraliy raes are independen which allows a sraighforward pricing of zero-coupon bonds, life annuiies, and erm life insurances wih he same forward ineres and moraliy raes. Norberg (2010 gives an example of dependen ineres and moraliy raes where he 2

13 Research conex and summary of research papers resuling forward raes canno be he same for zero-coupon bonds, life annuiies, and erm life insurances. However, i is an unsolved problem wheher he ineres and moraliy raes need o be independen in general or if here are dependen ineres and moraliy raes ha lead o consisen forward raes, which means in his conex ha all hree producs from above have he same forward raes. Since forward moraliy raes are increasingly popular, he concep is also used for muli-sae models (cf. Buchard, 2014; Norberg, The quesion of how o define a forward disabiliy rae is sudied by Norberg (2010. However, wha kind of dependency srucure wihin a disabiliy framework is implicily assumed when he forward raes shall be consisen is no discussed. For classical asse-liabiliy-managemen as well as for an inernal model in Solvency II i is indispensable o model he risk drivers by sochasic processes. While he asse-liabiliymanagemen is he basis for he decision making process of he insurance company s managemen, he sochasic models used for Solvency II deermine he amoun of SCR. Boh cases demonsrae how carefully and accuraely he models should be chosen (cf., e.g., Kouwenberg and Zenios, While here exis many sochasic models describing he developmen of ineres raes and socks, during he las wo decades i has become more and more popular o describe he moraliy rae by sochasic processes as well. Due o is simpliciy and is good fi for many counries, he parameric model proposed in he seminal paper from Lee and Carer (1992 somehow became he sandard. Of course, here are many exensions and oher models based on differen assumpions. For an overview of he differen models we refer o Piacco e al. (2009. Many insurance companies also sell proecion agains disabiliy and, consequenly, hey also need sochasic models for all ransiion raes wihin a disabiliy insurance framework. As menioned in he las secion, he moraliy raes of disabled individuals depend on he ime since disablemen. Hence, here is a need for exending moraliy models o incorporae duraional effecs and a naural choice is o exend he Lee-Carer model. Furhermore, he quesion is also how he parameers can be calibraed, wha economic meaning he parameers have, and how such a model can be forecas. In he course of decreasing ineres raes during he las decade, insurance companies sared o change heir sraegies: Insead of jus wriing as much business as possible, he profiabiliy of he differen insurance producs is given more aenion. This goes along wih he inroducion of many marke-consisen valuaion measures of insurance business and insurance companies (cf., e.g., Seyboh, One key facor ha decides on he profiabiliy of an insurance produc are safey margins. On he one hand, when he company adds high safey margins on op of he valuaion basis, he insurance company has a low loss probabiliy which resuls in a small SCR. The expeced surplus is also large, from which he company can forward a significan par o is shareholders. On he oher hand, high safey margins make he producs expensive so ha he company has a bad compeiive posiion. This area of ension is barely sudied in he lieraure, as safey margins are in general no widely discussed in lieraure. In paricular, in a general life insurance framework here does no exis a generally valid approach for he calculaion of safey margins for sysemaic biomeric risk, while here exis approaches for he unsysemaic biomeric risk and he biomeric esimaion risk (cf. Chrisiansen, 2012, 3

14 Research conex and summary of research papers For he applicabiliy of a new mehod i is essenial o demonsrae is pracicaliy for differen models and o analyze he properies of he resuling safey margins. To summarize, he following quesions are considered in he presen hesis: (1 Wha are he similariies, differences, and properies of he differen SCR definiions ha are discussed in lieraure? How can he SCR definiions be generalized o any poin in ime? (2 Is i possible o obain consisen forward moraliy and ineres raes for zero-coupon bonds, life annuiies, and erm life insurances wih dependen ineres and moraliy raes? Wha dependency srucure of he ransiion raes are implicily assumed when all relevan claims of a disabiliy insurance can be priced wih he same forward raes? (3 How can he model from Lee and Carer (1992 be exended o duraional effecs so ha i fis o he observed moraliy raes of disabled individuals? Wha are he main drivers of such a model and how can i be forecas ino he fuure? (4 How can risk-oriened safey margins be calculaed for sysemaic biomeric and financial risks wihin a semi-markov life insurance framework? Wha do such safey margins look like for differen insurance producs and how do hey perform in comparison o safey margins ha are jus ime-consan percenages of he bes esimae valuaion basis? These research quesions have no been addressed in he lieraure so far, and his disseraion aims o close his gap. The objecive of he firs paper is o analyze differen possible mahemaical inerpreaions of he SCR ha are in line wih he Solvency II direcive. Furhermore, he SCR should be generalized o any poin in ime so ha i is possible o give a sound definiion of he RM. The aim of he second paper is o discuss he dependency srucure of he underlying sochasic processes needed o obain consisen forward raes. This should be analyzed no only for he classical wo sae acive-dead model, bu also for disabiliy insurance. The hird paper seeks o find a sochasic model which represens he moraliy raes of disabled individuals and fis well o empirical daa of he public German disabiliy insurance. Addiionally, he daa should be analyzed o ge an impression of he mos imporan risk drivers of he moraliy raes (such as ime, duraion, and age. The objecive of he fourh paper is o derive risk-oriened safey margins for he sysemaic biomeric and financial risk in a join framework. Furhermore, he resuls should be analyzed for differen insurance producs and differen sochasic models o verify he reliabiliy of he proposed mehod. 4

15 Research conex and summary of research papers 3 Summary of research papers Research paper 1: Fundamenal definiion of he Solvency Capial Requiremen in Solvency II In he firs paper we se up a mahemaical framework o analyze he differen SCR definiions from lieraure and o demonsrae similariies, differences, and properies of he differen definiions. Addiionally, we discuss he definiion of he RM for which he SCR a fuure poins in ime is needed. I is a join work wih Marcus C. Chrisiansen and has been acceped for publicaion in he journal ASTIN Bullein. The paper has been presened a he Annual Congress of he German Insurance Science Associaion (2012 in Hanover, Germany, and a he AFIR Colloquium of he Inernaional Acuarial Associaion (2012 in Mexico Ciy, Mexico. Since he Solvency II direcive (cf. European Parliamen and he Council, 2009 defines he SCR only in words, here exis differen mahemaical inerpreaions of he SCR in he lieraure. However, for a harmonizaion of he insurance regulaion in Europe i is essenial o have a clear picure of he differen SCR definiions and heir relaionship o each oher. While mos papers only discuss he SCR a ime 0, Möhr (2011 and Ohlsson and Lauzeningks (2009 define he SCR a any poin in ime. Since Möhr (2011 does no specify he definiion of a condiional value a risk and Ohlsson and Lauzeningks (2009 define he fuure SCR only wihin a chain ladder framework, here is a need for a generally valid definiion of fuure SCRs. This definiion is he basis for a sound definiion of he RM. A he beginning, we presen wo verbal definiions of he SCR ha can be found in European Parliamen and he Council (2009 and assign hem he mahemaically well-defined definiions from lieraure. We add wo more definiions ha are reasonable wih regard o European Parliamen and he Council (2009 and analyze in wha follows he resuling five definiions. For he invesigaion we need a mahemaical modeling framework for he asses and liabiliies. Since he discoun facor in he SCR definiions urns ou o be essenial, we pu more effor ino modeling he asses han he liabiliies. One key poin is he quesion which asses he company would pay ou when i increases or reduces is asses. Hence, we inroduce a managemen sraegy funcion h ha gives he number of each asse ha is sold depending on he oal amoun by which he company wans o decrease is asses. By means of a simple example, we calculae he SCR as implied by each of he five definiions for an insurance company ha does no face any risk and see ha he wo SCR definiions used mos in lieraure have a SCR larger han zero, while he oher definiions yield a SCR of zero. For he generalizaion of he SCR o any poin in ime, we define a dynamic value a risk and prove ha i is well-defined. A similar definiion can be found in Kriele and Wolf (2012, bu hey assume ha he underlying probabiliy space has some specific srucure which we do no need. This allows us o generalize hree SCR definiions righ away o any poin in ime, 5

16 Research conex and summary of research papers since hey are based on he value a risk. For he oher wo definiions we firs ry o find a represenaion a ime zero ha is based on he value a risk so ha we can generalize hem again by means of he dynamic value a risk. I urns ou ha he managemen sraegy funcion h needs o be linear and all componens need o have he same sign, since oherwise he SCR a ime zero migh no exis or resul in oher unwaned properies. This resricion allows us o prove a represenaion of he wo remaining SCR definiions by means of a value a risk. One SCR definiion minimizes he value a risk wih respec o all managemen sraegy funcions h. Therefore, we show separaely ha his definiion is well-defined a any poin in ime. In he nex sep we compare he SCR definiions. Firs of all, we prove ha one definiion based on expeced values wih respec o a risk-neural measure is, under appropriae assumpions, equal o anoher SCR definiion. Furhermore, he SCR definiion ha minimizes he value a risk over all managemen sraegy funcions h is, under cerain condiions, a lower bound of he oher SCR definiions. Three definiions are of he form VaR (N s v(s, s + 1N s+1 F s, where he ne value N s is he difference of asses minus liabiliies and v is a one-year discoun facor ha disinguishes he hree definiions. We show ha wo definiions of his form are for all insurance companies and for all equivalen probabiliy measures equal if and only if he discoun facors are equal, and he discoun facors are equal for all financial markes if and only if he invesmen sraegies ha correspond o he discoun facors are equal. When an insurance company calculaes is SCR i can heoreically pay ou all own funds ha exceed he SCR. However, when he SCR is recalculaed wih he new amoun of own funds, he new SCR is in general differen o he original SCR. This procedure can be repeaed infiniely many imes. Thereby we assume ha he money is paid ou according o he managemen sraegy funcion h and ha he SCR is calculaed wih a definiion of he form from he las paragraph. We show ha under cerain condiions his ieraion converges o a unique fixed poin and ha his fixed poin is equal o he SCR from above, where he discoun facor corresponds o an invesmen according o he managemen sraegy funcion h. Addiionally, we show ha some definiions fulfill an invariance propery which means ha he SCR does no change when he asses are shifed following a specific sraegy. We apply his propery in an example o an insurance group wih wo subsidiary companies and demonsrae ha an invariance propery is no always advanageous. Finally, we discuss he definiion of he RM. For his, we do no only need he definiion of fuure SCRs, bu also a definiion of he SCR of a reference underaking as i is required in he Solvency II documenaion (cf. CEIOPS, We use he SCR definiion ha minimizes he value a risk wih respec o all managemen sraegy funcions h. Since we assumed above ha he managemen sraegy funcion h does no have mixed signs, we inroduce a minimal asse porfolio ha is in all componens smaller han a realisic porfolio would be. Now a shif from his porfolio is always posiive in all componens so ha a realisic minimal porfolio is considered in he minimizing SCR definiion from above. The resuling 6

17 Research conex and summary of research papers SCR corresponds o he SCR of a reference underaking and so we can presen a universally valid and mahemaically rigorous definiion of he RM. In summary, his paper answers he firs research quesion. We presen a mahemaical framework in which we compare and analyze he differen SCR definiions. Addiionally, we presen and prove many properies of he SCR definiions and show how he SCR of a reference underaking can be defined. Togeher wih he definiion of he SCR a fuure poins in ime, his is he basis for he definiion of he RM. This paper gives no only answers o unsolved research quesions wihin Solvency II, bu also connecs relaed papers. In conras o he RM inroduced in he Solvency II conex, we consider in he second paper a differen approach of deermining marke values. Research paper 2: On he forward rae concep in muli-sae life insurance In he second paper we analyze he dependency srucure ha is implicily assumed when forward raes are supposed o be equal for differen insurance producs. We sudy a wosae acive-dead model, a hree-sae acive-surrender-dead model, a disabiliy insurance, and a join life insurance. This paper is a join work wih Marcus C. Chrisiansen and has been acceped for publicaion in he journal Finance and Sochasics. The paper has been presened a he Acuarial Research Conference (2012 in Winnipeg, Canada, a he European Acuarial Journal Conference (2012 in Lausanne, Swizerland, a he workshop Perspecives on Acuarial Risks in Talks of Young Researchers (2013 in Ascona, Swizerland, a he Inernaional Congress on Insurance: Mahemaics and Economics (2013 in Copenhagen, Denmark, and a he Conference on Modeling, Analysis and Simulaion in Economahemaics (2014 in Ulm, Germany. So far, only few moraliy-linked securiies have been raded on he financial marke. Wih he inroducion of Solvency II he marke migh increase, since Solvency II requires valuaion of liabiliies wih marke values and i migh become more aracive o ransfer risk. One possibiliy o model marke values is by using of forward raes. This approach is widely spread for ineres and moraliy derivaives. Buchard (2014 and Norberg (2010 also define forward raes in muli-sae models. Since forward raes conain more informaion han spo raes, i is ofen advanageous o use forward rae models. Forward raes are an even more powerful ool when hey can be derived from one produc in order o price anoher produc. In general, hey are of special ineres when all forward raes resuling from differen produc designs are equal wihin one framework. In financial crises derivaives can become dependen ha were independen in regular marke siuaions. Consequenly, i is essenial for risk managers and for regulaors o undersand he dependency srucure ha is assumed wihin a framework. Therefore, we sudy he dependency srucure ha is implicily assumed when he forward raes are equal wihin a group of insurance producs. 7

18 Research conex and summary of research papers Firs of all, we discuss he differen definiions of forward moraliy raes ha can be found in lieraure. Based on hese definiions, we give a generalized definiion of forward raes in muli-sae models. The idea of his definiion is a subsiuion concep. This means ha he condiional expeced value under a pricing measure of he claim, which depends on he random ransiion and ineres raes, should be equal o he same formula of he claim, where he ransiion and ineres raes are replaced by he corresponding forward raes. This equaion should hold for all claims from a se M ha has o be specified. Consequenly, all producs from he se M can be priced using he resuling forward raes. Norberg (2010 proposes wo differen ways o define forward raes for muli-sae models, bu for boh definiions i is no clear how insurance claims can be priced wih hem. Buchard (2014 defines forward raes only for affine processes and uses he special srucure of such processes for he definiion of forward raes. In Norberg (2010 specific dependen ineres and moraliy raes are given, for which zero-coupon bonds, life annuiies, and erm life insurances canno imply he same forward raes, i.e. hese hree producs canno be included in he se M from above a he same ime in his paricular example. We sar from he opposie side and assume ha he forward raes are he same for all hree producs and wan o know which dependency srucure beween he ineres and moraliy raes his implies. We call such forward raes consisen. In general, we define ha forward raes are consisen when in he se M from above all sandardized producs necessary for rading, all sojourn paymens, and all ransiion paymens are included. For our invesigaion we se up a modeling framework and assume ha each ransiion and ineres rae is modeled by a one-dimensional diffusion process wih deerminisic sar value ha is eiher a Cox-Ingersoll-Ross process or a process where he drif and diffusion coefficien fulfill a measurabiliy condiion, a Lipschiz condiion, and have a linear growh bound. Under addiional condiions we prove ha differen mixed (condiional momens resul in independen processes. The key idea is o use he Taylor series, o plug in he definiion of he diffusion processes, and o analyze he asympoic behavior. The mixed momens are kep very general so ha hey can be applied o many examples. Addiionally, we prove ha anoher mixed momen leads o a special dependency srucure, where he difference of wo diffusion processes is independen of one of he processes. These heoreical resuls are now applied o differen insurance models. As a sar, we come back o he wo-sae acive-dead model where we consider a zero-coupon bond, a life annuiy, and a erm life insurance. Wihin our framework consisen forward raes imply ha he ineres and he moraliy raes need o be independen. Consequenly, i is a confirmaion of many papers which assume he independence, bu i could also be considered o be delicae since no oher dependency srucure is possible. In he nex sep, we add he possibiliy o surrender he conrac o he model and assume ha all corresponding producs are included in he model so ha we have consisen forward raes. Wih he general resuls from above, we can show again ha he ineres raes, he moraliy raes, and he surrender raes need o be independen of each oher. The siuaion is differen for a disabiliy model, where we do no allow for reacivaion. Here i urns ou ha for consisen forward raes he ineres raes 8

19 Research conex and summary of research papers need o be independen of all ransiions raes and ha he ransiion raes from acive o disabled, he ransiion raes from disabled o dead, and he difference of he ransiion raes from acive o dead and from disabled o dead need o be independen of each oher. This means ha he excess moraliy of acive individuals is independen of he basic moraliy level of disabled individuals. Of course, his is a severe resricion, since i limis he model choice significanly. Sill, i has o be acceped when assuming consisen forward raes for disabiliy insurance. As a las example, we consider a join life insurance, which can be spli ino wo disabiliy models. Consequenly, we ge similar resuls and he ineres raes again have o be independen of all ransiion raes and he differences of some moraliy raes have o be independen of oher moraliy raes. In summary, his paper provides an answer o he second research quesion and shows ha wihin an acive-dead model he ineres and moraliy raes need o be independen o obain consisen forward raes. For his invesigaion we prove general resuls ha can be applied o many oher insurance models. This paper complemens he firs paper, since i invesigaes he calculaion of marke values of insurance liabiliies by means of forward raes. More precisely, i gives imporan insighs for regulaors who should undersand he implied dependency srucure when dealing wih consisen forward raes. Research paper 3: Modeling and forecasing duraion-dependen moraliy raes The hird paper discusses exensions of he model from Lee and Carer (1992 o describe he moraliy rae of disabled individuals. The calibraion and discussion of hese exensions is based on a large daa se from FDZ-RV (2011 describing empirical moraliy raes of disabled individuals. The paper is a join work wih Marcus C. Chrisiansen and Lucia Teigiszerová and has been submied o he journal Compuaional Saisics & Daa Analysis (firs round. For he calculaion of he SCR in Solvency II wih an inernal model, for asse-liabiliy managemen, and for he calculaion of safey margins, sochasic models are needed. A seminal paper from Lee and Carer (1992 inroduces a parameric model for moraliy raes. There exis a lo of exensions and improvemens of his model in lieraure, bu no one has considered he exension by duraional effecs. Sill, as our daa and many papers from lieraure show, he duraion since disablemen srongly influences he moraliy raes (cf., e.g., Segerer, Consequenly, i is essenial for regulaors and risk managers o undersand and o quanify he influence of duraional effecs on he moraliy raes. In our paper, we discuss several exensions of he Lee-Carer model ha incorporae duraional effecs. Firs of all, we discuss he empirical daa from he FDZ-RV (2011 and show how we can esimae he moraliy raes from i. The daa represen 1% of he oal porfolio of pensioners a he end of he years 1993 o 2008 and 10% of he disconinuaions of pensioners due o moraliy in he years 1994 o We include in our sudy he variables age, calender ime, and duraion since disablemen. Unforunaely, he empirical daa do no exacly include he 9

20 Research conex and summary of research papers exposure and he number of deahs ha we need for he esimaion of he moraliy raes. To overcome his, we show in a hree-dimensional Lexis diagram how he moraliy raes can be esimaed from he daa. Thereby, i is assumed ha he moraliy raes are uniformly disribued wihin each observed group. We sar wih he calibraion of he classical Lee-Carer model o he daa which explains 11% of he empirical variance. When we use a simple model which includes only a parameer for he duraion and no parameer for age or calendar ime, his model explains 71% of he empirical variance. This shows ha he duraion has a significan influence on he moraliy raes of disabled individuals which is, in paricular, much higher han he influence of he age and he calendar ime. Furhermore, he Lee-Carer model is no appropriae o fi our daa in a reasonable way. This reinforces he necessiy of our paper. We discuss five exensions of he Lee-Carer model. For his, we add addiional parameers a differen posiions o he Lee- Carer model. We give condiions for he parameers of each model which make hem unique, bu sill allow ha each model can be ransformed so ha i fulfills he condiions. This also helps o compare he models wih each oher. The parameers of he differen models also give insighs ino he daa: The moraliy rae is increasing wih increasing age, decreasing wih increasing calendar ime, and decreasing wih increasing duraion. Furhermore, he moraliy improvemen is high for younger ages and low for older ages. In conras o his clear resul for he age parameer, here are significan flucuaions bu no sysemaic rend for he duraion. The proposed models explain beween 82% and 84% of he empirical variance. Since our daa are grouped ino 2240 classes and many classes are sparse, his resul is remarkable. Finally, we forecas one model exemplarily ino he fuure. Therefore, we describe he calendar ime parameer wih an ARIMA(0,1,0 model which allows o calculae he expeced value and quaniles of fuure moraliy raes. For simpliciy, we consider an example of one policyholder whose age and duraion is increasing wih increasing calendar ime. Thereby, we see ha in he firs years he duraion effec overlays all oher effecs. In he firs years he confidence inerval is geing wider, bu, surprisingly, i is geing narrower afer 15 years. This is he case, since he calendar ime has hardly any influence on older ages and consequenly, here is less uncerainy in he model. In summary, he paper gives an answer o he hird research quesion and discusses differen exensions of he Lee-Carer model o include duraional effecs. In his paper, we propose differen models, calibrae hem o empirical daa, inerpre he parameers, and show how a model can be forecas ino he fuure. As menioned before such models are no only ineresing on heir own, bu can be used for several applicaions. We will pick up one of hese models in he fourh paper and calculae wih i safey margins for sysemaic biomeric risk. However, hese models could also be used o calculae he SCR from he firs paper. 10

21 Research conex and summary of research papers Research paper 4: Safey margins for sysemaic biomeric and financial risk in a semi-markov life insurance framework In he fourh paper we propose a mehod for he calculaion of risk-oriened safey margins for sysemaic biomeric and financial risk. The mehod is inroduced in a universally valid semi-markov life insurance framework and is appropriaeness is demonsraed by hree differen examples. The paper has been submied o he journal Insurance: Mahemaics and Economics (firs round. Before he curren low-ineres period, he surplus resuling from he ineres raes was dominaing all oher surplus sources in mos life insurance companies and he biomeric surplus played only a minor role. Since he insurance companies are confroned wih low ineres raes, insurance producs like erm insurance and disabiliy insurance become more aracive. They imply a relaively low ineres and high biomeric risk. For conrolling hese risks, insurance companies add safey margins o he acuarial assumpions and he policyholder and he insurance company share he surplus which originaes from he difference of he occurred and he calculaed acuarial assumpions. This implies a delicae ask for he regulaor, since oo high safey margins lead o high surpluses and oo low safey margins o solvency problems. For insurance producs like disabiliy insurance, he surplus is passed on o he policyholder as a cash bonus so ha i is no only imporan o choose he oal safey margin carefully, bu also o conrol he safey margin in every inerval. For his, we follow a op down approach, inroduced by Bühlmann (1985, which means ha a general safey level for he whole conrac is fixed, hen in a second sep allocaed o single ransiions, poins in ime, and duraions. Pannenberg (1997 uses his approach o calculae safey margins for unsysemaic biomeric risk based on a yearly ime grid for an acive-dead model. Wihin a general muli-sae model Chrisiansen (2013 calculaes safey margins for unsysemaic biomeric risk and Chrisiansen (2012 for biomeric esimaion risk. Since safey margins for sysemaic biomeric risk are less discussed, we fill his gap and discuss hem in a semi-markov framework. Such safey margins are needed for he deducion of acuarial assumpions of firs order. However, he German Acuarial Sociey uses for he calculaion of he safey margins for sysemaic biomeric risk a rough approximaion and in many ables he safey margin is jus a ime-independen percenage on op of he bes esimae raes (cf., e.g., DAV-Unerarbeisgruppe Todesfallrisiko, 2009; Kolser e al., Firs of all, we give a shor inroducion o he semi-markov life insurance framework. We generalize he decomposiion of he difference of wo acuarial reserves calculaed wih differen acuarial assumpions ha is used in Norberg (2001 and Chrisiansen (2012. Norberg (2001 inroduces his decomposiion for he calculaion of he surplus. Since his became he sandard approach for he deducion of he surplus, we also use i laer on. This decomposiion yields a facorizaion by ransiion, ime, and duraion. Our goal is o calculae a safey margin for each ransiion, poin in ime, and duraion given a fixed solvency level. In paricular, he probabiliy ha he acual, random reserve is larger han he reserve calculaed wih he firs order acuarial assumpion should exceed he safey level. However, we 11

22 Research conex and summary of research papers do no know he firs order valuaion basis, since his is exacly wha we wan o calculae. Our approach is o add and subrac he reserve calculaed wih bes esimae assumpions and o decompose hen he difference of firs order and bes esimae reserve, as well as he difference of acual and bes esimae reserve. This gives us a decomposiion by ransiion, ime, and duraion. Furhermore, we assume ha he decomposiion of he difference of firs order and bes esimae valuaion basis is equal o a consan imes he value of a risk measure of he decomposiion of he difference of he acual and bes esimaion valuaion basis for each ransiion, poin in ime, and duraion. This is he crucial assumpion of he paper. We show how he firs order valuaion basis can be calculaed from his saring poin and how he consan has o be chosen o mach he given solvency level. One key resul is ha he consan can be calculaed before he firs order valuaion basis is known. The appropriaeness of his approach is demonsraed by a case sudy. We consider a Markovian acive-dead model wihou ineres rae, a Markovian muli-sae model for disabiliy insurance wih ineres rae, and a semi-markov model for he moraliy of disabled individuals. As risk measures we use expeced value, variance, sandard deviaion, value a risk, and ail value a risk. We compare our resuls wih a consan principle, where he firs order valuaion basis is jus a consan percenage of he bes esimae valuaion basis, since his approach is used in many life ables. The Markovian acive-dead model is kep very simple, bu gives firs insighs of he appropriaeness of our approach. To forecas he moraliy rae we use he model proposed and calibraed in Dahl and Møller (2006. While he loss probabiliy is quie consan in ime when he firs order valuaion basis is calculaed wih he sandard deviaion, value a risk, or ail value a risk as risk measure, i is significanly increasing in ime wih he consan principle. The variance and he expeced value seem o be inappropriae as risk measures in his conex, since he resuling loss probabiliies are srongly flucuaing. Furhermore, in one example where he policyholder ges an annuiy paymen in combinaion wih erm insurance, i is no possible o calculae a safey margin wih he consan principle ha has he required safey level. This is because he insurance company faces firs a longeviy risk and hen a moraliy risk. Consequenly, he sign of he safey margin calculaed wih he appropriae risk measures flips during he conrac which canno be reproduced by he consan principle. We consider a Markovian muli-sae model for disabiliy insurance wih and wihou ineres rae risk. For he projecion of he ransiion raes we use he mulivariae Lee-Carer model proposed in Chrisiansen e al. (2012 and for he ineres rae he model from Cox e al. (1985. In boh cases he proposed mehod implies quie consan loss probabiliies in ime for all risk facors wih he appropriae risk measures while he consan principle leads o srongly flucuaing loss probabiliies for he case wihou ineres rae risk. We do no apply he consan principle in he case wih ineres rae risk, since wih his mehod i is unclear how ransiion raes and he ineres rae can be compared. Ineresing insighs are also given by he empirical densiies of he insananeous surplus which are illusraed for seleced poins 12

23 Research conex and summary of research papers in ime. Here, we see ha he ransiion from acive o invalid has he highes risk and he ail is heavier han he one of he ineres rae. For he semi-markov invalid-dead model we pick up one of he models proposed in he hird paper. This model is embedded ino a disabiliy model wihou reacivaion and where he ransiion raes from acive o invalid and dead are deerminisic. The deerminisic par of he model is adoped from Haberman and Piacco (1999 and is needed o have a random poin in ime where he policyholder ges disabled so ha all possible combinaions of age and duraion can occur. The wo-dimensional safey margin does, of course, srongly depend on he model, bu he resuling loss probabiliies are again quie consan. This demonsraes he appropriaeness of he model also for he semi-markov case. In conras, he loss probabiliies wih he consan principle are again srongly flucuaing. In summary, his paper answers he fourh research quesion and presens a mehod o calculae safey margins for sysemaic biomeric and financial risk. In a case sudy, several examples are discussed showing wha he safey margins look like in applicaion. One of he srenghs of his paper is he generaliy of he framework so ha i can be even applied o semi-markov models. I is essenial o presen also an example for he semi-markov case which demonsraes how i can be applied and how good he resuls are. Therefore, he hird paper is indispensable, since i is he firs sochasic model for he moraliy of disabled individuals under consideraion of he duraion since disablemen. While in he firs paper we focused on he quesion of wha he correc SCR is, we end one sep afore and show how a conrac needs o be calculaed o have low loss probabiliies during he whole conrac. 13

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25 References Barrieu, P., Bensusan, H., El Karoui, N., Hillaire, C., Loisel, S., Ravanelli, C., and Salhi, Y. (2012. Undersanding, modelling and managing longeviy risk: key issues and main challenges. Scandinavian Acuarial Journal 2012(3: Bauer, D., Reuss, A., and Singer, D. (2012. On he Calculaion of he Solvency Capial Requiremen based on Nesed Simulaions. Asin Bullein 42(2: Bauer, D., Börger, M., Ruß, J., and Zwiesler, H.-J. (2008. The volailiy of moraliy. Asia- Pacific Journal of Risk and Insurance 3(1: Bingham, N. and Kiesel, R. (2004. Risk-neural valuaion: Pricing and hedging of financial derivaives. London: Springer. Buchard, K. (2014. Dependen ineres and ransiion raes in life insurance. Insurance: Mahemaics and Economics 55: Bühlmann, H. (1985. Premium calculaion from op down. Asin Bullein 15(2: Cairns, A., Blake, D., and Dowd, K. (2006. Pricing deah: Frameworks for he valuaion and securiizaion of moraliy risk. Asin Bullein 36(1: CEIOPS (2009. Final CEIOPS Advice for Level 2 Implemening Measures on Solvency II: Technical Provisions Aricle 86 (d Calculaion of he Risk Margin QIS 5 Technical Specificaions. hps : / / eiopa. europa. eu / fileadmin / x _ dam / files / consulaions / consulaionpapers/cp42/ceiops-l2-final-advice-on-tp-risk-margin.pdf. CEIOPS (2010. QIS 5 Technical Specificaions. hps://eiopa.europa.eu/fileadmin/x_dam/ files/consulaions/qis/qis5/qis5-echnical_specificaions_ pdf. Chrisiansen, M. C. (2012. Mulisae models in healh insurance. ASA Advances in Saisical Analysis 96(2: Chrisiansen, M. C. (2013. Safey margins for unsysemaic biomeric risk in life and healh insurance. Scandinavian Acuarial Journal 2013(4: Chrisiansen, M. C., Denui, M. M., and Lazar, D. (2012. The Solvency II square-roo formula for sysemaic biomeric risk. Insurance: Mahemaics and Economics 50(2: Cox, J., Ingersoll Jr, J., and Ross, S. (1985. An Ineremporal General Equilibrium Model of Asse Prices. Economerica: Journal of he Economeric Sociey 53(2: Dahl, M. (2004. Sochasic moraliy in life insurance: marke reserves and moraliy-linked insurance conracs. Insurance: Mahemaics and Economics 35(1: Dahl, M. and Møller, T. (2006. Valuaion and hedging of life insurance liabiliies wih sysemaic moraliy risk. Insurance: Mahemaics and Economics 39(2: DAV-Unerarbeisgruppe Todesfallrisiko (2009. Herleiung der Serbeafel DAV 2008 T für Lebensversicherungen mi Todesfallcharaker. Bläer der DGVFM 30(1:

26 Research conex and summary of research papers Doff, R. (2008. A Criical Analysis of he Solvency II Proposals. The Geneva Papers on Risk and Insurance - Issues and Pracice 33(2: Eling, M., Schmeiser, H., and Schmi, J. (2007. The Solvency II process: Overview and criical analysis. Risk Managemen and Insurance Review 10(1: European Parliamen and he Council (2009. Direcive of he European Parliamen and of he Council on he aking-up and pursui of he business of insurance and reinsurance (Solvency II. FDZ-RV (2011. SUF Demografiedaensaz Renenwegfall/-besand Forschungsdaenzenrum der Renenversicherung. Haberman, S. and Piacco, E. (1999. Acuarial Models for Disabiliy Insurance. Boca Raon: Chapman & Hall/CRC Press. Holzmüller, I. (2009. The Unied Saes RBC Sandards, Solvency II and he Swiss Solvency Tes: A Comparaive Assessmen. The Geneva Papers on Risk and Insurance - Issues and Pracice 34(1: Kolser, N., Loebus, H., and Mörlbauer, W. (1998. Neue Rechnungsgrundlagen für die Berufsunfähigkeisversicherung DAV Bläer der DGVFM 23(4: Kouwenberg, R. and Zenios, S. A. (2006. Sochasic programming models for asse liabiliy managemen. In Handbook of Asse and Liabiliy Managemen, Volume 1: Theory and Mehodology (S. A. Zenios & W. T. Ziemba eds: Amserdam: Norh Holland. Kriele, M. and Wolf, J. (2007. On marke value margins and cos of capial. Bläer der DGVFM 28(2: Kriele, M. and Wolf, J. (2012. Werorienieres Risikomanagemen von Versicherungsunernehmen. Berlin: Springer. Lee, R. D. and Carer, L. R. (1992. Modeling and Forecasing US Moraliy. Journal of he American saisical associaion 87(419: Maegebier, A. (2013. Valuaion and risk assessmen of disabiliy insurance using a discree ime rivariae Markov renewal reward process. Insurance: Mahemaics and Economics 53(3: Möhr, C. (2011. Marke-Consisen Valuaion of Insurance Liabiliies by Cos of Capial. Asin Bullein 41(2: Norberg, R. (2001. On Bonus and Bonus Prognoses in Life Insurance. Scandinavian Acuarial Journal 2001(2: Norberg, R. (2010. Forward moraliy and oher vial raes Are hey he way forward? Insurance: Mahemaics and Economics 47(2: Ohlsson, E. and Lauzeningks, J. (2009. The one-year non-life insurance risk. Insurance: Mahemaics and Economics 45(2: Pannenberg, M. (1997. Saisische Schwankungszuschläge für biomerische Rechnungsgrundlagen in der Lebensversicherung. Bläer der DGVFM 23(1: Piacco, E. (2012. Moraliy of Disabled People. Working Paper, available a SSRN Piacco, E., Denui, M., Haberman, S., and Olivieri, A. (2009. Modelling Longeviy Dynamics for Pensions and Annuiy Business. Oxford: Oxford Universiy Press. 16

27 Research conex and summary of research papers Rees, R., Gravelle, H., and Wambach, A. (1999. Regulaion of Insurance Markes. The Geneva Papers on Risk and Insurance Theory 24(1: Salzmann, R. and Wührich, M. (2010. Cos-of-Capial Margin for a General Insurance Liabiliy Runoff. Asin Bullein 40(2: Segerer, G. (1993. The acuarial reamen of he disabiliy risk in Germany, Ausria and Swizerland. Insurance: Mahemaics and Economics 13(2: Seyboh, M. (2009. DCF und MCEV/MCAV als werorieniere Bemessungsansäze für die Lebensversicherung. Working Paper, Universiy of Ulm. Sheldon, T. J. and Smih, A. D. (2004. Marke Consisen Valuaion of Life Assurance Business. Briish Acuarial Journal 10(03: Wührich, M., Bühlmann, H., and Furrer, H. (2010. Marke-Consisen Acuarial Valuaion. Berlin: Springer. 17